Two-phase gravity currents in porous media

Two-phase gravity currents in porous media
J. Fluid Mech. (2011), vol. 678, pp. 248–270.
c Cambridge University Press 2011
Two-phase gravity currents in porous media
M A D E L E I N E J. G O L D I N G†, J E R O M E A. N E U F E L D,
M A R C A. H E S S E A N D H E R B E R T E. H U P P E R T
Department of Applied Mathematics and Theoretical Physics, Institute of Theoretical Geophysics,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
(Received 15 October 2010; revised 20 January 2011; accepted 2 March 2011;
first published online 26 April 2011)
We develop a model describing the buoyancy-driven propagation of two-phase gravity
currents, motivated by problems in groundwater hydrology and geological storage
of carbon dioxide (CO2 ). In these settings, fluid invades a porous medium saturated
with an immiscible second fluid of different density and viscosity. The action of
capillary forces in the porous medium results in spatial variations of the saturation
of the two fluids. Here, we consider the propagation of fluid in a semi-infinite porous
medium across a horizontal, impermeable boundary. In such systems, once the aspect
ratio is large, fluid flow is mainly horizontal and the local saturation is determined
by the vertical balance between capillary and gravitational forces. Gradients in the
hydrostatic pressure along the current drive fluid flow in proportion to the saturationdependent relative permeabilities, thus determining the shape and dynamics of twophase currents. The resulting two-phase gravity current model is attractive because
the formalism captures the essential macroscopic physics of multiphase flow in porous
media. Residual trapping of CO2 by capillary forces is one of the key mechanisms that
can permanently immobilize CO2 in the societally important example of geological
CO2 sequestration. The magnitude of residual trapping is set by the areal extent
and saturation distribution within the current, both of which are predicted by the
two-phase gravity current model. Hence the magnitude of residual trapping during
the post-injection buoyant rise of CO2 can be estimated quantitatively. We show that
residual trapping increases in the presence of a capillary fringe, despite the decrease
in average saturation.
Key words: gravity currents, multiphase flow, porous media
1. Introduction
The buoyancy-driven flow of immiscible fluids within a porous medium is a
fundamental process that has applications to a host of engineering and environmental
systems. In all cases, the interfacial tension acting between the fluids gives rise to
a two-phase region over which the saturation, or pore volume fraction, of each
fluid can vary. The model problem considered in § 4 is applicable to perched aquifers,
which are formed when water in the subsurface accumulates on top of an impermeable
barrier above the main water table in the partially saturated vadose zone (Fetter 2001).
Another example is the migration, immobilization and reclamation of toxic substances
such as chlorinated solvents and trichloroethylene, often referred to collectively as
† Email address for correspondence: [email protected]
Two-phase gravity currents in porous media
dense non-aqueous phase liquids (DNAPLs). A related problem is the migration of
light non-aqueous phase liquids (LNAPLs) in the capillary zone above the water
table. In this case, three phases (air, water and LNAPL) flow in the pore space and
similar concepts to those presented here apply (Parker & Lenhard 1989; Bear et al.
1996; Bear & Ryzhik 1998). Here we consider a two-phase gravity current resulting
when a wetting phase, such as water, and a non-wetting phase, such as CO2 or
oil, are confined by an impermeable boundary in the porous medium and driven
predominately horizontally by the density difference between the fluids.
One particularly timely and pressing example of two-phase flows in porous media
is the geological storage of CO2 . Carbon capture and storage (CCS) is currently the
only proposed technological means to reduce CO2 emissions from fossil fuel power
plants (Metz et al. 2005). The CO2 is captured at source, compressed and injected
into porous geological formations at least 1 km underground. Potential storage sites
include saline aquifers, depleted oil reservoirs and unprofitable coal seams. Here
we focus on saline aquifers, which provide the largest potential storage volume. In
siliciclastic aquifers, which are mostly composed of silicate minerals, the aqueous brine
is typically the wetting phase and the injected supercritical CO2 is the non-wetting
phase. The supercritical CO2 is less dense than the ambient brine, so the buoyant CO2
rises as a plume until it reaches a flow barrier, at which point it spreads laterally as
a gravity current. The CO2 continues to spread until either it is able to rise towards
the surface, or it becomes permanently trapped by three main mechanisms listed here
in order of increasing time scales (Metz et al. 2005): residual trapping where CO2 is
immobilized in pore spaces by capillary forces once injection has ceased; solubility
trapping where CO2 dissolves into brine to form a denser fluid that sinks and mineral
trapping where CO2 precipitates as minerals.
The long-term security of geological CO2 storage is determined by the competition
between leakage and trapping mechanisms (Nordbotten et al. 2005; Woods & Farcas
2009; Neufeld, Vella & Huppert 2009). In their study of the distribution of existing
and abandoned wells in the oil-producing Alberta Basin, Gasda, Bachu & Celia
(2004) highlight the large number of potential leakage pathways located in otherwise
attractive CO2 storage sites. This emphasizes the importance of trapping mechanisms
that reduce the amount of CO2 that is free to potentially escape, and the dependence
of the security of long-term storage on trapping processes. Capillary forces can have
significant effects on the migration and trapping of the CO2 injected into geological
formations saturated with brine.
Drainage occurs when the non-wetting fluid displaces the wetting fluid within the
pore space and imbibition describes the opposite process. In both cases, a finite
fraction of the displaced fluid generally remains in the pores and may become
immobile. During drainage, the wetting phase remains within the pores as continuous
thin films coating grain surfaces. During imbibition, the remaining fraction of the
non-wetting phase is known as the residual saturation and is immobilized by capillary
forces in the form of isolated ganglia. The residual trapping of the non-wetting fluid
can be a problem in situations such as oil extraction or chemical spills because it
can be difficult to remove much of the non-wetting phase efficiently. In contrast,
capillary forces may have a positive influence on the long-term trapping of CO2
during geological storage. During injection of the non-wetting CO2 , capillary forces
both reduce the CO2 saturation within the current and increase the vertical extent of
the current. This increases the volume of reservoir and brine contacted by CO2 and
is therefore beneficial for dissolution and mineral trapping (Ennis-King & Paterson
2005; Riaz & Tchelepi 2006). When injection ceases, residual CO2 is trapped as the
M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert
current proceeds through the formation. The saturation of CO2 within the current
during the injection stage greatly affects the amount that is subsequently trapped,
and experimental work quantifies this dependence through initial–residual curves
(Pentland et al. 2008). Previous gravity current models have assumed a constant,
uniform CO2 saturation and hence a constant amount of residual trapping (Hesse,
Orr & Tchelepi 2008; Farcas & Woods 2009; MacMinn & Juanes 2009). Whilst these
models capture the dependence of residual trapping on the areal extent of the current,
the observed relationship between initial and residual saturations is not captured in
estimates obtained in this way.
Many studies of gravity currents in porous media use a sharp-interface
approximation, in which the regions within and without the current are fully saturated,
and the two fluids are separated by a macroscopically sharp interface (Huppert &
Woods 1995; Hesse et al. 2007; Gasda, Nordbotten & Celia 2009; Juanes, MacMinn
& Szulczewski 2010). Such models, which predict the evolution of the current height
h, were originally developed and tested for miscible flows in porous media and are
applicable to two-phase gravity currents only when capillary forces are negligible
compared to gravitational and viscous forces. Sharp-interface models have been
developed for gravity currents in confined geometries (h ∼ H , where H is the thickness
of the reservoir), where the flow of both fluids is non-negligible, and hence the viscosity
ratio plays a key role in their behaviour (Nordbotten & Celia 2006; Hesse et al. 2007;
Neufeld & Huppert 2008; Juanes et al. 2010). For sharp-interface gravity currents in
unconfined aquifers (h H ), there is negligible flow of the ambient fluid, and hence
models of their propagation are independent of the ratio of viscosities between the
fluids (Huppert & Woods 1995; Vella & Huppert 2006; Golding & Huppert 2010).
However, when the two fluids are different phases, capillary forces may lead to a
saturation transition zone that is not negligible, thus rendering the sharp-interface
assumption invalid. In the particular case of CO2 –brine systems, the potentially large
unstable viscosity contrast could further increase the size of the transition zone,
or capillary fringe. Furthermore, sharp-interface models do not capture the strong
reduction in relative permeability experienced by the non-wetting phase current, which
could greatly affect predictions for its spreading velocity (Bickle et al. 2007). This
provides an important motivation for the systematic development of a two-phase
gravity current model and its analysis.
In this paper, we consider the buoyancy-driven propagation of immiscible fluids
within a semi-infinite porous medium, taking explicit account of the effects of
capillarity on the propagation and saturation within the current. In § 2 we introduce
the standard formulation of two-phase flow in porous media (Pinder & Gray 2008).
We use the large aspect ratio of the current to show that vertical velocities are
negligible and that the saturation distribution is determined by vertical gravity–
capillary equilibrium (Lake 1996). Hence, in § 3 we derive a two-phase gravity current
model from the full governing equations. We find that the effect of capillary forces
on the behaviour of the current is controlled by two nonlinear functions, which
describe the dependence of the vertically integrated phase volume and flux on the
thickness of the current. In § 4 we apply our model to steady-state gravity currents
that form beneath discontinuous horizontal flow barriers during the buoyant rise of
CO2 through heterogeneous geological formations (Johnson, Nitao & Knauss 2004;
Green & Ennis-King 2010; Hesse & Woods 2010) and estimate the magnitude of
residual trapping. We conclude in § 5 with a discussion of the effect of two-phase
phenomena on the behaviour of buoyant currents within the subsurface and provide
an outlook for future research.
Two-phase gravity currents in porous media
2. The standard model of two-phase flow in porous media
We consider the invasion of a non-wetting fluid into a porous medium that is
initially saturated with a wetting fluid in a drainage process. In the following section,
we summarize the standard two-phase model, which is based on the concept of
macroscopic capillarity (Leverett 1941). The model considers the effects of interfacial
tension and the reduction in fluid flow caused by the convoluted interface of the two
fluids within the porous matrix on a scale much larger than the typical pore scale. Both
fluid phases are considered to be immiscible and incompressible Newtonian liquids of
constant, though unequal, density and viscosity. The action of capillary forces on the
pore scale results in only a partial displacement of the wetting phase, and so the two
phases co-exist within the same pore spaces. We denote quantities pertaining to the
wetting and non-wetting phases by subscripts w and n respectively. The saturation,
Si , is a macroscopic quantity that describes the average volume fraction, φi , of phase
i in a representative elementary volume (REV), normalized by porosity φ, i.e.
Si = φi /φ,
i = w, n.
This definition implies the saturation constraint
Sw + Sn = 1.
The irreducible wetting phase saturation, denoted by Swi , is the wetting phase
saturation that always remains within the pores during drainage and is assumed to
be constant. Two-phase flow models are therefore generally written in terms of an
effective non-wetting phase saturation, s, and wetting phase saturation 1 − s, defined
Sw − Swi
and 1 − s =
1 − Swi
1 − Swi
The capillary pressure caused by the surface tension at the interface between two
phases is expressed on the macroscopic scale by the difference in the pressure between
pn − pw = pc (s),
where the standard model assumes that the capillary pressure, pc , is a function of
saturation only.
In a drainage process, a minimum capillary pressure, called the capillary entry
pressure, must be achieved for the non-wetting phase to enter a pore. The magnitude
of the capillary entry pressure is inversely proportional to the size of the entrance to
the pore. This entry pressure can create flow barriers for the non-wetting phase in
regions where the porosity is reduced. As the capillary pressure increases, successively
smaller pores are invaded. Typical pc (s) curves for a primary drainage process, where
the non-wetting phase is invading the porous medium for the first time, are displayed
in figure 1(a) for rock types with different pore-size distributions. If the porous
medium has a narrow pore-size distribution (solid curve), once the capillary pressure
is above the entry threshold for the largest pore, a small increase will be sufficient for
the non-wetting phase to enter the majority of the remaining pores, and so the wetting
phase saturation decreases rapidly. A wider distribution of pore sizes will result in a
more gradual transition, as demonstrated in figure 1(a) (dash-dotted curve).
Two commonly used empirical models for capillary pressure curves are the Brooks–
Corey and van Genuchten models (Brooks & Corey 1964; van Genuchten 1980). The
derivation of the two-phase gravity current model presented here is independent of
the specific parameterization of the capillary pressure curve. However, we choose the
M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert
Λ =1
Λ = 10
Relative permeability
Capillary pressure
(a) Swi
1− s
1− s
Figure 1. (a) Typical capillary pressure curves for drainage in porous media with narrow
(solid curve) and wide (dash-dotted curve) pore-size distributions. (b) Non-wetting (+) and
wetting (×) phase relative permeability data for Ellerslie Sandstone in the Alberta Basin,
Canada, compared to typical model values with α = 2 and krn0 = 0.116 for the non-wetting
(dash-dotted curve) and β = 2 for the wetting (dashed curve) phases. Data source: Bennion &
Bachu (2005).
simpler Brooks–Corey curve to illustrate the model in order to minimize the number
of parameters and because it is the more appropriate curve for consolidated rocks
with a well-defined entry pressure. The Brooks–Corey capillary pressure function is
given by
pc = pe (1 − s)−1/Λ ,
where Λ is a fitting parameter related to the pore-size distribution. A small value of
Λ corresponds to a wide distribution of pore sizes, whereas in the limit Λ → ∞, all
pores are of the same size. Examples of this curve are plotted in figure 1(a) for Λ = 1
and 10. The parameter pe can be understood as the capillary pressure required for
the non-wetting phase to enter the largest pores.
Local mass conservation in each phase requires
(Si ) + ∇ · ui = 0, i = w, n,
where ui is the volumetric flux of phase i. The extension of Darcy’s law to two-phase
flow (Leverett 1941) assumes that
ui = −kλi [∇pi − ρi g],
i = w, n,
where k is the intrinsic permeability of the porous medium, g is the gravitational
acceleration, ρi is the density of phase i and pi is the pressure averaged over a REV
of phase i. The mobility of each phase,
λi (s) = kri (s)/µi ,
is inversely proportional to the dynamic viscosity, µi , and proportional to the relative
permeability, kri , of each phase, which is assumed to be a unique nonlinear function
of s only, during primary drainage. The presence of two phases alters both the pore
space geometry and microscopic boundary conditions for each fluid, thereby changing
the fluid flux in each phase which results from the application of a pressure gradient.
Two-phase gravity currents in porous media
To account for this on the macro-scale, the relative permeability, kri , of each phase is
a dimensionless scaling factor that multiplies the intrinsic permeability of the medium,
k (Leverett 1939).
An approximate relationship for kri (s) is commonly based on a power law (Corey
1954) of the effective saturation (2.3). Hence, we write the relative permeability of the
non-wetting phase as
krn = krn0 s α ,
and of the wetting phase as
krw = (1 − s)β ,
for some powers α and β, where krn0 < 1 is the end point relative permeability of
the non-wetting phase at its maximum saturation s = 1. When s = 1, the wetting
phase is at the irreducible saturation, and is still hydraulically connected in the sense
that Darcy flow could still occur via thin films. However, the effective permeability
of the wetting phase is approximately zero since the fluid films are too thin for
flow to be induced by the application of a typical pressure gradient. A review of
various models used for the constitutive relations for consolidated rocks is given by
Li & Horne (2006). Laboratory data for CO2 in Ellerslie standstone, obtained by
Bennion & Bachu (2005), are plotted in figure 1(b) along with the model values
with α = β = 2 and krn0 = 0.116, which show excellent agreement. In this paper we
consider flow in unconfined reservoirs for which fluid motion in the wetting phase
can be neglected. Therefore, we require an expression only for the non-wetting phase
relative permeability, and for this we set α = 2 where necessary to enable analytical
and numerical calculations. However, we emphasize that the model requires only
krn = krn (s) and is by no means restricted to this value of α.
Equations (2.2), (2.4) and (2.6)–(2.8), along with the constitutive relations (2.5),
(2.9) and (2.10), form a complete description of two-phase flow. The capillary entry
pressure, pe , and pore-size distribution parameter, Λ, in the Brooks–Corey capillary
pressure model, along with the relative permeability parameters α, β and krn0 , are
determined by fitting (2.5), (2.9) and (2.10) to experimental data. The irreducible
wetting phase saturation, Swi , embedded in the definition of effective saturation (2.3),
must also be determined experimentally.
3. Development of a two-phase gravity current model
We now consider the buoyancy-driven flow of non-wetting fluid along a horizontal
barrier and develop a vertically integrated model for a two-phase gravity current.
As the current propagates, the non-wetting phase displaces the wetting phase which
initially fills the pores in a primary drainage process. Fed by a constant flux per unit
width q, the gravity current propagates beneath an impermeable upper boundary in
an infinite reservoir initially fully saturated with a denser, wetting, fluid. This scenario
is applicable to CO2 spreading beneath an upper boundary of impermeable rock in a
CO2 storage reservoir as well as the migration of non-aqueous phase contaminants.
The current height, h(x, t), denotes the depth below which no non-wetting phase is
present and where the non-wetting phase saturation drops to zero, s = 0 (see figure 2).
The depth of the reservoir is much greater than the height of the current, and so
the horizontal pressure gradient and the velocity in the wetting phase are negligible
below the current. We additionally assume that the fluids are simply connected which
implies that, to leading order, there are no horizontal pressure gradients driving flow
of the wetting phase within the current.
M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert
h (x, t)
ρn, µn
φ, k
ρw, µw
Figure 2. Sketch showing a gravity current with a non-uniform saturation profile propagating
beneath an upper impermeable boundary into a porous medium saturated by fluid of a different
phase, density and viscosity. The plot on the left shows a representative profile of saturation
with depth.
3.1. Vertical gravity–capillary equilibrium and saturation profile
The dominant balance governing two-phase gravity currents can be found by applying
scaling arguments in a manner analogous to previous studies of single-phase gravity
currents (Huppert & Woods 1995, for example). For a current whose length L is much
greater than its height H , i.e. = H /L 1, local mass conservation, as expressed by
(2.6), implies that the vertical velocities are much less than the horizontal non-wetting
phase velocity. Thus, the pressure within each phase is nearly hydrostatic and given
= ρn g and
= ρw g.
Pressures pn and pw are plotted in figure 3(a), where the difference at the current
boundary z = h is equal to the capillary entry pressure pe . Consequently, (2.4) implies
= −ρg,
where ρ = ρw −ρn , which mathematically captures the balance between gravitational
and capillary forces within the current, known as vertical gravity–capillary equilibrium.
We integrate (3.2) between z and h to obtain the expression for capillary pressure as
a function of height
pc [h(x, t), z] = pe − ρg(z − h),
where we note that the capillary pressure at the edge of the current pc (z = h) = pe .
The assumption of gravity–capillary equilibrium is commonly invoked in studies
of multiphase flow (Yortsos 1995; Lake 1996, for example). In this study, it follows
from the large aspect ratio of the gravity current and the subsequent assumption
that pressure within each phase is hydrostatic. We expand on this assumption in
the Appendix and show that the time scale over which gravity–capillary equilibrium
is recovered during the motion of the current is much smaller than the time scale
for the motion itself. In this way, we show that the assumption remains valid as
the current propagates. We note that the assumption breaks down in areas where
saturation is very low (Lake 1996). In this region, which is located along the current
boundary, the relative permeability of the non-wetting phase is very small, leading to
potentially long time scales for recovering gravity–capillary equilibrium as the current
propagates. However, this region is thin, and we anticipate that the behaviour within
it has little influence on the overall motion of the current predicted by our vertically
integrated model.
Two-phase gravity currents in porous media
(a) 0
(b) 0
pn = pc + pw
Figure 3. (a) Graph illustrating the hydrostatic pressure within each fluid. The difference
between the non-wetting and wetting phase pressures at z = h is equal to the capillary entry
pressure pe . (b) Saturation profiles within the current are plotted for Λ = 1, 2 and 10. The
size of the capillary fringe or transition zone, where the non-wetting phase saturation varies
considerably, is indicated by arrows for each case.
Parker & Lenhard (1989) combine an expression for capillary pressure resulting
from the gravity–capillary equilibrium assumption in three-phase flow, analogous to
(3.3), with the Brooks–Corey model, to find the implied wetting phase saturation as
a function of vertical position. Here we treat the simpler two-phase case similarly by
equating (2.5) and (3.3), to obtain an expression for the non-wetting phase saturation,
s, within the gravity current as a function of h and z given by
s[h(x, t), z] = 1 − 1 +
where the length scale he is defined by
he ≡ pe /ρg.
The saturation profile given by (3.4), which is a function of h and z, is plotted in
figure 3(b). Physically, the capillary entry height, he , represents the height of a column
of non-wetting fluid such that the hydrostatic pressure equals the entry pressure of
the largest pore throat found in the porous medium. It is therefore a measure of the
relative strength of capillary forces to gravitational forces. The region in which the
non-wetting phase saturation varies considerably is referred to as the transition zone,
or capillary fringe. The width of the capillary fringe increases with the magnitude of
capillary forces, i.e. with the inverse of the pore-size distribution parameter 1/Λ (as
shown in figure 3b) and with the capillary entry height he . Bear et al. (1996) define
a similar height to he , which is often used as a characteristic length scale for the
capillary fringe in a three-phase system.
M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert
3.2. Vertically integrated equation for the current height profile
Vertical integration of (2.6) for the non-wetting phase between z = 0 and z = h yields
s dz − s(h)
un dz − [un ]z=h
+ [wn ]h0 = 0,
where ϕ = φ(1 − Swi ). This reduces to
s dz +
un dz = 0,
since s(h) = 0 and λn [s(h)] = 0 at the edge of the current, and therefore, by (2.7), the
non-wetting phase velocities un = wn = 0 at z = h. We impose the condition wn = 0
on the upper impermeable boundary at z = 0. This equation was derived by Bear &
Ryzhik (1998) to describe a DNAPL lens moving over an impermeable boundary.
The first integral in (3.7) is calculated directly using (3.4). We substitute (2.7) into
the second integral and therefore find that (3.7) becomes
−Λ ∂h
∂ ∂h h
λn dz = 0.
− kρg
ϕ 1− 1+
∂x ∂x 0
This can be further simplified by noting that the non-wetting phase saturation at the
upper boundary z = 0 can be expressed by the saturation function
s0 (h/ he ) ≡ s(h/ he , z = 0) = 1 − 1 +
Hence, we obtain the principal new equation that governs the flow of a two-phase
buoyancy-driven current,
ϕs0 (h/ he )
∂h ρg k̃ ∂
h F(h/ he ) = 0,
µn ∂x
where k̃ = kkrn0 . Here, the effect of the vertical saturation distribution on flow due to
variation in relative permeability is encapsulated in the flux function
λn [s(z)] dz
krn0 h 0
s0 (h/ he )
S α (1 − S)−(Λ+1)/Λ dS.
Λh 0
F(h/ he ) ≡
To obtain (3.12), we change variables from z to s (defined by (3.4)) by noting that
∂s/∂z = − (Λ/ he )(1 − s)(Λ+1)/Λ . It may be interesting to note that the integral in (3.12)
is, in fact, the incomplete beta function, defined as B(x : a, b) ≡ 0 ua−1 (1 − u)b−1 du,
with x = s0 (h/ he ), a = 1 + α and b = − 1/Λ. We are able to integrate (3.12) directly
Two-phase gravity currents in porous media
when α = 2 to find an analytical expression for F given by
⎧ 2
3s0 − 2s0
(Λ = 0.5)
⎪ (1 − s )2 − 2 log(1 − s0 )
he ⎨ s0 (2 − s0 )
+ 2 log(1 − s0 )
(Λ = 1)
F(h/ he ) =
h ⎪
1 − s0
(1 − s0 )−1/Λ 2 2Λ(s0 − Λ)
s0 +
(Λ =
0.5, 1),
1 − 2Λ
(Λ − 1)(1 − 2Λ)
where s0 = s0 (h/ he ), and we will use this in § 4.1.
The explicit framework encapsulated by (3.9), (3.10) and (3.12) represents a
significant advance in our understanding of the effects that two-phase phenomena
have on the propagation of gravity currents in porous media, accounting for the
non-uniform saturation distribution of the fluid in the current. Equation (3.10)
is the two-phase analogue of the sharp-interface models considered by a number
of authors (Huppert & Woods 1995). The two-phase behaviour is captured by
two functions of h/ he , defined in (3.9) and (3.12). The flux function, F, and
the saturation function, s0 , are investigated further in § 3.3. We show that in the
appropriate limits, h/ he → ∞ and Λ → ∞, the two-phase functions s0 (h/ he ) → 1 and
F(h/ he ) → 1, and thus the single-phase governing equation is recovered. It is worth
emphasizing the generality of (3.11), for which any function of saturation can be
supplied for the non-wetting phase mobility, λn . We use a power relation for the nonwetting phase relative permeability in this study to illustrate the two-phase behaviour
(Li & Horne 2006).
To complete the mathematical description of the problem, we specify two boundary
conditions. The flux at the origin can be written as
ρg k̃ ∂h
h F(h/ he ) = −q.
µn ∂x
The second boundary condition recognizes that the height of the current at the nose,
located at x = xN (t), is
h(xN ) = 0.
3.3. The flux function, F(h/ he ), and saturation function, s0 (h/ he )
The flux function F(h/ he ), defined by (3.12), accounts for the depth-integrated relative
permeability of the non-wetting phase and depends on the pore-size distribution
parameter Λ, the capillary entry height he and the power α in the constitutive
relationship (2.9). We focus on the effects of varying Λ and he in this paper in order
to constrain the parameter space, and because the value α = 2 is reasonable for many
CO2 /brine systems, as demonstrated in figure 1(b) (Bennion & Bachu 2005).
Figure 4(a) shows a graph of F(h/ he ) for Λ = 0.5, 1, 2 and 10. For finite values of
Λ, F is a monotonically increasing function of h/ he , which tends to 1 as h/ he → ∞.
This is due to the fact that when the current is much thicker than the capillary fringe,
the non-wetting fluid nearly saturates the pores in most of the current, and so capillary
effects become less important to its overall motion. The limit also corresponds to the
case of negligible capillary entry pressure, i.e. pe 1. When h is comparable to or
smaller than the capillary fringe, non-wetting phase saturations are lower, which
results in reduced local relative permeabilities and hence lower values of F.
M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert
(a) 1.0
s0 (h/he)
(b) 1.0
Λ = 10
Λ = 10
Figure 4. The flux function F(h/ he ), plotted in (a), and the saturation function s0 (h/ he ),
plotted in (b), are shown for Λ = 0.5, 1, 2 and 10.
The saturation function, s0 (h/ he ), plotted in figure 4(b) for Λ = 0.5, 1, 2 and 10,
indicates the thickness of the current with respect to the capillary fringe, which
is defined as the depth over which the non-wetting fluid saturation in the current
increases from s = 0 to some cut-off value s = sc ∼ 1. The width of the capillary fringe
is the value of h for which s0 (h/ he ) = sc . We can see clearly from figure 4(b) that the
capillary fringe width increases with 1/Λ and he .
Thus, for all finite values of Λ, the current approaches the single-phase result, where
F ≡ 1 and s0 ≡ 1, when h/ he 1. The rate of convergence to the single-phase limit
increases with Λ because the capillary fringe is thinner. In the limit Λ → ∞, which
describes a medium of uniform pore size with a single capillary entry pressure, there
are no capillary effects on the gravity current and s0 ≡ 1 and F ≡ 1 for all values of
h/ he .
Conversely, in the limit Λh/ he 1, the saturation function is approximately linear
in h, given by
s0 (h/ he ) ≈ Λh/ he ,
and F can be approximated as
F(h/ he ) ≈
α + 1 he
4. An application to geological CO2 storage
Many geological systems identified as potential sites for CO2 storage are
heterogeneous, containing nearly horizontal, discontinuous barriers to flow. Such
barriers may include shale layers, which have a greatly reduced permeability, are
regions of high capillary entry pressure, and are therefore effectively impermeable to
the non-wetting phase. Johnson et al. (2004) identified these ubiquitous shale layers
as having a crucial effect on migration and trapping of CO2 in storage reservoirs
such as Sleipner. More recently, Saadatpoor, Bryant & Sepehrnoori (2010) showed
that capillary barriers can have a dominant effect on CO2 migration and trapping.
The enhanced lateral dispersion due to discontinuous shales is important for residual
trapping because it causes the CO2 to contact a larger volume of the reservoir
(Johnson et al. 2004; Hesse & Woods 2010). The ‘inject low and let rise’ strategy
for geological CO2 storage (Kumar et al. 2005; Bryant, Lakshminarasimhan & Pope
2008) tries to maximize the dispersion and trapping of CO2 due to flow barriers, by
injecting it through a horizontal well at the base of the storage formation (figure 5).
Two-phase gravity currents in porous media
‘Inject low and let rise’
Steady state
Imbibition and residual trapping
Figure 5. Sketch showing the three stages of injecting CO2 into a storage reservoir. Figure (a)
depicts the early stages of injection where CO2 starts to rise through a geological formation
containing shale layers. In figure (b) the gravity currents that form beneath each shale layer
have reached a steady state. Figure (c) sketches the post-injection stage where CO2 is residually
trapped in regions previously occupied by the non-wetting phase gravity currents.
We envisage the following idealized scenario, shown schematically in figure 5.
Injection at the base of a heterogeneous formation begins with the rise of buoyant
CO2 through the permeable pore space (figure 5a). As the CO2 rises, it encounters
horizontal barriers to flow and spreads laterally (figure 5b), which has important
implications for the lateral dispersal and numerical upscaling of the reservoir, as
discussed in Hesse & Woods (2010) and Green & Ennis-King (2010). The CO2
perched beneath each barrier forms a steady-state gravity current. At the end of the
injection period (figure 5c), fluid continues to rise throughout the system, leaving
residual CO2 trapped in its wake. In this scenario, residual trapping occurs most
notably in the regions beneath flow barriers, previously occupied by steady-state CO2
plumes that can be modelled as gravity currents.
The volume of CO2 that is residually trapped during this final stage is of primary
importance for the long-term safe storage of CO2 within the subsurface because
it leads to rapid immobilization of CO2 after the end of injection. The quantity
trapped depends on two key factors, both of which are influenced by two-phase
phenomena: the volume of rock invaded by CO2 in the reservoir and the saturation
within the contacted region. Experimental data indicate that the residual saturation,
Sn,res , depends on the initial saturation of the non-wetting phase, Sn,init , at the time
of flow reversal (Pentland et al. 2008). Therefore, the amount of CO2 immobilized by
capillary forces and the rate of trapping depends on the saturation distribution of the
non-wetting CO2 at the end of the injection phase.
At leading order, the buoyant rise of CO2 through a series of flow barriers can be
understood by studying the steady current beneath a single layer (Hesse & Woods
2010). Therefore, we focus on the buoyancy-driven flow of non-wetting fluid beneath
a single barrier (figure 6). For continuous injection, we anticipate that the gravity
currents will reach a steady state after sufficient time, in which the flux into the
current equals the flux rising at the ends. In § 4.1 we investigate how capillary forces
affect the vertical extent and saturation distribution within a gravity current beneath
a single impermeable barrier. The height and saturation profiles obtained in § 4.1 are
then used as inputs to a trapping model, which captures the relationship between
initial and residual saturations. In this way, we are able to gain valuable insight into
how capillary forces affect the magnitude of residual trapping in the ‘inject low and
let rise’ strategy for CO2 storage.
M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert
Figure 6. Sketch of a steady-state two-phase gravity current beneath a barrier of finite extent.
Injection of a flux 2q at x = 0 is balanced exactly by vertical flow at x = ± L, resulting in a
steady-state current whose saturation we model with depth.
4.1. Steady-state two-phase current beneath a finite boundary
We consider a steady-state gravity current, under a barrier of finite length, 2L, where
L is much greater than the current height. Non-wetting fluid is injected with constant
flux 2q at the centre of the barrier and spreads to the edges at x = ± L, where it
rises vertically from the system and so the edges effectively act as sinks. Due to the
symmetry of the problem, we consider only half of the current sketched in figure 6
and solve for a current fed by a flux of magnitude q at x = 0. We see from (3.10) that
the profile of the steady state is given by
h F (h/ he ) = 0,
with boundary conditions
ρg k̃ dh
h F(h/ he ) = −q (x = 0),
µn dx
h = 0 (x = L).
Unlike the case for unbounded spreading, there exists a natural horizontal length
scale to our steady-state problem, L, and we define the corresponding vertical height
qµn L
H =
ρg k̃
introduced by Huppert & Woods (1995). We define dimensionless variables
x = x/L
= h/H,
but drop the primes for ease of notation in the subsequent analysis.
The existence of a third length scale, the capillary rise height he , is expressed in
terms of a Bond number
= H / he ,
which is the ratio of the hydrostatic pressure for a scale height H and the capillary
entry pressure. It therefore indicates the relative importance of capillary forces
compared to gravitational and viscous forces. The single-phase case is recovered
in the limit B → ∞ in which either the capillary entry pressure pe → 0 or the height
of the current is sufficiently large that the capillary fringe is small in comparison.
Two-phase gravity currents in porous media
Hence, after integration and application of the boundary condition at x = 0, the
dimensionless governing equation is
F (hB) = −1.
The remaining boundary condition is
h(x = 1) = 0,
and the flux function and saturation function are given by
s0 (hB)
F(hB) =
S α (1 − S)−(Λ+1)/Λ dS
BΛh 0
s0 (hB) = 1 − (1 + hB)−Λ
When α = 2, we can use the analytical expressions for the function F(h), (3.13), to
greatly simplify the governing equation for the current height. We make the expressions
dimensionless using (4.4), substitute into the steady-state governing equation (4.6),
integrate and rearrange, and hence obtain implicit algebraic expressions for h,
displayed here for the values Λ = 0.5 and 1 used in figures 7(c–f ),
⎪h2 + 6 h + 16 [1 − (1 + hB)3/2 ] + 2 (1 + hB) log(1 + hB) (Λ = 0.5),
3B 2
2(1 − x) =
⎩h2 + 6 −
(3 + 2hB) log(1 + hB)
(Λ = 1).
In a similar manner, equations for other values of Λ can be derived, but due to their
intricate nature, they are not displayed here. Hence, the steady-state height profile of
a two-phase gravity current is the solution to a nonlinear algebraic equation.
We note that the solution for the height profile in this steady-state problem in the
sharp-interface limit is
h = 2(1 − x),
which has been shown to fit experimental data in a Hele-Shaw cell very well (Hesse
& Woods 2010). It is worth noting that when B → ∞, the steady-state single-phase
solution is recovered.
4.1.1. The effect of pore geometry and capillary forces
By studying this steady-state problem, much insight can be gained into the general
influence of capillarity and pore-size distribution on two-phase gravity currents, as
captured in our model by parameters B and Λ respectively. The pore-size distribution
parameter Λ determines how quickly the saturation increases with distance from the
current boundary. A series of plots in figure 7 shows non-wetting phase saturation
profiles and distributions for Λ = 0.5, 1 and 10, where the Bond number B = 1 is held
constant. When Λ = 10, the current approaches the sharp interface limit. In this case,
figures 7(a) and 7(b) show that there is a small transition zone at the edge of the
current. As Λ decreases and capillary effects become more important (figures 7c–f ),
the transition zone becomes larger. As this happens, the average saturation decreases,
and consequently the current thickens in order to maintain the constant flux of
injected fluid.
M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert
s (x = 0, z)
z 1
Λ = 10
Λ = 10
Λ = 0.5
Λ = 0.5
Non-wetting phase saturation scale
Figure 7. Saturation profiles and distributions for B = 1, with Λ = 10 (a, b), Λ = 1 (c, d ) and
Λ = 0.5 (e, f ). Figures (a), (c) and (e) show the saturation at x = 0, and figures (b), (d ) and (f )
illustrate the shape and saturation within each current. In each case the solid line indicates
h(x), where s(h) = 0, and the dashed line indicates the sharp-interface limit.
The height of the current at the origin, h0 ≡ h(x = 0), as a function of Λ and B, is
plotted in figures 8(a) and 8(b) respectively. We use the current height at the origin
to give an indication of the areal extent of the current, which strongly influences
the extent of residual trapping that can occur, as discussed in § 4.2. Both graphs
demonstrate how increasing capillary effects, either by decreasing Λ or B, thicken the
current. The effect of varying B on the average saturation within a current where
Λ = 1 is summarized in figure 8(c).
It is not clear, a priori, how the opposing effects of a lower average saturation and
a larger areal extent will affect the total volume of non-wetting fluid contained within
the current. The graph in figure 8(d ) shows the total volume of the non-wetting phase
Two-phase gravity currents in porous media
(a) 5
(b) 7
(d) 1.15
Dimensionless volume of
non-wetting fluid
Average saturation
Figure 8. Figures (a) and (b) plot the current height at the origin, h0 , against Λ, with B = 1
and against B, with Λ = 1, respectively. Figures (c) and (d ) plot respectively the average
saturation and the total volume of non-wetting fluid contained within the steady-state current
as functions of B, with Λ = 1. The dashed lines indicate the values in the sharp-interface limit.
as a function of B, for Λ = 1. We see that the volume always increases with increasing
capillary forces.
We investigate in § 4.2 how both the volume of fluid contained in the currents and
the saturation distribution affect estimates for residual trapping within the system.
4.2. Residual trapping
Residual trapping occurs when ganglia of the non-wetting phase become disconnected
from the main bulk of the fluid and therefore immobilized during imbibition. It is
caused by surface tension acting within the pore geometry, making it impossible for
the wetting fluid to displace all of the non-wetting fluid. For detailed explanations of
the mechanisms on the pore scale which cause residual trapping, the reader is directed
towards Lenormand, Zarcone & Sarr (1983) and Pinder & Gray (2008).
At the macro-scale, the key experimental observation is the dependence of the
non-wetting phase residual saturation on the initial saturation at the time of flow
reversal (Land 1968). An initial–residual graph summarizing these data for a number
of two-phase-rock systems studied in the literature can be found in figure 2 in Pentland
et al. (2008). All the data sets have a common feature that a higher fraction of fluid
is trapped in regions where the initial saturation is lower. Several models have been
developed to represent these empirical measurements and a useful summary of these
results is contained in figure 6 of Pentland et al. (2008).
The magnitude of residual trapping in the gravity currents considered in § 4.1 is
primarily dependent on the areal extent of the current and the maximum saturation
reached during the drainage stage, i.e. the steady-state height and saturation profile.
This allows us to avoid modelling the details of the imbibition process when inferring
M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert
Trapped CO2/initial CO2
A =1
Figure 9. Fraction of fluid trapped residually, for Λ = 1, as a function of B estimated using
Land’s model with C = 1. The dashed line represents the estimate obtained by applying Land’s
model to a sharp-interface gravity current. Note that this limit is approached more quickly
with respect to increasing B for higher values of Λ.
the volume of the residually trapped non-wetting phase and instead apply a trapping
model directly to the steady-state saturations.
As an example, we use Land’s model (Land 1968) to estimate the fraction of fluid
that is trapped residually in the steady-state currents considered in § 4.1. Land’s model
relates initial and residual saturations by
Sn,res =
1 + CSn,init
where C is known as Land’s coefficient, and for application to a CO2 –brine–rock
system, has a value of order 1 (Juanes et al. 2006). We apply this relation across
the initial saturation profile calculated from the steady-state problem, integrate over
the height and length of the current and thus calculate how much fluid would be
trapped if the constant flux were turned off. Figure 9 presents estimates for the
fraction of initial fluid in the steady-state current that would be residually trapped
as a function of B, with Λ = 1. We compare this to estimates obtained using a sharp
interface model, with Sn,init = 1 everywhere and find that including capillary effects
can increase estimates for the fraction of CO2 residually trapped by more than 50 %.
We see this increase in the proportion of trapped fluid because the average saturation
in the currents is lower. Furthermore, the actual volume of fluid contained in a current
is larger with increased capillary forces, due to its greater areal extent (figure 8d ),
which means that a greater absolute volume of fluid is trapped.
5. Conclusions and discussion
One of the major difficulties of modelling the flow of CO2 during geological storage
is the range of scales over which processes occur. Full numerical simulations of flow
(Kumar et al. 2005; Mo et al. 2005; Ide, Jessen & Orr 2007) are able to incorporate
two-phase phenomena, but the computational intensity requires a compromise on
spatial resolution. On the other hand, sharp-interface models lose small-scale details
such as the saturation distribution but are able to model flow over much larger
distances (Nordbotten & Celia 2006; Hesse, Orr Jr & Tchelepi 2008; Juanes et al.
Two-phase gravity currents in porous media
2010). Many theoretical models of gravity currents in porous media to date have
assumed a sharp interface separating the fluid in the current from the ambient
(Huppert & Woods 1995; Hesse et al. 2007). This is appropriate in situations where
capillary forces are negligible compared to viscous and gravitational forces. However,
when the intruding and ambient fluids are different phases, capillary forces acting
between them can have a significant influence on their motion. In this study, we
considered the propagation of a gravity current of a non-wetting fluid, along an
impermeable horizontal boundary, into a porous medium initially saturated by a
wetting fluid. We assume that sufficient time has elapsed for the current to become
long and thin. We model the resulting non-uniform saturation distribution and
derived a vertically integrated, time-dependent equation that governs the height of
the gravity current. The two-phase governing equation is analogous to its single-phase
counterpart (Huppert & Woods 1995), with the two-phase effects incorporated by
two new functions describing the saturation at the impermeable boundary and the
vertically integrated flux through the current as a function of its height. We show that
our two-phase model recovers the sharp-interface case in appropriate parameter limits.
The two-phase framework is attractive because it captures the effects introduced by
surface tension between the two fluids, whilst maintaining the form of the sharpinterface model.
The two key features to consider when modelling two-phase flow on the macroscale are capillary pressure and relative permeability. The model developed in this
paper captures the general functional dependence of both features on saturation. We
illustrate this dependence by using the Brooks–Corey model for capillary pressure
and power law relations for relative permeability. The central assumption is that of
gravity–capillary equilibrium, where gravitational and capillary forces balance in the
vertical direction, which we show to be valid for long, thin gravity currents. Using
this assumption of gravity–capillary equilibrium, along with the capillary pressure as
a function of saturation, we obtained an explicit expression for the saturation profile
within the current. This profile depends primarily on the pore-size distribution,
characterized by Λ, and the ratio of capillary to gravitational forces, characterized by
he . We found that as both 1/Λ → 0 and he → 0, capillary forces become negligible and
the capillary fringe reduces to the sharp-interface limit. The effect of the saturation
profile on the flow of a two-phase gravity current was encapsulated in the flux
function F(h/ he ), which itself is a function of Λ and he . The value of the height
scale he in geological settings varies widely depending on the rock and fluid types.
For example, Bennion (2006) measured capillary entry pressures pe = 3.5 kPa in a
sandstone formation and pe = 493.6 kPa in a carbonate formation in the Alberta
Basin, Canada, which could potentially be suitable as sequestration sites. Using these
values, along with g = 9.8 m s−2 and representative densities ρw = 1020 kg m−3 and
ρn = 550 k gm−3 (Bickle et al. 2007), we calculate representative capillary entry heights
of he = 0.76 m and 107 m respectively.
We used our model of a two-phase gravity current to assess the magnitude
of residual trapping as a buoyant plume of CO2 rises through a heterogeneous
formation. By considering CO2 ponded beneath a single, finite barrier, we were able
to investigate how the main parameters influence the areal extent and non-uniform
saturation profile of a two-phase gravity current. We demonstrated how capillary
forces simultaneously thicken the current, i.e. increase its vertical extent, and reduce
the average saturation within it. These two effects have opposing influences on the
volume of fluid that accumulates in steady-state currents. We have shown that the
net result in all cases studied is an increase with capillary forces in the volume of
M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert
non-wetting fluid contained within a current, indicating that the increased vertical
extent dominates.
Residual trapping happens once injection of CO2 ceases and is one of the key
mechanisms enabling the success of long-term storage of CO2 as part of the
immense challenge of climate change mitigation. Some steps have been made towards
incorporating the effects of residual trapping in sharp interface models, by assuming
that the volume of fluid in the current decreases at a constant rate, controlled by a
trapping parameter (Kochina, Mikhailov & Filinov 1983; Hesse et al. 2008; Juanes
et al. 2010). With the two-phase model developed in this paper, we are able to
investigate the extent of residual trapping more accurately because we know the
height and saturation profiles at the end of the injection period. Whilst we use Land’s
trapping model to obtain estimates, it is important to note that any model relating
initial–residual saturations could be used once the saturation profile is known. The
essential feature that at lower saturations, the fraction of fluid trapped is higher is the
reason why, as capillary effects are increased and the two-phase currents have lower
average saturations, the fraction of fluid residually trapped is greater. Moreover, the
amount of fluid contained in the steady-state current at the start of imbibition is
larger when capillary forces are higher, so the overall volume of fluid trapped is also
higher, not just the fraction. This result is important given that residual trapping
post-injection is proportional to the volume of fluid per unit length in the steady
current (Hesse et al. 2008). Furthermore, the increased height means a greater volume
of rock is invaded by CO2 as capillary effects are increased, and it is therefore able to
contact a larger volume of brine, which could increase further trapping by dissolution
(Johnson et al. 2004).
Whilst we have considered steady-state gravity currents in this paper, the model
developed is fully time-dependent and therefore provides a theoretical framework in
which to investigate how capillary forces affect the motion of transient two-phase
gravity currents. For example, it is not clear how the reduced relative permeability
at the nose of the current, caused by the lower saturation when the height decreases
to zero, affects the steepness of the nose and the thickness and velocity of the
entire current. Other future work will extend the methods outlined here to two-phase
gravity currents propagating in a confined aquifer, where the viscosity ratio of the
fluids plays an important role. Our framework also holds the promise of being able to
model drainage and imbibition in a self-consistent manner, using differing capillary
pressure and relative permeability relationships, thus improving the assessment of
residual trapping. Furthermore, the simplifications introduced by the vertical gravity–
capillary equilibrium assumption allow efficient calculations of two-phase currents
over distances required at the basin scale during geological sequestration of CO2 . In
all these systems, and in a number of related areas, the incorporation of capillary
effects provides new insight into the dynamics of two-phase gravity currents.
The authors would like to gratefully acknowledge the artistic expertise of Mark
Hallworth. The research of M.J.G. is funded by the EPSRC, J.A.N. by the Leverhulme
Trust and Lloyds Tercentenary Foundation, M.A.H. partly by a David Crighton
Fellowship and the work of H.E.H. is partially supported by a Royal Society Wolfson
Research Merit Award.
Appendix. Justification for vertical gravity–capillary equilibrium
Vertical gravity–capillary equilibrium is the key assumption of the two-phase gravity
current model presented in this paper because it enables us to resolve the saturation
Two-phase gravity currents in porous media
distribution within the current. In a long, thin gravity current, where flow is assumed
to be predominately horizontal, we showed in § 3.1 that gravity–capillary equilibrium
is implied by assuming that each phase is in hydrostatic equilibrium. However, the
height of the current rises during propagation, so the saturation distribution, which
depends on current height, must keep re-equilibrating in order for the assumption
to remain valid. We find expressions for the time scale, tm , over which the current
height changes due to gravity-driven horizontal motion and the time scale, tc , required
for the saturation distribution to return to gravity–capillary equilibrium after a small
change in current height. By showing that the ratio of time scales rt = tc /tm is small, we
demonstrate that to a good approximation, gravity–capillary equilibrium is maintained
as the current propagates. It should be noted that the assumption breaks down at the
front of the current, where flow in the vertical direction is not negligible compared to
the horizontal motion. However, this is the case in all gravity current models which
consider long, thin currents and rely on the assumption of hydrostatic pressure. For
example, Huppert (1982) showed experimentally for viscous gravity currents that it
makes no difference to the overall motion of the current.
In order to find expressions for the time scales tm and tc , we vertically integrate
the non-wetting phase local mass conservation equation (2.6) between z and h. This
∂V (h, z)
un dz − [wn ]z = 0,
(A 1)
∂x z
where V (h, z) = z s(h, z
) dz
is the volume of fluid per unit length contained in
a column of current that has height h, and we have used s(z = h) = un (z = h) =
wn (z = h) = 0.
The time scale, tm , over which the current height changes due to the horizontal
velocity of the current fluid during propagation is obtained from the balance between
the first and second terms in (A 1). We make the assumption, as explained in
§ 3.1, that the current is in gravity–capillary equilibrium, and therefore we find that
∂V /∂t = s(h, z)∂h/∂t by using (3.4) and integrating directly. Darcy’s law (2.7) provides
an expression for horizontal velocity
un = −kλn (s)
= −ρgkλn (s) ,
(A 2)
where the second equality comes from the expression for the hydrostatic non-wetting
phase pressure, given by
pn (h, z) = pe + ρw gh − ρn g(h − z).
(A 3)
Note that this uses the condition pn = pe + ρw gh at the current boundary z = h. We
scale t ∼ tm , z, h ∼ H and x ∼ L, where H and L are the characteristic length scales
for the current height and horizontal extent, respectively, and write = H /L. Hence,
the first two terms of (A 1) yield the scaling
∼ ρgkλn (s) 2 ,
(A 4)
which we can rearrange to find an expression for the time scale, tm , given by
tm ∼
s −2
ρgk λn (s)
(A 5)
M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert
Now we consider a current at height h(x, t), which is in gravity–capillary equilibrium
and therefore has saturation distribution s(h, z), given by (3.4), and suppose that its
height increases by δh. The time scale tc indicates how quickly the recovery of vertical
gravity–capillary equilibrium is achieved due to the vertical velocity of the nonwetting phase and is therefore given by the balance between the first and third terms
in (A 1). Once gravity–capillary equilibrium is recovered, the saturation distribution
is s(h + δh, z), and therefore we find that
δV = V (h + δh, z) − V (h, z) = s(h, z)δh
to first order in δh. From (2.7) the vertical velocity, wn , is given by
− ρn g ,
wn = −kλn (s)
(A 6)
(A 7)
where the vertical gradient of the hydrostatic pressure pn (h + δh, z) after a change in
height of δh can be written as
pn (h + δh, h + δh) − pn (h + δh, 0)
(h + δh)
= ρn g
(A 8)
Hence, (A 7) becomes
(A 9)
We use (A 6) and (A 9) to scale ∂V /∂t ∼ s(h, z)δh/tc and wn ∼ ρn gkλn (s)δh/ h in (A 1)
to find
∼ ρn gkλn (s) ,
(A 10)
which can be rearranged to give
wn = −ρn gkλn (s)
tc ∼
ϕH s
ρn gk λn (s)
(A 11)
Thus, we find, using (A 5) and (A 11), that the ratio of time scales
rT ∼
ρ 2
(A 12)
For a long, thin gravity current, 1, and so we have shown that vertical gravity–
capillary equilibrium is recovered for a given change in current height on a time scale
much smaller than the time scale of the overall motion of the current. We therefore
conclude that the assumption of gravity–capillary equilibrium remains valid as the
two-phase current propagates through the porous medium.
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