Unified Fracture Design
Unified Fracture Design
Bridging the Gap Between Theory and Practice
Unified Fracture Design
Bridging the Gap Between Theory and Practice
Michael Economides
Ronald Oligney
Peter Valkó
Orsa Press
Alvin, Texas
Unified Fracture Design
Copyright © 2002 by Orsa Press
All rights, including reproduction by photographic or electronic process and
translation into other languages are fully reserved under the International
Copyright Union, the Universal Copyright Convention, and the Pan-American
Copyright Convention. Reproduction of this book in whole or in part in any
manner without written permission from the publisher is prohibited.
Orsa Press
P.O. Box 2569
Alvin, TX 77512
10 9 8 7 6 5 4 3 2 1
Library of Congress Cataloging-in-Publication Data
Economides, Michael J.
Unified fracture design : bridging the gap between theory and
practice / Michael Economides, Ronald Oligney, Peter Valkó
p. cm.
Includes bibliographical references and index.
ISBN 0-9710427-0-5 (alk. paper)
1. Oil wells—Hydraulic fracturing. I. Oligney, Ronald E.
II. Valkó, Peter, 1950- III. Title.
TN871.E335 2002
Printed in the United States of America
Hydraulic Fracturing for Production
or Injection Enhancement 1
Fractured Well Performance 5
Sizing and Optimization 7
Fracture-to-Well Connectivity 8
Tip Screenout Design 10
Net Pressure and Leakoff in the High Permeability Environment
Candidate Selection 12
Design Logic 14
Fracture Design Spreadsheet 14
How To Use This Book 17
Fracturing Crew
Well Stimulation as a Means to Increase
the Productivity Index 23
Well Performance for Low and Moderate Proppant Numbers
Fracturing Theory 39
Formal Material Balance: The Opening-Time Distribution Factor
Constant Width Approximation (Carter Equation II) 48
Power Law Approximation to Surface Growth 48
Detailed Leakoff Models 50
Perkins-Kern Width Equation 51
Khristianovich-Zheltov-Geertsma-deKlerk Width Equation
Radial (Penny-shaped) Width Equation 54
Fracturing of High Permeability Formations 57
Gravel Pack 60
High-Rate Water Packs
The Tip Screenout Concept 63
Net Pressure and Fluid Leakoff 66
Net Pressure, Closure Pressure, and Width in Soft Formations
Fracture Propagation 66
Fluid Leakoff and Spurt Loss as Material Properties: The Carter Leakoff
Model with Nolte’s Power Law Assumption 67
Filter Cake Leakoff Model According to Mayerhofer, et al. 68
Polymer-Invaded Zone Leakoff Model of Fan and Economides 70
Optimizing Fracture Geometry in Gas Condensate Reservoirs
Definitions and Assumptions 77
Case Study for the Effect of Non-Darcy Flow
Fracturing Materials 83
Calculating Effective Closure Stress
Fracture Width as a Design Variable
Proppant Selection 93
Fluid Selection 94
Fracture Treatment Design 101
Pump Time 110
Proppant Schedule 113
Departure from the Theoretical Optimum 118
TSO Design 119
Swab Effect Example 121
Perforations for HPF 122
Step-Rate Tests 123
Minifracs 125
Pressure Falloff Tests 126
Bottomhole Pressure Measurements
Fracture Design and Complications 129
Fracture Height Map 132
Practical Fracture Height Determination
A Typical Preliminary Design—Medium Permeability Formation: MPF01 137
Pushing the Limit—Medium Permeability Formation: MPF02 142
Proppant Embedment: MPF03 145
Fracture Design for High Permeability Formation: HPF01 148
Extreme High Permeability: HPF02 152
Low Permeability Fracturing: LPF01 157
Quality Control and Execution 163
Water Transfer and Storage 166
Proppant Supply 167
Slurrification and Blending 168
Pumping 169
Monitoring and QA/QC 172
Miscellaneous 174
Spotting the Equipment
Fluid Supply-to-Blender 177
Proppant Supply 177
Frac Pumps 177
Manifold-to-Well 178
Monitoring/Control Equipment and Support Personnel
Treatment Evaluation 185
Production Results 193
Evaluation of Real-Time HPF Treatment Data 194
Post-Treatment Well Tests in HPF 195
Validity of the Skin Concept in HPF 197
Assumptions 198
Restricted Growth Theory 199
Slopes Analysis Algorithms 201
The purpose of writing this book is to establish a unified design methodology for hydraulic fracture treatments, a long established well
stimulation activity in the petroleum and related industries. Few activities in the industry hold such potential to improve well performance
both profitably and reliably.
The word “unified” has been selected deliberately to denote both the
integration of all the highly diverse technological aspects of the process,
but also to dispel the popular notion that there is one type of treatment
that applies to low-permeability and another to high-permeability reservoirs. It is natural, even for experienced practitioners to think so
because traditional targets have been low-permeability reservoirs while
the fracturing of high-permeability formations has sprung from the
gravel pack, sand control practice.
The key idea is that treatment sizes can be unified because they
can be best characterized by the dimensionless Proppant Number,
which determines the theoretically optimum fracture dimensions at
which the maximum productivity or injectivity index can be obtained.
Technical constraints should be satisfied in such a way that the design
departs from the theoretical optimum only to the necessary extent. With
this approach, difficult topics such as high- versus low-permeability fracturing, extensive height growth, non-Darcy flow, and proppant embedment are treated in a transparent and unified way, providing the
engineer with a logical and coherent design procedure.
A design software package is included with the book.
The authors’ backgrounds span the entire spectrum of technical,
research, development, and field applications in practically all geographic and reservoir type settings. It is their desire that this book finds
its appropriate place in everyday practice.
Hydraulic Fracturing for
Production or Injection
This book has the ambition to do something that has not been done
properly before: to unite the gap between theory and practice in what
is arguably the most common stimulation/well completion technique
in petroleum production. Even more important, the book takes a new
and ascendant position on the most critical link in the sequence of
events in this type of well stimulation—the sizing and the design
of hydraulic fracture treatments.
Fracturing was first employed to improve production from marginal wells in Kansas in the late 1940s (Figure 1-1). Following an
explosion of the practice in the mid-1950s and a considerable surge
in the mid-1980s, massive hydraulic fracturing (MHF) grew to become
a dominant completion technique, primarily for low permeability reservoirs in North America. By 1993, 40 percent of new oil wells and
70 percent of gas wells in the United States were fracture treated.
With improved modern fracturing capabilities and the advent of
high permeability fracturing (HPF), which in the vernacular has been
referred to as “frac & pack” or variants, fracturing has expanded further to become the completion of choice for all types of wells in the
United States, but particularly natural gas wells (see Figure 1-2).
Unified Fracture Design
FIGURE 1-1. An early hydraulic fracture treatment, circa. 1949. (Source: Halliburton.)
The tremendous advantage in fracturing most wells is now largely
accepted. Even near water or gas contacts, considered the bane of fracturing, HPF is finding application because it offers controlled fracture
extent and limits drawdown (Mullen et al., 1996; Martins et al., 1992).
The rapid ascent of high permeability fracturing from a few isolated
treatments before 1993 (Martins et al., 1992; Grubert, 1991; Ayoub
et al., 1992) to some 300 treatments per year in the United States by
1996 (Tiner et al., 1996) was the start to HPF becoming a dominating optimization tool for integrated well completion and production.
Today, it is established as one of the major recent developments in
petroleum production.
The philosophy of this book hinges on the overriding commonality in fracture design that transcends the value of the reservoir permeability. There is a strong theoretical foundation to this approach,
which will be outlined in this book. Hence the title, Unified Fracture
Design, which suggests the connection between theory and practice,
but also that the design process cuts across all petroleum reservoirs—
low permeability to high permeability, hard rock to soft rock. And
indeed, it is common to all.
There is substantial room for additional growth of hydraulic fracturing in the worldwide petroleum industry, as well as other industries. It is estimated that hydraulic fracturing may add several hundred
thousand barrels per day from existing wells in a number of countries.
Hydraulic Fracturing for Production or Injection Enhancement
Fracs (Gas)
Fracs (Oil)
% Gas
% Oil
Percentage of Wells Treated
Number of Wells Treated
FIGURE 1-2. Fracturing as “completion of choice” in U.S. oil and gas wells. (Source:
This can be accomplished if the process is undertaken in a serious and
concerted way so that the economy of scale influences the cost of the
treatments and, hence, the overall economics.
There are two frequently encountered impediments to substantial
applications of hydraulic fracturing:
1. A widespread misunderstanding that the process is still only for
low permeability reservoirs (e.g., less than 1 md), or that it is
the last refuge for enhancing well production or injection performance, to be tried only if everything else fails. The latter
carries along with it an often unjustifiable phobia that hydraulic fracturing is dangerous, that it accelerates the onset of
water production, that it increases the water cut or affects zonal
isolation, and so on. The more serious associated problem is
that using fracturing as a last, at times desperate, resort implies
unplanned stimulation that may suffer from a number of problems (such as well deviation and inadequate perforating),
which, in turn, may almost guarantee disappointing results. A
final related problem is the notion that high permeability fracturing applies only to those reservoirs that need sand production control. This is clearly not the case, and reservoirs with permeabilities
of several hundred millidarcies are now routinely fractured.
2. At times, engineers in various companies outside of North America
may actually try fracturing, but a treatment is done so rarely
Unified Fracture Design
and so haphazardly that it is bound to be expensive, such that
the cost cannot be justified even if the incremental production
is substantial. Hydraulic fracturing is a massive operation with a
very large complement of equipment, complicated and demanding fluids and proppants, and a wide spectrum of ancillary and
people-intensive engineering and operational demands. Costs
assigned to individual, isolated jobs—e.g., one or two treatments
carried out every three to six months—are essentially prohibitive.
Coupled with an occasional job failure, sketchy and spotty
application of hydraulic fracturing is almost assured of economic
failure and the dampening of any desire to apply it further.
Virtually no petroleum operation carries such a differential price
tag among areas where it is applied in a widespread and massive way,
such as North America and offshore in the North Sea, and elsewhere. In
North America, over 60 percent of all oil wells and 85 percent of all
gas wells are hydraulically fractured, and the percentages are still
increasing. Yet, consider this: a 100-ton proppant treatment in the
United States, at the time of this writing, costs less than $100,000.
Exactly the same treatment, with the same equipment and the same
service company, for example in Venezuela or Oman, is likely to cost
at least $1 million, and it can cost as much as $2 million.
At the same time, virtually no other petroleum technology carries
a larger incremental asset. The hundreds-of-thousands to millions of
barrels per day of worldwide production increase that we project
assumes that the percentage of existing wells being hydraulically fractured approaches that of oil wells in the United States (60 percent),
and the incremental production realized from each well is just 25
percent over the pre-treatment state. The latter implies the very modest assumptions that all existing wells continue to produce, and that
fracturing would result in a very achievable average “skin” equal to
–2. In fact, the incremental production capacity from a massive stimulation campaign with adequate equipment and well-trained people is
likely to be much higher.
Hydraulic fracturing entails injecting fluids in an underground formation
at a pressure that is high enough to induce a parting of the formation.
Hydraulic Fracturing for Production or Injection Enhancement
Granulated materials—called “proppants,” which range from natural
sands to rather expensive synthetic materials—are pumped into the
created fracture as a slurry. They hold open, or “prop,” the created
fracture after the injection pressure used to generate the fracture has
been relieved.
The fracture, filled with proppant, creates a narrow but very conductive flow path toward the wellbore. This flow path has a very large
permeability, frequently five to six orders of magnitude larger than the
reservoir permeability. It is most often narrow in one horizontal
direction, but can be quite long in the other horizontal direction and
can cover a significant height. Typical intended propped widths in low
permeability reservoirs are on the order of 0.25 cm (0.1 in.), while
the length can be several hundred meters. In high permeability reservoirs, the targeted fracture width (deliberately affected by the design
and execution) is much larger, perhaps as high as 5 cm (2 in.), while
the length might be as short as 10 meters (30 ft).
In almost all cases, an overwhelming part of the production comes
into the wellbore through the fracture; therefore, the original nearwellbore damage is “bypassed,” and the pre-treatment skin does not
affect the post-treatment well performance.
Fractured Well Performance
The performance of a fractured well can be described in many ways. One
common way is to forecast the production of oil, gas, and even water
as a function of time elapsed after the fracturing treatment. However,
post-treatment production is influenced by many decisions that are not
crucial to the treatment design itself. The producing well pressure, for
example, may or may not be the same as the pre-treatment pressure,
and may or may not be held constant over time. Even if, for the sake
of evaluation, an attempt is made to set all well operating parameters
the same before and after the treatment, comparison over time is still
obfuscated by the accelerated nature of reservoir depletion in the presence of a hydraulic fracture.
Thus, in a preliminary sizing and optimization phase, it is imperative to use a simple performance index that describes the expected and
actual improvement in well performance due to the treatment.
In unified fracture design, we consider a very simple and straightforward performance indicator: the pseudo-steady state productivity
index. The improvement in this variable describes the actual effect of
the propped fracture on well performance. Realizing the maximum
Unified Fracture Design
possible pseudo-steady state productivity index, for all practical purposes, means that the fracture will not under-perform any other possible realization of the same propped volume, even if the well produces
for a considerable time period in the so-called “transient” regime.
While this statement might not appear plausible at first, the experienced production engineer will understand it by thinking of the transient flow period as a continuous increase in drainage area in which
the pseudo-steady state has already been established. Considerable
cumulative production can only come from a large drained area, and
hence that pseudo-steady state productivity index must be maximized,
which corresponds to the finally formed drainage area.
Fracture length and dimensionless fracture conductivity are the two
primary variables that control the productivity index of a fractured
well. Dimensionless fracture conductivity is a measure of the relative
ease with which produced fluids flow inside the fracture compared
to the ability of the formation to feed fluids into the fracture. It is
calculated as the product of fracture permeability and fracture width,
divided by the product of reservoir permeability and fracture (by convention, half-) length.
In low permeability reservoirs, the fracture conductivity is de facto
large, even if only a narrow propped fracture has been created and a
long fracture length is needed. A post-treatment skin can be as negative as –7, leading to several folds-of-increase in well performance
compared to the unstimulated well.
For high permeability reservoirs, a large fracture width is essential for adequate fracture performance. Over the last several years, a
technique known as tip screenout (TSO) has been developed, which
allows us to deliberately arrest the lateral growth of a hydraulic fracture
and subsequently inflate its width, exactly to affect a larger conductivity.
For a fixed volume of proppant placed in the formation, a well will
deliver the maximum production or injection rate when the dimensionless fracture conductivity is near unity. In other words, a dimensionless fracture conductivity around one (or more precisely, 1.6, as
shown in Chapter 3) is the physical optimum, at least for treatments
not involving extremely large quantities of proppant. Larger values of
the dimensionless fracture conductivity would mean relatively shorterthan-optimum fracture lengths and, thus, the flow from the reservoir
into the fracture would be unnecessarily restricted. Dimensionless fracture conductivity values smaller than unity would mean less-thanoptimum fracture width, rendering the fracture as a bottleneck to
optimum production.
Hydraulic Fracturing for Production or Injection Enhancement
There are a number of secondary issues that complicate the picture—
early time transient flow regime, influence of reservoir boundaries,
non-Darcy flow effects, and proppant embedment, just to mention a
few. Nevertheless, these effects can be correctly taken into account
only if the role of dimensionless fracture conductivity is understood.
It is possible that in certain theaters of operation the practical
optimum may be different than the physical optimum. In some cases,
the theoretically indicated fracture geometry may be difficult to
achieve because of physical limitations imposed either by the available equipment, limits in the fracturing materials, or the mechanical
properties of the rock to be fractured. However, aiming to maximize
the well productivity or injectivity is an appropriate first step in the
fracture design.
Sizing and Optimization
The term “optimum” as used above means the maximization of a
well’s productivity, within the constraint of a certain treatment size.
Hence, a decision on treatment size should actually precede (or go
hand-in-hand with) an optimization based on the dimensionless fracture conductivity criterion.
For a long time, practitioners considered fracture half-length as a
convenient variable to characterize the size of the created fracture. That
tradition emerged because it was not possible to independently change
fracture length and width, and because length had a primary effect on
productivity in low permeability formations. In unified fracture design,
where both low and high permeability formations are considered, the
best single variable to characterize the size of a created fracture is
the volume of proppant placed in the productive horizon, or “pay.”
Obviously, the total volume of proppant placed in the pay interval is always less than the total proppant injected. From a practical
point of view, treatment sizing means deciding how much proppant
to inject. When sizing the treatment, an engineer must be aware that
increasing the injected amount of proppant by a certain quantity x will
not necessarily increase the amount of proppant reaching the pay layer
by the same quantity, x. We will refer to the ratio of the two proppant
volumes (i.e., the volume of proppant placed in the pay interval
divided by the total volume injected into the well) as the volumetric
proppant efficiency.
By far the most critical factor in determining volumetric proppant
efficiency is the ratio of created fracture height to the net pay thickness.
Unified Fracture Design
Extensive height growth limits the volumetric proppant efficiency, and
is something that we generally try to avoid. (The possibility of intersecting a nearby water table is another important reason to avoid
excessive height growth.)
Actual selection of the amount of proppant indicated for injection
is primarily based on economics, the most commonly used criterion
being the net present value (NPV). As with most engineering activities, costs increase almost linearly with the size of the treatment, but
after a certain point, the revenues increase only marginally. Thus, there
is an optimum treatment size, the point at which the NPV of incremental revenue, balanced against treatment costs, is a maximum.
The optimum size can be determined if some method is available
to predict the maximum possible productivity increase achievable with
a certain amount of proppant. Unified fracture design makes extensive use of this fact, given that the maximum achievable productivity
increase is already determined by the volume of proppant in the pay.
Many of the operational details are subsumed by the basic decision
on treatment size, making possible a simple yet robust design process.
Therefore, we employ the concept of “volume of proppant reaching
the pay” or simply “propped volume in the pay” as the key decision
variable in the sizing phase of the unified fracture design procedure.
To handle it correctly, the amount of proppant indicated for injection
and the volumetric proppant efficiency must be determined.
Fracture-to-Well Connectivity
While the maximum achievable improvement of productivity is determined by the propped volume in the pay, several additional conditions
must be satisfied en route to a fracture that actually realizes this
potential improvement. One of the crucial factors is to establish an
optimum compromise between the length and width (or to depart from
the optimum only as much as necessary, if required by operational
constraints). As previously explained, the optimum dimensionless fracture conductivity is the variable that helps us to find the right compromise. However, another condition is equally important. It is related
to the connectivity of the fracture to the well.
A reservoir at depth is under a state of stress that can be characterized by three principal stresses: one vertical, which in almost all
cases of deep reservoirs (depth greater than 500 meters, 1500 ft) is
the largest of the three, and two horizontal, one minimum and one
maximum. A hydraulic fracture will be normal to the smallest stress,
Hydraulic Fracturing for Production or Injection Enhancement
leading to vertical hydraulic fractures in almost all petroleum production applications. The azimuth of these fractures is pre-ordained by
the natural state of earth stresses. As such, deviated or horizontal wells
that are to be fractured should be drilled in an orientation that agrees
with this azimuth. Vertical wells, of course, naturally coincide with
the fracture plane.
If the well azimuth does not coincide with the fracture plane, the
fracture is likely to initiate in one plane and then twist, causing considerable “tortuosity” en route to its final azimuth—normal to the
minimum stress direction. Vertical wells with vertical fractures or
perfectly horizontal wells drilled deliberately along the expected fracture plane result in the best aligned well-fracture systems. Other wellfracture configurations are subject to “choke effects,” unnecessarily
decreasing the productivity of the fractured well. Perforations and their
orientation may also be a source of problems during the execution of
a treatment, including multiple fracture initiations and premature
screenouts caused by tortuosity effects.
The dimensionless fracture conductivity in low permeability reservoirs
is naturally high, so the impact of choke effects from the phenomena
described above is generally minimized; to avoid tortuosity, point
source fracturing is frequently employed.
Fracture-to-well connectivity is considered today as a critical point
in the success of high permeability fracturing, often dictating the well
azimuth (e.g., drilling S-shape vertical wells) or indicating horizontal
wells drilled longitudinal to the fracture direction. Perforating is being revisited, and alternatives, such as hydro-jetting of slots, are considered by the most advanced practitioners. While some models
incorporate complex well-fracture geometries with choke and other
effects, the many uncertainties prevent us from predicting performance.
Rather, we are limited to explain the performance once post-treatment
well test and production information become available. In the design
phase, we try to make decisions that minimize the likelihood of such
unnecessary reductions in productivity.
Because high permeability fracturing has the most fertile possibility
for expansion in petroleum operations worldwide, key issues for this
Unified Fracture Design
type of well completion are described below. The purpose is to identify those features that distinguish high permeability fracturing from
conventional hydraulic fracturing.
Tip Screenout Design
The critical elements of high permeability fracturing treatment design,
execution and treatment behavioral interpretation are substantially
different than for conventional fracturing treatments. In particular, HPF
relies on a carefully timed “tip screenout,” or TSO, to limit fracture
growth and allow for fracture inflation and packing. The TSO occurs
when sufficient proppant has concentrated at the leading edge of the
fracture to prevent further fracture extension. Once the fracture growth
has been arrested (and assuming the pump rate is larger than the rate
of leakoff to the formation), continued pumping will inflate the fracture, i.e., increase the fracture width. Tip screenout and fracture inflation should be accompanied by an increase in net fracturing
pressure. Thus, the treatment can be conceptualized in two distinct
stages: fracture creation (equivalent to conventional designs) and fracture inflation/packing (after tip screenout).
Creation of the fracture and arrest of its growth (i.e., the tip screenout) is accomplished by injecting a relatively small “pad” of clean
fluid (no sand) followed by a “slurry” containing 1– 4 lbs of sand per
gallon of fluid (ppg). Once the fracture growth has been arrested,
further injection builds fracture width and allows injection of a highconcentration slurry (e.g., 10–16 ppg). Final areal proppant concentrations of 20 lbm/sq ft are possible. A usual practice is to retard the
injection rate near the end of the treatment (coincidental with opening the annulus to flow) to dehydrate and pack the fracture near the
well. Rate reductions may also be used to force a tip screenout in cases
where no TSO event is observed on the downhole pressure record.
Frequent field experience suggests that the tip screenout can be
difficult to model, affect, or even detect. There are many reasons for
this, including a tendency toward overly conservative design models
(resulting in no TSO), partial or multiple tip screenout events, and
inadequate pressure monitoring practices.
Accurate bottomhole measurements are imperative for meaningful treatment evaluation and diagnosis. Calculated bottomhole pressures are unreliable because of the sizeable and complex friction
pressure effects associated with pumping high proppant slurry concentrations through small diameter tubulars and service tool crossovers.
Hydraulic Fracturing for Production or Injection Enhancement
Surface data may indicate that a TSO event has occurred when the
bottomhole data shows no evidence, and vice versa.
Net Pressure and Leakoff in the
High Permeability Environment
The entire HPF process is dominated by net pressure and fluid leakoff
considerations. First, high permeability formations are typically soft
and exhibit low elastic modulus values, and second, the fluid volumes
are relatively small and leakoff rates high (high permeability, compressible reservoir fluids and non-wall building fracturing fluids).
While traditional practices applicable to design, execution, and evaluation in hydraulic fracturing continue to be used in HPF, these are
frequently not sufficient.
Net Pressure
Net pressure is the difference between the pressure at any point in the
fracture and the pressure at which the fracture will close. This definition implies the existence of a unique closure pressure. Whether the
closure pressure is a constant property of the formation or depends
heavily on the pore pressure (or rather on the disturbance of the pore
pressure relative to the long-term steady value) is an open question.
In high-permeability, soft formations it is difficult (if not impossible) to suggest a simple recipe to determine the closure pressure as
classically derived from shut-in pressure decline curves. Furthermore,
because of the low elastic modulus values, even small induced uncertainties in the net pressure are amplified into large uncertainties in the
calculated fracture width.
Fracture propagation, the availability of sophisticated 3D models
notwithstanding, is a very complex process and difficult to describe,
even in the best of cases, because of the large number and often competing physical phenomena. The physics of fracture propagation in soft
rock is even more complex, but it is reasonably expected to involve
incremental energy dissipation and more severe tip effects when compared to hard rock fracturing. Again, because of the low modulus
values, an inability to predict net pressure behavior may lead to a significant departure between predicted and actual treatment performance.
Ultimately, the classic fracture propagation models may not reflect even
the main features of the propagation process in high permeability rocks.
Unified Fracture Design
It is common practice for some practitioners to “predict” fracture
propagation and net pressure features ex post facto using a computer
fracture simulator. The tendency toward substituting clear models and
physical assumptions with “knobs”— e.g., arbitrary stress barriers,
friction changes (attributed to erosion if decreasing and sand resistance if increasing) and less than well understood properties of
the formation expressed as dimensionless factors—does not help to
clarify the issue. Other techniques are warranted and several are
under development.
Considerable effort has been expended on laboratory investigation of
the fluid leakoff process for high permeability cores. The results raise
some questions about how effectively fluid leakoff can be limited by
filtercake formation. In all cases, but especially in high permeability
formations, the quality of the fracturing fluid is only one of the
factors that influence leakoff, and often not the determining one. Transient fluid flow in the formation might have an equal or even larger
impact. Transient flow cannot be understood by simply fitting an empirical equation to laboratory data. The use of models based on
solutions to the fluid flow in porous media is an unavoidable step,
and one that has already been taken by many.
Candidate Selection
The utility of high permeability fracturing extends beyond the obvious productivity benefits associated with bypassing near-well damage
to include sand control. However, in HPF the issue is not mere sand
control, which implies most often mechanical retention of migrating
sand particles (and plugging), but rather sand deconsolidation control.
Increasingly, wellbore stability should be viewed in a holistic
approach with horizontal wells and hydraulic fracture treatments. Proactive well completion strategies are critical to wellbore stability and
sand-production control to reduce pressure drawdown while obtaining
economically attractive rates. Reservoir candidate recognition for the
correct well configurations is the key element. Necessary steps in
candidate selection include appropriate reservoir engineering, formation characterization, wellbore stability calculations, and the combining
of production forecasts with assessments of sand-production potential.
Hydraulic Fracturing for Production or Injection Enhancement
Complex Well-Fracture Configurations
Vertical wells are not the only candidates for hydraulic fracturing.
Figure 1-3 shows some basic single-fracture configurations for vertical and horizontal wells. Horizontal wells that employ conventional
or especially high permeability fracturing with the well drilled in the
expected fracture azimuth (accepting a longitudinal fracture) appear
to have, at least conceptually, a very promising prospect as discussed
in Chapter 5. However, a horizontal well intended for a longitudinal
fracture configuration would have to be drilled along the maximum
horizontal stress. And this, in addition to well-understood drilling problems, may contribute to long-term formation stability problems.
Figure 1-4 illustrates two multi-fracture configurations. A rather
sophisticated conceptual configuration would involve the combination
of HPF with multiple-fractured vertical branches emanating from a
horizontal “mother” well drilled above the producing formation. Of
course, horizontal wells, being normal to the vertical stress, are generally more prone to wellbore stability problems. Such a configuration would allow for placement of the horizontal borehole in a
competent, non-producing interval. There are other advantages to fracture treating a vertical section over a highly deviated or horizontal
section: multiple starter fractures, fracture turning, and tortuosity problems are avoided; convergence-flow skins (choke effects) are much less
of a concern; and the perforating strategy is simplified.
FIGURE 1-3. Single-fracture configurations for vertical and horizontal wells.
Unified Fracture Design
Multibranch Well with
Fractured Vertical Branches
(Horizontal "Parent" Well is
Drilled above the Reservoir)
Horizontal Well with Multiple
Transverse Fractures
FIGURE 1-4. Multibranched, multiple-fracture configurations for horizontal wells.
Design Logic
In unified fracture design, we consider treatment size, specifically
propped volume in the pay, as the primary decision variable. Once the
basic decision on size is made, the optimum length and width are
determined. These parameters are then revised in view of the technical constraints, and the target dimensions of the created fracture are
set. A preliminary injection schedule is calculated that realizes the
target dimensions and assures uniform placement of the indicated
amount of proppant. If the optimum placement cannot be realized by
traditional means, a TSO treatment is indicated. Even if the injected
amount of proppant is already fixed, the volumetric proppant efficiency
may change during the design process. It is extremely important that
the basic decisions be made in an iterative manner, but without going
into unnecessary details of fracture mechanics, fluid rheology, or reservoir engineering.
Fracture Design Spreadsheet
A simple spreadsheet, based on a transparent design logic, is an ideal
tool to make preliminary design decisions and a primary evaluation
of the executed treatment. The CD attached to the back cover of the
Hydraulic Fracturing for Production or Injection Enhancement
book contains such a spreadsheet, named HF2D. The HF2D Excel
spreadsheet is a fast 2D software package for the design of traditional
(moderate permeability and hard rock) and frac & pack (higher permeability and soft rock) fracture treatments.
Readers are strongly encouraged to use the spreadsheet while reading the book. By modifying various input parameters, an intuitive feel
for their relative importance in treatment design and final fractured
well performance can be rapidly acquired, an important but uncommon prospect in the era of complex 3D fracture simulators. The
spreadsheet will help readers make the most important decisions and
be aware of their consequences.
The attached spreadsheet is not necessarily intended as a substitute for more sophisticated software tools, but the rapid “back of the
envelope” calculations that it affords can provide substantive fracture
designs. In many cases, by virtue of restricting the analysis to important first-order considerations, the spreadsheet results are more robust
than those provided by highly involved 3D fracture simulators. It is
suggested that readers run parallel cases with one or more 3D simulators, if available, as an interesting exercise.
How To Use This Book
The purpose of this book is to transfer hydraulic fracturing technology and, especially, facilitate its execution. The various chapters supply information on candidate recognition, fracture treatment design,
execution and evaluation, materials selection, quality control, and
equipment specifications.
While the book includes late developments from some of the most
respected practitioners of hydraulic fracturing in the world—genuine
state-of-the-art technology—the entry point is deliberately low. That
is, the book can also serve as a very useful primer for those being
exposed to fracturing technology for the first time.
Chapters 1 through 10 provide a detailed narrative of the most important aspects across the spectrum of hydraulic fracturing activities.
Appendices A through G are reference material, including a glossary of fracturing terms; an extensive bibliography; data requirements
and user instructions for the included design software; standard quality control practices and forms; and example fracturing procedures.
Unified Fracture Design
The CD attached to the back cover of the book contains two
1. The HF2D Excel spreadsheet is a fast 2D software package for
the design of traditional (moderate permeability and hard rock)
and frac & pack (higher permeability and soft rock) fracture
2. The MF Excel spreadsheet is a minifrac (calibration test) evaluation package. Its main purpose is to extract the leakoff coefficient from pressure fall-off data.
Two industry-leading references are strongly recommended as addenda
to this book:
Hydraulic Fracture Mechanics, by Peter Valkó and Michael
Economides, addresses the theoretical background of this seminal technology. It provides a fundamental treatment of basic
phenomena such as elasticity, stress distribution, fluid flow, and
the dynamics of the rupture process. Contemporary design and
analysis techniques are derived and improved using a comprehensive and unified approach.
Stimulation Engineering Handbook, by John Ely, aptly covers
many issues of fracture treatment implementation and quality
control. This is a very hands-on book, intended to drive execution performance and quality control.
Other reference books that contain abundant information by dozens
of experts in the field include Petroleum Well Construction, edited by
Michael Economides, Larry Watters, and Shari Dunn-Norman; Reservoir Stimulation, Third Edition, by Michael Economides and Ken
Nolte; and the somewhat dated but classic volume, SPE Monograph
No. 12: Advances in Hydraulic Fracturing, edited by John Gidley,
Steve Holditch, Dale Nierode, and Ralph Veatch. While these books
provide historical perspective as well as in-depth discussion and opinions (some controversial) on various details of the fracturing process,
they are not recommended for a first reading because of the highly
technical language and compartmentalized style of presentation.
Which sections of the book that you will use—whether it’s a quick
review of the introductory material or a check of the glossary, reading
Hydraulic Fracturing for Production or Injection Enhancement
the chapter on fracturing fluids, only, or hands-on use of the
design theory and software—depends on your role in the fracturing operation.
Neither this book nor any other technology transfer mechanism is
useful apart from capable people. The following key personnel comprise the fracturing team and the targeted readership of this book.
Fracturing Crew
A fracturing crew is the absolute minimum and basic unit required for
a fracturing treatment. The crew may consist of anywhere from 7 to
15 people, depending on the number of pumping units and the monitoring capability on location. Many of these people are trained to do
multiple jobs, such as driving trucks, hooking up equipment, and
installing and maintaining the monitoring instruments.
In addition to being trained on each piece of equipment that they
will operate, each member of the fracturing crew should be conversant with the material in Chapter 10, On-Site Quality Control, and the
accompanying Appendix F, Standard Practices and QC Forms.
The key people in any fracturing operation, in order of critical
importance, are:
Frac-Crew Chief—Sometimes known as the field engineer, this is the
person responsible on-site for proper execution of the job. He is a
highly experienced person, either an engineer that has reverted into a
field service manager position, or a highly gifted operator who has
been promoted to the job. The crew chief directs fracturing operations
from the monitoring van (“frac van”) and has complete responsibility
for the operation, including safety. He communicates constantly by
two-way radio with all pumping, blender, and proppant storage operators.
He is certified to operate high pressure equipment. He understands the
fracture design and is responsible for its implementation. He has complete authority to continue or shut down a job. (Note that while the
pronoun “he” is used for clarity, there are several highly capable
women currently practicing as fracturing engineers.)
This is not a job that can be learned gradually in a start-up operation. This individual must be identified through a careful search
among qualified candidates. Extensive and relevant hands-on experience
in fracture execution is a must. The frac-crew chief should be highly
conversant with Unified Fracture Design in its entirety.
Desk Engineer—The desk engineer concept is practiced by many companies, within and external to the petroleum industry. Simply put, the
Unified Fracture Design
fracturing service company places one of its full-time staff permanently on location in each client producing company. The client is
responsible to furnish a space (desk) at which the external employee
(engineer) can sit and work, giving rise to the term desk engineer. This
constant accessibility and the cross-pollination of needs (producing
company) and capabilities (service company) can dramatically improve
the range and success of application of a technology, and could be
especially important for the rapid and necessarily massive introduction of hydraulic fracturing in a new operating area or country.
This individual will have the same aptitude as the frac-crew chief,
but typically with somewhat less experience. Like the frac-crew chief,
the desk engineer should become highly conversant with the entire
fracturing book.
QA/QC Chemist—Any fracturing operation requires a chemist who is
well versed in the chemistry and physics (rheology) of fracturing fluids
and additives. This person operates a specially outfitted laboratory. The
laboratory includes, in addition to all basic implements and working
spaces (e.g., hoods), a Fann 50 high-pressure/high-temperature viscometer and possibly a fluid shear-history simulator. The chemist should
have a background in polymer chemistry, or at least a good understanding of the subject matter, and should be trained in detecting the
quality of proppant (visually, with a 100-magnification microscope).
The chemist is the field quality assurance/quality control (QA/QC)
officer. Prior to the fracture treatment, he inspects the make-up
water, fluid additives, and proppant to ensure that they are appropriate and that they are of high quality. During the treatment, he makes
sure that the fracturing materials are blended in the correct proportions and at the proper time (e.g., in the case of delayed crosslinkers).
He continues to spot check and approve the proppant quality in realtime for the duration of the treatment.
It is almost entirely the responsibility of the QC/QA chemist to
understand Chapter 6 and Chapter 9 of this book, as well as Appendix F, and to revise them for company-specific needs. In addition, this
person should fully digest the Stimulation Engineering Handbook.
Fracture Design Engineer—As the title suggests, this individual is
responsible for design of the fracturing treatment. The fracture design
engineer must master the basics of hydraulic fracturing, as included in
Chapters 4 through 9, and should be proficient to run the included
fracture design software. Depending on the magnitude of the fracturing
activity, there could be several people trained to perform this task. In
Hydraulic Fracturing for Production or Injection Enhancement
small operations, the same person may double-up as the field engineer that performs real-time analysis of the treatment from the frac
van (Chapter 10).
The fracture design engineer must have an engineering background,
preferably petroleum engineering, and be dedicated to study the subtle
and sometimes complex aspects of fracture design. Experience in the
industry is desirable, but not necessary. With proper training, a gifted
person can start functioning properly after several jobs. Ultimately, the
fracturing engineer should be broadly conversant in fracture execution,
fracturing fluid chemistry, and well completions. He should be able
to make critical use of the additional literature recommended above.
Well Stimulation as a
Means to Increase the
Productivity Index
The primary goal of well stimulation is to increase the productivity
of a well by removing damage in the vicinity of the wellbore or by
superimposing a highly conductive structure onto the formation. Commonly used stimulation techniques include hydraulic fracturing, frac
& pack, carbonate and sandstone matrix acidizing, and fracture
acidizing. Any of these stimulation techniques can be expected to
generate some increase in the productivity index, which, in turn, can
be used either to increase the production rate or decrease the pressure drawdown. There is no need to explain the benefits of increasing the production rate. The benefits of decreased pressure drawdown
are less obvious, but include minimizing sand production and water
coning and/or shifting the phase equilibrium in the near-well zone to
reduce condensate formation. Injection wells also benefit from stimulation in a similar manner.
To understand how stimulation increases productivity, basic production and reservoir engineering concepts are presented below.
In discussing the productivity of a specific well, we think of a linear
relation between the production rate and the driving force (pressure drawdown),
Unified Fracture Design
q = J∆p
where the proportionality “constant” J is called the productivity
index (PI). During its lifespan, a well is subject to several changes in
flow conditions, but the two most important idealizations are constant
production rate,
∆p =
α1 Bqµ
2 πkh
and constant drawdown pressure,
2 πkh∆p
α1 Bµ
where k is the formation permeability, h is the pay thickness, B is the
formation volume factor, µ is the fluid viscosity, and α1 is a conversion constant (equal to 1 for a coherent system). Either the production rate (q) or the drawdown (∆p) are specified, and therefore used
to define the dimensionless variables. Table 3-1 lists some of the wellknown solutions to the radial diffusivity equation.
Because of the radial nature of flow, most of the pressure drop
occurs near the wellbore, and any damage in this region significantly
increases the pressure loss. The impact of damage near the well can
be represented by the skin factor, s, added to the dimensionless pressure in the expression of the PI:
Bµ( pD + s)
The skin is another idealization, capturing the most important aspect
of near-wellbore damage: the additional pressure loss caused by the
TABLE 3-1. Flow into an Undamaged Vertical Well
Flow Regime
pD (G1/qD)
1  1 
, where t D =
Ei −
φµct rw2
2  4t D 
Transient (infinite
acting reservoir)
pi – pwf
pD = −
Steady state
pe – pwf
p– – p
pD = ln(re /rw)
Pseudo-steady state
pD = ln(0.472re /rw)
Well Stimulation as a Means to Increase the Productivity Index
damage is proportional to the production rate. Even with best drilling
and completion practices, some kind of near-well damage is present
in most cases. The skin can be considered as the measure of the “goodness” of a well. Other mechanical factors, not caused by damage per
se may add to the skin effect. These may include bad perforations,
partial well penetration, or undersized well completion equipment, and
so on. If the well is damaged (or its productivity is less than the ideal
reference value for any other reason), the skin factor is positive.
Well stimulation increases the productivity index. It is reasonable
to look at any type of stimulation as an operation to reduce the skin
factor. With the generalization to negative values of skin factor, even
such stimulation treatments—which not only remove damage but also
superimpose some new or improved conductivity paths—can be put
into this framework. In the latter case, it is more correct to speak about
pseudo-skin factor, indicating that stimulation causes some structural
changes in the fluid flow path as well as removing damage.
As we explained in Chapter 1, crucial from the fracture design
viewpoint is the pseudo-steady state productivity index:
2 πkh
p − pwf α1 Bµ
where JD is called the dimensionless productivity index.
For a well located in the center of a circular drainage area, the
dimensionless productivity index in pseudo-steady state reduces to
JD =
 0.472 re 
ln 
 rw 
In the case of a propped fracture, there are several ways to incorporate the stimulation effect into the productivity index. One can use
the pseudo-skin concept,
 0.472 re 
ln 
 + sf
 rw 
or the equivalent wellbore radius concept,
JD =
JD =
 0.472 re 
ln 
 rw′ 
Unified Fracture Design
or one can just provide the dimensionless productivity index as a function of the fracture parameters,
JD =
function of drainage-volume geometry
and fracture parameters
All three options give exactly the same results (if done in coherent terms). The last option is the most general and convenient, especially if we wish to consider fractured wells in more general drainage
areas (not necessarily circular).
Many authors have provided charts and correlations in one form
or another to handle special geometries and reservoir types. Unfortunately,
most of the results are less obvious or difficult to apply in higher
permeability cases. Even for the simplest possible case, a vertical fracture intersecting a vertical well, there are quite large discrepancies (see,
for instance, Figure 12-13 of Reservoir Stimulation, Third Edition).
We consider a fully penetrating vertical fracture in a pay layer of
thickness h, as shown in Figure 3-1.
Note that in reality the drainage area is neither circular nor rectangular, however, for most drainage shapes these geometries are reasonable approximations. Using re or xe is only a matter of convenience.
The relation between the drainage area A, the drainage radius re and
the drainage side length, xe , is given by
A = re2 π = xe2
For a vertical well intersecting a rectangular vertical fracture that
penetrates fully from the bottom to the top of the rectangular drainage
2x f
2 re
FIGURE 3-1. Notation for fracture performance.
Well Stimulation as a Means to Increase the Productivity Index
volume, the performance is known to depend on the penetration ratio
in the x direction,
Ix =
2x f
and on the dimensionless fracture conductivity,
C fD =
kf w
kx f
where xf is the fracture half length, xe is the side length of the square
drainage area, k is the formation permeability, kf is the proppant pack
permeability, and w is the average (propped) fracture width.
The key to formulating a meaningful technical optimization problem
is to realize that the fracture penetration and the dimensionless fracture conductivity (through width) are competing for the same resource:
the propped volume. Once the reservoir and proppant properties and
the amount of proppant are fixed, one has to make the optimal compromise between width and length. The available propped volume puts
a constraint on the two dimensionless numbers. To handle the constraint easily, we introduce the dimensionless proppant number:
N prop = I x2 C fD
The proppant number as defined above is just a combination of
the other two dimensionless parameters: penetration ratio and dimensionless fracture conductivity. Substituting the definition of the penetration ratio and dimensionless fracture conductivity into Equation 3-13,
we obtain
N prop =
4k f x f w
4 k f x f wh
kxe2 h
2 k f Vprop
where Nprop is the proppant number, dimensionless; kf is the effective
proppant pack permeability, md; k is the formation permeability, md;
Vprop is the propped volume in the pay (two wings, including void
Unified Fracture Design
space between the proppant grains), ft3; and Vres is the drainage volume (i.e., drainage area multiplied by pay thickness), ft3. (Of course,
any other coherent units can be used, because the proppant number
involves only the ratio of permeabilities and the ratio of volumes.)
Equation 3-14 plainly reveals the meaning of the proppant number: it is the weighted ratio of propped fracture volume (two wings)
to reservoir volume, with a weighting factor of two times the proppantto-formation permeability contrast. Notice, only the proppant that
reaches the pay layer is counted in the propped volume. If, for
instance, the fracture height is three times the net pay thickness, then
V prop can be estimated as the bulk (packed) volume of injected
proppant divided by three. In other words, the packed volume of the
injected proppant multiplied by the volumetric proppant efficiency
yields the Vprop used in calculating the proppant number.
The dimensionless proppant number, Nprop , is by far the most
important parameter in unified fracture design.
Figure 3-2 shows JD represented in a traditional manner, as a function of dimensionless fracture conductivity, CfD , with Ix as a parameter. Similar graphs showing productivity increase are common in the
published literature.
Dimensionless Productivity Index, JD
lx = 1
y e = xe
2x f
Dimensionless Fracture Conductivity, CfD
FIGURE 3-2. Dimensionless productivity index as a function of dimensionless fracture
conductivity, with Ix as a parameter (McGuire-Sikora type representation).
Well Stimulation as a Means to Increase the Productivity Index
However, Figure 3-2 is not very helpful in solving an optimization problem involving a fixed amount of proppant. For this purpose,
in Figures 3-3 and 3-4, we present the same results, but now with the
proppant number, Nprop , as a parameter. The individual curves correspond to JD at a fixed value of the proppant number.
As seen from Figures 3-3 and 3-4, for a given value of Nprop , the
maximum productivity index is achieved at a well-defined value of
the dimensionless fracture conductivity. Because a given proppant number represents a fixed amount of proppant reaching the pay, the best
compromise between length and width is achieved at the dimensionless
fracture conductivity located under the peaks of the individual curves.
One of the main results seen from the figures is, that at proppant
numbers less than 0.1, the optimal compromise occurs always at CfD
= 1.6. When the propped volume increases, the optimal compromise
happens at larger dimensionless fracture conductivities, simply because
the dimensionless penetration cannot exceed unity (i.e., once a fracture reaches the reservoir boundary, additional proppant is allocated
only to fracture width). This effect is shown in Figure 3-4, as is the
absolute maximum achievable dimensionless productivity index of
1.909. The maximum value of PI, equal to 6/π, is the productivity
index corresponding to perfect linear flow in a square reservoir.
Dimensionless Productivity Index, JD
lx = 1
Np = 0.1
xe = ye
Np = 0.06
Np = 0.03
Np = 0.01
Np = 0.006
Np = 0.003
Np = 0.001
Np = 0.0006
Np = 0.0003
Np = 0.0001
Dimensionless Fracture Conductivity, CfD
FIGURE 3-3. Dimensionless productivity index as a function of dimensionless fracture
conductivity, with proppant number as a parameter (for Nprop < 0.1).
Unified Fracture Design
Dimensionless Productivity Index, JD
Ix = 1
xe = ye
Np = 100
Np = 60
Np = 30
Np = 10
Np = 6
Np = 3
Np = 1
Np = 0.6
Np = 0.3
Np = 0.1
Dimensionless Fracture Conductivity, CfD
FIGURE 3-4. Dimensionless productivity index as a function of dimensionless fracture
conductivity, with proppant number as a parameter (for Nprop > 0.1).
In medium and high permeability formations (above 50 md), it is
practically impossible to achieve a proppant number larger than 0.1.
For frac & pack treatments, typical proppant numbers range between
0.0001 and 0.01. Thus, for medium to high permeability formations,
the optimum dimensionless fracture conductivity is always CfDopt = 1.6.
In “tight gas” reservoirs, it is possible to achieve large dimensionless proppant numbers, at least in principle. Proppant numbers
calculated for a limited drainage area—and not questioning the portion of proppant actually contained in the pay layer—can be as high
as 1 to 10. However, in practice, proppant numbers larger than 1 may
be difficult to achieve. For large treatments, the proppant can migrate
upward, creating excessive and unplanned fracture height, or it might
penetrate laterally outside of the assigned drainage area.
The situation is more complex for an individual well in a larger
area. In this case, a (hypothetical) large fracture length tends to
increase the drained reservoir volume, and the proppant number decreases.
Ultimately, the large fracture is beneficial, but the achievable proppant
number remains limited.
In reality, even trying to achieve proppant numbers larger than
unity would be extremely difficult. Indeed, for a large proppant number, the optimum CfD determines an optimum penetration ratio near
unity. This can be easily seen from Figure 3-5, where the penetration
Well Stimulation as a Means to Increase the Productivity Index
Np = 100
Np = 30
Dimensionless Productivity Index, JD
Np = 10
xe = ye
Np = 6
Np = 3
Np = 1
Np = 0.6
Np = 0.3
Np = 0.1
Penetration Ratio, Ix
FIGURE 3-5. Dimensionless productivity index as a function of penetration ratio, with
proppant number as a parameter (for Nprop > 0.1).
ratio is shown on the x-axis. To place the proppant “wall-to-wall”
while keeping it inside the drainage volume would require a precision in the fracturing operation that is practically impossible
to achieve.
The maximum possible dimensionless productivity index for
Nprop = 1 is about JD = 0.9. The dimensionless productivity index of
an undamaged vertical well is between 0.12 and 0.14, depending on
the well spacing and assumed well radius. Hence, there is a realistic
maximum for the “folds of increase” in the pseudo-steady state productivity index (with respect to the zero skin case), i.e., 0.9 divided
by 0.13 is approximately equal to 7. Larger folds of increase are not
likely. Of course, larger folds of increase can be achieved with respect
to an originally damaged well where the pre-treatment skin factor has
a large and positive value.
Another common misunderstanding is related to the transient flow
period. Under transient flow, the productivity index (and hence the production rate) is larger than in the pseudo-steady state case. With this
qualitative picture in mind, it is easy to discard the pseudo-steady state
optimization procedure and to “shoot for” very high dimensionless fracture conductivities and/or to anticipate many more folds
of increase in the productivity. In reality, the existence of a transient
flow period does not change the previous conclusions on optimal
Unified Fracture Design
dimensions. Our calculations show that there is no reason to depart
from the optimum compromise derived for the pseudo-steady state
case, even if the well will produce in the transient regime for a considerable time (say months or even years). Simply stated, what is good
for maximizing pseudo-steady state flow is also good for maximizing
transient flow.
In the definition of proppant number, kf is the effective (or equivalent, as it is sometimes called) permeability of the proppant pack. This
parameter is crucial in design. Current fracture simulators generally
provide a nominal value for the proppant pack permeability (supplied
by the proppant manufacturer) and allow it to be reduced by a factor
that the user selects. The already-reduced value should be used in the
proppant number calculation.
There are numerous reasons why the actual (or equivalent) proppant
pack permeability will be lower than the nominal value. The main
reasons are as follows:
Large closure stresses crush the proppant, reducing the average
grain size, grain uniformity, and porosity.
Fracturing fluid residue decreases the permeability in the fracture.
High fluid velocity in the proppant pack creates “non-Darcy
effects,” resulting in additional pressure loss. This phenomenon
can be significant when gas is produced in the presence of a
liquid (water and/or condensate). The non-Darcy effect is caused
by the periodic acceleration-deceleration of the liquid droplets,
effectively reducing the permeability of the proppant pack. This
reduced permeability can be an order of magnitude lower than
the nominal value presented by the manufacturer.
During the fracture design, considerable attention must be paid to
the effective permeability of the proppant pack and to the permeability of the formation. Knowledge of the effective permeability contrast
is crucial, and cannot be substituted by qualitative reasoning.
Well Performance for Low
and Moderate Proppant Numbers
By low and moderate proppant numbers, we mean anything less
than 0.1. The most dynamic fracturing activities (frac & pack, for
example) fall into this category—making it extremely important from
a design standpoint.
Well Stimulation as a Means to Increase the Productivity Index
The optimum treatment design for moderate proppant numbers can
be simply and concisely presented in an analytical form. In the process,
we will show how the proppant number and dimensionless productivity index relate to some other popular performance indicators, such
as the Cinco-Ley and Samaniego pseudo-skin function and Prats’
equivalent wellbore radius. In fact, fracture designs based on these
related performance indicators are just the moderate (low) proppant
number limit of the more comprehensive unified fracture design.
Prats (1961) introduced the concept of equivalent wellbore radius
resulting from a fracture treatment. He also showed that, except for
the fracture extent, all fracture variables affect well performance only
through the combined quantity of dimensionless fracture conductivity. When the dimensionless fracture conductivity is high (e.g., greater
than 100), the behavior is similar to that of an infinite conductivity
fracture. The behavior of infinite conductivity fractures was studied
later by Gringarten and Ramey (1974). To characterize the impact of
a finite-conductivity vertical fracture on the performance of a vertical
well, Cinco-Ley and Samaniego (1981) introduced a pseudo-skin function which is strictly a function of dimensionless fracture conductivity.
According to the definition of pseudo-skin factor, the dimensionless pseudo-steady state productivity index can be given as
JD =
ln 0.472
+ sf
where sf is the pseudo-skin. In Prats’ notation the same productivity
index is described by
JD =
ln 0.472
where rw′ is the equivalent wellbore radius. Prats also used the relative equivalent wellbore radius defined by rw′ / x f .
In the Cinco-Ley formalism, the productivity index is described as
JD =
ln 0.472
where f is the pseudo-skin function with respect to the fracture half-length.
Unified Fracture Design
Table 3-2 shows the relations between these quantities.
The advantage of the Cinco-Ley and Samaniego formalism ( f-factor)
is that, for moderate (and low) proppant numbers, the quantity f
depends only on the dimensionless fracture conductivity. The solid line
in Figure 3-6 shows the Cinco-Ley and Samaniego f-factor as a function of dimensionless fracture conductivity.
Note that for large values of CfD, the f-factor expression approaches
ln(2), indicating that the production from an infinite conductivity fracture is equivalent to the production of π/2 times more than the production from the same surface arranged cylindrically (like the wall of
a huge wellbore). In calculations, it is convenient to use an explicit
expression of the form
f =
1.65 − 0.328u + 0.116u 2
1 + 0.18u + 0.064u 2 + 0.005u 3
where u = ln C fD
Because the relative wellbore radius of Prats can be also expressed
by the f-factor (see Table 3-2), we obtain the equivalent result:
1.65 − 0.328u + 0.116u 2 
= exp −
, where u = ln C fD
 1 + 0.18u + 0.064u + 0.005u 
The simple curve-fits represented by Equations 3-18 and 3-19 are
only valid over the range indicated in Figure 3-6. For very large values of CfD, one can simply use the limiting value for Equation 3-19,
which is 0.5, showing that the infinite conductivity fracture has a
productivity similar to an imaginary (huge) wellbore with radius xf /2.
Interestingly enough, infinite conductivity behavior does not mean
that we have selected the optimum way to place a given amount of
proppant into the formation.
TABLE 3-2. Relations Between Various Performance Indicators
xf 
f = s f + ln  
 rw 
r 
s f = ln  w 
 rw′ 
rw′ = rw exp[ − s f ]
rw′ = x f exp[ − f ]
= exp[ − f ]
= w exp[ − s f ]
Well Stimulation as a Means to Increase the Productivity Index
y = 0.5ln C fD + f
C fD,opt
FIGURE 3-6. Cinco-Ley and Samaniego f-factor and the y-function.
In this context (Nprop < 0.1), a strictly physical optimization problem
can be formulated: How should we select the length and width if the
propped volume of one fracture wing, Vf = w × h × xf , is given as a
constraint, and we wish to maximize the PI in the pseudo-steady state
flow regime. It is assumed that the formation thickness, drainage
radius, and formation and proppant pack permeabilities are known, and
that the fracture is vertically fully penetrating (i.e., hf = h).
Selecting CfD as the decision variable, the length is expressed as
 Vf k f 
xf = 
 C fD hk 
1/ 2
Substituting Equation 3-21 into 3-17, the dimensionless productivity
index becomes
JD =
+ 0.5 ln C fD + f
ln 0.472 re + 0.5 ln
Vf k f
where the only unknown variable is CfD. Because the drainage radius,
formation thickness, the two permeabilities, and the propped volume
are fixed, the maximum PI occurs when the quantity in parentheses,
Unified Fracture Design
y = 0.5 ln C fD + f
reaches a minimum. That quantity is also shown in Figure 3-6. Because
the above expression depends only on CfD, the optimum, CfD,opt = 1.6
is a given constant for any reservoir, well, and proppant volume.
This result provides a deeper insight to the real meaning of
dimensionless fracture conductivity. The reservoir and the fracture can
be considered as a system working in series. The reservoir can feed
more fluids into the fracture if the length is larger, but (since the
volume is fixed) this means a narrower fracture. In a narrow fracture,
the resistance to flow may be significant. The optimum dimensionless fracture conductivity corresponds to the best compromise between
the requirements of the two subsystems. Once it is found, the optimum
fracture half-length can be calculated from the definition of CfD as
 Vf k f 
xf = 
 1.6hk 
1/ 2
and consequently, the optimum propped average width should be
 1.6Vf k 
 hk f 
1/ 2
hx f
Notice that Vf is Vprop /2 because it is only one half of the propped
The most important implication of the above results is that there
is no theoretical difference between low and high permeability fracturing. In all cases, there exists a physically optimal fracture that
should have a CfD near unity. In low permeability formations, this
requirement results in a long and narrow fracture; in high permeability formations, a short and wide fracture provides the same dimensionless conductivity.
If the fracture length and width are selected according to the
optimum compromise, the dimensionless productivity index will be
J D,max =
0.99 − 0.5 ln N prop
Of course, the indicated optimal fracture dimensions may not be
technically or economically feasible. In low permeability formations,
Well Stimulation as a Means to Increase the Productivity Index
the indicated fracture length may be too large, or the extreme narrow
width may mean that the assumed constant proppant permeability is
no longer valid. In high permeability formations, the indicated large
width might be impossible to create. For more detailed calculations,
all the constraints must be taken into account, but, in any case, a
dimensionless fracture conductivity far from the optimum indicates that
either the fracture is a relative “bottleneck” (CfD << 1.6) or that it is
too “short and wide” (CfD >> 1.6).
The reader should not forget that the results of this section—
including the Cinco-Ley and Samaniego graph and its curve fit, the
optimum dimensionless fracture conductivity of 1.6, and Equation 3-26—
are valid only for proppant numbers less than 0.1. This can be easily
seen by comparing Figures 3-3 and 3-4. In Figure 3-3, the curves have
their maximum at CfD = 1.6, and the maximum JD corresponds to the
simple Equation 3-26. In Figure 3-4, however, where the proppant
numbers are larger than 0.1, the location of the maximum is shifted,
and the simple calculations based on the f-factor (Equation 3-18) or
on the equivalent wellbore radius (Equation 3-19) are no longer valid.
Optimization routines found on the CD that accompanies this book
are based on the full information contained in Figures 3-3 and 3-4,
and formulas developed for moderate proppant numbers are used only
in the range of their validity.
We wish to place a certain amount of proppant in the pay interval and
to place it in such a way that the maximum possible productivity
index is realized. The key to finding the right balance between size
and productivity improvement is in the proppant number. Since Vprop
includes only that part of the proppant that reaches the pay, and hence
is dependent on the volumetric proppant efficiency, the proppant number cannot be simply fixed during the design procedure.
In unified fracture design, we specify the amount of proppant
indicated for injection and then proceed as follows:
1. Assume a volumetric proppant efficiency and determine the
proppant number. (Once the treatment details are obtained, the
assumed volumetric proppant efficiency related to created fracture height may be revisited and the design process may be
repeated in an iterative manner.)
Unified Fracture Design
2. Use Figure 3-3 or Figure 3-4 (or rather the design spreadsheet)
to calculate the maximum possible productivity index, JDmax ,
and also the optimum dimensionless fracture conductivity,
CfDopt , from the proppant number.
3. Calculate the optimum fracture half-length. Denoting the volume of one propped wing (in the pay) by Vf , the optimum fracture half-length can be calculated as
 Vf k f 
xf = 
 C fD,opt hk 
1/ 2
4. Calculate the optimum averaged propped fracture width as
 C fD,opt Vf k 
 hk f
1/ 2
In the above two equations, Vf and h must correspond to each
other. If total fracture height is used for h, which is often denoted by
hf , then the proppant volume Vf must be the total propped volume of
one wing. However, if the selected Vf corresponds only to that portion of one wing volume that is contained in the pay layer, then h
should be the net thickness of the pay. The final result for optimum
length and width will be the same in either case. It is a better practice,
however, to use net thickness and net volume (contained in the pay)
because those variables are also used to calculate the proppant number.
Once reservoir engineering and economic considerations have dictated the fracture dimensions to be created, the next issue is how to
achieve that goal. From this point, design of the fracture treatment can
be viewed as adjusting treatment details (pumping time and proppant
schedule) to achieve the desired final fracture dimensions.
In the next chapter, we outline the mechanics of fracture creation
in some detail. This theoretical basis is needed before we can proceed
to design the fracture treatment, our ultimate goal.
Fracturing Theory
In the following, we briefly summarize the most important mechanical concepts related to hydraulic fracturing.
Elasticity implies reversible changes. The initiation and propagation
of a fracture means that the material has responded in an inherently
non-elastic way, and an irreversible change has occurred. Nevertheless, linear elasticity is a useful tool when studying fractures because
both the stresses and strains (except perhaps in the vicinity of the
fracture face, and especially the tip) may still be adequately described
by elasticity theory.
A linear elastic material is characterized by elastic constants that
can be determined in static or dynamic loading experiments. For an
isotropic material, where the properties are independent of direction,
two constants are sufficient to describe the behavior.
Figure 4-1 is a schematic representation of a static experiment with
uniaxial loading. The two parameters obtained from such an experiment are the Young’s modulus (E) and the Poisson ratio (ν). They are
Unified Fracture Design
ε xx =
ε yy =
σ xx
σ xx
ε xx
ε yy
ε xx
FIGURE 4-1. Uniaxial loading experiment.
calculated from the vertical stress (σxx) vertical strain (εxx ) and horizontal strain (ε yy), as shown in the figure.
Table 4-1 shows the interrelation of those constants most often
used in hydraulic fracturing. The plane strain modulus (Eⴕ ) is the only
elastic constant really needed in our equations.
In linear elastic theory, the concept of plane strain is often used
to reduce the dimensionality of a problem. It is assumed that the body
is infinite in at least one direction, and external forces (if any) are
applied parallel to this direction (i.e., “infinitely repeated” in every
cross section). In such case, it is intuitively obvious that the state of
strain also repeats itself infinitely.
TABLE 4-1. Interrelation of Various Properties of a Linear Elastic Material
E, ν
G, ν
E, G
2(1 + ν)
Young’s modulus, E
2G(1 + ν)
Poisson ratio, ν
E − 2G
1 − ν2
1− ν
4G 2
4G − E
Shear modulus, G
Plane strain modulus, Eⴕ
Fracturing Theory
Plane strain is a reasonable approximation in a simplified description of hydraulic fracturing. The main question is how to select the
plane. Two possibilities arise, and, in turn, this has given rise to two
different approaches to fracture modeling. The state of plane strain was
assumed in the horizontal plane by Khristianovitch and Zheltov (1955)
and by Geertsma and de Klerk (1969), while plane strain in the vertical plane (normal to the direction of fracture propagation) was
assumed by Perkins and Kern (1961) and Nordgren (1972).
Often, in the hydraulic fracturing literature, the term “KGD”
geometry is used interchangeably to the horizontal plane-strain assumption and “PKN” geometry is used as a substitute for postulating plane
strain in the vertical plane.
Exact mathematical solutions are available for the problem of a
pressurized crack in the state of plane strain. In particular, it is well
known that the pressurized line crack has an elliptical width distribution (Sneddon, 1973):
w( x ) =
4 p0 2
c − x2
where x is the distance from the center of the crack, c is the halflength (the distance of the tip from the center) and p0 is the constant
pressure exerted on the crack faces from inside. From Equation 4-1,
the maximum width at the center is
w0 =
indicating that a linear relationship is maintained between the crack
opening induced and the pressure exerted. When the concept of pressurized line crack is applied for a real situation, p0 is substituted with
the net pressure, pn , defined as the difference of the inner pressure
and the minimum principal stress acting from outside, trying to close
the fracture (Hubbert and Willis, 1957; Haimson and Fairhurst, 1967).
Fracture mechanics has emerged from the observation that any
existing discontinuity in a solid deteriorates its ability to carry loads.
A (possibly small) hole may give rise to high local stresses compared
to the ones being present without the hole. The high stresses, even if
they are limited to a small area, may lead to the rupture of the material. It is often convenient to look at material discontinuities as stress
concentrators which locally increase the otherwise present stresses.
Unified Fracture Design
Two main cases must be distinguished. If the form of discontinuity is smooth (e.g., a circular borehole in a formation), then the maximum stress around the discontinuity is higher than the virgin stress
by a finite factor, which depends on the geometry. For example, the
stress concentration factor for a circular borehole is three.
The situation is different in the case of sharp edges, such as at
the tip of a fracture. Then the maximum stress at the tip becomes
infinite. In fracture mechanics, we have to deal with singularities. Two
different loadings (pressure distributions) of a line crack result in two
different stress distributions. Both cases may yield infinite stresses at
the tip, but the “level of infinity” is different. We need a quantity to
characterize this difference. Fortunately, all stress distributions near
the tip of any fracture are similar in the sense that they decrease
according to r –1/2, where r is the distance from the tip. The quantity
used to characterize the “level of infinity” is the stress intensity factor, KI , defined as the multiplier to the r –1/2 function. For the idealization of a pressurized line crack with half-length, c, and constant
pressure, p0, the stress intensity factor is given by
K I = p0 c1/ 2
In other words, the stress intensity factor at the tip is proportional to
the constant pressure opening up the crack and to the square root of
the crack half-length (characteristic dimension).
According to the key postulate of linear elastic fracture mechanics (LEFM), for a given material there is a critical value of the stress
intensity factor, KIC , called fracture toughness. If the stress intensity
factor at the crack tip is above the critical value, the crack will propagate; otherwise it will not. Fracture toughness is a useful quantity for
safety calculations, when the engineer’s only concern is to avoid fracturing. In well stimulation, where the engineer’s primary goal is to
create and propagate a fracture, the concept has been found somewhat controversial because it predicts that less and less effort is necessary to propagate a fracture with increasing extent. In the large scale,
however, the opposite is usually true.
Fluid materials deform continuously (in other words, flow) without
rupture when subjected to a constant stress. Solids generally will assume
Fracturing Theory
a static equilibrium deformation under the same stresses. Crosslinked
fracturing fluids usually behave as viscoelastic fluids. Their stressstrain material functions fall between those of pure fluids and solids.
From our point of view, the most important property of fluids is
their resistance to flow. The local intensity of flow is characterized
by the shear rate, γ̇ , measured in 1/s. It can be considered as the rate
of change of velocity with the distance between sliding layers. The
stress emerging between the layers is the shear stress, τ. Its dimension is force per unit area (in SI units, Pa). The material function
relating shear stress and shear rate is the rheological curve. This information is necessary to calculate the pressure drop (actually, energy dissipation) for a given flow situation, such as flow in pipe or flow
between parallel plates.
Apparent viscosity is defined as the ratio of stress to shear rate.
Generally, the apparent viscosity varies with shear rate, except in
the case of a Newtonian fluid—a very specific fluid in which the viscosity is a constant. The rheological curve and the apparent viscosity
curve contain the same information and are used interchangeably.
Figure 4-2 shows typical rheological curves, and Table 4-2 lists some
commonly used rheological constitutive equations.
Shear Stress, τ
Shear Rate, γ̇
FIGURE 4-2. Typical rheological curves.
Unified Fracture Design
TABLE 4-2. Commonly Used Rheological Constitutive Equations
τ = µγ˙
τ = Kγ˙ n
Power law
τ = τ y + µ p γ˙
Bingham plastic
τ = τ y + Kγ˙ n
Yield power law
The model parameters vary with chemical composition, temperature
and, to a lesser extent, many other factors including shear history. In
the case of foams, the volumetric ratio between the gas and liquid
phases plays an important role (Reidenbach, 1985; Winkler, 1995).
Most fracturing gels exhibit significant shear thinning (i.e., loss
of viscosity with increasing shear rate). A constitutive equation that
captures this primary aspect of their flow behavior is the Power law
model. The flow behavior index, n, usually ranges from 0.3 to 0.6.
All fluids exhibit some finite limiting viscosity at high shear rates.
The build-up of very high apparent viscosity at low shear might be
approximated by the inclusion of a yield stress for certain fluids.
Many fluids demonstrate what appears to be Newtonian behavior at
low shear rates.
Much of the current rheology research focuses on building more
realistic apparent viscosity models that effectively incorporate each of
the previously mentioned characteristics as well as the nonlinear, timedependent viscoelastic effects of crosslinked gels.
A rheological model is used to predict the pressure losses (gradient)
associated with an average fluid flow velocity in a given physical
geometry. The equations of motion have been solved for the standard
rheological models in the most obvious geometries (e.g., flow in circular tubes, annuli, and between thin-gap parallel plates). The solution is often presented as a relation between average linear velocity
(flow rate per unit area) and pressure drop. In calculations, it is convenient to use the equivalent Newtonian viscosity (µe), that is, the
viscosity that would be used in the equation of the Newtonian fluid
to obtain the same pressure drop under the same flow conditions.
While apparent viscosity (at a given local shear rate) is the property
of the fluid, equivalent viscosity depends also on the flow geometry
and carries the same information as the pressure drop solution. For more
Fracturing Theory
complex rheological models, there is no closed-form solution (neither
for the pressure drop nor for the equivalent Newtonian viscosity), and
the calculations involve numerical root-finding.
Of particular interest to hydraulic fracturing is the laminar flow
in two limiting geometries. Slot flow occurs in a channel of rectangular cross section when the ratio of the longer side to the shorter side
is extremely large. Limiting ellipsoid flow occurs in an elliptic cross
section with extremely large aspect ratio. The former corresponds to
the KGD geometry and the latter to the PKN geometry.
Table 4-3 gives the solutions commonly used in hydraulic fracturing calculations. The most familiar equation, valid for Newtonian
behavior, is presented first. Then an equivalent viscosity is given for
the Power law fluid. The equivalent viscosity can be used in the
Newtonian form of the pressure drop equation. Notice that the equivalent viscosity depends on the average velocity (u avg) and on the
geometry of the flow channel (in case of slot flow, on the width, w;
in case of elliptical cross section, on the maximum width, w0). It is
interesting to note that the equation for laminar flow of a Power law
fluid in the limiting ellipsoid geometry has not been derived. The
solution presented here can be obtained by analogy considerations (for
details, see Valkó and Economides, 1995).
The friction pressure associated with pumping fracturing fluids
through surface lines and tubulars cannot be calculated directly using
the classic turbulent flow correlations. Special relations have to be
applied to account for the drag reduction phenomena caused by the
long polymer chains. Rheological behavior also plays an important
role in the proppant carrying capacity of the fluid (Roodhart, 1985;
Acharya, 1986).
TABLE 4-3. Pressure Drop and Equivalent Newtonian Viscosity
Rheological model
τ = µγ˙
Power law
τ = Kγ˙ n
Slot flow
∆p 12µuavg
µe =
2 n−1  1 + 2 n 
n −1
Kw1− n uavg
3  n 
Ellipsoid flow
∆p 16µuavg
µe =
2 n−1  1 + ( π − 1)n 
1− n n −1
 Kw0 uavg
π 
Unified Fracture Design
The polymer content of the fracturing fluid is partly intended to impede the loss of fluid into the reservoir. This phenomenon is envisaged as a continuous build-up of a thin layer of polymer (the filter
cake), which manifests an ever-increasing resistance to flow through
the fracture face. The actual leakoff is determined by a coupled system that includes not only the filter cake, which is one element, but
also flow conditions in the reservoir.
A fruitful approximation dating back to Carter, 1957 (cf. appendix to Howard and Fast, 1957), is to consider the combined effect of
the different phenomena as a material property. According to this
concept, the leakoff velocity, vL , is given by the Carter I equation:
vL =
where CL is the leakoff coefficient (length/time1/2 ) and t is the time
elapsed since the start of the leakoff process. The integrated form of
the Carter equation is
= 2C L t + S p
where V Lost is the fluid volume that passes through the surface AL
during the time period from time zero to time t. The integration constant, SP, is called the spurt loss coefficient. It can be considered as
the width of the fluid body passing through the surface instantaneously at the very beginning of the leakoff process. Correspondingly, the term 2CL t can be considered as the leakoff width. (Note
that the factor 2 is an artifact of the integration. It has nothing to do
with the “two wings” and/or “two faces” introduced later.) The two
coefficients, CL and SP , can be determined from laboratory tests or,
preferably, from evaluation of a fracture calibration test.
Formal Material Balance: The Opening-Time
Distribution Factor
Consider the fracturing treatment shown schematically in Figure 4-3.
The volume Vi injected into one wing during the injection time te
consists of two parts: the volume of one fracture wing at the end of
Fracturing Theory
2q i
FIGURE 4-3. Notation for material balance.
pumping (Ve) and the volume lost (leakoff volume). The subscript e
denotes that a given quantity is being measured or referenced at the
end of pumping. Note that all the variables are defined with respect
to one wing. The area Ae denotes the surface of one face of one fracture wing. Fluid efficiency ηe is defined as the fraction of the fluid
remaining in the fracture: ηe = Ve / Vi. The average width, w , is
defined by the relation, V = Aw .
A hydraulic fracturing operation may last from tens-of-minutes up
to several hours. Points on the fracture face near the well are “opened”
at the beginning of pumping while the points near the fracture tip are
younger. Application of Equation 4-5 necessitates the tracking of the
opening-time of the different fracture face elements.
If only the overall material balance is considered, it is natural to
rewrite the injected volume as the sum of the fracture volume, leakoff
volume, and spurt volume using the formalism,
Vi = Ve + K L 2Ae C L te + 2 Ae S p
where the variable K L is the opening-time distribution factor. It
reflects the history of the evolution of the fracture surface, or rather
the distribution of the opening-time, hence the name. In particular, if
all the surface is opened at the beginning of the injection, then KL
Unified Fracture Design
reaches its absolute maximum, KL = 2. The fluid efficiency is the ratio
of the created volume to the injected volume. Dividing both volumes
by the final fracture area, we can consider fracture efficiency as the
ratio of the created width to the would-be width, where the would-be
width is defined as the sum of the created and lost widths.
Therefore, another form of Equation 4-6 is
ηe =
we + 2K L C L te + 2 S p
showing that the term 2K L C L te can be considered as the “leakoff
width,” and the term 2Sp as the “spurt width.” Equation 4-7 can be
rearranged to obtain the opening-time distribution factor in terms of
fluid efficiency and average width at the end of pumping:
KL = −
C L te
2 C L te 2ηe C L te
Note that these relations are independent of the actual shape of the
fracture face or the history of its evolution.
Constant Width Approximation (Carter Equation II)
In order to obtain an analytical solution for constant injection rate,
Carter considered a hypothetical case in which the fracture width
remains constant during the fracture propagation (the width “jumps”
to its final value in the first instant of pumping). Then a closed form
expression can be given for the fluid efficiency in terms of the two
leakoff parameters and the width:
ηe =
we we + 2 S p 
exp(β 2 ) erfc(β) +
− 1
4 πCL te
π 
where β =
2CL πte
we + 2 S p
Power Law Approximation to Surface Growth
A basic assumption postulated by Nolte (1979, 1986) leads to a
remarkably simple form of the material balance. He assumed that the
fracture surface evolves according to a power law,
Fracturing Theory
AD = t Dα
where AD = A/Ae and tD = t/te , and the exponent α remains constant
during the entire injection period. Nolte realized that, in this case, the
opening-time distribution factor is a function of α only. He represented
the opening-time distribution factor and its dependence on the exponent
of fracture surface growth using the notation g0(α) and presented g0
for selected values of α. A simple expression first obtained by Hagel
and Meyer (1989) can be used to obtain the value of the opening-time
distribution factor for any α:
g0 (α ) =
παΓ (α )
Γ (α + 2 / 3)
where Γ(α) is the Euler gamma function.
In calculations, the following approximation to the g0 function
might be easier to use:
g0 (α ) =
2 + 2.06798 α + 0.541262 α 2 + 0.0301598 α 3
1+1.6477 α + 0.738452 α 2 + 0.0919097 α 3 + 0.00149497 α 4
Nolte assumed that the exponent remains between 0.5 and 1. With
this assumption, the factor KL lies between 4/3 (1.33) and π/2 (1.57),
indicating that for two extremely different surface growth histories,
the opening-time distribution factor varies less than 20 percent. Generally, the simple approximation K L = 1.5 should provide enough
accuracy for design purposes.
Various practitioners have related the exponent α to fracture
geometry, fluid efficiency at the end of pumping, and fluid rheological behavior. None of these relations can be considered as proven
theoretically, but they are reasonable engineering approximations,
especially because the effect of the exponent on the final results is
limited. Our recommendation is to use α = 4/5 for the PKN, α = 2/3
for the KGD, and α = 8/9 for the radial model. These exponents can
be derived from the no-leakoff equations shown later in Table 4-4.
Numerically, the original constant-width approximation of Carter
and the power law surface growth assumption of Nolte give very similar results when used for design purposes. The g0-function approach does,
however, have technical advantages when applied to the analysis of
calibration treatments.
Unified Fracture Design
Detailed Leakoff Models
The bulk leakoff model is not the only possible interpretation of the
leakoff process. Several mechanistic models have been suggested in
the past (Williams, 1970 and Settari, 1985; Ehlig-Economides, et al.,
1994; Yi and Peden, 1994; Mayerhofer, et al., 1995). The total pressure difference between the inside of a created fracture and a far point
in the reservoir is written as the sum,
∆p(t ) = ∆p face (t ) + ∆p piz (t ) + ∆pres (t )
where ∆pface is the pressure drop across the fracture face dominated
by the filter cake, ∆ppiz is the pressure drop across a polymer-invaded
zone and ∆pres is the pressure drop in the reservoir. Depending on their
significance under the given conditions, one or two terms may be
neglected. While the first two terms are connected to the leakoff rate
at a given time instant, the reservoir pressure drop is transient. It
depends on the entire history of the leakoff process, not only on its
instant intensity.
The detailed leakoff models hold an advantage in that they are
based on physically meaningful parameters, such as permeability and
filter cake resistance, and they allow for explicit pressure-dependent
simulation of the leakoff process. However, the application of these
models is limited by the complexity of the mathematics involved and
by the extra input they require.
Engineering models for the propagation of a hydraulically induced
fracture combine elasticity, fluid flow, material balance, and (in some
cases) an additional propagation criterion. Given the fluid injection
history, a model should predict the evolution with time of the fracture dimensions and the wellbore pressure.
For design purposes, an approximate description of the geometry
might be sufficient, so simple models that predict fracture length and
average width at the end of pumping are very useful. Models that predict these two dimensions—while the third one, fracture height, is fixed—
are referred to as 2D models. If the fracture surface is postulated to
propagate in a radial fashion, that is, the height is not fixed, the model
is still considered to be 2D (the two dimensions being fracture radius
and width).
Fracturing Theory
A further simplification occurs if we can relate fracture length and
width, neglecting the details of leakoff for now. This is the basic
concept of the early so-called “width equations.” It is assumed that
the fracture evolves in two identical wings, perpendicular to the minimum principal stress of the formation. Because the minimum principal
stress is usually horizontal (except for very shallow formations), the
fracture will be vertical.
Perkins-Kern Width Equation
The PKN model assumes that the condition of plane strain holds in
every vertical plane normal to the direction of propagation; however,
unlike the rigorous plane-strain situation, the stress and strain state are
not exactly the same in subsequent planes. In other words, the model
applies a quasi-plane-strain assumption, and the reference plane is
vertical, normal to the propagation direction. Neglecting the variation
of pressure along the vertical coordinate, the net pressure, pn , is
considered as a function of the lateral coordinate x. The vertically constant pressure at a given lateral location gives rise to an elliptical cross
section. Straightforward application of Equation 4-1 provides the
maximum width of the ellipse as
w0 =
2h f pn
Perkins and Kern (1961) postulated that the net pressure is zero
at the tip of the fracture, and they approximated the average linear
velocity of the fluid at any location based on the one-wing injection
rate (qi ) divided by the cross-sectional area. They obtained the pressure loss equation in the form,
πw03h f
Combining Equations 4-14 and 4-15, and integrating with the zero
net pressure condition at the tip, they obtained the width profile:
w0 ( x ) = w w , o  1 − 
xf 
1/ 4
where the maximum width of the ellipse at the wellbore (see Figure 4-4) is given by
Unified Fracture Design
FIGURE 4-4. Basic notation for Perkins-Kern differential model.
 µqi x f 
ww, 0 = 3.57
 E′ 
1/ 4
In reality, the flow rate in the fracture is less than the injection
rate, not only because part of the fluid leaks off, but also because the
increase of width with time “consumes” another part of the injected
fluid. In fact, what is more or less constant along the lateral coordinate
at a given time instant, is not the flow rate, but rather the flow velocity,
uavg. However, repeating the Perkins-Kern derivation with a constant
flow velocity assumption has very little effect on the final results.
Equation 4-17 is the Perkins-Kern width equation. It shows the
effect of the injection rate, viscosity, and shear modulus on the width,
once a given fracture length is achieved. Knowing the maximum width
at the wellbore, we can calculate the average width, multiplying it by
a constant shape factor, γ:
w = γww, 0 , where γ =
π4 π
= = 0.628
45 5
The shape factor contains two elements. The first one is π/4, which
takes into account that the vertical shape is an ellipse. The second
element is 4/5, which accounts for lateral variation in the maximum width.
Fracturing Theory
In the petroleum industry, a version of Equation 4-17 with a
slightly different constant is used more often, and is referred to as the
Perkins-Kern-Nordgren (PKN) width equation (Nordgren, 1972):
 µqi x f 
ww, 0 = 3.27
 E′ 
1/ 4
Khristianovich-Zheltov-Geertsma-deKlerk Width Equation
The first model of hydraulic fracturing, elaborated by Khristianovich
and Zheltov (1955), envisioned a fracture with the same width at any
vertical coordinate within the fixed height, hf . The underlying physical hypothesis is that the fracture faces slide freely at the top and
bottom of the layer. The resulting fracture cross section is a rectangle.
The width is considered as a function of the coordinate x. It is determined from the plane-strain assumption, now applied in the (every)
horizontal plane. The Khristianovich and Zheltov model contained
another interesting assumption: the existence of a non-wetted zone near
the fracture tip. Geertsma and deKlerk (1969) accepted the main
assumptions of Khristianovich and Zheltov and reduced the model into
an explicit width formula. The KGD width equation is
336 
ww = 
 π 
µqi x 2f 
 E ′h 
f 
1/ 4 
1/ 4
 µqi x 2f 
= 3.22
 E ′h f 
1/ 4
In this case, the shape factor, relating the average width to
the wellbore width, has no vertical component. Then, because of the
elliptical horizontal shape, we obtain
w = γww , where γ =
= 0.785
Daneshy’s (1978) extension of the KGD model considers a nonconstant pressure distribution along the fracture length, and a nonNewtonian fracturing fluid whose properties can change with time and
temperature. Numerical computations yield the specific leakoff, increase
in width, and flow rate at points along the fracture length during fracture extension.
For short fractures, where 2xf < h f , the horizontal plane-strain
assumption (KGD geometry) is more appropriate, and for 2xf > hf ,
Unified Fracture Design
the vertical plane-strain assumption (PKN geometry) is physically
more sound. Interestingly, for the special case when the total fracture
length and height are equivalent, the two equations give basically the
same average width and, hence, fracture volume.
Radial (Penny-shaped) Width Equation
This situation corresponds to horizontal fractures from vertical wells,
vertical fractures extending from horizontal wells, or when fracturing
relatively thick homogeneous formations—from a limited perforation
interval in all cases. While the computations of fracture width are
sensitive to how the fluid enters the fracture (a true point source would
give rise to infinite pressure), a reasonable model can be postulated
by analogy, which results in the same average width as the PerkinsKern equation when R f = xf = hf / 2.
The result is
 µqi R f 
w = 2.24
 E′ 
1/ 4
The real significance of the simple models presented in this section is the insight they provide—helping us to consider the effect of
input data on the evolving fracture. Additional insight can be gained
by comparing the fracture geometry and net pressure behavior of
the models. Table 4-4 provides a direct side-by-side comparison of the
basic fracture models (no-leakoff case).
The last row in Table 4-4 deserves particular attention. For the
no-leakoff case, net pressure increases with time for the Perkins-Kern
model, but decreases with time for the other two models. This is a
well-known result that raises some questions. For example, in massive hydraulic fracturing, the net treating pressure most often increases
with time, so net pressures derived from the Geertsma-deKlerk and
radial models are of limited practical value. A more startling (and less
well-known) observation is that the net pressures provided by the
Geertsma-deKlerk and radial models are independent of injection rate.
The KGD (and radial) view implies that when the fracture extent
becomes large, very low net pressures are required to maintain a certain width. While this is a consequence of linear elasticity theory and
the way that the plane-strain assumption is applied, it leads to absurd
Fracturing Theory
TABLE 4-4. No-Leakoff Solutions of the Basic Fracture Models
Perkins and Kern
x f = c1t 4 / 5
x f = c1t 2 / 3
 q3E′ 
c1 = c1′  i 4 
 µh f 
625 
c1′ = 
 512 π 3 
1/ 5
1/ 5
= 0.524
c2′ =  2 
 π 
16 
c1′ = 
 21π 3 
1/ 6
1/ 6
= 0.539
1/ 5
 q 3µ 
c2 = c2′  i 3 
 E ′h f 
= 3.04
c2′ =  3 
 π 
 q3E′ 
c1 = c1′  i 
 µ 
c1′ = 0.572
1/ 6
1/ 6
= 2.36
 q 3µ 2 
c2 = c2′  i 2 
 E′ 
w = γw w
w = γw w,0
γ = 0.628
γ = 0.785
γ = 0.533
pn,w = c3t 1/ 5
pn,w = c3t −1/ 3
pn,w = c3t −1/ 3
c3′ =  2 
π 
1/ 4
1/ 9
c2′ = 3.65
w = γw w,0
 E ′ 4 µ qi2 
c3 = c3′ 
 hf 
1/ 9
w w,0 = c2 t 1/ 9
w w = c2 t 1/ 3
1/ 5
R f = c1t 4 / 9
 q3E′ 
c1 = c1′  i 3 
 µh f 
w w,0 = c2 t 1/ 5
 q2µ 
c2 = c2′  i 
 E ′h f 
Geertsma and deKlerk
1/ 5
= 1.52
c3 = c3′ ( E ′ 2 µ )
1/ 3
c3′ =  
 16 
1/ 3
= 1.09
c3 = c3′ ( E ′ 2 µ )
1/ 3
c3′ = 2.51
results in the large scale. It is safe to say that the PKN model captures
the physical fracturing process better than the other two models.
While many investigations have been performed during the last
half century, the same ingredients must always appear in the “mix”
of any suggested fracture model: material balance, relating injection
rate and fracture volume; linear elasticity, relating fracture width to
fracture extent; and fluid mechanics, relating width and pressure loss
along the fracture. Additionally, an explicit fracture propagation criterion may be or may not be present.
Fracturing of High
Permeability Formations
As recently as the early 1990s, hydraulic fracturing was used almost
exclusively for low permeability reservoirs. The large fluid leakoff and
unconsolidated sands associated with high permeability formations
would ostensibly prevent the initiation and extension of a single, planar
fracture with sufficient width to accept a meaningful proppant volume.
Moreover, such fracture morphology, even if successfully created and
propped, would be incompatible with the defined needs of moderate
to high permeability reservoirs, that is, large conductivity (width).
A key breakthrough tied to the advance of high permeability
fracturing (HPF) is the tip screenout (TSO), which arrests lateral
fracture growth and allows for subsequent fracture inflation and
packing. The result is short but wide to exceptionally wide fractures.
While in traditional, unrestricted fracture growth an average fracture
width of 0.25 in. would be considered normal, in TSO treatments,
widths of one inch or even larger are commonly expected.
The role of hydraulic fracturing has expanded to encompass oil
wells with permeabilities greater than 50 md and gas wells with over
5 md of permeability (Table 5-1). These wells clearly require a TSO
design. Because of these developments, hydraulic fracturing has
Unified Fracture Design
TABLE 5-1. Fracturing Role Expanded
k < 0.5 md
k < 5 md
0.5 < k < 5 md
5 < k < 50 md
k > 5 md
k > 50 md
captured an enormous share of all well completions, and further gains
are certain, only tempered by the economy of scale affecting many
petroleum provinces. In places such as the United States and Canada,
hydraulic fracturing is poised to be applied to almost all petroleum
wells drilled, as was shown in Figure 1-2.
It is interesting that HPF, which is often referred to as frac & pack
or fracpac, did not necessarily originate as an extension of hydraulic
fracturing—although HPF borrowed heavily from established techniques—but rather as a means of sand production control.
In controlling the amount of sand production to the surface, there
are two distinctly different activities that can be done downhole: sand
exclusion and sand deconsolidation control. Sand exclusion refers to
all filtering devices such as screens and gravel packs. Gravel packing,
the historically preferred well completion method to remedy sand production, is one such technique. These techniques do not prevent sand
migration in the reservoir, so fines migrate and lodge in the gravel
pack and screen, causing large damage skin effects. Well performance
progressively deteriorates and often is not reversible with matrix
stimulation treatments. Attempts to stem the loss in well performance
by increasing the pressure drawdown often aggravates the problem
further and may potentially lead to wellbore collapse.
A more robust approach is the control of sand deconsolidation,
(i.e., prevention of fines migration at the source). It is widely perceived
that the use of HPF accomplishes this by mating with the formation
in its (relative) undisturbed state and reducing fluid velocities or “flux”
at the formation face.
There are actually three factors that contribute to sand deconsolidation: (1) pressure drawdown and the “flux” created by the
resulting fluid production, (2) the strength of the rock and integrity
of the natural cementation, and (3) the state of stress in the formation. Of these three, the only factor that can be readily altered is the
distribution of flow and pressure drawdown. By introducing formation
Fracturing of High Permeability Formations
fluids to the well along a more elongated path (e.g., a hydraulic
fracture or horizontal well), it is entirely possible to reduce the fluid
flux and, in turn, control sand production.
Consider a simple example by assuming a well that penetrates a
100 ft thick reservoir. If the well has a diameter equal to 1 ft, then
the area for incoming radial flow in an open hole completion would
be about 300 ft2. However, for a fracture half-length of 100 ft, the
area of flow would be (2 × 100 × 100 × 2) 40,000 ft2. (Note: the
second 2 accounts for the two walls of the fracture.) Remember that
in a fractured well almost all fluid flow would be from the reservoir
into the fracture, and then along the fracture into the well. For the
same production rate, this calculation suggests the fluid flux in
a fractured well would be less than 1/100th the fluid flux in an
unfractured well.
Of course, not a great deal can be done to affect the state of stress
or formation competence. The magnitude of earth stresses depends
primarily on reservoir depth and to some extent pressure, with the
situation becoming more complicated at depths of 3,000 ft or less.
Pressure maintenance with gas or water flooding may be counterproductive unless maintenance of reservoir pressure allows economic
production at a smaller drawdown. Various innovations have been
suggested to remedy incompetent formations or improve on natural
cementation—for example, by introducing complex well configurations
or various exotic chemical treatments—but there is little that can be
done to control this factor either.
In light of the discussion above, it should not be surprising that
HPF has replaced gravel packs in many petroleum provinces susceptible to sand production, especially in operations where more
sophisticated engineering is done. As with any stimulation technique
that results in a productivity index improvement (defined as the
production rate divided by the pressure drawdown), it is up to the
operator to allocate this new productivity index either to a larger rate
or a lower drawdown, or any combination of the two.
HPF indicates a marked departure from the heritage of gravel
packing, incorporating more and more from hydraulic fracture technology. This trend can be seen, for instance, in the fluids and proppants
applied. While the original fracpack treatments involved sand sizes and
“clean” fluids common to gravel packing, the typical proppant sizes
for hydraulic fracturing (20/40 mesh) now dominate. The increased
application of crosslinked fracturing fluids also illustrates the trend.
Unified Fracture Design
For this reason, the terminology of “high permeability fracturing,”
or HPF, seems more appropriate than fracpack, and is used throughout
this book.
In the following section, HPF is considered in a semi-quantitative
light in view of competing technologies. This is followed by a discussion of the key issues in high permeability fracturing, including
design, execution, and evaluation.
Gravel Pack
Gravel pack refers to the placement of gravel (actually, carefully
selected and sized sand) between the formation and the well in order
to filter out (retain) reservoir particles that migrate through the porous
medium. A “screen” is employed to hold the gravel pack in place.
This manner of excluding reservoir fines from flowing into the well
invariably causes an accumulation of fines in the near-well zone and
a subsequent reduction in the gravel pack permeability (i.e., damage
is caused).
The progressive deterioration of gravel pack permeability (increased
skin effect) leads, in turn, to a decline in well production. Increasing
pressure drawdown to counteract production losses can result in
accelerated pore-level deconsolidation and additional sand production.
Any productivity index relationship (e.g., the steady-state expression for oil) can be used to demonstrate this point:
pe − pwf
 0.472 re
141.2 Bµ ln
+ s
Assuming k = 50 md, h = 100 ft, B = 1.1 res bbl/STB, µ = 0.75 cp
and ln re /rw = 8.5, the productivity indexes for an ideal (undamaged),
a relatively damaged (e.g., s = 10), and a typical gravel packed well
(e.g., s = 30) would be 5, 2.3, and 1.1 STB/d/psi, respectively. For a
drawdown of 1,000 psi, these productivity indexes would result in
production rates of 5,000, 2,300, and 1,100 STB/d, respectively. Clearly,
the difference in production rates between the ideal and gravel packed
wells can be considerable and very undesirable.
Fracturing of High Permeability Formations
Consider for a moment the use of high permeability fracturing
under the same scenario. This technology combines the advantages of
propped fracturing to bypass the near-wellbore damage and gravel
packing to provide effective sand control. Figure 5-1 is the classic
presentation (compare Figure 3-6) of the equivalent skin effect (Cinco
and Samaniego, 1978) in terms of dimensionless fracture conductivity,
CfD (= kf w/ kxf ), and fracture half-length, xf .
It can be seen from Figure 5-1 that even with a hydraulic fracture
of less than optimum conductivity (e.g., CfD = 0.5) and short fracture
length (e.g., xf = 50 ft), the skin effect, sf (again using rw = 0.328 ft),
would be equal to –3.
A negative skin effect equal to –3 applied to Equation 5-1 yields
a productivity index of 7.7 STB/d/psi, more than a 50 percent increase
over the ideal PI and seven times the magnitude of a damaged gravelpacked well. Even with a damaged fracture (e.g., leakoff-induced
damage as described by Mathur et al., 1995) and a skin equal to –1,
the productivity index would be 5.6 STB/d/psi, a five-fold increase
over a damaged gravel-packed well.
This calculation brings forward a simple, yet frequently overlooked, issue. Small negative skin values have a much greater impact
on well performance than comparable magnitudes (absolute value) of
sf + ln(xf /rw), y
sf + l n (xf /rw)
C fD,opt
FIGURE 5-1. Pseudoskin factor for a vertical well intersected by a finite conductivity fracture.
Unified Fracture Design
positive skin. Furthermore, in the example calculation here, a five-fold
increase in the productivity index suggests that the production rate
would increase by the same amount if the drawdown is held constant.
Under an equally possible scenario, the production rate could be held
constant and the drawdown reduced to one-fifth its original value. Any
other combination between these two limits can be envisioned.
The utility of high permeability fracturing is, thus, compelling—
not just for production rate improvement, but also for the remedy of
undesirable drawdown-dependent phenomena.
High-Rate Water Packs
Empirical data reported by Tiner et al. (1996), as distilled and presented in Table 5-2, support the frequent notion that high-rate water
packs have an advantage over gravel packs, but do not afford the
productivity improvement of HPF. This improvement over gravel packs
is reasonable by virtue of the additional proppant placed in the perforation tunnels.
While not shown in the table, the performance of these completions over time is also of interest. It is commonly reported that
production from high-rate water packs (as in the case of gravel packs)
deteriorates with time. By contrast, Stewart et al. (1995), Mathur et al.
(1995), and Ning et al. (1995) all report that production may progressively improve (skin values decrease) during the first several
months following a HPF treatment.
Two of the most important developments in petroleum production in
the last 15 years are horizontal wells and high permeability fracturing. Considerable potential is possible by combining the two.
TABLE 5-2. Skin Values Reported by Tiner et al. (1996)
Gravel Pack
+5 to +10 excellent
+40 and higher are reported
High-Rate Water Pack
+2 to +5 reported
0 to +2 normally
0 to –3 in some reports
Fracturing of High Permeability Formations
Horizontal wells can be drilled either transverse or longitudinal to
the fracture azimuth. The transverse configuration is appropriate for
low permeability formations and has been widely used and documented
in the literature. The longitudinally fractured horizontal well warrants
further attention, specifically in the case of high permeability formations. HPF often results in hydraulic fractures with low dimensionless
conductivities. Yet, such fractures installed longitudinally in horizontal
wells in high permeability formations can have the net effect of
installing a (relative) high conductivity streak in an otherwise limited
conductivity flow conduit. Using a generic set of input data, Valkó
and Economides (1996) showed discounted revenues for 15 cases that
demonstrate this point.
Table 5-3 shows that for a given permeability, the potential for
the longitudinally fractured horizontal well is always higher than that
of a fractured vertical well and, with realistic fracture widths, may
approach the theoretical potential of an infinite conductivity fracture.
Furthermore, the horizontal well fractured with 10-fold less proppant
(CfD = 0.12) still outperforms the fractured vertical well for k = 1 and
10 md, and is competitive at 100 md. The longitudinal configuration
may provide the additional benefit of avoiding excess breakdown
pressures and tortuosity problems during execution.
The Tip Screenout Concept
The critical elements of HPF treatment design, execution, and interpretation are substantially different than for conventional fracture
TABLE 5-3. Discounted Revenue in US$ (1996) Millions
k = 1 md
k = 10 md
k = 100 md
Vertical well
Horizontal well
Fractured vertical well, Cf D = 1.2
Fractured horizontal well, CfD = 1.2
Infinite-conductivity fracture (upper bound
for both horizontal and vertical well cases)
Unified Fracture Design
treatments. In particular, HPF relies on a carefully timed tip screenout
to limit fracture growth and to allow for fracture inflation and packing.
This process is illustrated in Figure 5-2.
The TSO occurs when sufficient proppant has concentrated at the
leading edge of the fracture to prevent further fracture extension. Once
fracture growth has been arrested (and assuming the pump rate is
larger than the rate of leakoff to the formation), continued pumping
will inflate the fracture (increase fracture width). This TSO and
fracture inflation is generally accompanied by an increase in net
fracture pressure. Thus, the treatment can be conceptualized in two
distinct stages: fracture creation (equivalent to conventional designs)
and fracture inflation/packing (after tip screenout).
Figure 5-3 after Roodhart et al. (1994) compares the two-stage
HPF process with the conventional single-stage fracturing process.
Creation of the fracture and arrest of its growth (tip screenout) is
accomplished by injecting a relatively small pad and a 1–4 lbm/gal
sand slurry. Once fracture growth has been arrested, further injection
builds fracture width and allows injection of higher concentration (e.g.,
10–16 lbm/gal) slurry. Final areal proppant concentrations of 20 lbm/ft2
are possible. The figure also illustrates the common practice of
retarding injection rate near the end of the treatment (coincidental with
opening the annulus to flow) to dehydrate/pack the well annulus and
near-well fracture. Rate reductions may also be used to force tip
screenout in cases where no TSO event is observed on the downhole
pressure record.
Fracture Inflation
Packed Fracture
FIGURE 5-2. Width inflation with the tip screenout technique.
Fracturing of High Permeability Formations
Injection Rate
Injected Slurry
Fracture Creation
Fracture Inflation
and Packing
Fracturing Fluid and Proppant Concentrations in Fracture:
Pad Injection
Slurry Injection
- End of Job for Conventional Design After FIP
FIGURE 5-3. Comparison of conventional and HPF design concepts.
The tip screenout can be difficult to model, affect, or even detect.
There are many reasons for this, including a tendency toward overly
conservative design models (resulting in no TSO), partial or multiple
tip screenout events, and inadequate pressure monitoring practices.
It is well accepted that accurate bottomhole measurements are
imperative for meaningful treatment evaluation. Calculated bottomhole
pressures are unreliable because of the dramatic friction pressure
effects associated with pumping high sand concentrations through
small diameter tubulars and service tool crossovers. Surface data may
indicate that a TSO event has occurred when the bottomhole data
shows no evidence, and vice versa. Even in the case of downhole
pressure data, there has been some discussion of where measurements
should be taken. Friction and turbulence concerns have caused at least
one operator to conclude that bottomhole pressure data should be
collected from below the crossover tool (washpipe gauges) in addition
to data collected from the service tool bundle (Mullen et al., 1994).
The detection of tip screenout is discussed further in Chapter 10
along with the introduction of a simple screening tool to evaluate
bottomhole data.
Unified Fracture Design
Net Pressure and Fluid Leakoff
The entire HPF process is dominated by net pressure and fluid leakoff
considerations, first because high permeability formations are typically
soft and exhibit low elastic modulus values, and second, because the
fluid volumes are relatively small and leakoff rates high (high permeability, compressible reservoir fluids, and non-wall-building fracturing fluids). Also, as described previously, the tip screenout design itself
affects the net pressure. While traditional practices applicable to
design, execution, and evaluation in MHF continue to be used in HPF,
these are frequently not sufficient.
Net Pressure, Closure Pressure,
and Width in Soft Formations
Net pressure is the difference between the pressure at any point in the
fracture and that of the fracture closure pressure. This definition
involves the existence of a unique closure pressure. Whether the
closure pressure is a constant property of the formation or depends
heavily on the pore pressure (or rather on the disturbance of the pore
pressure relative to the long term steady value) is an open question.
In high permeability, soft formations it is difficult (if not impossible)
to suggest a simple recipe to determine the closure pressure as classically derived from shut-in pressure decline curves (see Chapter 10).
Furthermore, because of the low elastic modulus values, even small,
induced uncertainties in the net pressure are amplified into large
uncertainties in the calculated fracture width.
Fracture Propagation
Fracture propagation, the availability of sophisticated 3D models
notwithstanding, presents complications in high permeability formations, which are generally soft and have low elastic modulus values.
For example, Chudnovsky (1996) emphasized the stochastic character
of this propagation. Also, because of the low modulus values, an
inability to predict net pressure behavior may lead to a significant
departure between predicted and actual treatment performance.
It is now a common practice to “predict” fracture propagation and
net pressure features using a computer fracture simulator. This trend
of substituting clear models and physical assumptions with “knobs”—
such as arbitrary stress barriers, friction changes (attributed to erosion,
Fracturing of High Permeability Formations
if decreasing, and sand resistance, if increasing) and less-than-well
understood properties of the formation expressed as dimensionless
“factors”—does not help to clarify the issue.
Considerable effort has been expended on laboratory investigation of
the fluid leakoff process for high permeability cores. A comprehensive
report can be found in Vitthal and McGowen (1996) and McGowen
and Vitthal (1996). The results raise some questions about how effectively
fluid leakoff can be limited by filter cake formation.
In all cases, but especially in high permeability formations, the
quality of the fracturing fluid is only one of the factors that influence
leakoff, and often not the determining one. Transient fluid flow in the
formation might have an equal or even larger impact. Transient flow
cannot be understood by simply fitting an empirical equation to
laboratory data. The use of models based on solutions to the fluid flow
equation in porous media is an unavoidable step.
In the following, three models are considered that describe fluid
leakoff in the high permeability environment. The traditional Carter
leakoff model requires some modification for use in HPF as shown.
(Note: While this model continues to be used across the industry,
it is not entirely sufficient for the HPF application.) An alternate,
filter cake leakoff model has been developed based on the work by
Mayerhofer, et al. (1993). The most appropriate leakoff model for high
permeability formations may be that of Fan and Economides (1995),
which considers the series resistance caused by the filter cake, the
polymer-invaded zone, and the reservoir. While the Carter model is in
common use, the models of Mayerhofer, et al. and Fan and Economides
represent important building blocks and provide a conceptual framework
for understanding the key issue of leakoff in high permeability fracturing.
Fluid Leakoff and Spurt Loss as Material Properties: The
Carter Leakoff Model with Nolte’s Power Law Assumption
There are two main schools of thought concerning leakoff. The first
considers the phenomenon as a material property of the fluid/rock
system. The basic relation (called the integrated Carter equation, given
also in Chapter 4) is given in consistent units as
Unified Fracture Design
= 2CL t + S p
where AL is the area and VL is the total volume lost during the time
period from time zero to time t. To make use of material balance, the
term VL must be described. For rigorous theoretical development, VL
is the volume of liquid entering the formation through the two created
fracture surfaces of one wing. The integration constant, Sp , is called
the spurt loss coefficient and is measured in units of length. It can be
considered as the width of the fluid body passing through the surface
instantaneously at the very beginning of the leakoff process, while
2 C L t is the width of the fluid body following the first slug. The
two coefficients, CL and Sp , can be determined from laboratory or
field tests.
As discussed in more detail in Chapter 4, Equation 5-2 can be
visualized assuming that the given surface element “remembers” when
it has been opened to fluid loss and has its own “zero” time that is
likely different from that of other elements along the fracture surface.
Points on the fracture face near the well are opened at the beginning
of pumping while the points at the fracture tip are younger. Application
of Equation 5-2 or its differential form necessitates tracking the
opening time for different fracture-face elements, as discussed in
Chapter 4.
The second school of thought considers leakoff as a consequence
of flow mechanisms in the porous medium, and employs a corresponding mathematical description.
Filter Cake Leakoff Model According to Mayerhofer, et al.
The method of Mayerhofer, et al. (1993) describes the leakoff rate
using two parameters that are physically more realistic than the leakoff coefficient: (1) filter cake resistance at a reference time and
(2) reservoir permeability. It is assumed that these parameters (R0, the
reference resistance at a reference time t0, and kr , the reservoir permeability) have been identified from a minifrac diagnostic test. In
addition, reservoir pressure, reservoir fluid viscosity, porosity, and total
compressibility are assumed to be known.
Total pressure gradient from inside a created fracture out into the
reservoir, ∆p, at any time during the injection, can be written as
∆p(t ) = ∆p face (t ) + ∆p piz (t ) + ∆pres (t )
Fracturing of High Permeability Formations
where ∆pface is the pressure drop across the fracture face dominated
by the filtercake, ∆ppiz is the pressure drop across a polymer invaded
zone, and ∆pres is the pressure drop in the reservoir. This concept is
shown in Figure 5-4.
In a series of experimental works using typical hydraulic fracturing
fluids (e.g., borate and zirconate crosslinked fluids) and cores with less
than 5 md of permeability, no appreciable polymer invaded zone was
detected. This simplifying assumption is not valid for linear gels such
as HEC (which do not form a filter cake) and may break down for
crosslinked fluids at higher permeabilities (e.g., 200 md). Yet, at least
for crosslinked fluids in a broad range of applications, the second term
in the right-hand side of Equation 4-21 can reasonably be ignored, so
∆p(t ) = ∆p face (t ) + ∆pres (t )
The filter cake pressure term can be expressed as a function of,
and is proportional to, R0 , the characteristic resistance of the filter
cake. The transient pressure drop in the reservoir can be re-expressed
as a series expansion of pD , a dimensionless pressure function describing the behavior (unit response) of the reservoir. Dimensionless time,
tD , is calculated with the maximum fracture length reached at time
Filter Cake
Distance from
Fracture Center
FIGURE 5-4. Filter cake plus reservoir pressure drop in the Mayerhofer et al. (1993)
Unified Fracture Design
tn. And rp is introduced as the ratio of permeable height to the total
height (hp / hf ).
With rigorous introduction of these variables and considerable
rearrangement (not shown), an expression for the leakoff rate can be
written that is useful for both hydraulic fracture propagation and
fracture-closure modeling:
∆p(t n ) −
qn =
n −1
µr 
− qn−1 pD (t Dn − t Dn−1 ) + ∑ (q j − q j −1 ) pD t Dn − t Dj −1 
πkr rp h f 
j =1
+ r D Dn Dn−1
πkr rp h f
2rp An te
This expression allows for the determination of the leakoff rate
at any time instant, tn , if the total pressure difference between the
fracture and the reservoir is known, as well as the history of the
leakoff process. The dimensionless pressure solution, pD t Dn − t Dj −1 ,
must be determined with respect to a dimensionless time that takes
into account the actual fracture length at tn .
The model can be used to analyze the pressure fall-off subsequent
to a fracture injection (minifrac) test, as described by Mayerhofer,
et al. (1995). The method requires more input data than the similar
analysis based on Carter leakoff, but it offers the distinct advantage
of differentiating between the two major factors in the leakoff process,
filter cake resistance and reservoir permeability.
Polymer-Invaded Zone Leakoff Model
of Fan and Economides
The leakoff model of Fan and Economides (1995) concentrates on the
additional resistance created by the polymer-invaded zone.
The total driving force behind fluid leakoff is the pressure difference between the fracture face and the reservoir, pfrac – pi , which
is equivalent to the sum of three separate pressure drops—across the
filter cake, the polymer-invaded zone, and in the reservoir:
p frac − pi = ∆pcake + ∆pinv + ∆pres
The fracture treating pressure is equivalent to the net pressure plus
fracture closure pressure (minimum horizontal stress).
When a non-cake building fluid is used, the pressure drop across
the filter cake is negligible. This is the case for many HPF treatments.
Fracturing of High Permeability Formations
The physical model of this situation (i.e., fluid leakoff controlled
by polymer invasion and transient reservoir flow) is depicted in Figure 5-5. The polymer invasion is labeled in the figure as region 1,
while the region of reservoir fluid compression (transient flow) is
denoted as 2.
By employing conservation of mass, a fluid flow equation, and an
appropriate equation of state, a mathematical description of this fluid
leakoff scenario can be written. As a starting point, Equation 5-7
describes the behavior of a Power law fluid in porous media:
1− n
∂ 2 p nφµ eff ct  1  ∂p
 u
∂x 2
where ct is the system compressibility, k is the formation permeability,
u is the superficial flow rate, n is the fluid flow behavior index, φ is
1− n
K′ 
9 +  (150 kφ) 2 is the fluid
the formation porosity, and µ eff =
12 
effective viscosity (Kⴕ is the power law fluid consistency index).
Combining the description of the polymer-invaded zone and the
reservoir, the total pressure drop is given by Fan and Economides
(1995) as
FIGURE 5-5. Fluid leakoff model with polymer invasion and transient reservoir flow.
Unified Fracture Design
π φη 
p frac − pr =
2 k 
+ µ r α 2 e 
 η  
erf 
 4α1  
η 
 η 
4α 2 
 4α 2  
η 
4α1 
and a2 =
1− n
nφµ eff
 u
At given conditions, Equation 5-8 can be solved iteratively for the
parameter η (not to be confused with fluid efficiency). Once the value
of η is found for a specified total pressure drop, the leakoff rate is
calculated from
where a1 =
 η 1
q L = A 
 2φ  t
In other words, the factor η/(2φ) can be considered a pressuredependent apparent leakoff coefficient.
In gas condensate reservoirs, a situation emerges very frequently that
is tantamount to fracture face damage. Because of the pressure gradient that is created normal to the fracture, liquid condensate is formed,
which has a major impact on the reduction of the relative permeability to gas. Such a reduction depends on the phase behavior of the fluid
and the penetration of liquid condensate, which in turn, depends on
the pressure drawdown imposed on the well. These phenomena cause
an apparent damage that affects the performance of all fractured wells,
but especially those with high reservoir permeability.
Wang, et al. (2000) presented a model that predicts the fractured
well performance in gas condensate reservoirs, quantifying the effects
of gas permeability reduction. Furthermore, they presented fracture
treatment design for condensate reservoirs. The distinguishing feature
Fracturing of High Permeability Formations
primarily affects the required fracture length to offset the problems
associated with the emergence of liquid condensate.
Gas relative permeability curves were derived using a pore-scale
network model and are represented by a weighted linear function of
immiscible and miscible relative permeability curves:
krg = fkrgI + (1 − f )krgM
where krg is the gas relative permeability, and f is a weighing factor
that is a function of the capillary number,
f =
1/ b
1+  c 
 a
The numerical values for a and b are 1.6 × 10 –3 and 0.324,
respectively, and Nc is the capillary number, defined as
Nc =
In Equation 5-12, k is the permeability, ∇p is the pressure gradient,
and σ is the interfacial tension. The conventional relative permeability
for capillary dominated (immiscible) flow in Equation 5-10, krgI , is
defined as
 Sg 
 1 − Swi 
where Sg is the gas saturation, Swi is the connate water saturation, and
ng is a constant equal to 5.5. The relative permeability function in the
limit of viscous dominated (miscible) flow, krgM , is defined as
krgM =
1 − Swi
Recall that Cinco and Samaniego (1981) provided an expression of the fracture face skin effect that is additive to the dimensionless pressure for the finite conductivity fracture performance:
s fs =
πbs  k
 − 1
2 x f  ks 
where bs is the penetration of damage and ks is the damaged permeability.
Unified Fracture Design
An analogy can readily be made for a hydraulically fractured gas
condensate reservoir. Liquid condensate that drops out normal to the
fracture face can also result in a skin effect, in this case reflecting a
reduction in the relative permeability to gas. The penetration of
damage would be the zone inside which liquid condensate exists (i.e.,
the dew point pressure establishes the boundary).
The permeability ratio reduces to the ratio of the relative permeabilities, and because at the boundary kr g is equal to 1, Equation 5-15 becomes simply,
s fs =
2x f
 1
 k − 1
 rg 
Optimizing Fracture Geometry
in Gas Condensate Reservoirs
In gas condensate reservoirs, the fracture performance is likely to be
affected greatly by the presence of liquid condensate, tantamount to
fracture face damage. An assumption for the evaluation is that the
reservoir pressure at the boundary of this “damaged” zone must be
exactly equal to the dew point pressure.
For any fracture length and a given flowing bottomhole pressure
inside the retrograde condensation zone of a two-phase envelope, the
pressure profile normal to the fracture phase and into the reservoir will
delineate the points where the pressure is equal to the dew point
pressure. From this pressure profile, the distribution of fracture face
skin can be determined. The depth of the affected zone is determined
from Equation 5-16, the modified Cinco-Ley and Samaniego expression. An additional necessary element is the relative permeability
impairment given by the correlation presented in Equations 5-10 to 5-14.
Two example case studies are presented below. The first represents
a reservoir with 5 md permeability and a gas condensate with a dew
point pressure of 2,545 psi. The flowing bottomhole pressure is 1,800
psi. First, a standard hydraulic fracture optimization—ignoring the
effects of the fracture face skin—using a proppant number, Nprop , equal
to 0.02, results in an expected dimensionless fracture conductivity of
1.6 and a fracture half-length of 220 ft for a 4,000 ft square reservoir.
(The value of the proppant number, assuming kf = 50,000 md, h = 50 ft,
ρp = 165 lb/ft3 and φp = 0.4, implies a proppant mass approximately
equal to 80,000 lbm.) The dimensionless productivity index would be 0.35.
Fracturing of High Permeability Formations
A series of simulations based on the work of Wang, et al. shows
the maximum productivity index that can be achieved when the gas
condensate skin is introduced, and indicates appropriate changes to the
fracture design. The fracture length is progressively increased, while
the proppant number (i.e., the mass of proppant injected) is held
constant. This, of course, causes an unavoidable reduction in the
fracture conductivity, even while maximizing the productivity index.
The results, shown in Figure 5-6, indicate an optimum fracture half-length of 255 ft (16 percent increase from the zero-skin
optimum) and an optimum dimensionless conductivity of 1.2 instead
of 1.6. Much more significant is the drop in the optimum productivity
index to 0.294.
Meeting the expected zero-skin productivity index of 0.35 would
necessitate raising the proppant number to approximately 0.045—and
more than double the required mass of proppant.
For a much higher permeability reservoir (200 md)—again, ignoring
the fracture face skin initially—the same calculation results in an
optimum fracture half-length equal to 35 ft (Cf D = 1.6). The proppant
number for this case is 0.0005 (for the same 80,000 lbm of proppant).
The corresponding dimensionless productivity index is 0.21.
Figure 5-7 is the optimization for the fracture dimensions with gas
condensate damage, showing an optimum half-length of 45 ft (a 30
Productivity, lr
Fracture Half Length, xf
FIGURE 5-6. Optimized fracture geometry in a gas-condensate reservoir (k = 5 md).
Unified Fracture Design
Productivity Index, JD
Fracture Half Length, xf
FIGURE 5-7. Optimized fracture geometry in a gas-condensate reservoir (k = 200 md).
percent increase over the zero-skin optimum). The new optimum Cf D
is 1 and the corresponding productivity index is 0.171.
Here the impact of gas condensate damage on the productivity
index expectations and what would be needed to counteract this effect
is far more serious. The required proppant number would be 0.003—
suggesting 6 times the mass of proppant originally contemplated!
In most cases, such a fracture treatment would be highly impracticable, so the expectations for well performance would need to be pared
down considerably.
Non-Darcy flow is another important issue that deserves specific
consideration in the context of HPF. Non-Darcy flow in gas reservoirs
causes a reduction of the productivity index by at least two mechanisms.
First, the apparent permeability of the formation may be reduced
(Wattenbarger and Ramey, 1969) and second, the non-Darcy flow may
decrease the conductivity of the fracture (Guppy et al., 1982).
Consider a closed gas reservoir producing under pseudosteady-state
conditions, and apply the concept of pseudoskin effect determined by
dimensionless fracture conductivity.
Fracturing of High Permeability Formations
Definitions and Assumptions
Gas production is calculated from the pseudosteady-state deliverability
( )] ×
πkhTsc m( p ) − m pwf
psc T
kr , app
 0.472 re  
kr  f1 C fD, app + ln
 x f  
where m(p) is the pseudopressure function, kf,app is the apparent permeability of the proppant in the fracture, and kr,app is the apparent
permeability of the formation. (All equations in this subsection are
given for a consistent system of units, such as SI.) The function f was
introduced by Cinco-Ley and Samaniego (1981) and was presented in
Chapter 3 as
f1 (C fD ) = s f + ln
1.65 − 0.328u + 0.116u 2
1 + 0.18 ln u + 0.064u 2 + 0.005u 3
where u = lnCfD.
The apparent dimensionless fracture conductivity is defined by
C fD, app =
k f , app w
kr , app x f
The apparent permeabilities are flow-rate dependent; therefore, the
deliverability equation becomes implicit in the production rate.
Proceeding further requires a model of non-Darcy flow. Almost
exclusively, the Forcheimer equation is used:
dp µ
= v + βρ v v
dx k
where ν = qa /A is the Darcy velocity and β is a property of the porous
A popular correlation was presented by Firoozabadi and Katz
(1979) as
k 1.2
where c = 8.4 × 10 –8 m1.4 (= 2.6 × 1010 ft –1 md1.2).
Unified Fracture Design
To apply the Firoozabadi and Katz correlation, we write
cρ v 
1  βkρ v 
= µv 1 +
 = µv 1 + 0.2 
µ 
k  k µ
showing that
cρ v
1 + 0.2
k µ
The equation above can be used both for the reservoir and for the
fracture if correct representative linear velocity is substituted. In the
following, it is assumed that h = hf .
A representative linear velocity for the reservoir can be given in
terms of the gas production rate as
4hx f
where qa is the in-situ (actual) volumetric flow rate; hence, for the
reservoir non-Darcy effect,
 cρqa 
 cρv 
 0.2  = 
 k µ  r  2 hµ  2 x f kr0.2
A representative linear velocity in the fracture can be given in
terms of the gas production rate as
2 hw
Thus, for the non-Darcy effect in the fracture, one can use
 cρqa  1
 cρv 
 0.2  = 
 k µ  f  2 hµ  wk 0f .2
The term ρqa is the mass flow rate and is the same in the reservoir
and in the fracture; cρqa is expressed in terms of the gas production
rate as
cρqa cρa γ g
q = c0 q
2 hµ
2 hµ
where q is the gas production rate in standard volume per time, γg is
the specific gravity of gas with respect to air, and ρa is the density of
Fracturing of High Permeability Formations
air at standard conditions. The factor c 0 is constant for a given
reservoir-fracture system.
The final form of the apparent permeability dependence on production rate is
 kapp 
 =
 k  r 1 + c0 q
2 x f kr0.2
for the reservoir and
 kapp 
 =
 k  f 1 + c0 q
wk 0f .2
for the fracture. As a consequence, the deliverability equation becomes
( )] ×
πkhTsc m( p ) − m pwf
psc T
 0.472re  
c0 q
1 + 2 x k 0.2   f1 C fD,app + ln x
f r
kf w
wk 0f .2
kr x f 1 + c0 q
2 x f kr0.2
C fD, app
The additional skin effect, sND , appearing because of non-Darcy
flow, can be expressed as
 0.472re   
 0.472re  
c0 q  
 f C fD,app + ln
 −  f1 C fD + ln
s ND = 1 +
0.2  1
2 x f kr  
 x f   
 x f  
( )
The additional non-Darcy skin effect is always positive and depends on
the production rate in a nonlinear manner.
Equations 5-31 and 5-33 are of primary importance to interpret
post-fracture well testing data and to forecast production. If the
mechanism responsible for the post-treatment skin effect is not understood clearly, the evaluation of the treatment and the production
forecast might be severely erroneous.
Unified Fracture Design
Case Study for the Effect of Non-Darcy Flow
As previously discussed, non-Darcy flow in a gas reservoir causes a
reduction of the productivity index by at least two mechanisms. First,
the apparent permeability of the formation may be reduced, and
second, the non-Darcy flow may decrease the fracture conductivity.
In this case study, the effect of non-Darcy flow on production rates
and observed skin effects is investigated.
Reservoir and fracture properties are given in Table 5-4.
A simplified form of Equation 5-30 in field units is
p 2 − pwf
1424µZT 
 0.472 re  
c0 q  
kr h
1 + 2 x k 0.2   f1 C fD, app + ln x
 
f r 
where c0 =
cρa γ g
must be expressed in ft-md0.2/MMSCF/day. In the
2 hµ
given example, c0 = 73 ft-md0.2/MMSCF/day and
C fD, app
 k f w  1 + c0 r q
 kf w 
2 x f kr0.2
 k x  1+ c q
 r f
 kr x f  1 + c0 q
wk f
where c0r = 2.34 × 10 –3 m3/s = 7.67 × 10 –2 (MSCF/day)–1
c0f = 6.14 × 10 –1 m3/s = 2.78 × 102 (MSCF/day)–1
Therefore, in field units
C fD, app = 1.39
1 + 0.76q
1 + 280 q
4000 2 − pwf
(1 + 0.76q)[ f1 (C fD, app ) + 3.16]
The non-Darcy component of the skin effect can be calculated as
s ND = (1 + 0.00076q ) f1 C fD, app + 3.16 − 4.619
The results are shown graphically in Figures 5-8 to 5-10.
Fracturing of High Permeability Formations
TABLE 5-4. Data for Fractured Well in Gas Reservoir
pw, psi
q, MMscf/d
FIGURE 5-8. Inflow performance of fractured gas reservoir, non-Darcy effect from
Firoozabadi-Katz correlation.
Unified Fracture Design
q, MMscf/d
FIGURE 5-9. Additional skin effect from non-Darcy flow in the fracture.
q, MMscf/d
FIGURE 5-10. Observable pseudoskin, the resulting effect of fracture with non-Darcy
flow effects.
It is apparent that the effect of the fracture (negative skin on the
order of –3) is hidden by the positive skin effect induced by non-Darcy
flow. The zero or positive observable skin effect, while directly
attributable to the (inevitable) effect of non-Darcy flow, might be
interpreted as an unsuccessful HPF job.
Fracturing Materials
Materials used in the fracturing process include fracturing fluids,
fluid additives, and proppants. The fluid and additives act together, first
to create the hydraulic fracture, and second, to transport the proppant
into the fracture. Once the proppant is in place and trapped by the
earth stresses (“fracture closure”), the carrier fluid and additives are
degraded in-situ and/or flowed back out of the fracture (“fracture
cleanup”), establishing the desired highly-productive flow path.
Proppants and chemicals constitute a large share of the total cost
to fracture treat a well. The relative value of fracturing materials and
pumping costs for treatments performed in the United States are
estimated as follows: 45 percent for pumping (pump rental and horsepower charges), 25 percent for proppants, 20 percent for fracturing
chemicals, and 10 percent for acid.
Materials and proppants used in hydraulic fracturing have undergone tremendous changes since the first commercial fracturing treatment was performed in 1949 with a few sacks of coarse sand and
gelled gasoline as the carrier fluid.
Unified Fracture Design
The fracturing fluid transmits hydraulic pressure from the pumps to
the formation, which creates a fracture, and then transports proppant
(hence the name carrier fluid) into the created fracture. The invasive
fluids are then removed (or cleaned up) from the formation, allowing
the production of hydrocarbons. Factors to consider when selecting the
fluid include availability, safety, ease of mixing and use, viscosity
characteristics, compatibility with the formation, ability to be cleaned
up from the fracture, and cost.
Fracturing fluids can be categorized as (1) oil- or water-base,
usually “crosslinked” to provide the necessary viscosity, (2) mixtures
of oil and water, called emulsions, and (3) foamed oil- and water-base
systems that contain nitrogen or carbon dioxide gas. Oil-based fluids
were used almost exclusively in the 1950s. By the 1990s, more than
90 percent of fracturing fluids were crosslinked water-based systems.
Today, nitrogen and carbon dioxide systems in water-based fluids are
used in about 25 percent of fracture stimulation jobs.
Table 6-1 lists the most common fracturing fluids in order of
current usage. The choice of which crosslinking method to use is based
TABLE 6-1. Crosslinked Fluid Types
Gelling Agent
pH Range
B, non-delayed
Guar, HPG
70–300 °F
B, delayed
Guar, HPG
70–300 °F
Zr, delayed
150–300 °F
Zr, delayed
70–250 °F
Zr, delayed
200–400 °F
Zr-a, delayed
70–275 °F
Ti, non-delayed
100–325 °F
Ti, delayed
100–325 °F
Al, delayed
70–175 °F
Sb, non-delayed
Guar, HPG
60–120 °F
a—compatible with carbon dioxide
Fracturing Materials
on the capability of a fluid to yield high viscosity while meeting cost
and other performance requirements.
Viscosity is perhaps the most important property of a fracturing
fluid. Guar gum, produced from the guar plant, is the most common
gelling agent used to create this viscosity. Guar derivatives called
hydroxypropyl guar (HPG) and carboxymethyl-hydroxypropyl guar
(CMHPG) are also used because they provide lower residue, faster
hydration, and certain rheological advantages. For example, less gelling
agent is required if the guar is crosslinked.
The base guar or guar derivative is reacted with a metal that
couples multiple strands of gelling polymer. Crosslinking effectively
increases the size of the base guar polymer, increasing the viscosity
in the range of shear rates important for fracturing from 5- to 100fold. Boron (B) is often used as the crosslinking element, followed
by organometallic crosslinkers such as zirconium (Zr) and titanium
(Ti), and to a lesser extent antimony (Sb) and aluminum (Al).
Foams are especially useful in water-sensitive or depleted (low
pressure) reservoirs (Chambers, 1994). Their application minimizes
fracture face damage and eases the clean-up of the wellbore after
the treatment.
Gelling agent, crosslinker, and pH control (buffer) materials define the
specific fluid type and are not considered to be additives. Fluid
additives are materials used to produce a specific effect independent
of the fluid type. Table 6-2 lists commonly used additives.
Biocides control bacterial contamination. Most waters used to
prepare fracturing gels contain bacteria that originate either from
contaminated source water or the storage tanks on location. The
bacteria produce enzymes that can destroy viscosity very rapidly.
Bacteria can be effectively controlled by raising the pH to greater than
12, adding bleach, or employing a broad-spectrum biocide.
Fluid loss control materials provide spurt loss control. The material
consists of finely ground particles ranging from 0.1 to 50 microns. The
most effective low-cost material is ground silica sand. Starches, gums,
resins, and soaps can also be used, with the advantage that they allow
some degree of post-treatment cleanup by virtue of their solubility in
water. Note that the guar polymer itself eventually controls leakoff,
once a filter cake is established.
Unified Fracture Design
TABLE 6-2. Fracturing Fluid Additives
gal or lbm added
per 1,000 gallons
of clean fluid
0.1–1.0 gal
Prevents guar polymer decomposition by bacteria
Fluid loss
10–50 lbm
Decreases leakoff of fluid during fracturing
0.1–10 lbm
Provides controlled fluid viscosity reduction
Friction reducers
0.1–1.0 gal
Reduces wellbore frictional pressure loss while
0.05–10 gal
Reduces surface tension, prevents emulsions, and
changes wettability
Foaming agents
Clay control
1–10 gal
Provides stable foam with nitrogen and carbon
1–3% KCl
Provides temporary or permanent clay (water
Breakers reduce viscosity by reducing the size of the guar polymer,
thereby having the potential to dramatically improve post-treatment
cleanup and production. Table 6-3 summarizes several breaker types
and application temperatures.
Surfactants prevent emulsions, lower surface tension, and change
wettablilty (i.e., to water wet). Reduction of surface tension allows
improved fluid recovery. Surfactants are available in cationic, nonionic,
and anionic forms, and are included in most fracturing treatments.
Some specialty surfactants provide improved wetting and fluid recovery.
Foaming agents provide the surface-active stabilization required
to maintain finely divided gas dispersion in foam fluids. These ionic
materials also act as surfactants and emulsifiers. Stable foam cannot
be prepared without a surfactant for stabilization.
Clay control additives produce temporary compatibility in waterswelling clays. Solutions containing 1 to 3 percent KCl or other salts
are typically employed. Organic chemical substitutes are now available,
which are used at lower concentrations.
The type of additives and concentrations used depend greatly on
the reservoir temperature, lithology, and fluids. Tailoring of additives
for specific applications and advising clients is a main function of the
QA/QC chemist.
Fracturing Materials
TABLE 6-3. Fracturing Fluid Breakers
60–200 °F
Efficient breaker; limit use to pH less than 10
Encapsulated enzyme
60–200 °F
Allows higher concentrations for faster breaks
Persulfates (sodium,
120–200 °F
Economical; very fast at higher temperatures
Activated persulfates
70–120 °F
Low temperature and high pH applications
Encapsulated persulfates
120–200 °F
Allows higher concentrations for faster breaks
High temperature
200–325 °F
Used where persulfates are too fast-acting
Because proppants must oppose earth stresses to hold open the fracture
after release of the fracturing fluid hydraulic pressure, material strength
is of crucial importance. The propping material must be strong enough
to bear the closure stress, otherwise the conductivity of the (crushed)
proppant bed will be considerably less than the design value (both the
width and permeability of the proppant bed decrease). Other factors
considered in proppant selection are size, shape, composition, and, to
a lesser extent, density.
The two main categories of proppants are naturally occurring sands
and manmade ceramic or bauxite proppants. Sands are used for lowerstress applications, in formations approximately 8,000 ft and (preferably, considerably) less. Manmade proppants are used for high-stress
situations in formations generally deeper than 8,000 ft. For high
permeability fracturing, where a high conductivity is essential, using
high-strength proppants may be justified at practically any depth.
There are three primary ways to increase fracture conductivity:
(1) increase the proppant concentration, that is, to produce a wider
fracture, (2) use a larger (and hence, higher permeability) proppant,
or (3) employ a higher-strength proppant, to reduce crushing and
improve conductivity. Figures 6-1, 6-2, and 6-3 illustrate the three
methods of increasing conductivity through proppant choice.
Unified Fracture Design
3.0 lb/ft 2
Conductivity, md-ft
2.0 lb/ft 2
1.0 lb/ft 2
0.5 lb/ft 2
Closure Stress, psi
FIGURE 6-1. Fracture conductivity for various areal proppant concentrations (20/40 mesh).
Conductivity, md-ft
2.0 lb/ft 2
Closure Stress, psi
FIGURE 6-2. Fracture conductivity for various mesh sizes.
Figure 6-4 is a selection guide for popular proppant types based
on the dominant variable of closure stress.
Calculating Effective Closure Stress
In the course of proppant selection, it is necessary to estimate the
magnitude of the closure stress acting on the proppant. The most
Fracturing Materials
20/40 Ceramic
2.0 lb/ft 2
Conductivity, md-ft
20/40 Bauxite
20/40 Ottawa
Closure Stress, psi
FIGURE 6-3. Fracture conductivity for various proppants.
Closure Stress, 1,000 psi
FIGURE 6-4. Proppant selection guide.
Unified Fracture Design
common equation used to estimate the closure stress (i.e., the minimum
horizontal stress at depth in the reservoir) is known as Eaton’s equation. It is commonly given in the form:
Sh =
( Sv − p p ) + p p
1− ν
where ν is the Poisson ratio, Sv is the absolute vertical stress, and pp
is the reservoir pore pressure. It is worthwhile to understand the
forward development of this relationship.
The absolute vertical stress, Sv , is essentially equal to the force
exerted by the weight of the overburden per unit area. Formally, it is
the integral of the formation density of the various layers overlying
the reservoir. In practice, this value is found to range from 0.95 to
1.1 psi per foot of depth, and in the (typical) absence of specific
information, is taken to be equal to 1 psi/ft.
To obtain the effective vertical stress (i.e., the weight of the
overburden supported by the rock matrix), the total vertical stress must
be reduced by an amount equal to the reservoir pore pressure, giving
σ v = Sv − α p p
where the coefficient α, called Biot’s constant or the poroelastic
constant, is added to the pore pressure term to account for the fact
that reservoir fluids are locally free to move out of the control volume
under consideration (not a closed box). This situation is depicted in
Figure 6-5.
Biot’s constant is typically a value between 0.7 and 1, but most
often is taken as unity in order to simplify the already rather approximate calculation.
Now, we know that the longitudinal strain that results when a
linear elastic solid is placed under a uniaxial load translates to a lateral
strain according to classic mechanics of materials, that is, the two
quantities being related by (in fact defining) the Poisson ratio of the
solid, ν = ∂ex / ∂ez. In a similar way, the vertical stress created by the
soil layers overlying an oilfield will induce a horizontal stress in the
reservoir rock (through the solid matrix). The magnitude of this
horizontal stress is calculated by:
σh =
1− ν
where σh is of course the effective horizontal stress.
Fracturing Materials
Effective Stress = Total Stress – α (Pore Pressure)
Pore Fluid
α ~ 0.7 to 1.0
FIGURE 6-5. Poroelasticity.
Combining Equations 6-2 and 6-3 and rearranging slightly,
Sh =
( Sv − α p p ) + α p p
1− ν
which, taking Biot’s constant to be equal to 1, yields the common form
of Eaton’s equation.
Now, it is important to recognize that, unless the producing bottomhole pressure in the fracture treated well is drawn down to somewhere
near zero, the entire burden of this horizontal stress will not be borne
by the proppant. Another important observation is that the horizontal
stress in the reservoir is itself a function of reservoir pore pressure,
so the closure stress on the proppant is nominally reduced with
reservoir depletion.
Fracture Width as a Design Variable
A great deal has been published concerning optimum fracture dimensions in HPF. While there are debates regarding the optimum, fracture
Unified Fracture Design
width is largely regarded as more important than fracture length. Of
course, this is an intuitive statement and only recognizes the first
principle of fracture optimization: higher permeability formations
require higher fracture conductivity to maintain an acceptable value
of the dimensionless fracture conductivity, CfD .
A “rule of thumb” is that fracture length should be equal to 1/2
of the perforation height (thickness of producing interval). Hunt et al.
(1994) showed that cumulative recovery from a well in a 100 md
reservoir with a 10 ft damage radius is optimized by extending a fixed
8,000 md-ft conductivity fracture to any appreciable distance beyond
the damaged zone. This result implies that there is little benefit to a
50 ft fracture length compared to a 10 ft fracture length. Two observations may be in order. First, the Hunt et al. evaluation is based on
cumulative recovery. Second, their assumption of a fixed fracture
conductivity implies a decreasing dimensionless fracture conductivity
with increasing fracture length (i.e., less than optimal placement of
the proppant).
It is generally true that if an acceptable Cf D is maintained—this may
require an increase in areal proppant concentration from 1.5 lbm/ft2,
which is common in hard-rock fracturing, to 20 lbm/ft2 or more—
additional length will provide additional production. As explained in
Chapter 3, the optimum fracture conductivity of 1.6 corresponds to
the best compromise between the capacity of the fracture to conduct
and the capacity of the reservoir to deliver fluids. This applies to high
permeability and low permeability formations alike.
The problem, in practice, has been that fracture extent and width
are difficult to influence separately. Historically, once a fracturing fluid
and injection rate are selected, the fracture width evolves with increasing length according to strict relations (at least in the well-known PKN
and KGD design models). Therefore, the key decision variable has
been the fracture extent. After the fracture extent is determined, the
width is calculated as a consequence of technical limitations (e.g.,
maximum realizable proppant concentration). Knowledge of the leakoff
process helps to determine the necessary pumping time and pad volume.
The tip screenout technique has brought a significant change to
this design philosophy. Through TSO, fracture width can be increased
without increasing the fracture extent. Now we have a very effective
means to design and execute fractures that satisfy the optimum condition.
The ultimate decision requires optimizing the mass of proppant
based on economics or, in cases where total fluid and proppant
Fracturing Materials
volumes are physically limited (e.g., in offshore environments), optimizing placement of a finite proppant volume
Proppant Selection
The primary and unique issue relating to proppant selection for high
permeability fracturing, beyond maintaining a high permeability at any
stress, is proppant sizing. While specialty proppants such as intermediate
strength and resin-coated proppants have certainly been employed in HPF,
the majority of treatments are pumped with standard graded-mesh sand.
When selecting a proppant size for HPF, the engineer faces competing
priorities: size the proppant to address concerns with sand exclusion,
or use maximum proppant size to ensure adequate fracture conductivity.
As with equipment choices and fluids selection, the gravel-packing
roots of frac & pack are also evident when it comes to proppant
selection. Engineers initially focused on sand exclusion and a gravel
pack derived sizing criteria such as that proposed by Saucier (1974).
Saucier recommends that the mean gravel size (Dg50 ) be five to six
times the mean formation grain size (Df 50). The so-called “4-by-8 rule”
implies minimum and maximum grain-size diameters that are distributed around Saucier’s criteria (i.e., Dg,min = 4Dg 50 and Dg,max = 8Dg50 ,
respectively). Thus, many early treatments were pumped with standard
40/60 mesh or even 50/70 mesh sand. The somewhat limited conductivity of these gravel pack mesh sizes under in-situ formation
stresses is not adequate in many cases. Irrespective of sand mesh size,
frac & packs tend to reduce concerns with fines migration by virtue
of reducing fluid flux at the formation face.
The current trend in proppant selection is to use fracturing-size
sand. A typical HPF treatment now employs 20/40 proppant (sand).
Maximizing the fracture conductivity can itself help prevent sand
production by virtue of reducing drawdown. Results with the larger
proppant have been encouraging, both in terms of productivity and
limiting or eliminating sand production (Hannah et al., 1993).
It is interesting to note that the topics of formation competence
and sanding tendency, major issues in the realm of gravel pack technology, have not been widely studied in the context of HPF. It seems
that in many cases HPF is providing a viable solution to completion failures in spite of the industry’s primitive understanding of (soft)
rock mechanics.
Unified Fracture Design
This move away from gravel pack practices toward fracturing-type
practices is common to many aspects of HPF with the exception (so
far) of downhole tools, and it seems to justify the migration in our
terminology from frac & pack to high permeability fracturing. The
following discussion of fluid selection is consistent with this perspective.
Fluid Selection
Fluid selection for HPF has always been driven by concerns with
damaging the high permeability formation, either by filter cake buildup
or (especially) polymer invasion. Most early treatments were carried
out using HEC, the classic gravel pack fluid, as it was perceived to
be less damaging than guar-based fracturing fluids. While the debate
has lingered on and while some operators continue to use HEC fluids,
the fluid of choice is increasingly borate-crosslinked HPG.
Based on a synthesis of reported findings from several practitioners, Aggour and Economides (1996) provide a rationale to guide
fluid selection in HPF. Their findings suggest that if the extent of
fracturing fluid invasion is minimized, the degree of damage (i.e.,
permeability impairment caused by filter cake or polymer invasion)
is of secondary importance. They employ the effective skin representation of Mathur et al. (1995) to show that if fluid leakoff penetration is small, even severe permeability impairments can be tolerated
without exhibiting positive skin effects. In this case, the obvious
recommendation in HPF is to use high polymer concentration, crosslinked fracturing fluids with fluid-loss additives, and an aggressive
breaker schedule. The polymer, crosslinker, and fluid-loss additives
limit polymer invasion, and the breaker ensures maximum fracture
conductivity, a critical factor which cannot be overlooked.
Experimental work corroborates these contentions. Linear gels
have been known to penetrate cores of very low permeability (1 md
or less) whereas crosslinked polymers are likely to build filter cakes
at permeabilities two orders of magnitude higher (Roodhart, 1985;
Mayerhofer et al., 1991). Filter cakes, while they may damage the
fracture face, clearly reduce the extent of polymer penetration into the
reservoir normal to the fracture face. At extremely high permeabilities,
even crosslinked polymer solutions may invade the formation.
Cinco-Ley and Samaniego (1981) and Cinco-Ley et al. (1978)
described the performance of finite-conductivity fractures and delineated
three major types of damage affecting this performance.
Fracturing Materials
Reduction of proppant pack permeability resulting from either
proppant crushing or (especially) unbroken polymer chains leads
to fracture conductivity impairment. This can be particularly
problematic in moderate to high permeability reservoirs. Extensive progress in breaker technology has dramatically reduced
concerns with this type of damage.
Choke damage refers to the near-well zone of the fracture, which
can be accounted for by a skin effect. This damage can result from
either over-displacement at the end of a treatment or by fines
migration during production. In the latter case, one can envision
fines from the formation or proppant accumulating near the well
but within the fracture.
Fracture face damage implies permeability reduction normal to
the fracture face, including permeability impairments caused by
the filter cake, polymer-invaded, and filter cake-invaded zones.
Composite Skin Effect
Mathur et al. (1995) provide the following representation for effective
skin resulting from radial wellbore damage and fracture face damage:
sd =
(b1 − b2 )kr − b1 
b2 kr
2  b1k3 + x f − b1 k2 b1k1 + x f − b1 kr x f 
Figure 6-6 depicts the two types of damage accounted for in sd
(i.e., fracture-face and radial wellbore damage).
The b- and k- terms are defined graphically in Figure 6-7 and
represent the dimensions and permeabilities of various zones included
in the finite conductivity fracture model of Mathur et al.
The equivalent damage skin can be added directly to the undamaged
Cinco and Samaniego fracture skin effect to obtain the total skin,
st = sd + s f
Parametric Studies
Aggour and Economides (1996) employed the Mathur et al. model
(with no radial wellbore damage) to evaluate total skin and investigate
the relative effects of different variables. Their results related the total
skin in a number of discrete cases to (1) the depth of fluid invasion
normal to the fracture face and (2) the degree of permeability reduction
Unified Fracture Design
Radial Damage
Fracture Damage
FIGURE 6-6. Fracture face damage.
Radial Fluid-Invaded Zone
FIGURE 6-7. Fluid invaded zones.
Fracturing Materials
in the polymer-invaded zone. A sample of their results (for xf = 25 ft,
Cf D = 0.1, and kf = 10 md), expressed initially in terms of damage
penetration ratios, b2 /xf , and permeability impairment ratios, k2 /kr , are
re-expressed in real units in Table 6-4. Under each of these conditions,
the total skin is equal to zero.
These results suggest that for a (nearly impossible) 2.5 ft penetration of damage, a positive skin is obtained only if the permeability
impairment in the invaded zone is more than 90 percent. For a damage
penetration of 1.25 ft, the permeability impairment would have to be
over 95 percent to achieve positive skins. If the penetration of damage
can be limited to 0.25 ft, even a 99 percent permeability reduction in
the invaded zone would not result in positive skins. At a higher
dimensionless conductivity equal to 1, even higher permeability impairments can be tolerated without suffering positive skins. Thus, if the
fracturing fluid leads to a clean and wide proppant pack, penetration
and damage to the reservoir can be tolerated.
It is also clear from this work that the extent of damage normal
to the fracture face is more important than the degree of damage. If
fluid invasion can be minimized, even 99 percent damage can be
tolerated. The importance of maximizing Cf D is also illustrated; certainly, a good proppant pack should not be sacrificed in an attempt to
minimize the fracture face damage.
This points toward the selection of appropriate fracturing fluids:
Linear gels by virtue of their considerable leakoff penetration
are not recommended.
Crosslinked polymer fluids with high gel loadings appear to be
much more appropriate.
Aggressive breaker schedules are imperative.
Filter cake building additives may also be considered to minimize the spurt loss and total leakoff.
TABLE 6-4. Fluid Invasion Damage Tolerated for Zero Skin
Depth of Fluid Invasion
Normal to Fracture Face
Permeability Reduction in
Invaded Zone
2.5 ft
1.25 ft
0.25 ft
Source: Aggour and Economides (1996).
Unified Fracture Design
Work by Mathur et al. (1995) and Ning et al. (1995) further
support the conclusion that fracture face damage should not significantly alter long-term HPF performance. The Mathur et al. study
of Gulf Coast wells assumed a linear cleanup of the fracture and
observed an improvement of the production rate at early time. The
Ning et al. study of gas wells in Alberta, Canada, showed that fracture
conductivity has the greatest effect on long-term production rates,
whereas the effects of polymer invasion were minimal.
Experiments in Fracturing Fluid Penetration
McGowen et al. (1993) presented a series of experiments showing the
extent of fracturing fluid penetration in cores of various permeabilities.
Fracturing fluids used were 70 lbm per 1,000-gal HEC and 30 or 40
lb m per 1,000 gal borate-crosslinked HPG. Filtrate volumes were
measured in ml per cm2 of leakoff area (i.e., cm of penetration) for
a 10 md limestone and 200 and 1,000 md sandstones at 120°F
and 180°F.
Several conclusions can be drawn from the work:
Crosslinked fracturing fluids are far superior to linear gels in
controlling fluid leakoff. For example, 40 lbm per 1,000 gal
borate-crosslinked HPG greatly outperforms 70 lbm per 1,000
gal HEC in 200 md core at 180°F.
Linear gel performs satisfactorily in 10 md rock but fails dramatically at 200 md. Even aggressive use of fluid loss additives (e.g.,
40 lbm per 1,000 gal silica flour) does not appreciably alter the
leakoff performance of HEC in 200 md core.
Increasing crosslinked gel concentrations from 30 to 40 lbm
per 1,000 gal has a major impact on reducing leakoff in
200 md core. Crosslinked borate maintains excellent fluid loss
control in 200 md sandstone and performs satisfactorily even at
1,000 md.
This experimental work strongly corroborates the modeling results
of Aggour and Economides (1996) and points toward the use of higherconcentration crosslinked polymer fluids with, of course, an appropriately designed breaker system.
Fracturing Materials
Viscoelastic Carrier Fluids
HEC and borate-crosslinked HPG fluids are the dominant fluids currently employed in HPF. However, there is a third class of fluid that
deserves to be mentioned, the so-called viscoelastic surfactant, or VES
fluids. There is little debate that these fluids exhibit excellent rheological properties and are non-damaging, even in high permeability
formations. The elegance of VES fluids is that they do not require the
use of chemical breaker additives; the viscosity of this fluid conveniently breaks (leaving considerably less residue than polymer-based
fluids) either when it contacts formation oil or condensate or when
its salt concentration is reduced. Brown et al. (1996) present typical
VES fluid performance data and case histories.
The vulnerability of VES fluids is in their temperature limitation
and much higher costs per unit volume. The maximum application
temperature for VES fluids has only recently been extended from 130°F
up to 240°F.
VES fluids have great potential when considered in a holistic
manner: treatments may cost more than polymer fluids, but the resulting appropriately sized fracture could be a far superior producer.
Fracture Treatment
Fracture treatment goes well beyond the sizing of a fracture, as important as that is for production enhancement, to include the calculation
of a pumping schedule that will realize the goals set for the treatment.
This chapter also includes discussion of pre-treatment diagnostics that
are often incorporated with fracture treatments to determine or at least
place bounds on parameters that are critical to the design procedure
and execution.
The microfracture stress test (“microfrac”) determines the magnitude
of the minimum principal in-situ stress of a target formation. The test
usually involves the injection of pressurized fluid into a small, isolated zone (4 to 15 ft, 1.2 to 4.6 m) at low injection rates (1 to 25
gal/min, 0.010 to 0.095 m3/min). The minimum principal in-situ stress
can be determined from the pressure decline after shut-in or the pressure increase at the beginning of an injection cycle. The fracture closure
pressure and fracture reopening pressure provide good approximations
for the minimum principal in-situ stress.
Unified Fracture Design
The most important test on location before the main treatment is
known as a “minifrac,” or a fracture calibration test. The minifrac is
a pump-in/shut-in test that employs full-scale pump rates and relatively
large fluid volumes, on the order of thousands of gallons. Information gathered from a minifrac includes the closure pressure, pc , net
pressure, entry conditions (perforation and near-wellbore friction), and
possibly evidence of fracture height containment. The falloff portion
of the pressure curve is used to obtain the leakoff coefficient for
a given fracture geometry. Figure 7-1 illustrates the strategic locations on a typical pressure response curve registered during the calibration activities.
A minifrac design should be performed along with the initial treatment design. The design goal for the minifrac is to be as representative as possible of the main treatment. To achieve this objective,
sufficient geometry should be created to reflect the fracture geometry
of the main treatment and to obtain an observable closure pressure
from the pressure decline curve. The most representative minifrac
would have an injection rate and fluid volume equal to the main treatment, but this is often not practical. In reality, several conflicting
design criteria must be balanced, including minifrac volume, created
2. Propagation
Clock Time
Injection Rate
Bottomhole Pressure
1. Formation
3. Instantaneous
4. Closure Pressure
from Fall-off
5. Reopening
6. Closure Pressure
from Flowback
7. Asymptotic
8. Closure Pressure
from Rebound
FIGURE 7-1. Key elements on minfrac pressure response curve.
Fracture Treatment Design
fracture geometry, damage to the formation, a reasonable closure time,
and the cost of materials and personnel.
Fracture closure is typically determined from one or more constructions of the pressure decline curve while taking into consideration
any available prior knowledge (e.g., that obtained from microfrac
tests). Some of the most popular plots used to identify fracture closure pressure are:
pshut-in vs. t
pshut-in vs.
pshut-in vs. g-function (and variations)
log ( pISIP – pshut-in )
The origin and use of these various plots is sometimes more intuitive than theoretical, which can lead to spurious results. The theoretical basis and limitations of pressure decline analysis must be
understood in the context of individual applications. An added complication is that temperature and compressibility effects may cause
pressure deviations. In this case, temperature-corrected decline curves
can be generated to permit the normal interpretations of the different
plot types (Soliman, 1984).
The original concept of pressure decline analysis is based on the
observation that the rate of pressure decline during the closure process contains useful information on the intensity of the leakoff
process (Nolte, 1979, Soliman and Daneshy, 1991). This stands in
contrast to the pumping period, when the pressure is affected by many
other factors.
If we assume that the fracture area has evolved with a constant
exponent α and remains constant after the pumps are stopped, at time
(te + ∆t) the volume of the fracture is given by
Vt e + ∆t = Vi − 2Ae S p − 2Ae g( ∆t D ,α ) C L te
where the dimensionless delta time is defined as
∆t D = ∆t/te
and the two-variable function g(∆tD ,α) can be obtained by integration.
Its general form is given by (Valkó and Economides, 1995):
Unified Fracture Design
4α ∆t D + 2 1 + ∆t D × F  , α; 1 + α; 1 + ∆t D
 2
g ∆t D , α =
1 + 2α
 (7-3)
The function F[a, b; c; z] is the “Hypergeometric function” available in the form of tables or computing algorithms. For computational
purposes (e.g., the included MF Excel spreadsheet for minifrac analysis),
the g-function approximations given in Table 7-1 are useful.
Dividing Equation 7-1 by the area, the fracture width at time ∆t
after the end of pumping is given by
wt e + ∆t =
− 2 S p − 2CL te g( ∆t D ,α )
Hence, the time variation of the width is determined by the g(∆t D ,α) function, the length of the injection period, and the leakoff coefficient, but
is not affected by the fracture area.
The decrease of average width cannot be observed directly, but
the net pressure during closure is already directly proportional to the
average width according to
pnet = S f w
simply because the formation is described by linear elasticity theory
(i.e., Equation 4-2). The coefficient Sf is the fracture stiffness, expressed
in Pa/m (psi/ft). Its inverse, 1/Sf , is called the fracture compliance.
For the basic fracture geometries, expressions of the fracture stiffness
are given in Table 7-2.
TABLE 7-1. Approximation of the g-Function for Various Exponents ␣
1.41495 + 79.4125 d + 632.457 d 2 + 1293.07 d 3 + 763.19 d 4 + 94.0367 d 5
g d ,  =
 5
1. + 54.8534 d 2 + 383.11 d 3 + 540.342 d 4 + 167.741 d 5 + 6.49129 d 6
1.47835 + 81.9445 d + 635.354 d 2 + 1251.53 d 3 + 717.71 d 4 + 86.843 d 5
g d ,  =
 3  1. + 54.2865 d + 372.4 d 2 + 512.374 d 3 + 156.031 d 4 + 5.95955 d 5 − 0.0696905 d 6
1.37689 + 77.8604 d + 630.24 d 2 + 1317.36 d 3 + 790.7 d 4 + 98.4497 d 5
g d ,  =
 9  1. + 55.1925 d + 389.537 d 2 + 557.22 d 3 + 174.89 d 4 + 6.8188 d 5 − 0.0808317 d 6
Fracture Treatment Design
TABLE 7-2. Proportionality Constant, Sf and Suggested
␣ for Basic Fracture Geometries
πh f
πx f
3πE ′
16 R f
The combination of Equations 7-4 and 7-5 yields the following
(Nolte, 1979):
S f Vi
p =  pC +
− 2 S f S p  − 2 S f CL te × g( ∆t D ,α )
Equation 7-6 shows that the pressure falloff in the shut-in period
will follow a straight line trend,
p = bN − m N × g( ∆t D ,α )
if plotted against the g-function (i.e., transformed time, Castillo, 1987).
The g-function values should be generated with the exponent α considered valid for the given model. The slope of the straight line, mN ,
is related to the unknown leakoff coefficient by
CL =
− mN
2 te s f
Substituting the relevant expression for fracture stiffness, the
leakoff coefficient can be estimated as given in Table 7-3. This table
shows that the estimated leakoff coefficient for the PKN geometry does
not depend on unknown quantities because the pumping time, fracture height, and plain strain modulus are assumed to be known. For
the other two geometries considered, the procedure results in an estimate of the leakoff coefficient that is strongly dependent on the fracture extent (xf or Rf ).
From Equation 7-6 we see that the effect of the spurt loss is concentrated in the intercept of the straight line with the g = 0 axis:
Sp =
b − pC
− N
2 Ae
2S f
Unified Fracture Design
TABLE 7-3. Leakoff Coefficient and No-Spurt Fracture
Extent for Various Fracture Geometries
πh f
Leakoff coefficient, CL
4 te E ′
xf =
Fracture Extent
(− m N )
2 E ′Vi
πh 2f (bN − pC )
πx f
2 te E ′
xf =
(− m N )
8R f
3π t e E ′
(− m N )
3E ′Vi
E ′Vi
Rf = 3
8(bN − pC )
πh f (bN − pC )
As suggested by Shlyapobersky (1987), Equation 7-9 can be used
to obtain the unknown fracture extent if we assume there is no spurt
loss. The second row of Table 7-3 shows the estimated fracture
extent for the three basic models. Note that the no-spurt-loss assumption results in an estimated fracture length for the PKN geometry, but
this value is not used to obtain the leakoff coefficient. For the KGD
and radial models, fracture extent is calculated first and then used to
interpret the slope (i.e., to determine CL ). Once the fracture extent and
the leakoff coefficient are known, the lost width at the end of pumping can be easily obtained from
w Le = 2g0 (α ) C L te
The fracture width is
we =
− w Le
x f hf
for the two rectangular models and
we =
Rf π /
− w Le
for the radial model.
Often the fluid efficiency is also determined:
ηe =
we + w Le
Note that the fracture extent and the efficiency are state variables,
which is to say that they will have different values in the minifrac
Fracture Treatment Design
and main treatment. Only the leakoff coefficient is a model parameter that can be transferred from the minifrac to main treatment, but
even then some caution is needed in its interpretation. The bulk leakoff
coefficient determined from the above method is “apparent” with
respect to the fracture area. If we have information on the permeable
height, hp , and it indicates that only part of the fracture area falls
into the permeable layer, the apparent leakoff coefficient should be
converted into a “true” value that corresponds to the permeable area
only. This is done by simply dividing the apparent value by rp (see
Equation 7-14).
While adequate for many low permeability treatments, the outlined
procedure might be misleading for higher permeability reservoirs. The
conventional minifrac interpretation determines a single effective fluid
loss coefficient, which usually slightly overestimates the fluid loss
when extrapolated to the full job volume (Figure 7-2).
This overestimation typically provides an extra factor of safety in
low permeability formations to prevent a screenout. However, this
same technique applied in high permeability, or when the differential
pressure between the fracture and the formation is high, can significantly overestimate the fluid loss for wall-building fluids (Figure 7-3,
Dusterhoft, 1995).
Overestimating fluid leakoff can be highly detrimental when the
objective is to achieve a carefully timed tip screenout. In this case,
Volume Per Area, mL/cm2
Fluid-Loss Fuction
Predicted by Minifrac
Pumping Time
Main Job
Pumping Time
Time, min 0.5
FIGURE 7-2. Fluid leakoff extrapolated to full job volume, low permeability.
Unified Fracture Design
Volume Per Area, mL/cm2
Fluid-Loss of Crosslinked
Gel, High-Permeability Rock
Actual Fluid Loss
Fluid Loss from
Minifrac Analysis
Pumping Time = 16 min
Time, min 0.5
FIGURE 7-3. Overestimation of fluid leakoff extrapolated to full job volume, high
modeling both the spurt loss and the combined fluid loss coefficient
by performing a net pressure match in a 3D simulator is an alternative to
classical falloff analysis. This approach is illustrated in Figure 7-4.
Note that the incorporation of more than one leakoff parameter
(and other adjustable variables) increases the degrees of freedom.
Volume Per Area, mL/cm2
Spurt Volume
Actual Fluid Loss
Fluid Loss from
Minifrac Analysis
Pumping Time = 16 min
Spurt Time
Time, min 0.5
FIGURE 7-4. Leakoff estimate based on a net-pressure match in a 3D simulator (Source:
Dusterhot et al., 1995).
Fracture Treatment Design
While a better match of the observed pressure can usually be achieved,
the solution often becomes non-unique (i.e., other values of the same
parameters may provide a similar fit).
We ended Chapter 3 by delineating a certain design logic: for a given
amount of proppant reaching the pay layer, we can determine the
optimum length (and width). One of the main results was that, for low
or moderate proppant numbers (relatively low proppant volumes and/or
moderate-to-high formation permeabilities), the optimal compromise
occurs at CfD = 1.6.
When the formation permeability is above 50 md, it is practically
impossible to achieve a proppant number larger than 0.1. Typical
proppant numbers for HPF range from 0.0001 to 0.01. Thus, for
moderate and high permeability formations, the optimum dimensionless fracture conductivity is always Cf Dopt = 1.6.
In “tight gas” it is possible to achieve large dimensionless proppant
numbers, at least in principle. If we assume a limited drainage area
and do not question whether the proppant actually reaches the pay
layer, a dimensionless proppant number equal to 1 or even 5 can be
calculated. However, proppant numbers larger than one are not likely
in practice.
When the propped volume becomes very large, the optimal compromise happens at larger dimensionless fracture conductivities
simply because the fracture penetration ratio cannot exceed unity
(i.e., fracture length becomes constrained by the well spacing or limits
of the reservoir).
A crucial issue in the design is the assumed fracture height. The
relation of fracture height to pay thickness determines the volumetric
proppant efficiency. The actual proppant number depends on that part
of the proppant that is placed into the pay. It is calculated as the
volume of injected proppant multiplied by the volumetric proppant
efficiency. Therefore, strictly speaking, an optimum target length can
be obtained only if the fracture height is already known. In the following, we assume that the fracture height is known. Later we will
return to this issue.
Unified Fracture Design
Pump Time
Armed with a target length and assuming that hf , Eⴕ, qi , µ, CL , and
Sp are known, we can design a fracture treatment. The first problem
is to determine the pumping time, t e , using the combination of
a width equation and material balance. The first part of a typical
design procedure is shown in Table 7-4. Notice that the injection rate,
qi , refers to the slurry (not clean fluid) injected into one wing.
Techniques used to refine KL are delineated in Tables 7-5 to 7-7.
If the permeable height, hp , is less than the fracture height, it is
convenient to use exactly the same method, but with “apparent” leakoff
and spurt loss coefficients. The apparent leakoff coefficient is the
“true” leakoff coefficient (the value with respect to the permeable
layer) multiplied by the factor rp , defined as the ratio of permeable to
fracture surface (cf. Figures 7-5 and 7-6).
TABLE 7-4. Determination of the Pumping Time
1. Calculate the wellbore width at the end of pumping from the PKN (or any
 µqi x f 
other) width equation: w w,0 = 3.27
 E′ 
1/ 4
(or rather the non-Newtonian
form shown later)
2. Convert wellbore width into average width: we = 0.628w w,0
3. Assume an opening time distribution factor, KL = 1.5 (techniques to refine
this value are described below)
4. Solve the following equation for te:
qi t
− 2 K L C L t − ( we + 2 S p ) = 0 (Quadratic Equation for x =
hf x f
t as the new unknown, a simple quadratic equation must be solved:
at + b t + c = 0 where
a = i ; b = − 2 K L C L ; c = −( we + 2 S p )
hf x f
5. Calculate injected volume: Vi = qi te , and fluid efficiency: ηe =
h f x f we
Fracture Treatment Design
Refinement of KL using the Carter II Equation
TABLE 7-5.
Calculate an improved estimate of KL from:
KL = −
where ηe =
C L te
2 C L te
2η e C L t e
we we + 2 S p 
2CL πte
− 1 and β =
exp(β ) erfc(β) +
4 π CL2 te
we + 2 S p
If KL is near enough to the previous guess, stop; otherwise, iterate by repeating the
material balance calculation using the new estimate of KL .
TABLE 7-6.
Refinement of KL by Linear Interpolation According to Nolte
Estimate the next KL from
K L = 1.33ηe + 1.57(1 − ηe ) ,
where ηe =
we x f h f
If KL is near enough to the previous guess, stop; otherwise, iterate by repeating the
material balance calculation using the new estimate of KL .
TABLE 7-7. KL from the ␣ Method
Assume a power law exponent α (Table 7-2) and calculate KL = g0(α) using equations in Table 7-1. Use the obtained KL instead of 1.5 in the material balance. (Note
that this is not an iterative process.)
For the PKN and KGD geometries, it is the ratio of permeable to
the fracture height,
rp =
while for the radial model it is given by
rp =
x (1 − x 2 )0.5 + arcsin( x )] where x =
2 Rf
Unified Fracture Design
2q i
A = h fx f
FIGURE 7-5. Ratio of permeable to total surface area, KGD, and PKN geometry.
2q i
Rf 2 π
FIGURE 7-6. Ratio of permeable to total surface area, radial geometry.
Fracture Treatment Design
There are several ways to incorporate non-Newtonian behavior into
the width equations. A convenient procedure is to add one additional
equation connecting the equivalent Newtonian viscosity with the flow
rate. Assuming power law fluid behavior, the equivalent Newtonian
viscosity can be calculated for the average cross section using the
appropriate entry from Table 4-3. After substituting the equivalent
Newtonian viscosity into the PKN width equation, we obtain
ww , 0 =
9.15 2 n + 2
× 3.98 2 n + 2
n 1− n
1 + 2.14n  2 n + 2 K 2 n + 2  i h f x f  2 n + 2
 E′ 
Proppant Schedule
Given the total pumping time and slurry volume, a stepwise pump
schedule (more specifically, a proppant addition schedule, or just
proppant schedule) is still needed that will yield the designed, propped
fracture geometry.
Fluid injected at the beginning of the job without proppant is
called the “pad.” It initiates and opens up the fracture. Typically, 30
to 60 percent of the fluid pumped during a treatment leaks off into
the formation while pumping; the pad provides much of this necessary extra fluid. The pad also generates sufficient fracture length and
width to allow proppant placement. Too little pad results in premature bridging of proppant and shorter-that-desired fracture lengths. Too
much pad results in excessive fracture height growth and created fracture length. For a fixed slurry volume, excessive pad may result in
a final propped length that is considerably shorter than the created
(desired) fracture length. Even if the fluid loss were zero, a minimum
pad volume would be required to open sufficient fracture width to
admit proppant. Generally, a fracture width equal to three times the
proppant diameter is felt to be necessary to avoid bridging.
After the specified pad is pumped, the proppant concentration of
the injected slurry is ramped up step-by-step until a maximum value
is reached at end of the treatment.
Figure 7-7 conceptually illustrates the proppant distribution in the
fracture after the first proppant-carrying stage. Most fluid loss occurs
in the pad, near the fracture tip. However, some fluid loss occurs along
the fracture, and in fact, fluid loss acts to dehydrate the proppant-laden
stages. Figure 7-8 shows the concentration of the initial proppant stage
climbing from 1 up to 3 lbm of proppant per gallon of fluid (ppg) as
Unified Fracture Design
1 lb/gal
At Time the First Proppant
Stage is Injected
FIGURE 7-7. Beginning of proppant distribution during pumping.
3 lb/gal
2 lb/gal
3 lb/gal
1 lb/gal
to 3 lb/gal
At Intermediate Time
FIGURE 7-8. Evolution of slurry proppant distribution during pumping.
the treatment progresses. Later stages are pumped at higher initial
proppant concentrations because they suffer less fluid leakoff (i.e.,
shorter exposure time and reduced leakoff rates near the well).
Figure 7-9 completes the ideal sequence in which the pad is
depleted just as pumping ends and the first proppant stage has concentrated to a final designed value of 5 ppg. The second proppant stage
has undergone less dehydration, but also has concentrated to the same
Fracture Treatment Design
2 to
5 lb/gal
3 to 5 lb/gal
4 to 5 lb/gal
1 lb/gal
to 5 lb/gal
At End of Pumping
FIGURE 7-9. Proppant concentration in the injected slurry.
final value. If done properly, the entire fracture is filled with a uniform proppant concentration at the end of the treatment.
If proppant bridges in the fracture prematurely during pumping, a
situation known as a “screen-out,” the treating pressure will rise rapidly to the technical limit of the equipment. In this case, pumping must
cease immediately (both for the safety of personnel on location and
to avoid damaging the equipment), effectively truncating the treatment
before the full proppant volume has been placed. Making things worse,
the treatment string is often left filled with sand, which then requires
incremental rig time and expense to clean out.
TSO designs for highly permeable and soft formations are specifically intended to screen out. In this case, it is often possible to
continue pumping and inflate the fracture width without exceeding the
pressure limits of the equipment because these formations tend to be
highly compliant.
While more sophisticated methods are available to calculate the
ramped proppant schedule, the simple design technique given in
Table 7-8 using material balance and a prescribed functional form (e.g.,
power law, Nolte 1986) is satisfactory.
One additional parameter must be specified: ce , the maximum
proppant concentration of the injected slurry at the end of pumping.
The physical capabilities of the fracturing equipment being used provides one limit to the maximum proppant concentration, but rarely
should this be specified as the value for ce . Ideally, the proppant schedule should be designed to result in a uniform proppant concentration in the fracture at the end of pumping, with the value of the
Unified Fracture Design
TABLE 7-8. Proppant Schedule
1. Calculate the exponent of the proppant concentration curve:
1 − ηe
1 + ηe
2. Calculate the pad volume and the time needed to pump it:
Vpad = εVi
t pad = εte
3. The required proppant concentration (mass per unit of injected slurry volume)
curve is given by the following:
 t − t pad 
c = ce 
 ,
 te − t pad 
where ce is the maximum end-of-job proppant concentration in the injected slurry.
4. Convert the proppant concentration from mass per unit of injected slurry volume into mass added per unit volume of base fluid (or “neat” fluid), denoted by
ca , and usually expressed in ppga (pounds added per gallon added of neat fluid).
concentration equal to ce . Therefore, the proppant concentration, ce ,
at the end of pumping should be determined from material balance:
M = ηe ceVi
where Vi is the volume of slurry injected in one wing, ηe is the fluid
efficiency (or more accurately, slurry efficiency), and M is the mass
of injected proppant (one wing).
According to Nolte (1986), the schedule is derived from the requirement that (1) the whole length created should be propped; (2) at
the end of pumping, the proppant distribution in the fracture should
be uniform; and (3) the proppant schedule should be of the form of a
delayed power law with the exponent, ε, and fraction of pad being
equal (Table 7-8). More complex proppant scheduling calculations
attempt to account for the movement of the proppant both in the lateral and the vertical directions; variations of the viscosity of the slurry
with time and location (due to temperature, shear rate and changes in
solid content); width requirements for free proppant movement; and
other phenomena (Babcock et al. 1967, Daneshy 1974, Shah 1982).
Note that in the above schedule the injection rate qi refers to the
slurry (not clean fluid) injected into one wing. The obtained proppant
mass M also refers to one wing.
Fracture Treatment Design
Continuing our previous example, assume that the target fracture
length (152.4 m or 500 ft) was obtained from the requirement to place
optimally M = 8,760 kg (19,400 lbm) of proppant into each wing.
Using Equation 7-17, we obtain that ce = 875 kg/m3 (7.3 lbm/gal). Note
that this is still expressed in mass per slurry volume. This means that
12.5 lbm of proppant must be added to one gallon of neat fracturing
fluid (i.e., the added proppant concentration is 12.5 ppga).
The conversion from mass/slurry-volume to mass/neat-fluidvolume is
ca =
where ρp is the density of the proppant material.
In our example the fluid efficiency is 19.3 percent, so the proppant
exponent and the fraction of pad volume is ε = 0.677. Therefore, the
pad injection time is 27.8 min, and after the pad, the proppant concentration of the slurry should be continuously elevated according to
, where c is in kg/m3 and t is in seconds,
the schedule: c = 875( t –795
, where c is in lbm/gal of slurry volume and t is
or c = 7.3( t –1327.3.8 )
in minutes. The obtained proppant curve is shown in Figure 7-10.
Proppant Concentration
in Injected Slurry, kg/m3, or lbm /gal
Pumping Time, min
FIGURE 7-10. Evolution of proppant distribution during pumping.
Unified Fracture Design
At the end of pumping, the proppant concentration is equal to ce
everywhere in the fracture. Thus, the mass of proppant placed into one
wing is M = Ve × ce = ηe × Vi × ce , or in our case, M = 8,760 kg
(19,400 lbm). The average propped width after closure can be determined
if the porosity of the proppant bed is known. Assuming φp = 0.3, the
propped volume is Vp = M/[(1 – φp)ρp], or in our case, 6.0 m3. The
average propped width is wp = Vp /(xf × hf ), that is, 2 mm (0.078 in.).
A quick check of the dimensionless fracture conductivity, substituting the propped width, shows that CfD = (60 × 10–12 × 0.002)/
(5 × 10–16 × 152) = 1.6, as it should be for a treatment with a relatively low proppant number.
In the above example, we assumed that the optimum target length and
width can be realized without any problem. Of course, it is possible that
certain physical or technical constraints (e.g., maximum possible proppant
concentration in the slurry) do not allow optimal placement.
Departure from the Theoretical Optimum
In case of conflict, the design engineer has several options. One possibility is to overcome technical limitations by, for example, choosing another type of fluid, proppant and/or equipment.
More often, however, we choose to depart from the theoretical
optimum. The art of fracture design is to depart from the theoretical
optimum dimensions, but in a reasonable manner and only as much
as necessary. In practical terms, this means that the optimum fracture
length or pad volume should be reduced or increased by a “factor.”
For low permeability formations, the first design attempt often
results in very long but narrow fracture. Because there is a certain
minimum propped width that is required to maintain continuity of the
fracture (e.g., 3 times the proppant diameter), the design engineer
should reduce the target length—multiplying it by a factor of 0.5
or sometimes even 0.1. In a careful design procedure, the engineer departs from the theoretical optimum only as much as necessary to satisfy another technical limitation, such as a required minimum width.
In high permeability formations, the first attempt may result in a
short fracture with insufficient conductivity (width). This portends a
move from the conventional to TSO design, which can produce extremely
large fracture widths.
Fracture Treatment Design
TSO Design
It is the tip screenout or TSO design which clearly differentiates high
permeability fracturing from conventional massive hydraulic fracturing. While HPF introduces other identifiable differences (e.g., higher
permeability, softer rock, smaller proppant volumes, and so on), it is
the tip screenout that makes these fracturing treatments unique. Conventional fracture treatments are designed to propagate laterally and
achieve TSO at the end of pumping. In high permeability fracturing,
pumping continues beyond the TSO to a second stage of fracture width
inflation and packing. It is this two-stage treatment that gives rise to
the vernacular of frac & pack. The conventional and HPF design concepts were illustrated and compared in Figures 5-3 and 5-4.
Early TSO designs commonly called for 50 percent pad (similar
to conventional fracturing) and proppant schedules that rampedup aggressively; then it became increasingly common to reduce
the pad to 10 to 15 percent of the treatment and extend the 0.5 to
2 lbm/gal stages (which combined may constitute 50 percent of the
total slurry volume, for example). Notionally, this was intended to
“create width” for the higher concentration proppant addition (e.g., 12
to 14 lbm/gal).
In our design model (included HF2D Excel spreadsheet), the TSO
design procedure differs from the conventional procedure in one
basic feature: it uses a “TSO criterion” to separate the lateral fracture
propagation period from the width inflation period. This criterion is
based on a “dry-to-wet” average width ratio, that is, the ratio of dry
width (assuming only the “dry” proppant is left in the fracture) to wet
width (dynamically achieved during pumping). According to our
assumptions, the screen-out occurs and arrests fracture propagation
when the dry-to-wet width ratio reaches a critical value.
After the TSO is triggered, injection of additional slurry only
serves to inflate the width of the fracture. Thus, it is important to
schedule the proppant such that the critical dry-to-wet width ratio is
reached at the same time (pumping time) that the created fracture
length matches the optimum fracture length. With the TSO design,
practically any width can be achieved—at least in principle. In addition, the first part of any TSO design very much resembles a traditional design, only the target length is reached in a relatively short
time, and the dry-to-wet width ratio must reach its critical value during this first part of the treatment.
Unified Fracture Design
We suggest a critical dry-to-wet ratio of 0.5 to 0.75 as the TSO
criterion (representing quite dehydrated sand in the fracture). Unfortunately, there is no good theoretical or practical method to refine this
value. Engineering intuition and previous experience are critical to
judging whether a significant arrest of fracture propagation is even
possible in a given formation.
There is also no clear procedure to predict if TSO width inflation
will be possible in a given formation, though rock mechanics laboratory
investigations can suggest the answer. The formation needs to be “soft
enough”; in other words, the elasticity modulus cannot be too high.
On the other hand, soft formations are often unconsolidated, lacking
significant cohesion between the formation grain particles. The main
technical limitation to keep in mind is the net pressure, which increases
during width inflation. The design engineer should be prepared to depart
from the theoretical optimum placement if necessary to keep the fracture
treating pressure below critical limits imposed by the equipment.
Another consideration in TSO design is that the created fracture
must bypass the assumed damaged region near the wellbore. As such,
the design should specify a minimum target length, even if the theoretical optimum calls for a shorter fracture. Often the minimum length is
on the order of 50 ft, while the nature of the damage and the length
of the perforated interval may dictate other values. Note that this
departure from the optimum again can be realized by specifying
the “multiply optimum length by a factor” parameter in the provided
design software.
Anecdotal observations related to real-time HPF experiences are abundant
in the literature and are not the focus of an engineering-operations text
such as this. However, some observations related to treatment execution are in order:
Most treatments are pumped using a gravel pack service tool
in the “circulate” position with the annulus valve closed at the
surface. This allows for live annulus monitoring of bottomhole
pressure (annulus pressure + annulus hydrostatic head) and realtime monitoring of the progress of the treatment.
Fracture Treatment Design
When there is no evidence of the planned TSO on the real-time
pressure record, the late treatment stages can be pumped at a
reduced rate to effect a tip screenout. Obviously, this requires
reliable bottomhole pressure data and direct communication by
the frac unit operator.
Near the end of the treatment, the pump rate is slowed to gravel
packing rates and the annulus valve is opened to begin circulating a gravel pack. The reduced pump rate is maintained until
tubing pressure reaches an upper limit, signaling that the screencasing annulus is packed.
Because very high proppant concentrations are employed, the
sand-laden slurry used to pack the screen-casing annulus must
be displaced from surface with clean gel, well before the end
of pumping. Thus, proppant addition and slurry volumes must
be metered carefully to ensure there is sufficient proppant left
in the tubing to place the gravel pack (i.e., to avoid overdisplacing proppant into the fracture).
Conversely, if an HPF treatment sands out prematurely (i.e.,
with proppant in the tubing), the service tool can be moved into
the “reverse” position and the excess proppant circulated out.
Movement of the service tool from the squeeze/circulating position to the reverse position can create a sharp instantaneous
drawdown effect and should be done carefully to avoid swabbing unstabilized formation material into the perforation tunnels
and annulus.
Swab Effect Example
The following simple equation, given by Mullen et al. (1994) can be
used to convert swab volumes into oilfield unit flow rates:
qs = 2, 057
where qs is the instantaneous swab rate in bbl/day, Vs is the swabbed
volume of fluid in gal, tm is the time of tool movement in seconds,
and 2,057 is the conversion factor for gal/sec to bbl/day.
Unified Fracture Design
The volume of swabbed fluid is calculated from the service tool
diameter and the length of stroke during which the sealed service tool
does not allow fluid bypass. The average swab volume of a 2.68 in.
service tool is 2.8 gal when the service tool is moved from the squeeze
position to the reverse-circulation position. Assuming a rather normal
movement time of 5 sec, this represents an instantaneous production
rate of 1,100 bbl/day.
Perforations for HPF
It is widely agreed that establishing a conductive connection between
the fracture and wellbore is critical to the success of HPF, but no
consensus or study has emerged that gives definitive direction.
With an eye toward maximizing conductivity and fluid flow rate,
many operators shoot the entire target interval with high shot density
and large holes (e.g., 12 shots per foot with “big hole” charges). Other
operators—more concerned with multiple fracture initiations, near-well
tortuosity, and perforations that are not packed with sand—take the
extreme opposite approach, perforating just the middle of the target
interval with a limited number of 0° or 180° phased perforations.
Arguments are made for and against underbalanced versus overbalanced perforating: underbalanced may cause formation failure and
“sticking the guns;” overbalanced eliminates a cleanup trip but may
negatively impact the completion efficiency.
Solvent or other scouring pills are commonly circulated to the
bottom of the workstring and then reversed out to remove scale, pipe
dope, or other contaminants prior to pumping into the formation. Several hundred gallons (e.g., 10 to 25 gallons per foot) of 10 to 15 percent HCl acid will then be circulated or bullheaded down to the
perforations and be allowed to soak (i.e., to improve communication
with the reservoir by cleaning up the perforations and dissolving
debris in the perforation tunnel). Some operators are beginning to
forego the solvent and acid cleanup (obviously to reduce rig time and
associated costs). Their presumption is that the damaging material
is pumped deep into the formation and will not seriously impact
well performance.
There are several features unique to high permeability fracturing which
make pre-treatment diagnostic tests and well-specific design strategies
Fracture Treatment Design
highly desirable if not essential: fracture design in soft formations is
very sensitive to leakoff and net pressure; the controlled nature of the
sequential tip screenout/fracture inflation and packing/gravel packing
process demands relatively precise execution strategies; and the treatments are very small and typically “one-shot” opportunities. Furthermore, methods used in hard-rock fracturing to determine critical
fracture parameters a priori (e.g., geologic models, log and core data,
or Poisson ratio computational models based on poroelasticity) are of
limited value or not yet adapted to the unconsolidated, soft, high permeability formations.
There are three tests (with variations) that form the current basis
of pre-treatment testing in high permeability formations: step-rate tests,
minifrac tests, and pressure falloff tests.
Step-Rate Tests
The step-rate test (SRT), as implied by the name, involves injecting
clean gel at several stabilized rates, beginning at matrix rates and
progressing to rates above fracture extension pressure. In a high permeability environment, a test may be conducted at rate steps of 0.5,
1, 2, 4, 8, 10, and 12 barrels per minute, and then at the maximum
attainable rate. The injection is held steady at each rate step for a
uniform time interval (typically 2 or 3 minutes at each step).
In principle, the test is intended to identify the fracture extension
pressure and rate. The stabilized pressure (ideally bottomhole pressure)
at each step is classically plotted on a Cartesian graph versus injection rate. Two straight lines are drawn, one through those points that
are obviously below the fracture extension pressure (dramatic increase
in bottomhole pressure with increasing rate), and a second through
those points that are clearly above the fracture extension pressure
(minimal increase in pressure with increasing rate). The point at which
the two lines intersect is interpreted as the fracture extension pressure.
The dashed lines on Figure 7-11 illustrate this classic approach.
While the conventional SRT is operationally simple and inexpensive, it is not necessarily accurate. A Cartesian plot of bottomhole
pressure versus injection rate, in fact, does not generally form a
straight line for radial flow in an unfractured well. Simple pressure
transient analysis of SRT data using desuperposition techniques shows
that with no fracturing the pressure versus rate curve should exhibit
upward concavity. Thus, the departure of the real data from ideal
behavior may occur at a pressure and rate well below that indicated by
the classic intersection of the straight lines (see Figure 7-11).
Unified Fracture Design
Injection Rate
FIGURE 7-11. Ideal SRT—radial flow with no fracturing.
The two-SRT procedure of Singh and Agarwal (1988) is more fundamentally sound. However, given the relatively crude objectives of
the SRT in high permeability fracturing, the conventional test procedure and analysis may be sufficient.
The classic test does provide an indication of several things:
Upper limit for fracture closure pressure (useful in analysis of
minifrac pressure falloff data).
Surface treating pressure that must be sustained during fracturing (or whether sustained fracturing is even possible with a
given fluid).
Reduced rates that will ensure no additional fracture extension
and packing of the fracture and near-wellbore with proppant
(aided by fluid leakoff).
Perforation and/or near wellbore friction, which is seldom a
problem in soft formations with large perforations and high shot
Casing pressure that can be expected if the treatment is pumped
with the service tool in the circulating position.
Fracture Treatment Design
A step-down option to the normal SRT is sometimes used specifically
to identify near-wellbore restrictions (tortuosity or perforation friction).
This test is done immediately following a minifrac or other pump-in
stage. By observing bottomhole pressure variations with decreasing
rate, near-wellbore restrictions can be immediately detected (i.e.,
bottomhole pressures that change only gradually as injection rate is
reduced sharply in steps is indicative of no restriction).
Following the SRT, a minifrac should be performed to tailor the HPF
treatment with well-specific information. This is the critical diagnostic test. The minifrac analysis and treatment design modifications can
typically be done on-site in less than an hour.
Concurrent with the rise of HPF, minifrac tests, and especially the
use of bottomhole pressure information, have become much more
common. Otherwise, the classic minifrac procedure and primary outputs as described in the preceding section (i.e., determination of fracture closure pressure and a bulk leakoff coefficient) are widely applied
to HPF, this in spite of some rather obvious shortcomings.
The selection of closure pressure, a difficult enough task in hard
rock fracturing, can be arbitrary or nearly impossible in high permeability, high-fluid-loss formations. In some cases, the duration of the closure period is so limited (one minute or less) that the pressure signal
is masked by transient phenomena. Deviated wellbores and laminated
formations (common in offshore U.S. Gulf Coast completions), multiple
fracture closures, and other complex features are often evident during
the pressure falloff. The softness of these formations (i.e., low elastic
modulus) means very subtle fracture closure signatures on the pressure decline curve. Flowbacks are not used to accent closure features
because of the high leakoff and concerns with production of unconsolidated formation sand.
New guidelines and diagnostic plots for determining closure pressure in high-permeability formations are being pursued by various
practitioners and will eventually emerge to complement or replace the
standard analysis and plots.
The shortcomings of classic minifrac analysis are further exposed
when used (commonly) to select a single effective fluid loss coefficient for the treatment. As described above, in low permeability formations this approach results in a slight overestimation of fluid loss
and actually provides a factor of safety to prevent screenout. In high
Unified Fracture Design
permeability formations, the classic approach can dramatically underestimate spurt loss (zero spurt loss assumption) and overestimate
total fluid loss. This uncertainty in leakoff behavior makes the controlled timing of a tip screenout very difficult. Entirely new procedures based on sound fundamentals of leakoff in HPF (as outlined in
Chapter 5) are ultimately needed. The traditional practice of accounting for leakoff with a bulk leakoff coefficient is simply not sufficient
for this application.
Pressure Falloff Tests
A third class of pre-treatment diagnostics for HPF has emerged that
is not common to MHF: pressure falloff tests. Owing to the high formation permeability, common availability of high quality bottomhole pressure data, and multiple pumping and shut-in cycles, matrix
formation properties including kh and skin can be determined from
short duration pressure falloff tests using the appropriate transient flow
equation. Chapman et al. (1996) and Barree et al. (1996) propose prefrac or matrix injection/falloff tests that involve injecting completion
fluid below fracturing rates for a given period of time, and then analyzing the pressure decline using a Horner plot.
The test is performed using standard pumping equipment and poses
little interruption to normal operations. A test can normally be completed within one hour or may even make use of data from unplanned
injection/shut-in cycles.
The resulting permeability certainly relates to fluid leakoff as
described in Chapter 5 and allows the engineer to better anticipate
fluid requirements. An initial skin value is useful in “benchmarking”
the HPF treatment and for comparison with post-treatment pressure
transient analysis.
Bottomhole Pressure Measurements
A discussion of pre-treatment diagnostic tests requires a discussion of
the source of pressures used in the analysis. Implicit to the discussion is that the only meaningful pressures are those adjacent to the
fracture face, whether measured directly or translated to that point.
There are at least four different types of bottomhole pressure data,
depending on the location at which the real data are taken:
Fracture Treatment Design
Calculated bottomhole pressure—implies bottomhole pressure
calculated from surface pumping pressure.
Deadstring pressure—open annulus, bottomhole pressure deduced
knowing density of fluid in annulus; tubing may also be used as
dead string when treatment is pumped down the casing.
Bundle carriers in the workstring—measured downhole, but above
the service tool crossover.
Washpipe data—attached to washpipe below service tool crossover.
Washpipe pressure data is the most desirable for HPF design and
analysis based on its location adjacent to the fracture and downstream
of all significant flowing pressure drops. Workstring bundle carrier data
can introduce serious error in many cases because of fluid friction
generated through the crossover tool and in the casing-screen annulus. Without detailed friction pressure corrections that account for
specific tool dimensions and annular clearance, there is a possibility
for a significant departure between washpipe and workstring bundle
carrier pressures. Deadstring pressures are widely used and considered
acceptable by most practitioners; some others suggest that redundant
washpipe pressure data has shown that the deadstring can mask subtle
features of the treatment. The use of bottomhole transducers with realtime surface readouts is suggested in cases where a dead string is not
feasible or when well conditions (e.g., transients) may obscure important information.
Reliance on bottomhole pressures calculated from surface pumping pressure is not recommended in HPF. The combination of heavy
sand-laden fluids, constantly changing proppant concentrations, very
high pump rates, and short pump times makes the estimation of friction pressures nearly impossible.
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