Study on the influence law of well location and water injection displacement on heat extraction performance of EGS

The enhanced geothermal system (EGS) is the key to improving the heat production efficiency of hot dry rock (HDR). Due to the existence of EGS, the reservoir produces significant heterogeneity, so the optimal design of well location and injection rate has an important influence on the heat recovery effect of an EGS. To solve this problem, a calculation model for the heat output of an EGS under multifield coupling is established in this paper, and the influence of well location and injection rate on the heat recovery effect under different scales of EGS is deeply analyzed. The results show that the production well location and injection well displacement have a great influence on the heat recovery effect under different fracture‐network scales. The larger the size of the fracture network, the larger the effective utilization area of reservoir heat recovery and the better the effect of heat recovery. The closer the production well is to the fracture network and in the direction of the fracture network, the larger the effective utilization area of reservoir heat recovery is, and the better the effect of heat recovery is. The higher the injection well displacement is, the larger the effective utilization area of reservoir heat recovery is, and the better the heat recovery effect is. The research results have important guiding significance for the optimization of the scale, well placement, and water injection displacement of the EGS in the HDR reservoir.

them, the buried depth of the HDR reservoir is about 3-10 km, and the temperature of the stratum rock mass is about 150-650°C. [4][5][6] The HDR reservoir is mostly granite rock, and there is no water or steam in the rock mass. It is the most potential and valuable part of geothermal energy.
The heat recovery of HDR requires the circulation of cold water injection to recover heat. Because the compact of HDR is very high, to achieve a better heat recovery effect, it is necessary to establish an enhanced geothermal system (EGS) through hydraulic fracturing. The shape and scale of the EGS have a significant influence on the effect of heat recovery. The concept of EGS was first proposed by Los Alamos Laboratory in the United States in the 1970s, followed by the development of the first EGS 7 heat reservoir in Fenton Hill, New Mexico in 1977, France, the UK, Japan, Australia, and other countries have also carried out EGS research. 8 In 1987, France started experimental research on EGS in the Soultz region. The Soultz project successfully built artificial heat storage on a commercial scale, which has played a positive role in the commercialization of EGS projects around the world. 9 The larger the scale of the EGS, the better the heat recovery, but due to the limitations of economic and engineering conditions, the EGS cannot blindly pursue the expansion of scale. The effect of heat recovery can be further improved under the limited scale of the EGS by optimizing well location and injection rate. In particular, the existence of EGSs leads to significant heterogeneity of the reservoir, and the influence of well location and injection rate on heat production is more prominent, making this optimization more necessary. Single vertical fracture model (SVFM) is the most basic model to study the characteristics of EGS, Cheng et al. 10 showed that the SVFM could fully reflect the flow and heat recovery process. A mathematical model of fluid flow-heat transfer in a single fracture in a reinforced geothermal system was established. 11 The model studied the effects of fracture wall permeability and fracture width on the fluid front morphology in the fracture, but the model did not consider the effects of temperature and fluid pressure on rock porosity and permeability. The equivalent permeability method is used to establish a continuous model, which simulates the injection-production process of the EGS by equating a larger permeability to the fracture unit. 12 However, this method can only obtain the overall reservoir temperature, and cannot reflect the influence of fracture morphology on the injection-production process. The research results are limited in guiding the fine development of geothermal reservoirs. Wu et al. 13 get approximate solutions that are found for a mathematical model developed to predict the heat extraction of a closed-loop geothermal system which consists of two vertical wells (one for injection and the other for production) and one horizontal well which connects the two vertical wells. On a large scale, thermalfluid-solid coupling thermal recovery simulation of strata with a single fracture was carried out to study the exploitation of geothermal reservoirs under the condition of one injection and one production. 14 However, the interior of the EGS is often crisscrossed by fractures, with few single fractures. The granite core was cut to construct artificial cracks and the laboratory test was carried out to determine the influence of cracks on seepage and heat transfer. 15 However, due to the small scale of the core, the law obtained has not been applied to large-scale digital models. A series of natural fracture or discrete fracture-network models are established, and the numerical simulation of reservoir injection and production under multifield coupling is carried out. 16 The simulation results show that fractures play a crucial role in the heat recovery of EGS, and are the main fluid flow channels and heat transfer pathways. However, the influence of well location and injection well displacement on the heat recovery effect is not considered in the study. By analogy with the multilinkage mechanism, this paper studies the influence of fracture morphology on reservoir heat transfer and seepage field by using a method similar to finite element analysis. 17 This research method is novel, but only studies the case of a single hydraulic fracture. In fact, to improve the reconstruction effect, complex fractures with multiple branches are often formed by fracturing. The isolated and unconnected natural fractures in the formation also play a certain role in heat transfer and seepage, and these factors are not taken into account. The fluid flow-heat transfer analysis model of vertical multifracture in EGS was proposed for the first time, and the results show that fracture aperture, initial fluid temperature, fluid density, and initial rock temperature have significant effects on instantaneous heat extraction efficiency. 18 This paper studies the influence of injected working fluid temperature on the heat recovery performance of EGS. 19 However, the model adopted in this study does not include fractures and does not involve the coupling between multiple physical fields. In fact, the porosity and permeability of HDR reservoirs are very low, and most fluid migration occurs in fractures during the extraction process, so the research results have certain limitations. To study this problem, a calculation model for heat recovery of EGS under multifield coupling is established in this paper, and the influence of different well locations and injection rates on the heat recovery effect is analyzed in depth. It provides guidance for the optimization of well placement location and water injection displacement under the different scales of EGS in HDR reservoirs.

| HEAT PRODUCTION MECHANISM AND GOVERNING EQUATION OF EGS
After fracturing, the HDR reservoir has a certain heat production capacity and becomes an EGS. The main purpose of EGS is to extract heat from the ground more efficiently. A common method is to inject cold water and heat the reservoir to produce hot water. This process is similar to the water injection of petroleum but more complicated is the obvious temperature variation effect in the process of injection and production of HDR reservoir, which involves the coupling of multiple physical fields. The governing equation of energy change is the key to establish the coupling model of multiple physical fields.
The temperature field of the HRD reservoir is composed of the temperature field of the reservoir rock and the temperature field of the fluid in the fracture. The heat transfer process in an HRD reservoir is mainly composed of the thermal diffusion of reservoir rock, thermal diffusion of fluid in fracture, and convective heat transfer on fracture wall. Due to the extremely low effective porosity and permeability of granite, there is almost no fluid flow in the matrix, so it can be considered that there is no convective heat transfer caused by seepage in the matrix for the sake of the simplification model. The heat transfer process of the HRD reservoir is shown in Figure 1.
According to the energy conservation theorem of reservoir rock, the heat change caused by the heat diffusion of the matrix in the body can be expressed as where Q 1 is the heat change caused by the heat diffusion of matrix, J; K r is the heat conduction coefficient of reservoir rock, J/(msK); T r is the temperature of reservoir rock, K. The heat change caused by convective heat transfer on the fracture wall in the control body can be expressed as where Q 2 is the heat change caused by convective heat transfer on the crack wall, J; A is the crack surface area, m 2 ; h rw is the convective heat transfer coefficient on the crack wall, J/(msK); T w is the temperature of the fluid in the crack, K.
By combining (1) and (2), the change of heat in the body over time can be controlled by where ρ r is the density of rock, kg/m 3 ; c r is the specific heat capacity of rock, J/(kgK).
The change in heat caused by fluid diffusion in a crack can be expressed as where Q 3 is the heat change caused by fluid diffusion in cracks, J; K w is the heat conduction coefficient of fluid in fracture, J/(msK). The heat change caused by fluid inflow and outflow in a fracture can be expressed as where Q 4 is the heat change caused by fluid inflow and outflow in crack, J; u w is the flow velocity of fluid in fracture, m/s; T in is the temperature of the fluid at the fracture inlet, K; T out is the temperature of the fluid at crack outlet, K.
By combining Equations (3)- (5), it can be obtained that the change of heat in the body over time can be expressed as where ρ w is the fluid density in the crack, kg/m 3 ; c w is the specific heat capacity of fluid in fracture, J/(kgK). Darcy's law, which considers gravity when studying fluid flow in underground rocks, can be expressed as where v is the seepage velocity, m/s; k is the rock permeability, m 2 ; g is the acceleration of gravity, m/s 2 ; p is the pore pressure, Pa; ρ 1 is the fluid density, kg/m 3 . The opening and expansion of fractures can be controlled by several factors, including temperature, stress, and fluid seepage. The equations that govern these factors and their effects on fracture opening and expansion are as follows: The temperature control equation for fracture opening is given by where h Δ f is the change in fracture aperture, m; α is the coefficient of thermal expansion of the rock, 1/K; T Δ r is the change in temperature, K; and L f is the length of the fracture, m.
Fracture permeability is the ability of a fracture to transmit fluids. It can be characterized by the following equation: where k f is the fracture permeability, C f is the fracture compressibility, h f is the fracture aperture, m; μ f is the fluid viscosity, Pas; and L f is the length of the fracture, m. The stress formula for the opening and expansion of fractures can be expressed by the Griffith equation as follows: where σ crit is the minimum stress intensity required for crack opening, γ is the surface energy of the rock, E is the elastic modulus of the rock, a is the length of the crack, and ν is Poisson's ratio of the rock. The multiphysical field coupling can accurately describe the changing process of each physical quantity and synchronously calculate the chain reaction caused by the change. In this paper, the full coupling method is adopted, each physical quantity and physical field reach equilibrium in a time step, and the full coupling energy of the transient problem is more accurately described.
The EGS is considered to have reached a steady state before injection and production, that is, the physical quantity no longer changes with time. However, water injection and extraction break this balance, causing changes in the formation pore pressure field, temperature field, and seepage field. Changes in the pressure field will lead to changes in formation rock effective stress, resulting in changes in porosity, permeability, and other parameters, which further affect fluid flow and heat transfer. The change in temperature field will produce thermal stress, make the rock skeleton and fluid viscosity, density, and other seepage physical characteristics with temperature change, affect the fluid in the rock seepage state. The change of seepage field will cause intense heat exchange in the reservoir, resulting in the change of rock temperature, the change of heat balance state under the initial condition, and the increase of pore pressure after fluid injection, the effective stress in the reservoir will be reduced. The overall temperature of the reservoir decreases after the production of water from the production well, which also causes the change of multiple physical fields.
The changes in reservoir expansion and fluid pressure and seepage caused by water injection can be expressed by Equation (11): where φ is the formation porosity, dimensionless; ρ is the fluid density, kg/m 3 ; u is the Darcy flow velocity vector, m/s; Q m is the change in overall reservoir density per unit time, kg/(m s) 3 ; k is the permeability, D; μ is the fluid viscosity, Pas; p is the fluid pressure gradient, Pa.

The stress balance equation in the formation is
where G is the shear modulus, Pa; u i is the displacement component, m; ν is Poisson's ratio, dimensionless; E is the modulus of elasticity, Pa; α B is the Biot coefficient; α T is the thermal expansion coefficient of rock mass, 1/K; K′ is the bulk elastic modulus of porous media; F i is the physical force per unit volume in the i direction, Pa; K α T ′ T i is the thermal stress term; α P B i is the action term of water pressure (permeability under pore pressure).

| NUMERICAL SIMULATION OF HEAT PRODUCTION IN HDR RESERVOIR
To accurately describe the coupling process of various physical fields in the heat production process and obtain the most accurate simulation results, the multiphysical field coupling of the injection-production process of the EGS was numerically simulated in this paper. The traditional discrete fracture network only analyzes a single natural fracture, and there are often a lot of natural fractures in HDR reservoirs. Therefore, the three groups of fracture networks used in this paper are obtained by using the finite element method and secondary development for hydraulic fracturing numerical simulation. The main simulation method is to describe the distribution of natural fractures and the formation process of the fracture network by discretizing a continuous formation and inserting it into a prefabricated fracture path. Three groups of fractures are shown in the diagram. Group (a) represents strata with a completely random distribution of natural fractures, Groups (b) and (c) represent strata with dense fractures and sparse fractures in a "checkerboard" distribution of natural fractures, respectively. The multiphysical field fully coupled simulation of the injection-production process in the above three fracture networks also considers the influence of natural fracture distribution and hydraulic fracturing effect on the final exploitation degree of the HDR reservoir. Compared with the discrete fracture-network results, it is closer to the engineering practice and has more clear guiding significance (Figure 2).
Since the solution of the model does not require the refinement of the mesh, the calculation amount of the original size can be borne, so the size of the model is scaled up to 1 km × 1 km, and the internal cracks are scaled up in equal proportions. During the injection-production process, the fracture opening of each fracture in the fracture network is consistent. The method of reservoir exploitation is "one injection and four production," and the injection well is located at the center point of the model, which is also the fracturing point. The boundary condition of the model is the normal displacement on the constrained boundary, and the boundary is considered to be impervious. The four production wells are located on the four vertices. The injection well adopts the method of constant flow injection, the production well is constant pressure production, and the selected working liquid is water. The rock thermal conductivity, specific heat capacity, density, and fluid density, specific heat capacity are constant and do not change in the whole process of heat recovery. The parameters used in the model are shown in Table 1. The reliability of the model can be verified by the analytical solution of the SVFM.  The heat extraction under three groups of fracturenetwork conditions was simulated. Figure 3 shows the distribution nephogram of formation temperature during injection and production of Group (a) fracture network. The results show that by 100 years, the temperature in the area around the fracture has dropped to 20°C, which is not valuable for exploitation, but the temperature in the area far away from the fracture is still 200°C, which is unexploited. This indicates that the current production conditions are not suitable and should be adjusted to allow sufficient heat to be extracted from the entire area. Figures 4 and 5 show the temperature distribution nephogram in the process of fracture-network injection and extraction in Groups (b) and (c). The comparison of formation temperature distribution in the three groups of fracture network shows that in Figure 6, the formation temperature changes under different fracture-network conditions have certain differences. This distribution difference is caused by the different morphologies of the fracture network, and the low-temperature area is always consistent with the fracture morphology. The temperature distribution of Group (b) fracture network is the most complex, covering the widest area, and the heat recovery effect is the best. After 100 years of extraction, the low-temperature zone of Group (b) is greater than that of the other two groups.
T A B L E 1 Parameters used in simulation.
F I G U R E 4 Formation temperature variation in Group (b) fracture-network extraction process.

| Influence of well location on heat recovery effect
It can be seen from the above simulation results that the shape of the fracture network has a great influence on the heat production effect and the fully mined area, but the fracturing results are often difficult to control. To achieve a good heat recovery effect in strata with different degrees of fracturing, it is intended to improve the heat recovery effect of the reservoir and increase the fully mined area by optimizing the location of production wells.
In EGS, convection heat transfer is the main form of heat transfer, and fractures are the main fluid migration channels. Therefore, the geothermal in early exploitation mainly comes from around the fracture surface, and the seepage in the matrix gradually plays a dominant role after the temperature near the fracture is reduced to an unrecoverable degree. The formation pressure around the fracture is high, which is the area with the largest seepage velocity. Therefore, production wells should be located near the fractures. The specific method to optimize the production well is to move the well that is not on the fracture network toward the fracture network, so that it is closer to the fracture wall. The well locations before and after optimization under different fracturenetwork conditions are shown in Figure 7 The red dot in the figure is the well location before optimization, and the blue dot is the well location after optimization. Figure 8 and Table 2 show the fully mined area ratio to the total area under the optimized three groups of the fracture network and the specific value of fully mined area ratio before and after the optimized production well location. Figure 9 shows Produced heat before and after the optimized production well location under the three groups of fracture-network conditions. The results show that the fully mined area of the optimized three groups of fracture network has increased, but the extent of improvement is different. Group (a) has the least improvement, mainly because Group (a) fracture network is too simple, and the main factor restricting the degree of geothermal exploitation is the scale of the fracture network. Therefore, the effect of improving heat recovery by optimizing the location of production wells is not obvious; Group (c) has the best optimization effect because the fracture network of Group (c) has formed a certain scale, but the original well location is far away from the fracture, it is difficult to effectively communicate with the fracture, optimized the location of well is closer to the fracture, so it plays a good heat recovery effect. The optimization effect of group b is medium and flat, the main reason is that the scale of the fracture network of Group (b) is large enough, and the main factor restricting its fully mined area is the extraction time and other factors, rather than the distance between the production well and the fracture. Therefore, although the optimization effect of heat recovery has been improved, the effect is not obvious. Figure 10 shows the distribution of fully mined areas in 100 years of production under the three groups of optimized fracturenetwork conditions. The results show that with the optimized production well location, the fully mined area is distributed more evenly, and the fully mined area shows a trend of extending around the injection well, which is more favorable for productivity optimization in the later stage of production.

| Influence of water injection displacement on heat recovery effect
To further improve geothermal productivity and fully exploit geothermal resources, it is necessary to further optimize injection well displacement after optimizing the production well location to improve the fully mined area. In this part, based on 5 L/min displacement, the displacement F I G U R E 7 Well locations of production wells before and after optimization under different fracture network conditions. F I G U R E 8 Comparison of fully mined areas before and after the optimized production well location under the three groups of fracturenetwork conditions.
T A B L E 2 Specific value of fully mined area ratio before and after the optimized production well location. is gradually increased, and the displacement is, respectively, taken as 9, 11, 13, and 15 L/min for comparison. The fully mined area of the three groups of fracture networks under different displacements is shown in Figure 11 and Table 3. The results show that, under the three groups of fracture-network conditions, increasing the water injection well displacement can greatly improve the degree of heat recovery, among which the fully mined area of the group 100 years after injection with 15 L/min is 2.75 times that of injection with 5 L/min, and the improvement effect is extremely obvious. Groups (b) and (c) also had a significant improvement. Under the condition of three groups of the fracture network, the fully mined area and the average temperature drop curve under different water injection displacements at 100 years of extraction are shown in Figures 15  and 16. Figure 17 shows the produced heat of three F I G U R E 9 Produced heat before and after the optimized production well location under the three groups of fracture-network conditions. F I G U R E 10 Comparison of fully mined areas in 100 years of production under the three groups of optimized fracture-network conditions. groups of fracture networks under different injection displacements at 100 years of exploitation. For Group (a), the effect of heat recovery is significantly improved by using large displacement, and the relationship between the proportion of fully mined area and the injection displacement is an approximately straight line, the average temperature eventually falls the lowest. The reason for analysis is that Group (a) has the simplest fracture network, and the influence of fracture scale on heat recovery is not obvious, which is mainly reflected in the influence of displacement. As far as conditions permit, the maximum injection displacement can be selected to improve the extraction degree; the fracture network of Groups (b) and (c) is relatively complex, and the impact of displacement on them is less than that of Group (a), average temperatures fell faster in the first 30 years than in Group (a). The growth rate of the proportion of fully mined area decreases obviously when the displacement is 9 L/min. Further increasing the injection displacement can still improve the degree of exploitation of the reservoir, but the effect is not obvious, and it is not advisable to continue increasing the injection displacement from the perspective of cost.

| CONCLUSION
In this paper, the basis on simulating the fracture network of different scales formed by hydraulic fracturing in the early stage, the injection and production simulation of EGS based on multiphysical field coupling was carried out, and the influence of well location and injection well displacement on the heat recovery effect under different fracture-network scales was studied. The following conclusions are obtained: (1) Fluid migration in the EGS is mainly in the fracture, convection heat transfer is the main heat transfer mode, the larger the size of the fracture network, the faster the growth of the heat recovery rate. The shape and the mined area are closely related to the shape and area of the fracture network in the early injection-production period, but in the late injection-production period, it is more likely to be affected by the location of the production well. (2) The location of the production well has a significant impact on the overall heat recovery effect of the reservoir. The well located on the strike of the fracture network can significantly affect the area and shape of the reservoir temperature drop zone. When the production well is not on the strike of the fracture but close to the fracture, it can also have a significant impact on the reservoir temperature drop zone. Production well layout should consider the strike of the fracture network. (3) The higher the flow rate of the injection well, the larger the area of the inland thermal reservoir will be fully exploited at the same time. However, for different scale fracture networks, the influence degree of increasing the displacement is different, increasing the injection displacement has a greater impact on heat recovery in reservoirs with relatively simple fracture networks, so it should be optimized according to the actual situation.