Numerical Modeling of Porous Flow in Fractured Rock and Its Applications in Geothermal Energy Extraction

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1 Journal of Earth Science, Vol. 6, No., p , February 05 ISSN X Printed in China DOI: 0.007/s Numerical Modeling of Porous Flow in Fractured Rock and Its Applications in Geothermal Energy Extraction Yucang Wang, Shimin Wang*, Sheng Xue, Deepak Adhikary. Earth Science and Resource Engineering, CSIRO, Kenmore Qld 4069, Brisbane, Australia. Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing 00049, China ABSTRACT: Understanding the characteristics of hydraulic fracture, porous flow and heat transfer in fractured rock is critical for geothermal power generation applications, and numerical simulation can provide a powerful approach for systematically and thoroughly investigating these problems. In this paper, we present a fully coupled solid-fluid code using discrete element method (DEM) and lattice Boltzmann method (LBM). The DEM with bonded particles is used to model the deformation and fracture in solid, while the LBM is used to model the fluid flow. The two methods are two-way coupled, i.e., the solid part provides a moving boundary condition and transfers momentum to fluid, while the fluid exerts a dragging force to the solid. Two widely used open source codes, the ESyS_Particle and the OpenLB, are integrated into one code and paralleled with Message Passing Interface (MPI) library. Some preliminary D simulations, including particles moving in a fluid and hydraulic fracturing induced by injection of fluid into a borehole, are carried out to validate the integrated code. The preliminary results indicate that the new code is capable of reproducing the basic features of hydraulic fracture and thus offers a promising tool for multiscale simulation of porous flow and heat transfer in fractured rock. KEY WORDS: discrete element method, lattice Boltzmann method, hydraulic fracturing, geothermal energy extraction, multiscale modelling. 0 INTRODUCTION Geothermal energy is clean, renewable, and steadily supplied by the Earth. The use of geothermal energy can help in both meeting the increasing energy needs over the world and protecting the environment of our planet. Among the various uses of geothermal energy, geothermal power generation is of particular importance. As the capacity factor of geothermal power plants can be up to 95% (much higher than wind power, solar power, and the direct usage of geothermal energy), geothermal power can serve as the base load of a county s power grid, and is also capable of cogeneration of electricity and heat (Pang et al., 0). In China, the potential of geothermal power is enormous, but the operating geothermal power capacity has not increased during the past two decades. Geothermal energy extraction process typically involves the injection of high pressure fluid to exchange heat from rocks at depths. In order to effectively and economically extract geothermal energy from the Earth, natural and artificial fractures are commonly used as the working fluid passage. Generally hydraulic fracturing is used to break the rock and increase permeability for the fluid flow. Although hydraulic *Corresponding author: smwang@ucas.ac.cn China University of Geosciences and Springer-Verlag Berlin Heidelberg 05 Manuscript received February 7, 04. Manuscript accepted May 9, 04. fracturing is a mature technique in the oil industry, there remain some issues which are not fully understood. The challenges in this process include how to stimulate and sustain the flow of fluid through the geothermal field and how to generate an efficient hydraulic subsurface heat exchanger system. In-depth research is needed to offer satisfactory prediction for types of fracturing as well as flow rates with respect to different operating conditions, such as the fluid viscosity, propellants used, pump rates, etc.. Therefore, understanding the characteristics of hydraulic fracture, porous flow and heat transfer in fractured rock is critical for geothermal power generation applications, including conventional and enhanced geothermal systems. Due to limitations of existing technologies and high costs, the mechanically-hydrodynamically-thermally coupled behaviors of fractured rocks cannot be studied solely relying on the in-situ geophysical measurements and laboratory tests, while numerical simulation provides an alternative approach for systematically and thoroughly investigating the characteristics of porous flow and heat transfer in fractured rocks. In this paper, we mainly focus on a microscopic model based on combining the discrete element method (DEM) and lattice Boltzmann method (LBM) to provide two-way coupled simulations for solid deformation and frature as well as thermal fluid flow. The DEM has emerged as a powerful numerical tool to model granular flow. Some DEM models allow particles to be bonded and the bonds to break, explicitly modeling microscopic fracturing events. Using DEM, many problems which are highly dynamic with large deformations and a large Wang, Y. C., Wang, S. M., Xue, S., et al., 05. Numerical Modeling of Porous Flow in Fractured Rock and Its Applications in Geothermal Energy Extraction. Journal of Earth Science, 6(): 0 7. doi:0.007/s

2 Numerical Modeling of Porous Flow in Fractured Rock and Its Applications in Geothermal Energy Extraction number of frequently changing contacts can be modeled easily. On the other hand, the LBM is a powerful tool to model fluid flow. It is based on the kinetic gas theory, which simulates fluid flows by tracking the evolution of fluid particle distribution. Some advantages of LBM over the classical Navier-Stokes approach include its ease in implementation, paralleling, and handling boundary conditions of complicated geometries. In this study, an integrated code coupling DEM and LBM is developed by combining two well developed open source codes: the ESyS_Particle and the OpenLB. The coupled approach is tested by some preliminary D simulations, including particles moving in a fluid and hydraulic fracturing induced by injection of fluid into a borehole. DISCRETE ELEMENT METHOD The discrete element method (Cundall and Strack, 979) is based on the concept that the modeled material can be represented as a collection of discrete solid particles interacting with one another at their contacts. The precise nature of the interaction depends on the scale of interest and the details of the simulation. At each time step, the calculations performed in DEM alternate between integrating equations of motion for each particle and applying the force-displacement law at each contact, through which the contact forces are updated based on the relative motions between two particles and their relevant contact stiffness. One kind of DEM allows particles to be bonded so that tensile forces can be transmitted. Fracturing is represented explicitly as broken bonds, which form and coalesce into macroscopic fractures. The bonded DEM model is often used to model wave propagation and rock fractures. The ESyS_Particle is an open source DEM software developed at the University of Queensland, Australia. Written in C++ and Python, the ESyS_Particle is designed for execution on parallel supercomputers based on spatial domain decomposition via the Message Passing Interface (MPI) (Abe et al., 004). The ESyS_Particle has been extended by the first author to include single particle rotation and a full set of interactions between particles (Wang, 009; Wang and Alonso-Marroquin, 009; Wang and Mora, 009; Wang et al., 006). The other major features of the ESyS_Particle include the explicit representation of particle orientations using unit quaternion, complete interactions (six kinds of independent relative movements are transmitted between two 3D interacting particles, see Fig. ) and a new method of decomposing the relative rotations between two rigid bodies such that the resulting torques and forces can be uniquely determined. Theoretical studies have been done to investigate the relationship between particle-scale stiffness and macroscopic elastic parameters (Wang and Mora, 008). The ESyS_Particle has been utilised in the study of rock fracture and earthquake dynamics (Mora and Place, 994, 993). The details of the ESyS_Particle code can be found in open literature (Wang et al., 0, 006; Wang, 009; Wang and Alonso-Marroquin, 009; Wang and Mora, 009, 008). Here two examples are given to show the capability of the code in modeling rock fracturing. Figure shows simulations of brittle crack extension in rock-like materials under impact by a hard ball. The brittle rock is modeled by bonded particles, and bonds between particles can break if a criterion is reached, explicitly modeling microscopic fracturing events. When the ball hits the rock, cracks induced by the impact are generated from the hit area and progressively propagated away. In Fig. b, the colors represent horizontal displacement of particles. A crushing area near the impact can be seen where the material is highly fragmented. Several groups of meridian cracks radiated from the impact area (color changes indicate displacement disconinuity) in vetical planes are observed. Figure 3 presents f s X Figure. Six kinds of interactions between bonded particles. f r is normal force, f s and f s are shear forces, τ t is twisting torque, and τ b and τ b are bending torques. (a) (b) τ b Figure. Simulation of brittle crack extension of rock-like materials under impact by a hard ball. (a) D; (b) 3D. Z f r τ t τ b f s Y

3 Yucang Wang, Shimin Wang, Sheng Xue and Deepak Adhikary (a) (b) (c) Figure 3. Progressive fracture development under bi-axial compression. Dog-eared shaped breakouts are formed. the results for a simulation of typical progressive borehole breakout development in the case of σ h /σ H =0., where σ H (in vertical direction) and σ h are, respectively, the maximum and minimum principle stresses. Two groups of conjugate cracks, dominated by shear fractures, start and intersect at both sides of the borehole surface, and then release the crushed grains (Fig. 3a). With the removal of some particles and continuous loading, stresses concentrate at the corner of the previous fault intersection, inducing several groups of shear fracture bands, and producing another major layer of fractures. Finally at both sides of the wall, a group of shear conjugate fractures develops progressively, forming a dog eared shape (Figs. 3b and 3c). Such dog eared fracture patterns have been widely observed in laboratory tests and in-situ surveys. LATTICE BOLTZMANN METHOD In contrast to the conventional computational fluid dynamics (CFD) techniques that solve macroscopic Navier- Stokes equations, LBM is built on a mesoscopic scale in which fluid is described by a group of discrete particles that propagate along a regular lattice and collide with each other. LBM solves the fluid particle distribution function f. The completely discretized equation, with the time step t and space step x, is given by BGK model (Chen and Doolen, 998) eq f e Δt, t Δt) () where τ denotes the lattice relaxation time, e α is the discrete lattice velocity in direction α, x i is a point in the discretized physical space, and f α eq is the equilibrium distribution function (see Eq. 4). Equation is usually solved in the following two steps (collision step and streaming step) ~ eq f, t Δt) () ~ f e Δt, t Δt), t Δt) (3) where f ~ represents the post-collision state. LBM uses the Eulerian approach to describe the motion of fluid particles; therefore the fixed Euler grids are adopted. A square and nine-velocity lattice, the DQ9 model (Fig. 4), is used in this study. The equilibrium distribution function is of the form (DQ9) f w ( ) c c c e u e u u u (4) eq 4 where c= x/ t is the lattice speed, ρ is the lattice fluid density, u is the macroscopic velocity, and w α is the weighting factor given by w 4 / 9, 0, /9,,3,5,7, /36,,4,6,8. The macroscopic quantities such as density, momentum, fluid pressure and viscosity can be obtained by 8 0 where c s c / 3 f, u 8 0 f s s (5) e, p c, v ( / ) c Δt is the speed of sound in this model. 3 TWO-WAY COUPLING OF DEM AND LBM In this study, two open source codes, the OpenLB and the ESyS_Particle, are used in coupling of LBM and DEM. At each time step, the fluid and solid components run in a separate and independent fashion, while the two codes are integrated into one combined system to facilitate a fully coupled solution. In order to implement such coupling, the following issues need to be considered: moving boundary conditions for a curved shape for LBM which handles the momentum transfer from solid particle to fluid and the reflection of fluid at the solid boundaries, as well as force and momentum transfer from fluid to solid particles. The most widely used boundary condition in LBM is no-slip bounce-back condition in which an incoming fluid particle from a fluid node is reflected back to the node it comes from, and velocity or pressure boundary condition (Zou and He, 997) in which velocity or density along the boundary node is fixed. These boundary conditions are only applicable in the case of non-moving objects and straight boundaries. For moving objects inside the fluid it is necessary to adapt the moving boundary conditions. Curved boundary treatments provide a means of improving the computational accuracy of the stair-shaped approximation used in LBM simulations. Due to the arbitrary position and curved surfaces of the solid particles, (6)

4 Numerical Modeling of Porous Flow in Fractured Rock and Its Applications in Geothermal Energy Extraction 3 the particle surface can intersect the link between two nodes of LBM at an arbitrary distance with a ratio of x f xw 0, (7) x x f b where x f and x b denote the lattice node on the fluid side of the boundary and that on the solid side. When δ=0, x w =x f, the boundary wall locates at the point x f ; when δ=, x w =x b, the boundary wall locates at the point x b. The reflected distribution function at node x f can be calculated using an interpolation scheme (Yu et al., 003) f ( xf, t Δ t) ( ) ( xf, t Δ t) ( xb, tδ t) f ( xf, t Δ t) 6 w we uw / c where w α is the same weight factor as in Eqs. 4 and 5, ρ w is the fluid density at node x f, u w is the velocity of the solid particle, and x f =x f e α t (Fig. 5). The last term in Eq. 8 represents the momentum transferred from solid particle to fluid. The fluid force acting on the solid particle surface can be obtained using the formula 9 F ( b, ) ( f, Δ ) (Δ ) /Δ xb (8) f e f x t f x t t x t (9) where the first summation is taken over all solid boundary nodes at x b adjacent to the fluid nodes and the second summation is taken over all possible lattice directions pointing from x b towards all possible neighboring fluid nodes around the solid node x b. This force is added to the solid particle force in DEM code. Hence, a combination of Eq. 8 and Eq. 9 enables fully two-way coupled solid-fluid simulations. To implementing this boundary condition, a single interpolation is used for both δ<0.5 and δ 0.5, which is found to be very stable. 4 PRELIMINARY RESULTS To assess the performance of the coupled DEM-LBM approach, two simple preliminary numerical results are presented here. At this stage, validations of large scale simulations and detailed comparisons with physical experiments have not yet 6 3 e 3 e 7 e 6 e Figure 4. A widely used -D 9-velocity lattice model (DQ9). The cell in the center is stationary, representing fluid particles at rest. The fluid particles are only allowed to move along the eight moving directions to neighboring cells or to rest in this cell. e e 5 e 8 e 5 x f e ~ x f Boundary wall e ~ e x w x b Figure 5. The moving curved wall boundary condition. The curved line represents the boundary wall between a solid particle and the fluid. x f and x f are the lattice nodes on the fluid side of the boundary; and x b is the node on the solid side; x w denotes the intersection of the wall with lattice link.

5 4 Yucang Wang, Shimin Wang, Sheng Xue and Deepak Adhikary been carried out since the coupled code is just integrated. These small scaled models, however, can be used as a qualitative display of the capability and potential of the coupled approach. Figure 6 shows the modeling results of solid particle motion in a fluid. Three particles bonded as a single rigid body with unbroken bonds. These particles are driven by the fluid flow in a long pipe. A boundary condition of a given parabolic velocity profile is specified for the inlet (the left boundary), while a constant pressure condition is applied at the outlet (the right boundary) to facilitate a Poiseuille-type flow. The motion and rotation of the rigid body, as well as vortex behind it, are clearly observed. In this simulation, Reynolds number Re is 00 and the solid particle size is 5 times greater than the grid size of the fluid. This example verifies the algorithm of the two-way coupling of solid and fluid, including moving boundary condition and dragging forces. Figures 7 to 9 show a D simulation of a hydraulic fracturing process. In this test, rock is modeled by 7 37 DEM particles with a hole in the middle. Particles with sizes ranging from 0. to unit are bonded by brittle elastic bonds. When the stress in a bond is large enough, the bond breaks, explicitly modeling a micro-fracturing event. The release of stress on the broken bond will transfer stress to the neighboring bonds, possibly causing them to break in a later time. In this way, a progressive failure process is modeled by extensions of micro-cracks. The fluid flow is simulated using 0 0 LBM grids. The size of the fluid grid is half of the minimum solid particle radius. Four motionless walls provide constant displacement boundary conditions, and the friction between walls and particles is set to zero to minimize influence of the boundary. Constant pressure boundary conditions are set for the LBM fluid. Fluid pressure in the center of the hole is increased slowly, modeling the injection of water. A pressure increase is realized by adding a source term on the right-hand side of Eq., resulting in f e Δt, t Δt ) f, t ) f, t ) f eq, t ) Δf, t ) (0) where Δfα(xi,t) in each direction is proportional to fα(xi,t) and Figure 6. Three particles bonded as a rigid body and move in the fluid. Particles are driven by the drag forces of fluid flow. Colors represent fluid scalar velocity, with blue for low and red for high. Figure 7. Snapshots of hydraulic fracture. Fluid pressure in the centre of the hole is increased slowly, modeling the injection of water. Colors represent solid particle displacement (blue for low and red for high) in horizontal direction. Fracture initiation and propagation due to hydraulic loading is clearly seen.

6 Numerical Modeling of Porous Flow in Fractured Rock and Its Applications in Geothermal Energy Extraction 5 Figure 8. Fracturing event distribution. Tensile fractures are dominant in the process of the crack initiation and extension. Figure 9. Fluid pressure (blue for low and red for high) during hydraulic fracturing. Only those LBM grids covered by fluid are visualized, representing the process of cracking and fluid flowing in the hydraulically generated fracturing tunnels. the increase of fluid pressure in one time step is determined by Δ p cs 8 0 Δ f. Figure 7 shows several snapshots of fracture initiation and propagation due to hydraulic loading. Different colors in this figure represent the different magnitudes of solid particle displacement in horizontal direction. The cracks are observed to start from the surface of the borehole, caused by the increasing fluid pressure on the surface of the borehole, and propagate inside the rock with the continuous breakage of bonds and infiltration of pressured fluid. Figure 8 plots evolution of the micro-fracturing of bonds. The sizes of the event represent fracturing energy released. It is clearly seen that tensile fractures are dominant in the beginning of the crack initiation. The fluid pressure is visualized in Fig. 9. Since there are many LBM grids ( 0 0), these plots can rather be interpreted as the presence and extension of the fluid. In the initial state, fluid only occupies the borehole area. With the occurrence of fractures, fluid is driven by pressure into the cracks, and then exerts further pressure on the crack surface, which results in more fractures generated. Although the current simulation is still rudimental and of small scale, and the influence of input parameters, such as particle-scale stiffness, fracturing parameters, confining pressure and injection rate, on the fracturing behavior is not investigated, the simulation includes the major mechanisms of hydraulic fracturing process: (i) mechanical deformation and fracturing induced by the fluid pressure; (ii) flow of fluid within the fracture; and (iii) fracture propagation. Furthermore, there are no specific assumptions about how fluid flows inside cracks and how a solid breaks, i.e., they are treated in a straightforward way in the new coupled model. 5 DISCUSSION Although the simulation results presented above remain un-calibrated against laboratory and in-situ testing, they do reproduce the basic features of hydraulic fracturing and demonstrate the promising applications of the new modeling approach. The coupled code requires significant improvements to realize its full potentials. Firstly, the current coupled code is still D and needs to be extended to 3D, which should be straightforward as both the ESyS_Particle and the OpenLB are 3D codes and the coupled code is structured to have provisions for 3D coupling. Secondly, thermal effect is required to achieve a full hydro- and mechanical-coupling capability. Currently, the ESyS_Particle has included thermal effects, including heat transfer, thermal expansion and friction generated heat, and the OpenLB also has this capability. Once these improvements have been incorporated, the fully developed model is capable of efficiently simulating the dynamics and evolution of real deep geothermal reservoir systems. The existing numerical models for extracting geothermal energy are almost all based on the continuum approach (Hunsweck et al., 03; Secchi and Schrefler, 0; Zhang et al., 009). Although these models have been shown to be valuable in geothermal energy exploration, as reported in a number of case studies in the literature (Zhou and Hou, 03; Gong et al., 0; Blocher et al., 00; Bataille et al., 006), they are incapable of capturing the full physics of the modeled fractured rocks. As fractures in rock normally involve various geometric

7 6 Yucang Wang, Shimin Wang, Sheng Xue and Deepak Adhikary scales, the continuum models may be adequate to represent the large scale fractures, but definitely not for complicated micro-fracture systems. A physically sound and computational affordable numerical approach for modeling the fractured rocks is in need but not available yet. We belive multiscale numerical modeling is an ideal approach to investigate the mechanism and characteristics of porous flow and heat transfer in fractured rock as applied to the real world problems of geothermal power generation. In the mesoscale simulation, the DEM can be adopted to model the deformation in solid as well as the formation, enlargement, connection and closure behaviors of microfractures, while the LBM can be adopted to model the fluid flow in pores and fractures; the combination of these two methods provides a fully coupled simulation of solid deformation, fluid flow as well as heat conduction and convection. Based on the analysis and integration of the mesoscale modeling results, quantitative relations of parameters like permeability and thermal dispersion in fractured rock as functions of pressure, deviatoric stress and temperature can be determined, and then the macroscale constitutive relations of the porous flow and heat transfer in fractured rock can be derived, which form the basis for constructing continuum models to simulate the macroscale hydro-thermal behaviors of fractured rocks. Such multiscale modeling appoach may be employed to simulate the porous flow and heat transfer processes between injection and production wells in geothermal power plants, to study the effects of injection pressure, injection rate and natural convection on the geothermal reservoir temperature and its serving life, and to investigate the optimal distribution of the injection and production wells as well as the technological strategy for wisely and economically extracting deep geothermal resources. 6 CONCLUSIONS This paper presents a two-way coupled DEM-LBM scheme based on two open source codes, the ESyS_Particle for DEM and the OpenLB for LBM. The ESyS_Particle solves solid particle motion, interaction between particles, and fracturing of bonds, while the OpenLB is responsible for fluid motion on the fixed Euler grids. The two-way coupling is achieved through drag forces applied to DEM particles by fluid, moving boundaries, and momentum transfer from solid particles to fluid grids. The preliminary results of D simulations for particles moving in a fluid and hydraulic fracturing induced by injection of fluid into a borehole indicate that the new code is capable of solving coupled fluid-solid interaction problems and reproducing the basic features of hydraulic fracturing. As a result, the numerical approach proposed in this study offers a promising tool for multiscale simulation of porous flow and heat transfer in fractured rock. ACKNOWLEDGMENTS This study was financially supported by the CSIRO Earth Science and Resource Engineering Geothermal Energy Capability Development Fund, the Huainan Coal Mining Group in China, and the National Natural Science Foundation of China (No ). It was also supported by the NCI National Facility at the ANU and ivec at the UWA through the use of advanced computing resources. Two open source codes, the ESyS_Particle and the OpenLB, are used in this study. We thank the developers of these codes. 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