Energy 93 (2015) 1173e1188 Contents lists available at ScienceDirect Energy journal homepage: www.elsevier.com/locate/energy Energy analysis for guiding the design of well systems of deep Enhanced Geothermal Systems Mengying Li, Noam Lior* University of Pennsylvania, Department of Mechanical Engineering and Applied Mechanics, Philadelphia, PA 19104-6315, USA a r t i c l e i n f o a b s t r a c t Article history: Received 7 August 2015 Received in revised form 23 September 2015 Accepted 24 September 2015 Available online xxx The focal objective of this work is to calculate the energy consumption for constructing the EGS (Enhanced Geothermal Systems) wells, to examine the energy (heat and power) performance of such well systems, and to propose and evaluate several ways for improving that performance. A model was developed to compute the pressure and temperature ﬁelds of the geoﬂuid ﬂowing in the production and injection wells to be able to calculate the ﬂow pumping energy consumption, and the heat gain/loss during its ﬂow in/out of the enhanced reservoir, for wells up to 10 km deep. The total well construction energy consumption was calculated to be 19.40 TJ/(km of well) for the considered well conﬁgurations, and increases approximately linearly with the ﬂow cross section area of the well. Several ways to improve the energy performance of the wells, by increasing the heat output of the production wells and decreasing the required power for pumping the geoﬂuid were evaluated: (1) increasing the number of injection/production wells to reduce the pressure drop in each, (2) increasing the ﬂow cross section of the injection/projection well, and (3) adding thermal insulation to the circumference of the production wells (to reduce the geoﬂuid heat loss to the rock). Most of these methods were found to indeed increase the power output of the geothermal system but have increased the construction energy requirement somewhat more. More energy efﬁcient drilling methods and materials of lower embodied energy can lead to a higher EROI (energy return on investment). The EROI of the recommended EGS well system designs ranged from 33.8 to 286.2. © 2015 Elsevier Ltd. All rights reserved. Keywords: EGS (Enhanced Geothermal System) Geothermal well drilling Flow and heat transfer in geothermal wells Energy reduction for constructing EGS well systems Well-construction embodied energy EROI (energy return on investment) 1. Introduction Geothermal energy is abundant, practically renewable, and has a long-term potential estimated to be more than 200,000-fold of current world energy demand . In the continental United States, over 99% of the geothermal heat within depths of about 10 km is available in HDR (hot dry rock) where little natural ﬂuid exists and the rock has low permeability [1,2]. One characteristic of an HDR resource is the geothermal gradient (G C/km), which quantiﬁes the temperature increase with resource depth. To extract the geothermal heat, a large volume of the hot rock (reservoir) must be fractured in the high geothermal gradient region (Fig. 1). The fractures are created by means of hydro-fracturing and etc.  to allow liquid ﬂow through the hot rock. Injection and production wells are drilled to connect the fractures to allow liquid circulation between the reservoir and the surface structures. Cold ﬂuid * Corresponding author. 220 S. 33rd Street, 212 Towne Building, Philadelphia, PA 19104, USA. Tel.: þ1 215 898 4803. E-mail address: email@example.com (N. Lior). http://dx.doi.org/10.1016/j.energy.2015.09.113 0360-5442/© 2015 Elsevier Ltd. All rights reserved. (“geoﬂuid”, usually water) is injected into this reservoir to be heated and brought back to the surface for direct use or for generating electricity in a power plant. The injection wells (IW-s), production wells (PW-s), fractured reservoir and surface power plant constitute a typical EGS (Enhanced Geothermal System) where IW-s and PW-s are essential components of an EGS system, which connect the engineered reservoir with the surface power plant. To bring as much thermal energy to the surface while incurring least energy consumption, the design goals for IW-s and PW-s should be to minimize: (1) The energy consumption for constructing the wells. (2) The heat losses from the geoﬂuid to surrounding rock during its ﬂow through the PW-s. (3) The pumping energy consumption for driving the geoﬂuid ﬂow through these wells. The focal objective of this work is therefore to calculate the energy consumption of constructing the EGS wells (Section 2), and 1174 M. Li, N. Lior / Energy 93 (2015) 1173e1188 The contribution of this work is to examine quantitatively the energy input and output for typical EGS wells in an EGS system (the analysis of the power generation plant and engineered reservoirs can be found in Refs.  and ), and examine their sensitivity to the number of IW/PW, ﬂow cross section of IW/PW and adding insulation to PW-s, for ultimately proposing highest net energy output well system design. Here the energy input is the energy consumption for constructing the wells. The energy output is the electricity generated by the EGS power plant minus the geoﬂuid circulating pumping energy. The temperature and pressure changes of the geoﬂuid ﬂowing through the wells are calculated by a ﬁnite difference method. The detailed contributions of this work are: Fig. 1. Schematic diagram of an EGS system [1,3]. to model the pressure and temperature ﬁelds of the geoﬂuid ﬂowing in the wells to be able to calculate the ﬂow pumping energy consumption, and the heat gain/loss during its ﬂow in/out of the engineered reservoir (Section 3). Since deeper reservoirs typically contain more heat and that at higher temperature, they require longer and deeper wells that negate some of the beneﬁt because they require more energy for their construction and also incur higher heat and pressure losses. It thus clearly poses a design optimization problem focused on maximizing overall energy performance, addressed here by varying the reservoir depth, number of wells, well ﬂow cross sections and the best energy performance design are proposed (Section 4). While costs are of paramount importance, this study was focused on the energy analysis because (1) energy perspective analyses are lacking in the published literature while cost analyses are widely available [1,4,5]. (2) Knowledge of energy quantities can be translated relatively easily to costs if the price of energy, which ﬂuctuates wildly, is known. Nine cases were calculated: 3 depths each at 3 geothermal gradients, as shown in Table 1, assuming in all that the ground surface temperature is Ts ¼ 15 C . The average PW bottom temperature Tb,pw (assumed to be equal to the reservoir temperature) is: Tb;pw ¼ 15 þ Z$G; (1) where Z is the well depth (km) and G is the geothermal gradient ( C/km). When the geoﬂuid is injected into the EGS subsurface system (wells and reservoirs), it is preferred to keep it in the liquid phase throughout the subsequent heating and recovery process, rather than allow it to ﬂash or otherwise evaporate, as explained in Refs.  and . To accomplish this, the pressure in the subsurface system must be maintained high enough to keep the hot liquid below its boiling point. (1) Our calculations extend to well depths of 10 km. (2) A simpliﬁed mathematical model was developed and used to compute the EGS well construction energy consumption. (3) A model for ﬂow and heat transfer in EGS wells was developed to include both subcritical and supercritical geoﬂuid and density changes in wells. (4) Methods that improve the performance of IW/PW were examined. They include the possibility to add more IW/PW, to enlarge the ﬂow cross section areas of IW/PW and to add thermal insulation to the PW-s. 2. Energy consumption for constructing EGS wells The drilling process of EGS wells is in many ways similar to that used for the gas and oil. After exploration to determine the location, the drilling rig is set in place. The drilling rig creates a hold to the ground and provides the torque to rotate the drill bits and power to circulate drilling mud (ﬂuid to lubricate and cool the bits and remove the cuttings while drilling). The rig also provides monitoring equipment along the well to do measurements and diagnosis needed for the well drilling process. In the traditional drilling method, rock in the drill's path is broken into small pieces under the high pressure applied by the drill bits. A hole of desired depth is created typically in a few months. There are also proposals for novel drilling methods like jet drilling and thermal drilling as introduced in Refs.  and . After creating the hole, it needs to be cased and cemented to complete the well (Fig. 2). Casing is to install permanent hardware (usually metal tubes) inside the borehole to maintain its integrity and isolate it from its surrounding environment. After casing, the gaps between casings and the borehole are ﬁlled with cement . The geoﬂuid in EGS applications ﬂows inside the inner diameter of the casing. The well system of deep EGS consists of at least one IW and one PW. More than one PW per IW is common. For example, there are doublets, one IW for one PW; triplets, two PW-s for each IW; quartet, three PW-s with a single IW and ﬁve-spots, and so on . After reaching the desired depth, the wells must typically be deviated to a horizontal section into the reservoir region (Fig. 1) to eventually stimulate (fracture) it. The deviated well arrangement offers a larger drilling window along the created horizontal part, and could create nearly parallel vertical fracture [1,11]. The casing in Table 1 Average PW bottom temperatures Tb,pw for the depths and geothermal gradients considered in this work. Geothermal gradient G, C/km 40 Well depth Z, km Average PW bottom temperature Tb,pw, C 5 215 60 7.5 315 10 415 5 315 80 7.5 465 10 615 5 415 7.5 615 10 815 M. Li, N. Lior / Energy 93 (2015) 1173e1188 1175 Fig. 2. Conﬁgurations of (a) 5 km wells and (b) 7.5 and 10 km wells [1,11]. In (b), the numbers in parenthesis represents 10 km wells. the deviated part of the IW in the reservoir region is perforated to allow hydro-fracturing . As discussed in Ref. , the length of the deviated part is a product of the chosen number of fractures in the reservoir and the distance between them. That inter-fracture distance was found in Ref.  to be 120 m to maintain the heat extraction decay from the rock within 10% during 40 years of operation and to make the thermal interference between adjacent fractures negligible. The length of the deviated part, shown in Table 2 is calculated based on the recommended number of fractures from Ref.  (under certain assumptions). Since the number of fractures is highly dependent on how fast the heat extraction from the reservoir decay one can tolerate that beyond the scope of this work, in the following analysis, we only consider the vertical part of the wells. In this study, the borehole is considered to be drilled using PDC (polycrystalline diamond compact) and roller cone hybrid bits  in a traditional drilling method. The energy for drilling EGS wells consists of three parts: the energy consumed to create the borehole and operating casing and cementing (drilling operation), the embodied energy of well casings and cement and other miscellaneous such as the embodied energy of the drilling mud. 2.1. Well conﬁgurations Studies in Ref.  identiﬁed that the need for an increasing number of casing strings or liners with depth is a reason for the signiﬁcant increase of drilling costs with depth. A liner is a special casing string that does not extend to the top of the wellbore, but instead is anchored or suspended from inside the bottom of the previous casing string . Ref.  provides the recommended number of casing strings or liners of EGS wells for different depths, and the information is reproduced in Table 3. For wells over 6 km deep, liners instead of casing strings can be used to reduce the casing cost . We therefore chose 4 casing strings/liners for the 5 km well, and 6 casing strings/liners for the 7.5 km and 10 km wells, and assumed that wells with the same number of casing strings/liners have the same well conﬁgurations , as shown in Fig. 2 (b), and that the 7.5 km and 10 km wells have the same conﬁgurations. PW-s are of same conﬁguration design as corresponding IW-s of the same depth. The number of PW-s per IW depends on the temperature and pressure changes of the geoﬂuid transporting in them (Section 4). 2.2. Energy consumption of the drilling operation According to Ref. , making a 6 km well (including borehole creation and casings/cement operation) requires 8475 kg/day of diesel fuel. The speciﬁc chemical energy of the diesel fuel is 45.3 MJ/ kg , thus the daily chemical energy consumption is E_ dr ¼ 0.384 TJ/day. The energy is for rotating the drill bits and operating the drilling mud pump, as well as for installing the casings and cementing. For different types of soil and underground rock stresses, the rate of penetration (ROP, m/day) would differ, and so would the time required to drill wells of the same depth. For typical drilling conditions considered in Ref. , the ROP for different borehole diameters is listed in Table 4, which, incidentally, also shows that it is not directly proportional to the borehole diameter. The total drilling time is around 1.554 times of borehole creation time, which takes into consideration the additional time needed for installing the casings and for cementing . The total energy consumption of the drilling operation can thus be calculated as: Table 2 Length of the horizontal part of wells for the depths and geothermal gradient considered in this work. Well depth Z, km 5 Geothermal gradient G, C/km Recommended number of fractures  Length of horizontal part of well, km 40 30 3.60 7.5 60 23 2.76 80 19 2.28 40 10 1.20 10 60 8 0.96 80 7 0.84 40 5 0.60 60 4 0.48 80 4 0.48 1176 M. Li, N. Lior / Energy 93 (2015) 1173e1188 where Mcm is the mass of cement (kg), rcm is the density of cement (1522 kg/m3 ), Dwb is the diameter of wellbore (m), D is the diameter of casing (m) and Lc is the length of casings/cement well segment (m). Table 5 presents the embodied energy of casing and cement for 5 km, 7.5 km and 10 km wells. Column 3 presents the unit casing weight obtained from Refs.  and , columns 4e6 present the casings/cement length of well segments, columns 7e9 present the calculated casing mass, which equals to the length of the casing multiplied by the corresponding unit weight, columns 10e12 present the calculated cement mass, using Eq. (4). The embodied energy of casing and cements are also presented in Table 5. The total energy consumption of well construction is: Table 3 Number of recommended casing strings or liners for EGS wells (from Ref. ). Depth, km 1.5 2.5 3 4 5 5 6 6 7.5 10 No. of Casing Strings 4 4 4 4 4 5 5 6 6 6 Table 4 Energy consumption of borehole creation for 5 km, 7.5 km and 10 km wells. Hole diameters Dwb, inch (Fig. 2) Well depth, km 3600 2600 17-1/200 12-1/400 8-1/200 Sum Energy consumed, TJ (Eq. (2)) Edr ROP, m/day  33.5 83.8 83.8 62.5 45.7 Drilling length Ldr, m (Fig. 2) 5 0 112.5 2375 2500 0 4987.5 42.9 7.5 168.75 1687.5 1875 2625 1125 7481.25 70.3 10 225 2250 2500 3500 1500 9975 93.7 Ewell ¼ Edr þ Bcs þ Bcm þ Eother : (5) where Eother (TJ) represents the embodied energy of drilling mud, the energy consumption of measurements and diagnosis during drilling and are obtained from Ref. . 2.4. Well construction analysis results and discussion E_ L ¼ dr dr : ROP (2) The construction energy consumptions for wells of depths 5, 7.5 and 10 km, calculated by using Eqs. (2)e(5) are shown in Fig. 3. The drilling energy consumption and casings' embodied energy compose over 93.1% of the total energy consumption while the cement embodied energy and the miscellaneous energy consists less than 6.9%. The associated embodied energy requirement was found to be about 43.0%e49.8% of the total energy requirement. The well construction energy increases nearly linearly with depth (Fig. 3). The speciﬁc energy consumption is ewell ¼ 19.4 TJ/(km of well). To validate our calculations, we compared our results with those in Ref.  and the comparisons are listed in Table 6, which show that they are within ±7%. It is noteworthy that our calculations are simpler, more detailed and easier to duplicate. A dimensionless parameter A* is deﬁned to indicate the relative ﬂow cross section area of a well compared to the wells in Fig. 2, where A* ¼ 2 indicates that the ﬂow cross section area of the well is doubled (i.e. the diameter is enlarged √2 times) and A* ¼ 3 indicates that the ﬂow cross section area of the well is tripled (the diameter is enlarged √3 times). For A* ¼ 1, the energy requirement to construct a well is presented in Fig. 3. For A* ¼ 2, the volume of rock that must be removed is doubled and the casing and cement material weight is doubled. So the energy consumed to construct a well is approximately doubled. In a similar way, for A* ¼ 3, the energy consumed to construct a well is approximately tripled. To reduce the heat exchange between geoﬂuid in the wells and surrounding rock, the cement can be replaced by a more thermal insulating material, such as ceramic insulation. For the magnesia where Ldr is the drilling distance (m). The energy consumptions for borehole creation we calculated using Eq. (2) are listed in Table 4. Columns 1 and 3e5 of the table present the diameter and length of the considered well segments (Fig. 2) and column 2 presents the ROP values obtained from Ref. . 2.3. The embodied energy of well casings and cement Typically, the well casings are made of stainless steels to resist corrosion from the geoﬂuid, and the cementing material is Portland cement . The speciﬁc embodied energy of stainless steel is bcs ¼ 56.7 MJ/kg and of Portland cement is bcm ¼ 5.5 MJ/kg . The unit casing weights are found from the American Petroleum Institute casing dimensions chart  and reference , where the casing thickness is dependent on its diameter. The casing stainless steels embodied energy is thus: Bcs ¼ bcs Mcs (3) where Mcs is the mass of casing (kg). The embodied energy of the cement is, Bcm ¼ bcm Mcm ¼ bcm rcm p 2 Dwb D2 Lc ; 4 (4) Table 5 Embodied energy of casing and cement for 5 km, 7.5 km and 10 km deep wells. Column 1 2 3 4 Hole diameters Dwb, inch (Fig. 2) Casing diameter D, inch (Fig. 2) Unit casing weight, kg/m [11,17] Casing length/Cementing length Lc, m (Fig. 2) Casing mass Mcs, ton (Casing length unit casing weight) Cement mass Mcm, ton (Eq. (4)) 5 7.5 10 5 7.5 10 5 7.5 10 0 125 2500 2500 0 187.5 1875 3750 2625 1125 250 2500 5000 3500 1500 0.0 24.7 267.9 281.3 0.0 573.9 32.5 55.7 371.1 401.8 295.3 95.6 1219.5 69.1 74.2 494.8 535.8 393.8 127.5 1626.0 92.2 0.0 26.6 232.4 110.7 0.0 369.7 2.03 57.2 398.9 348.6 116.2 20.2 941.1 5.18 76.3 531.9 464.8 154.9 26.9 1254.8 6.90 Well depth, km 3000 3600 2600 2600 17-1/200 13-5/800 12-1/400 9-5/800 8-1/200 700 Sum Embodied energy, TJ (Eqs. (3)e(4)) 460.73 251.17 131.08 79.62 47.62 5 6 7 8 9 10 11 12 M. Li, N. Lior / Energy 93 (2015) 1173e1188 1177 3.1. The mathematical model for ﬂow and heat transfer of geoﬂuid The assumptions are: Fig. 3. Itemized and total (numbers on top of each bar) well construction energy consumption as a function of well depth. considered here the thermal conductivity (km ¼ 0.07 W/(m K) ) is 4.12-fold lower than that of Portland cement (kcm ¼ 0.29 W/(m K) ). The magnesia is much more expensive than cement, and its speciﬁc embodied energy is bm ¼ 45 MJ/kg , which is much higher than that of Portland cement (bcm ¼ 5.5 MJ/kg ). The embodied energy of insulation material is calculated similarly to Eq. (4) by, Bm ¼ bm Mm ¼ bm rm p 4 D2wb D2 Lm (6) where rm is the density of the insulation material, 3400 kg/m . 3 3. Modeling and analysis of ﬂow and heat transfer in the wells The underground system of deep EGS consists of at least one IW and one PW, and the stimulated reservoir. Section 2 provides the model for calculating the energy consumption for well construction. In pursuit of our objective (described in Section 1), the energy gains/losses of the geoﬂuid and geoﬂuid pumping energy demand are calculated. Energy gains/losses of geoﬂuid are due to heat gain/ loss from the IW/PW, and the geoﬂuid pumping energy demand is proportional to the pressure changes. A model is therefore developed here to calculate the pressure and temperature changes of the geoﬂuid ﬂowing in the IW-s and PW-s. Knowledge of the pressure and temperature proﬁle of the geoﬂuid in the wells allows the determination of these energy components, and the effects of adding IW/PW, of enlarging their ﬂow cross section areas and of adding thermal insulation to them. This produces the information needed for recommending a highest net energy output design of geothermal well systems. Table 6 Well construction energy consumption results validation. (1) For the purpose of this study it is not necessary to consider interactions between IW-s and PW-s, which are also typically far apart. (2) The pressure is kept high enough to prevent the geoﬂuid from evaporating. (3) The time dependences of geoﬂuid properties such as temperature, pressure, density, velocity and etc. are not included in the governing equations but are introduced in the dimensionless temperature parameter TD (see Eqs. (21) and (22)). The heat transfer between the geoﬂuid and the surrounding rock is therefore varying with time, and so are thus the geoﬂuid temperature, pressure, velocity, and density and other ﬂuid properties. (4) The EGS geoﬂuid is pure water devoid of gases and minerals. Water injected into an EGS reservoir often picks up various minerals and gases, but the associated complexity that is associated with their consideration in heat transfer analysis, accompanied by insufﬁcient experimental information on their amount in various EGS systems, and since typically their content in the geoﬂuid is low in EGS system, led most of the past studies to assume that the geoﬂuid is pure [1,22e27]. Setting the mass ﬂow rate of the injection geoﬂuid to m_ iw , the mass ﬂow rate of geoﬂuid in each PW is: m_ pw ¼ This work, TJ Ref. , TJ Difference, % 5 7.5 10 80.2 149.1 199.5 84.0 139.5 208.3 4.57 6.85 4.24 (7) where m_ iw=pw is mass ﬂow rate of the geoﬂuid in a well, kg/s, rwl is the geoﬂuid loss rate in the reservoir (assumed to be 10% ) and Npw is the number of PW-s per IW. So Npw ¼ 1 for doublet wellconﬁguration, Npw ¼ 3 for quartet, etc. Considering that the resistance pressure loss SKV2/2 is minor compared to pipe friction loss S2fLV2/D  for deep EGS wells where L >> DK/4f, the total pressure gradient during ﬂuid ﬂow is the sum of gravity, friction and the ﬂow acceleration gradients [25,29]: 2 dP V r dV þ grHfD þ rV ¼ 0; dz 2D dz (8) where the minus sign before fD applies for IW-s and the plus sign applies for PW-s, and, P e the geoﬂuid pressure in wells, Pa; z e the vertical position (along g) of the geoﬂuid, m; g e the gravitational acceleration, here 9.8 m/s2; r e the geoﬂuid density in the wells, r ¼ r (P,T), kg/m3 and T is the geoﬂuid temperature, C; fD e the Darcy friction factor; V e the average ﬂow velocity of the geoﬂuid in the wells, m/s: V¼ Well depth, km ð1 rwl Þm_ iw ; Npw 4m_ iw=pw prD2 (9) Using Chen's explicit equation for Darcy friction factor fD in pipes to simplify the calculation (this equation was reported to satisfy nearly the whole related range of Reynolds numbers (4 103 to 4 108) ): 1178 M. Li, N. Lior / Energy 93 (2015) 1173e1188 2 ε=D 5:0452 log10 L fD ¼ 2 log10 3:7065 Re ðε=DÞ1:1098 7:149 0:8981 þ L¼ Re 2:8257 1 1 ln DDwb Dwb D U ¼ þ : h 2 1 kcm 2 f (12) (10) where: hf e convection heat transfer coefﬁcient between the geoﬂuid and the casing inside surface, W/(m2∙K) : where: ε e casing inner roughness factor, m; Re e Reynolds number, Re ¼ VD/n ¼ Re (P,T), where n is the kinematic viscosity of the geoﬂuid, n ¼ n (P,T), m2/s; hf D ¼ 0:23Re0:8 Pr0:4 k (13) The speciﬁc enthalpy of the geoﬂuid in the wells can be calculated from the energy balance equation : dh dV Q_ þgþV H ¼ 0; dz dz m_ k e thermal conductivity of the geoﬂuid, k ¼ k (P,T), W/(m∙K); Pr e Prandtl number of the geoﬂuid, Pr ¼ Pr (P, T); kcm e thermal conductivity of cement, here 0.29 W/(m∙K) . (11) The minus sign in front of Q_ applies to IW-s and the plus sign applies to PW-s, and h e speciﬁc enthalpy of geoﬂuid, h ¼ h (P,T), J/kg; Q_ e the heat transfer rate from surrounding rock to the geoﬂuid per unit depth of wells, W/m. It is positive when the geoﬂuid gains heat from the surrounding rock (in IW-s) while it is negative when it loses heat to the surroundings (in PW-s). Using the heat transfer e electric resistance analogy (as shown in Fig. 4(b)), the heat transfer between the geoﬂuid and surrounding rock has two thermal resistances in series: (1) Combined heat transfer from the geoﬂuid to the wellbore/rock interface 1/U1, and (2) conduction heat transfer from the wellbore/rock interface to the undisturbed rock 1/U2 (the rock at some distance from wells where the temperature is undisturbed by the operation of wells). The overall heat transfer coefﬁcient from geoﬂuid to wellbore/ rock interface U1 is calculated as: The thickness of casings is neglected here because it is less than 1% of the casing diameter , and since they are made of stainless steel, which has high heat conductivity relative to cement or rock (the ratio of conductivities of steel and cement is over 55 ). Therefore, the thermal resistance of the casings is neglected. Assuming the temperature of the surrounding rock is a function of time t and radial location r, an energy balance of the surrounding rock can be expressed in cylindrical coordinates as , 1 vTr v2 Tr 1 vTr ¼ 2 þ ; ar vt r vr vr (14) where: ar e Thermal diffusivity of the surrounding rock, here 7.69 107 m2/s ; Tr e temperature of surrounding rock at time t and location r, W/(m∙K); t e well operation time, s; r e distance from center of the well, m. Fig. 4. (a) Sketch of well segment in the ﬁnite difference simulation scheme, and (b) the heat transfer process in the ith interval of an IW . M. Li, N. Lior / Energy 93 (2015) 1173e1188 1179 The initial condition is, Tr ðr; 0Þ ¼ Tr;∞ (15) Based on the geothermal gradient, G, at the well site, the temperature of the undisturbed rock is, Tr;∞ ¼ Ts zðG=1000Þ: (16) The boundary conditions are, vTr Q_ ¼ pkr Dwb vr r¼Dwb =2 vTr ¼0 vr r¼þ∞ (17) where kr is the thermal conductivity of surrounding rock, here 1.61 W/(m∙K) ; Deﬁning a dimensionless temperature of the surrounding rock as, TD ¼ 2pkr Twb Tr;∞ _ Q (18) The continuity of heat ﬂow yields , 2pkr Q_ ¼ pDwb U1 ðTwb TÞ ¼ Twb Tr;∞ TD (19) Eliminating Twb in (19), we have, Q_ ¼ 2pDwb U1 kr Tr;∞ T ; TD Dwb U1 þ 2kr (20) As derived in Refs.  and , an algebraic equation for TD represents the solutions of Eqs. (14)e(17) quite accurately, pﬃﬃﬃﬃﬃi h TD ¼ ln e0:2tD þ 1:5 0:3719etD tD (21) where tD is the dimensionless well operation time : tD ¼ 4ar t ; D2wb (22) As plotted in Fig. 5, TD increases with tD, indicating that a decreasing amount of heat Q_ is transferred to the geoﬂuid in the wells from the rock because the geoﬂuid temperature gradually rises due to the heat transfer from the surrounding rock, the rock temperature drops as it transfers its heat to the geoﬂuid, and consequently the temperatures of the geoﬂuid and rock approach each other over time until a steady state is reached. 3.2. The solution method, range and validation A ﬁnite difference method is used to solve Eqs. (8) and (11), modeling a well composed of N longitudinal intervals, as shown in Fig. 4 (a). Fig. 4 (b) (modiﬁed from Ref. ) depicts the heat exchange with the surroundings in the ith interval of an IW. The geoﬂuid ﬂow and heat ﬂow directions are reversed in PW-s. Within each interval i, the geoﬂuid is considered to be incompressible, with a constant density ri ¼ r(Pi,Ti). All other geoﬂuid thermal properties in each depth (z) interval i are evaluated at Ti and Pi. Eqs. (8) and (11) are thus expressed as : 2 2 2 Piþ1 Pi V iþ1 V i ðz zi ÞV i þ gðziþ1 zi ÞHfDi iþ1 þ ¼0 ri 2 2Di 2 2 V V i Q_ i ðziþ1 zi Þ H hiþ1 hi þ gðziþ1 zi Þ þ iþ1 ¼0 2 m_ (23) Fig. 5. The dimensionless temperature TD as a function of the dimensionless well operating time tD The coupled ﬁnite difference Eq. (23) was solved by using the Engineering Equation Solver program (EES ). Simultaneous solution of Eq. (23) successively for each interval i leads to the determination of Piþ1 and Tiþ1 from known Pi and Ti. The surrounding rock is assumed to be granite. The geoﬂuid properties are calculated in EES using its ﬂuid property database ‘Steam_IAPWs’ . All other used values are tabulated in the Nomenclature section at the end. For validation of our analytical method, we ﬁrst note that the conservation Eqs. (8) and (11) are consistent with the widely used ones in geothermal wellbore models GWELL, GWNACL, and HOLA [37,38]. Further validation, of the numerical results, was done by examining the effect of grid size on the resulting mass, pressure, and enthalpy balance, and choosing then a grid at which further size decreases have a negligible effect. For the validation, we chose the case of one 7.5 km deep PW with a geothermal gradient of 60 C/km. The geoﬂuid mass ﬂow rate was assumed to be 100 kg/s. The geoﬂuid at the wellhead is either saturated liquid or supercritical ﬂuid at 23 MPa. The calculation procedure for the PW is shown in Fig. 6, where Tc is the critical temperature of pure water. The geoﬂuid temperature at the PW well-bottom is assumed to be equal to the average reservoir temperature of 465 C (Table 1). The calculation is to ﬁnd the PW head geoﬂuid temperature Th,pw and pressure Ph,pw, as well as the well-bottom pressure Pb,pw. The resulting sensitivity of the numerical solution after 1 year of operation to grid size, where the size here is the well segment length, is shown in Fig. 7 (a). It can be seen that well segments length of 50 m is small enough to make the numerical solution essentially grid-size independent, and we therefore chose a well segments length of 25 m for all the computations in this study. The numerical solutions were also tested by examining the magnitudes of the absolute mass, pressure, and enthalpy residuals of the governing Eq. (23). V pr D2 Rm;i ¼ m_ pm i i i 4 2 2 2 P ðziþ1 zi ÞV i iþ1 Pi V iþ1 V i Rp;i ¼ þ gðziþ1 zi Þ þ fDi þ ; ri 2 2Di 2 2 V V 1 Q_ i ðziþ1 zi Þ Rh;i ¼ hiþ1 hi þ gðziþ1 zi Þ þ iþ1 þ 2 m_ pw (24) 1180 M. Li, N. Lior / Energy 93 (2015) 1173e1188 Fig. 6. The calculation procedure diagram for the sample PW. Fig. 7. (a) Grid size dependence of the PW wellhead temperature Th,pw and well bottom pressure Pb,pw; (b) numerical calculation residuals of ﬂow in wells at the operation time of t ¼ 1 yr. Substituting all the values in Eq. (24) with the numerical calculation results, gives the residuals at each well segment i, presented in Fig. 7(b), which shows that the absolute residuals are all smaller than 104, indicating a satisfactory and converged solution. 3.3. The time dependence of the results Considering for example a 7.5 km deep PW with a geothermal gradient of 60 C/km and geoﬂuid ﬂow rate of 100 kg/s the time variation of the wellhead temperature Th,pw and well bottom pressure Pb,pw are shown in Fig. 8. The changes are rapid within the ﬁrst year and then markedly slow down. The 40-year timeaveraged wellhead temperature and well bottom pressure were calculated to be 381.5 C and 46.5 MPa, respectively, which is approximately equal to the values after the operating time of 5 years. The results presented in the following sections are for the 40year time-averaged values. 3.4. Results for injection wells As described in Section 3.2, a 25 m long well segment was chosen for the ﬁnite difference analysis. The geoﬂuid in the IW-s is kept as a supercooled liquid. The dynamic pressure change of geoﬂuid in IW is deﬁned as, DPd;iw ¼ Pb;iw Ph;iw rh;iw gZ (25) where rh,iw is the density (in kg/m3) of the geoﬂuid at IW head and it is a function of the IW head temperature Th,iw. For the considered Th,iw ¼ 35 C, 70 C and 105 C, the rh,iw is 994.0 kg/m3, 977.8 kg/m3 and 955.0 kg/m3 , respectively. The temperature change of geoﬂuid in IW is deﬁned as, M. Li, N. Lior / Energy 93 (2015) 1173e1188 1181 enthalpy (caused by reduced geoﬂuid and rock heat transfer area), resulting in a slightly increased geoﬂuid temperature. Regression of the resulting data produces the following approximate equations to express the well bottom pressure Pb,iw and temperature Tb,iw as a function of the studied parameters: Pb;iw yPh;iw þ rh;iw gZ C1 m_ A* C2 C3 (27) Tb;iw yTh;iw þ C4 Z$G þ C5 m_ þ C6 where C1 ¼ 4.87 107 sC2/kgC2, C2 ¼ 3.13, C3 ¼ 0.9 MPa, C4 ¼ 2.26 102, C5 ¼ 4.87 107 s/kg and C6 ¼ 16.84 C. 3.5. Results for production wells Fig. 8. Time variation of wellhead temperature Th,pw and well bottom pressure Pb,pw in a 7.5 km deep PW with a geothermal gradient of 60 C/km. DTiw ¼ Tb;iw Th;iw (26) Fig. 9 presents the results of a sensitivity analysis of DPd,iw and DTiw to six parameters, the well depth Z, geothermal gradients G, geoﬂuid ﬂow rate m_ iw , well relative ﬂow cross section area A*, geoﬂuid injection temperature Th,iw and geoﬂuid injection pressure Ph,iw. Their chosen values are presented in Table 7. The following conclusions can be drawn from the results presented in Fig. 9: (1) The dynamic pressure change DPd,iw rises with decreased ﬂow rate m_ iw and increased well relative ﬂow cross section area A* because of decreased friction pressure loss in both cases. The well depth Z, geothermal gradient G, injection temperature and Th,iw and pressure Ph,iw have negligible effect on the dynamic pressure change (the symbols in the plots of Fig. 9 overlap). (2) The temperature change DTiw rises with increased well depth Z and geothermal gradient G because hotter rock transfers heat to the geoﬂuid. The temperature change rises with decreased ﬂow rate m_ iw because the heat gained from rock per unit geoﬂuid mass increases. It also slightly rises with the decreased well relative ﬂow cross section area A* because the decrease of well bottom geoﬂuid pressure (caused by friction) over-compensates the decrease of well bottom geoﬂuid The PW bottom temperature is assumed to be equal to the average temperature of the reservoir (Table 1). The pressure of the geoﬂuid is continuous in the EGS system, as shown in the explanatory schematic Fig. 10. The PW head pressure Ph,pw is the lowest in the subsurface system, which needs to be kept high enough to maintain the geoﬂuid in saturated liquid (Ph,pw ¼ Psat (Th,pw)) or supercritical state (Ph,pw ¼ 23 MPa, ~1 MPa above the critical pressure of water). The arrows in Fig. 10 indicate the ﬂow direction of the geoﬂuid. The pressure change to be provided by the injection pump is: DPip ¼ DPpw DPiw þ DPr þ DPpp ; (28) where: DPpw e the geoﬂuid pressure drop in PW-s, DPpw ¼ Pb,pw Ph,pw, MPa, calculated using Eq. (23). DPiw e the geoﬂuid pressure change in the IWs, MPa. DPiw ¼ Pb,iw Ph,iw , calculated using Eq. (27). DPr e the geoﬂuid pressure drop in the fractured reservoir, MPa. It is a function of reservoir depth, geothermal gradient and the characteristics of reservoir fractures including the number of fractures, fracture radius, fracture width and permeability . DPpp e the geoﬂuid pressure drop in the power plant, MPa. For ﬂash and supercritical power plants analyzed by us in Ref. , the pump within the power plant is to bring the geoﬂuid to the pressure at which it enters the plants. For a binary power plant, the geoﬂuid exchanges heat with a secondary working ﬂuid in a Fig. 9. (a) Geoﬂuid dynamic pressure change and (b) temperature change in IW with respect to the well depth Z, geothermal gradients G, geoﬂuid ﬂow rate m_ iw , well relative ﬂow cross section A*, geoﬂuid injection temperature Th,iw and geoﬂuid injection pressure Ph,iw. 1182 M. Li, N. Lior / Energy 93 (2015) 1173e1188 Table 7 Chosen values for the IW DPd,iw and DTiw sensitivity analysis. Upper values Base values lower values Well depth Z, km Geothermal gradients G, C/km Flow rate m_ iw , kg/s Well relative ﬂow cross section area A* Injection temperature Th,iw, C Injection pressure Ph,iw, MPa 11.3 7.5 3.8 90.0 60.0 30.0 166.7 111.1 55.6 1.5 1.0 0.5 105.0 70.0 35.0 15.0 10.0 5.0 heat exchanger. The pressure drop in the heat exchanger is relatively small and assumed to be negligible . Therefore, DPpp ¼ 0. (29) The energy output (in TJ) from the EGS well system is the net electricity output of the geothermal power plant minus the required pumping energy due to the frictions in the wells, over its lifetime, ! DPpw DPiw m_ iw Epp FLf ; rip hp Eout ¼ (30) where: Epp e the net power output from the surface power plant, MJ. It is an increasing function of PW head temperature Ph,pw (equivalent to power plant inﬂow geoﬂuid temperature) and is ﬁtted using the results presented in our paper , as follows: 2:6466 Epp ¼ 5:0 105 Th;pw Epp Npw X m_ pw ; Th;pw 2 200 C; Tc 1 Npw X ¼ 937:78 ln Th;pw 4815:5 m_ pw ; Th;pw 2 Tc ; 800 C 1 (31) Fig. 10. Explanatory schematic of the pressure proﬁle in the EGS system. rip e the density of the geoﬂuid ﬂowing through the injection pump. Assumed to be 977.8 kg/m3  because the geoﬂuid injection temperature is approximately 70 C ; hp e the energy efﬁciency of the injection pump, assumed here to be 80%; F e the capacity factor of the surface geothermal power plant, assumed to be 70% ; Lf e the life of EGS well system, assumed to be 40 years . The 40-yrs-averaged PW head temperature Th,pw and pressure drop DPpw as well as IW pressure drop DPiw are used in Eqs. (30) and (31) to calculate Eout. To increase the energy output Eout, the design of PW-s should either increase the PW head geoﬂuid temperature Th,pw or decrease the pressure drop in PW-s DPpw (Eq. (30)). The geoﬂuid pressure drop in PW-s, DPpw, for a given ﬂow cross section area can be reduced by reducing the mass ﬂow rate in a well to reduce the friction pressure loss. To attain a chosen overall hot geoﬂuid extraction rate, the number of such wells needs to be increased. Another way to reduce the friction pressure loss is to increase the ﬂow cross section area of the wells for the same ﬂow rate. The PW head geoﬂuid temperature Th,pw can be increased by reducing the heat loss from the hot geoﬂuid to the surrounding rock, e.g., by replacing the cement between the casing outer surface and the rock by thermal insulation materials, such as ceramic insulation (that has high temperature resistance). 3.5.1. Effects of increasing the number of PW-s The geoﬂuid pressure drop in PW-s, DPpw, and temperature at the PW head, Th,pw, for different numbers of PW-s and whether insulation is incorporated are plotted in Fig. 11, for different well depths and geothermal gradients. As mentioned, the total mass ﬂow rate of the geoﬂuid in the PW-s was assumed to be 100 kg/s. The pressure drop DPpw decreases with increased geothermal gradients because of the increased buoyancy effect when the geoﬂuid becomes hotter. Fig. 11 also shows that insulation has little effect on the pressure drop, while the number of PW-s has large effect on the pressure drop. For deeper wells (7.5 km) with larger geothermal gradients (60 C/km), adding more PW-s can reduce the pressure drop by over 60%. The insulation could increase Th,pw but the degree of improvement is less than 10 C. Adding more PW-s will increase Th,pw because reduced geoﬂuid velocity reduces the convective heat loss (Eq. (13)) to the surrounding rock. 3.5.2. Effects of increasing the ﬂow cross section area of PW-s As mentioned, another way to reduce the pressure drop in a PW is to increase its ﬂow cross section area. While this may be difﬁcult due to the cost and even technology limitations, we analyze this option here to examine the potential resulting improvements. The following results are calculated by solving Eq. (23) with enlarged ﬂow cross section areas. The computed geoﬂuid pressure drop in PW-s, DPpw, and PW head geoﬂuid temperature Th,pw for PW-s with different cross section areas and whether insulation is incorporated are plotted in Fig. 12 for different well depths and geothermal gradients. As M. Li, N. Lior / Energy 93 (2015) 1173e1188 1183 Fig. 11. The geoﬂuid pressure drop DPpw in PW-s (top row) and the geoﬂuid temperature Th,pw at PW head (bottom row) of different numbers of PW-s and whether insulation is incorporated. The black symbols: Npw ¼ 1, blue symbols: Npw ¼ 2, red symbols: Npw ¼ 3. The crosses ‘x’: no insulation is incorporated, circles ‘o’: the insulation is incorporated. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) shown in Fig. 12, the effect of insulation on the pressure drop was found to be very small, but the ﬂow cross section areas of PW-s have large effect on the pressure drop. For deeper wells (7.5 km) with larger geothermal gradients (60 C/km), enlarging the ﬂow cross sections areas of PW-s was found here to reduce the pressure drop by over 60%. The insulation could increase Th,pw but by less than 8 C. Enlarging the ﬂow cross section areas of PW-s increases Th,pw because reduced geoﬂuid velocity reduces the convective heat loss (Eq. (13)) to the surrounding rock. Comparison of Fig. 11 to Fig. 12 shows that the effects of enlarging the well ﬂow cross section area and those of drilling more wells, on PW pressure drop and PW head temperature, are similar. account the increase of the energy output Eout (TJ) as a result of such changes as well as the corresponding energy input Ein (TJ) needed to construct them, and this are calculated below, noting that the energy input of preparing the drilling site as well as setting the drilling rigs have however not been included. The total energy input (Section 2.4) to construct the EGS well system is, Ein ¼ Npw þ 1 Ewell þ Npw Im ðBm Bcm Þ; where: Ewell e energy consumed to construct one well, TJ, which is given in Section 2.4. Im e indicator of whether the cement of PW-s is replaced with insulation material to reduce heat loss to surrounding rock; Im ¼ 1 when cement is replaced with insulation material while Im ¼ 0 when cement is not replaced with insulation material. 4. Design guidance for well system The design guidance for well systems is based on net-energy and EROI (energy return on investment) performance criteria, as presented in this section. (32) The net energy output of EGS well system is listed in Table 8 and deﬁned as, 4.1. Net-energy-based design guidance for PW systems Improving system energy performance by increasing the number of PW-s or increasing their ﬂow cross section areas, or by insulating them from the surrounding rock, consumes more energy to construct. Suitable thermal insulation materials (such as magnesia) are much more expensive than cement, and their speciﬁc embodied energy is much higher than that of Portland cement (Section 2.4). An optimal energy-based design would thus take into Enet ¼ Eout Ein (33) The chosen performance criterion for comparison is the relative j net energy increment for case j, ðDEnet ÞR , deﬁned as, j DEnet j R ≡ base Enet Enet base Enet (34) 1184 M. Li, N. Lior / Energy 93 (2015) 1173e1188 Fig. 12. The geoﬂuid pressure drop in PW-s DPpw (top row) and the geoﬂuid temperature Th,pw at PW head (bottom row) of different PW ﬂow cross section areas and whether insulation is incorporated. The black symbols: A* ¼ 1, the blue symbols: A* ¼ 2 and the red symbols: A* ¼ 3. The crosses ‘x’: no insulation is incorporated and the circles ‘o’: the insulation is incorporated. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) where the chosen base design is D0, which is a single PW with A* ¼ 1 using Portland cement casing. Cases D1 to D9 correspond to designs with different numbers of PW-s, different ﬂow cross sections of PW-s and replacement of the Portland cement by magnesia insulation. Comparison in Table 8 of the results for D1 vs. D6, D2 vs. D7, D4 vs. D8, and D5 vs. D9, shows that enlarging the well cross sections raises the net energy output only by 0.05%e2.43% more than adding more wells for the same total geoﬂuid output ﬂow rate (assumed to be 100 kg/s). In this comparative study of 10 different PW system designs, the highest net energy output designs characterized by largest positive (DEnet)R are shown in bold underlined font in Table 8 and are listed in Table 9. Use of the magnesia insulation increases the energy output from the 40 C/km resource with 5 km deep wells, the 60 C/ km resource with 5 km and 10 km deep wells, and the 80 C/km resource with 7.5 km and 10 km deep wells, because the heat loss from the geoﬂuid to surrounding rock in PW-s is signiﬁcant. For the resource with 40 C/km geothermal gradient, the highest net energy output design of PW-s system is single 10 km PW with cross section doubled as compared to Fig. 2, and without magnesia insulation. For a resource with 60 C/km gradient, the recommended design is single 10 km PW with tripled cross section, and with magnesia insulation. For a resource with 80 C/km gradient, the recommended design is single 7.5 km PW with cross section tripled, and with magnesia insulation. Table 9 presents the corresponding geoﬂuid pressure drop in the PW-s DPpw, the geoﬂuid temperature at PW head Th,pw, and the geoﬂuid temperature drop in the PW-s, DTpw (DTpw ¼ Tb,pw Th,pw) of the highest net energy output designs. The asterisked numbers in Table 9 indicate when the produced geoﬂuid is supercritical. Table 9 shows that when the geothermal gradient is 40 C/km for 5 kme10 km deep wells, and 60 C/km for the 5 km deep well, the produced geoﬂuid is at saturated liquid phase, and under the other conditions the produced geoﬂuid is in the supercritical phase. The pressure drop in PW-s DPpw, decreases with higher geothermal gradient because of the increased assistance from buoyancy of the geoﬂuid. The values of DPpw are much smaller for supercritical geoﬂuid than for saturated geoﬂuid because of its larger density variation. The temperature drop in PW-s DTpw increases with increased well depths and geothermal gradients because of the corresponding increased area and consequent heat loss to the surrounding rock, and larger acceleration energy. The acceleration energy component in Eq. (11) (the 3rd term) is larger because the velocity increases due to decreased density of the geoﬂuid for hotter resources. The temperature at PW head, Th,pw, is the geoﬂuid inﬂow temperature to the power plant, from which the net electricity generation from power plant can be calculated using Eq. (31). 4.2. EROI-based design guidance for PW systems The EROI (energy return on investment), deﬁned as the life-time ratio of the energy output of an energy conversion system relative to the energy required to construct it, is an important and widely used performance criterion, so we have calculated it for the EGS well systems considered in this study. Speciﬁcally for the well system j, the EROI is deﬁned as: M. Li, N. Lior / Energy 93 (2015) 1173e1188 1185 Table 8 Net energy output (Enet) and relative ((DEnet)R) differences for 10 designs over the lifetime of wells (40 years). Bolded underlined numbers show the maximal positive Enet and (DEnet)R. Asterisked numbers show the maximal positive Enet and (DEnet)R for a given geothermal gradient. Geothermal gradient G, C/km 40 Well depth Z, km 5 Design# D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 Npw 1 2 3 1 2 3 1 1 1 1 Im 0 0 0 1 1 1 0 0 1 1 A* 1 1 1 1 1 1 2 3 2 3 D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 1 2 3 1 2 3 1 1 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 2 3 2 3 60 7.5 10 Net energy output Enet, PJ 6.39 16.50 29.99 17.10 6.29 16.32 29.90 16.92 6.14 16.07 29.49 16.70 6.41 16.48 29.96 17.11 6.26 16.27 29.83 16.96 6.11 15.89 29.37 16.74 6.34 16.51 30.18** 17.05 6.26 16.31 29.97 17.08 6.31 16.44 30.08 17.05 6.23 16.19 29.80 16.92 Relative net energy output increments (DEnet)R, % 0.00 0.00 0.00 0.00 1.54 1.14 0.27 1.02 3.85 2.64 1.65 2.33 0.24 0.16 0.10 0.09 2.00 1.41 0.52 0.80 4.36 3.71 2.05 2.12 0.79 0.02 0.63* 0.30 2.10 1.19 0.06 0.13 1.25 0.42 0.32 0.26 2.57 1.90 0.64 1.02 j EROIj ¼ Eout j where Eout is the power output from the EGS system j, as calculated j from Eq. (30), and Ein is the energy input needed to construct the well system j, as calculated by Eq. (32). The relationship between EROI and (DEnet)R for each Design j can be derived from Eqs. (33)e(35) as: EROIj ¼ base @ Enet j DEnet R þ1 j Ein 1 Aþ1 7.5 10 5 7.5 10 72.61 76.24 77.11 72.54 76.22 77.11 77.33 78.19 77.26 78.16 97.51 103.30 103.69 97.67 103.69 104.34 105.14 106.06 105.44 106.48** 68.53 68.57 68.45 68.53 68.52 68.40 68.61 68.54 68.59 68.50 99.10 104.06 104.42 99.26 104.41 104.95 105.58 106.30 105.80 106.57 126.49 130.28 130.04 126.83 130.86 130.84 132.03 132.56 132.45 133.08** 0.00 5.00 6.21 0.09 4.97 6.20 6.51 7.69 6.41 7.65 0.00 5.94 6.34 0.17 6.34 7.01 7.82 8.77 8.13 9.20* 0.00 0.07 0.12 0.00 0.00 0.19 0.12 0.02 0.10 0.03 0.00 5.01 5.37 0.16 5.36 5.91 6.54 7.27 6.77 7.54* 0.00 3.00 2.81 0.27 3.46 3.44 4.38 4.80 4.71 5.22 The EROI was found to decrease with the increase of the number of PW-s and with the increase of their ﬂow cross section areas, because the required energy input Ein of design D0 is substantially lower than for the other designs, while its energy output Eout is relatively just a little lower. For example, doubling the number of PWs or doubling the cross section of a PW will approximately double the required energy input Ein, but the corresponding increase of energy output Eout is less than 8%. Replacement of regular cement by Magnesia insulation decreases the EROI for the same reason. Table 10 shows that the simplest design, D0, with a single production well of the smallest considered ﬂow cross section and no thermal insulation has the highest EROI in comparison with all the proposed and analyzed alternatives D1 e D9. While many of these alternative designs raise the well systems energy output Eout, the energy input Ein required to construct them was found to increase relatively more. It is noteworthy, however, that employment of more energy efﬁcient drilling methods and materials of lower embodied energy can lead to designs employing the proposed improvement approaches that would also have a larger EROI. (35) j Ein 0 80 5 (36) base is a selected positive constant, Eq. (36) shows that the SinceEnet EROIj of EGS well system j increases with the ratio of the relative net j energy increment for case j as ðDEnet ÞR , which represents an energy efﬁciency for that system relative to the base-case in the energy comparison sample (Table 8), to the total energy input to construct j j j this EGS well system j, Ein , thus from Eq. (36) as½ðDEnet ÞR þ 1=Ein . This explains why the values of EROIj (Table 10) were found to j decrease even when the values of ðDEnet ÞR (Table 8) were positive, j simply the value of the energy input Ein in all these cases rose by j more than the rise in ½ðDEnet ÞR þ 1. To avoid misunderstanding, it is noteworthy that the EROI and the ‘relative net energy increment’ (DEnet)R, used in Section 4.1, have different deﬁnitions and purposes: the former is absolute for a given well system, while the latter is comparative among a selected sample of different well systems. 5. Conclusions The objective of this study is to examine quantitatively the energy input and output for typical EGS well systems up to 10 km deep and by developing models and using them to analyze various Table 9 The highest net energy output designs of PW system for different well depths and geothermal gradients. Asterisked numbers indicates that the geoﬂuid is supercritical. Geothermal gradient G, C/km 40 Well depth Z, km 5 7.5 10 5 7.5 10 5 7.5 10 # of PWs Im A* DPpw, MPa DTpw, C Th,pw, C 1 1 1 42.9 7.7 207.3 1 0 2 55.3 21.7 293.3 1 0 2 57.3 53.6 361.4 1 1 1 36.5 15.0 300.0 1 0 3 11.6 83.7 381.3* 1 1 3 9.2 68.1 546.9* 1 0 2 23.4 39.1 375.9* 1 1 3 6.6 51.3 563.7* 1 1 3 7.5 62.8 752.2* 60 80 1186 M. Li, N. Lior / Energy 93 (2015) 1173e1188 Table 10 Energy return on investment EROI for 10 designs over the lifetime of wells (40 years). Geothermal gradient G, C/km 40 Well depth Z, km 5 60 7.5 10 Design# Npw Im A* Energy return on investment, EROI D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 1 2 3 1 2 3 1 1 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 2 3 2 3 40.8 27.1 20.2 33.8 21.2 15.3 27.3 20.5 21.3 15.6 56.3 37.5 27.9 43.5 27.0 19.4 37.9 28.3 27.2 19.7 76.2 51.0 38.0 58.8 36.6 26.4 51.4 38.6 36.9 26.8 ways to improve the well-system energy performance. The main results include: The well construction energy consumption was found to be 19.40 TJ/(km of well) for wells with ﬂow cross sections recommended in Ref. , and it increases approximately linearly with the increase of the well ﬂow cross section area. The associated embodied energy requirement was found to be about 43.0%e49.8% of the total energy requirement. A correlation equation was developed to express the injection wells' bottom pressure Pb,iw as a function of the injection pressure Ph,iw, well depth Z, mass ﬂow rate m_ iw and well relative ﬂow cross section area A*. A correlation equation was developed to express the IW bottom temperature Tb,iw as a function of the injection temperature Th,iw, well depth Z, geothermal gradient G and mass ﬂow rate m_ iw . The pressure drop in a PW decreases with increased geothermal gradients because of the increased ﬂow-supporting buoyancy effect when the geoﬂuid becomes hotter. Adding insulation to PW-s has little effect on the pressure drop in PW but can increase the wellhead temperature by 8e10 C. The number of PW-s and their ﬂow cross section areas have a strong effect on the geoﬂuid pressure drop in a PW: e.g., for deeper wells (7.5 km) with larger geothermal gradients (60 C/km), adding more PW-s or enlarging their ﬂow cross section areas can reduce the pressure drop by over 60%. Adding more PW-s or enlarging their ﬂow cross section areas will increase Th,pw because the reduced geoﬂuid velocity reduces the convective heat loss to the surrounding rock. Constructing wells with larger ﬂow cross section areas has only 0.05%e2.43% more net energy output than constructing more wells. It is advantageous to add insulation to PW-s when (1) G ¼ 40 C/ km, Z ¼ 5 km, (2) G ¼ 60 C/km, Z ¼ 5 km or 10 km and (3) G ¼ 80 C/km, and Z ¼ 7.5 km or 10 km. Single 10 km deep production wells with doubled ﬂow cross section are recommended for a 40 C/km resource. Single 10 km deep production wells with tripled ﬂow cross section and insulation are recommended for a 60 C/km resource, and single 7.5 km deep production wells with tripled ﬂow cross section and insulation are recommended for an 80 C/km resource. Supercritical geoﬂuid will be produced for (1) G ¼ 60 C/km, Z ¼ 7.5 km or 10 km and (2) G ¼ 80 C/km, and Z 5 km. The EROI was found to decrease with the increase of the number of PW-s, with the increase of their ﬂow cross sections and the addition of insulation, because the energy input in all these cases rose relatively more than did the resulting energy output. Employment of more energy efﬁcient drilling methods and materials of lower embodied energy can, however, lead to 80 5 7.5 10 5 7.5 10 107.6 71.3 53.1 88.6 55.6 40.3 71.9 54.2 55.9 40.7 244.5 171.4 130.3 188.1 122.7 90.2 173.9 132.1 124.4 91.4 245.4 173.6 130.9 189.5 124.9 91.3 176.7 133.9 127.0 93.1 428.2 286.0 214.4 351.6 221.5 161.6 286.2 214.7 221.7 161.8 333.3 233.6 176.1 257.0 167.7 122.3 237.0 179.2 169.9 124.2 318.0 218.7 164.0 245.7 157.4 114.2 221.6 167.1 159.3 116.2 designs employing the proposed improvement approaches that would also have a larger EROI. The design with a single PW, smallest cross section with no insulation had the largest EROI The considered EGS well systems have EROI of 33.8e286.2. Nomenclature and abbreviations All the values of thermal properties listed below are evaluated at the average temperature in the considered temperature range. Parameters related to the construction energy consumption of wells (Section 2) Symbol, A* Bcm Bcs Bm bcm bcs bm D Dwb E_ dr Edr Eother Ewell ewell Im Lc Ldr Mm Mcm Mcs Z rcm rm name, value and unit relative ﬂow cross section area embodied energy of the cement, TJ embodied energy of the casings, TJ embodied energy of the magnesia insulation, TJ speciﬁc embodied energy of cement, 5.5 MJ/kg  speciﬁc embodied energy of casing, 56.7 MJ/kg  speciﬁc embodied energy of magnesia insulation, 45.0 MJ/kg  the diameter of casing, m the diameter of wellbore, m daily chemical energy consumption, 0.384 TJ/day[11,14] energy consumption of creating boreholes, TJ the other energy consumption of drilling, TJ energy consumption of constructing wells, TJ speciﬁc energy consumption of wells, TJ/km indicator of whether cement is replaced with magnesia insulation casing length/cementing length, m drilling distance, m mass of magnesia insulation, kg mass of cement, kg mass of casing, kg well depth, km density of cement, 1522 kg/m3  density of magnesia insulation, 3400 kg/m3  Parameters related to the ﬂow and heat transfer of geoﬂuid in wells (Section 3) Symbol, name, value and unit F capacity factor of surface power plant, 70%  fD the Darcy friction factor M. Li, N. Lior / Energy 93 (2015) 1173e1188 G g h hf k kcm km kr Lf m_ iw=pw N P Pr Q_ Re Rh Rm Rp r rwl T Tc TD tD Ts Tr Tr,∞ Twb U1 V z ar DPpp ε hp n r rip t the average geothermal gradient, C/km the gravitational acceleration, 9.8 m/s2 the speciﬁc enthalpy of geoﬂuid, J/kg convection heat transfer coefﬁcient between geoﬂuid and casing inside surface, W/(m2∙K) thermal conductivity of geoﬂuid, W/(m∙K) thermal conductivity of cement, 0.29 W/(m∙K)  thermal conductivity of magnesia insulation, 0.07 W/ (m∙K)  thermal conductivity of surrounding rock, 1.61 W/(m∙K)  lifetime of wells, 40 yrs  mass ﬂow rate of geoﬂuid in each IW/PW, kg/s the number of longitudinal intervals of a wellbore the geoﬂuid pressure in wells, Pa Prandtl number of geoﬂuid heat transfer rate from surrounding rock to the geoﬂuid per unit length of wells, W/m the Reynolds number absolute residual of energy conservation equation, J/kg absolute residual of mass ﬂow rate conservation equation, kg/s absolute residual of pressure conservation equation, Pa distance from center of the well, m the geoﬂuid mass loss rate in reservoir, 10%  the geoﬂuid temperature in wells, C the critical temperature of pure water, 374.1 C  dimensionless temperature dimensionless well operation time the ground surface temperature, 15 C  the temperature of rock, C the temperature of undisturbed rock, C the temperature of wellbore/rock interface, C overall heat transfer coefﬁcient, W/(m2∙K) the average ﬂow velocity of geoﬂuid in wells, m/s the vertical position of geoﬂuid, m thermal diffusivity of surrounding rock, 7.69 107 m2/s  geoﬂuid pressure drop in surface power plant, 0 Pa  the inner casing roughness factor, 0.2 103 m  energy efﬁciency of injection pump, 80% (our assumption) the kinematic viscosity of geoﬂuid, m2/s the geoﬂuid density in wells, kg/m3 geoﬂuid density through injection pump, 977.8 kg/m3 wells operation time, yrs Parameters related to the design guidance for well system (Section 4) Ein Enet Epp Eout Npw Pb,iw Pb,pw Ph,iw Ph,pw Tb,iw Tb,pw Th,iw Th,pw (DEnet)R energy input to construct the well system, TJ energy output of the EGS well system, TJ net power output from the surface power plant, MJ net electricity output from the EGS system over its lifetime, TJ number of PW-s per IW the geoﬂuid pressure at IW bottom, Pa the geoﬂuid pressure at PW bottom, Pa the geoﬂuid pressure at IW head, Pa the geoﬂuid pressure at PW head, Pa the geoﬂuid temperature at IW bottom, C the geoﬂuid temperature at PW bottom, C the geoﬂuid temperature at IW head, C the geoﬂuid temperature at PW head, C relative increased net energy output, % DPd,iw DPip DPiw DPpw DPr DTiw rh,iw 1187 dynamic pressure change in IW, MPa geoﬂuid pressure increase in injection pump, MPa geoﬂuid pressure drop in IW, MPa geoﬂuid pressure drop in PW, MPa geoﬂuid pressure drop in reservoir, MPa geoﬂuid temperature increase in IW, MPa density of geoﬂuid in at IW head, kg/m3 Abbreviations EES Engineering Equation Solver EGS Enhanced Geothermal System EROI energy return on investment HDR hot dry rock IW injection well PDC polycrystalline diamond compact PW production well ROP rate of penetration References  Tester JW, Anderson B, Batchelor A, Blackwell D, DiPippo R, Drake E, et al. 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