Analysis of Fracture Network Response to Fluid Injection

PROCEEDINGS, Fourtieth Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, January 26-28, 2015 SGP-TR-204 Analysis of Fracture Network Response to Fluid Injection Moien Farmahini-Farahani, Ahmad Ghassemi Mewbourne School of Petroleum & Geological Engineering, The University of Oklahoma, Norman, OK, 73019, U.S.A Keywords: Displacement discontinuity method, Fast multipole method, Stochastic large scale fracture network, slip weakening ABSTRACT` Geothermal as well as oil and gas are often produced from designed or naturally fractured reservoirs. Numerical modeling is usually required to predict and control behavior of fractures under complex processes, and often the DDM has shown to be an effective approach. In order to predict behavior of large-scale natural fracture networks using DDM an approximation technique such as fast multipole method can be used to efficiently calculate the interaction between fractures in optimal time of within O(N), where N is number of unknown. Differences of flow conditions in the reservoir carry potential change of a joint to slip or microearthquake (MEQ) when the shear stress surpasses a failure criterion. In this paper, Fast Multipole Displacement Discontinuity Method (FMDDM) is employed to simulate flow and micro-earthquake (MEQ) in a stochastic fracture network. Stress and displacement fields subjected to induced pressure by injection and production are calculated and joint deformation and the potential for MEQ is assessed using slip weakening conditions and a failure criterion. Expectedly, the orientation of a fracture with respect to the stress field plays an important role in its slip and concomitant seismicity. 1. INTRODUCTION The deformation of natural fracture network is a crucial aspect of hydraulic stimulation of unconventional resources. Usually, slip on fractures increases reservoir permeability and results in micro-seismicity during the injection/productions. Changes in the flow pressure and induced deformation in the fracture networks might cause a joint to separate or slip with micro-earthquakes (MEQ) occurrence. A realistic model such as slip weakening model better represents joint deformation. These features can be effectively captured using the Displacement Discontinuity Method. The Displacement Discontinuity Method has some advantages over other methods such as the finite element method namely in the DDM, instead of meshing the whole field, only fracture surface (boundary) are meshed. This means lower number of unknowns or lower number of equations that needs to be solved. However, since the influences of all fracture elements are usually required to be calculated, the equation system matrix is non-symmetric and fully populated. In other words, the conventional DDM is not computational efficient for large fracture systems. Fast approximation techniques such as the Fast Multipole Method (FMM) (Liu 2009, Liu and Nishimura 2006), Hierarchical Matrices (H-Matrices) (Benedetti et al. 2008), wavelet method (Beylkin et al. 1991), panel clustering (Aimi et al. 2005), or fast Fourier transformation methods (Phillips and White 1997) can significantly improve the computational cost and required memory for boundary element problems. Among them FMM and H-Matrices are most popular methods and comparisons of these two reveal that (Brunner et al. 2010, Forster et al. 2003, Buchau et al. 2003): 1- FMM requires less memory in comparison to H-Matrices; 2- Building the system matrix is much faster with FMM, since H-Matrices (using adaptive cross approximation) requires computation of the low-rank approximants for entire system matrix. However, for the FMM only computation of near-field interaction for the sparse matrix is required; 3- H-Matrices only requires multiplication approximants with vector which makes ACA much faster than FMM in which expansion series and transformations needs to be calculated. For a problem with high number of iteration H-Matrices is recommended; 4- For FMM, once the coefficient matrix is built, field points can be calculated efficiently. For H-Matrices, an integration over all elements for each field point is required and it makes process longer than FMM. Recently, the combination of FMM and DDM (FM-DMM) has been successfully applied to accelerate the calculation time for large-scale fracture networks in reservoir geomechanics problems (Verde and Ghassemi 2013a, b). In this work we improve and enhance the FMDDM (Verde and Ghassemi 2013a, b) to examine and analyze stress and displacement of large-scale natural fracture networks by considering shear slip and associated potential mirco-earthquakes. The shift in state from joint to hydraulic fracture (open crack) is considered in the investigation of fracture deformation. The FMDDM approach is verified by comparing results with an analytical solution and conventional DDM code. Simulation examples involving as many as 10,000 degrees of freedoms are presented to highlight the adaptability of the FMDDM method and hydraulic natural fracture deformation, and induced micro earthquakes. 2. STRESS AND DISPLACEMENT ANALYSIS USING THE FAST MULTIPOLE DISPLACEMENT DISCONTINUITY METHOD (FMDDM) Details of modeling approaches for discontinuity displacement, fluid flow and friction resistance for fracture deformation, pressure change, and MEQ are briefly discussed below. 2.1. The Displacement and Flow Modeling Displacement Discontinuity Method (DDM) is an indirect boundary element method suitable for analysis of fracture deformation (change in aperture size) in a medium containing multiple fractures (Crouch and Starfield 1983). The DDM is based on the analytical solution to the problem of a finite line segment centered in the x-y plane with constant normal and shear discontinuities in displacement (See Fig. 1a). 1

Figure 1: (a) A two-dimensional fracture segment with constant discontinuity displacement components D x and D y. For calculation of displacement over a fracture, the fracture is divided into N elemental segments (see Fig. 1b). Induced normal and shear stresses on ith fracture segment calculated as the sum of effects of all N elements (Crouch and Starfield 1983). In natural fractures with in-situ stress, fluid flow pressure, and fracture stiffness set of equations can be rearranged as: (1) Where,,, and are boundary influence coefficients, and are the crack shear and normal stiffness, respectively. Normal stiffness is estimated by a hyperbolic equation as a function of the maximum closure and initial stiffness (Bandis et al. 1983), is dilation angle which accounts for change of the normal displacement by change in the shear displacement. The approach employed by (Zhou and Ghassemi 2011) is used in this paper to couple flow, displacement, and stress and estimate micro earthquakes (MEQ). In addition, changes in fracture permeability because of shear slip are taken into account. Aperture change for each element is expressed as: (2) where, is increment of the normal effective stress, is the dilation induced aperture change generated by slip. For pressurized fracture modeling, each element can be in three different states of slip, stick, or separation. Since state of each element might change from one time step (mining step) to another step, its state must be determined at every time step. Under different state conditions, different right-hand sides for DDM should be considered to calculate displacement discontinuity. If the state of element is separation, the element pressurized (P) is more than in-situ stress acting on the surface of the element, thus, the element is open under the load. This means zero stiffness in both normal and shear directions. Equation (3) is adopted to judge for separation of an element: where, is cohesion and is effective friction angle element surface which is summation of friction angle and dilation angle. Next is to determine whether a close element is in state of stick or slip. Here, Mohr-Coulomb criterion (Eq. (4)) is used where, is the effective normal stress and is the yield stress. If the shear stress of an element exceeds the Mohr-Coulomb criterion, the element is yielding and its state is changed to slip. To calculated the displacement for such an element, its is set to zero and acting shear stress on the element is set to. In the case that an element is not surpassing Mohr-Coulomb criterion, its state is stick. In this state, normal stiffness is estimated through the proposed equation by Barton-Bandis (Bandis, Lumsden, and Barton 1983) and shear stiffness is constant equal to the initial value. Shear strength of fracture can be roughly characterized using slip-weakening model in which shear strength is proportional to the normal stress and weakens with increasing slip (Templeton and Rice 2008). By considering Slip-Weakening Model (SWM), the friction angle decreases from peak ( ) value to residual value ( ) and the cohesion decays from the initial (intact) value to zero. To determine resistant parameters and, only plastic value of shear displacement ( ) is measured. If plastic deformation is less than critical displacement discontinuity ) the value of friction angle and cohesion are linearly estimated as a function of : (3) (4) (5) 2

If, then and. And the value of shear resistance can then be calculated from Eq. (4). The seismic moment caused by slip displacement and estimate of the magnitude of earthquake produced by the seismic moment can be estimated as (McGarr et al. 1979): (6) (7) where G is shear modulus of the rock. Injection or production leads to change of pressure in fluid flow inside a fracture network and consequently change of stress over the fracture surface and then an aperture-size change. To calculate flow through the interconnected fractures mass balance and Darcy s law are combined as: (8) where, is fracture permeability, is fracture aperture, is segment length, is fluid viscosity, is production rate per thickness, and is fluid compressibility. In Eq. (8), is net flow rate in a fracture, is change of fluid volume due to expansion/compression, and represents change of fluid volume by fracture deformation. Equations (1) and (8) are coupled to solve fracture deformation and fluid flow inside the fracture network. Equation (1) is set on N-body system of equation which involves evaluation of all pair-wise interaction of elements (over fractures). It can be written as: (9) where is a kernel function. Direct computation of requires O(N 2 ) step and memory to store coefficient and depending on the solver and up to O(N 3 ) operation to solve, where N is number of unknowns. Therefore, for large scale (realistic) problems this drawback become prominent and computational time increases quadratically. One solution is to employ approximation techniques. For instance, Fast Multipole Method (FMM) is an approximation technique which decreases computational effort and required memory to O(N). 3. Fast Multipole-Displacement Discontinuity Method (FM-DDM) Fast Multipole Method (FMM) is an approximation method which accelerates the solution of DDM despite decrease in required memories. The FMDDM simultaneous solves the changes of the normal (n) and shear (s) displacement discontinuities and porepressure (Verde and Ghassemi 2013a, b) (see Fig. 1). The approximation approach is accomplished by categorizing the influence points into near-field and far-field interaction, based on the distance of interaction points. The normal and shear stress due to the interactions among all fractures can be expressed mathematically as: (10) where the near-field interactions are calculated through the conventional DDM furmulations, and the far-field influences that involves most of the algebraic products, are calculated efficiently using FMM to reduce the computational cost (time) proportional to N. Classical FMM requires approximation of kernel based on analytical expansions (Greengard and Rokhlin 1987) which make the appraoches a kernel-dependent appraoch. Few kernel-independent (black-box) FMMs exist which allow constructing a fast method emplying only numerical values of kernel (Fong and Darve 2009). One of these technique is singular value decomposion (SVD). A heirarical tree (or quadtree in 2D) is constructed and computational domain is decomposed to subdivision to to identify near field and far field at each level. To approximate influence of a cluster of source elements over fractures the center of cluster is considered as concentration point and influences of such elements changes based on the distance between field and source positions. A general algorithm for the combined near-field and far-field calculation using the FMM involves the following steps: 1. Discretiz the boundary like a conventional BEM problem; 2. Construct the heirachical tree by subdiving the larger clusters (parents) to smaller clusters (children) until only a specific number of elements are in the cluster; 3- Compute influence of clusters on far clusters (upward pass); 4- Compute influence of far cluster on clusters (Downward pass); 5- Compute combined influence of all clusters on every other cluster. This step is called evaluation in which right-hand side of Eq. (7) is formed; and 6- Solve the systems of equation through an iterative solver such as GMRES. An efficient preconditioner is essential for faster convergence and boosting computer efficiency. More details and variations of the FMM have been described previously by other authors (Fong and Darve 2009, Greengard and Rokhlin 1987, Liu and Nishimura 2006). 3

4. CASE STUDIES 4.1. Verification Two sample cases are presented for the verification of the Fast Multipole Displacement Discontinuity Method (FMDDM). Mechanical and environmental modeling parameters for the both cases are the same. Shear modulus is 15 GPa, Poisson s ratio is 0.25, isotropic in-situ stress is 10 MPa, and pore pressure is 15 MPa. 4.1.1. A Series of Collinear Periodic Pressurized Fractures The FMDDM code is verified by comparing its predictions with an analytical solution. The displacement of a set of evenly spaced collinear periodic fractures based on an analytical solution is (Hwu 1991): where, and (10) Where, is periodicity and is length of each fracture (see Fig. 2). Figure 2: schematic configuration of collinear periodic fractures. For this case periodicity is 150 m and length of each fracture is 50 m with total number of elements is 500 (1000 DOFs). The normal displacement from FMDDM model is compared with analytical solution as well as a conventional DDM model (the conventional DDM code is taken from appendix A of (Crouch and Starfield 1983)). The result is shown in Fig. 3. Figure 3: Comparison and verification of FMDDM with analytical solution as well as a conventional DDM. The average error of the FMDDM in comparison with analytical solution is ~7.1%, but the conventional DDM shows ~14.0% error with the analytical solution which shows that the FMDDM can calculate displacement accurately. 4.1.2. A pressurized fracture network The configuration of the fractures is shown in Fig. 5a. This arrangement is consist of two parallel vertical fractures with a separation distance of 30 m and three parallel horizontal fractures separated by 15 m. The number of elements is 792 with 1584 DOFs. In order to prevent rigid body motion of the interior domain, fractures are set not to intersect. 4

Figure 4: (a) Geometry of fractures for verification of FMDDM with conventional DDM; (b) Comparison of the FMDDM with a conventional DDM. The comparison of the normal displacement for central fractures (y=0) is shown in Fig. 4b. As it can be seen, the FMDDM results match very well with the conventional DDM. In this case the average error is ~0.04%. In almost all cases that the FMDDM is compared with the conventional DMM, accuracy of the FMDDM is very high as well and also, the computational time is lower. 4.2. Simulation Cases 4.2.1. Natural fracture under injection and production In this case a natural fracture network of 10 large cracks, which have different orientations and lengths, is simulated under constant injection and production. Changes of fracture aperture, pressure and friction angle are presented. The geometry of the fracture network as well as the injection and production wells are shown in Fig. 5. Figure 5: Geometry of the fractures network, arrows indicate injection and production wells. The fracture network is in equilibrium under in-situ stress of = 8 and = 10 MPa and initial pore pressure of 5 MPa. Initial shear and normal stiffness are 500 and 25 GPa/m, respectively. Intact friction angle and residual friction angle are 30 and 26, respectively with cohesion of zero. Water is injected at the injection well at a rate of 4.5 10-6 m 3 /s (per unit thickness of the rock mass) and the production rate is half of the injection rate. Compressibility of water and its viscosity are 4.2 10-10 Pa -1 and 2.03x10-4 Pa.s, respectively. Higher injection rate causes pressure increase inside the fractures and consequently fractures closer to the injection well face higher pressure and fractures closer to the production well face lower pressure. Such a result is shown in Fig. 6a. Pressure difference between the injection and production well is as low as 0.55 kpa which means pressure inside the fracture does not significantly affect difference in the aperture size. The aperture size is slightly higher where fractures are not connected to another fracture and leads to accumulation of water and increase in the aperture size (see Fig. 6b). 5

Figure 6: (a) pressure change from initial pore pressure to pressure at the end of the simulation; (b) Aperture of the fracture network. Depending on the orientation and induced normal and shear stress on the surface of an element of a fracture, shear stress might surpass the failure criterion (in this simulation Mohr-Coulomb) and yield. Based on an idealization of the slip-weakening model, the friction angle is a linear function of shear displacement and plastic displacement. Figure 7a shows fractures that their shear stress exceed MC (Mohr-Coulomb) criterion in blue. These fractures will have lower friction angle closer to residual friction angle, which is 26 in this case, depending on the plastic and shear displacement based on the Eq. (5). Figure 7b shows friction of different fractures. Please note that fracture in red in Fig. 7a have the friction angle of intact friction angle which is 30. Figure 7: (a) Status of fracture based on the MC criterion; (b) Friction angle of elements over fractures at the end of the simulation. Fractures at different orientations and positions face different total shear and normal stress. As the pressure inside a fracture increases the total normal stress decreases at the same time shear stress over surface of the fracture is almost constant. In this situation, the fracture might meet MC criterion (, cohesion of zero) and yields. Note that since the pressure inside one fracture is nearly uniform, when one element of the fracture meets the MC, it is highly likely that all the other elements of the fracture meet the MC; thus we say the fracture meets MC instead of an element over the fracture. After yielding, friction angle decreases toward the residual friction angle according to slip weakening modeling (see Eq. (5)). Figure 8a shows shear stress vs. effective normal stress of middle element of the indicated sample fracture in Fig. 7a during the last three time steps. The circled numbers in Fig. 8a refer to the separate steps used to show different time steps (#1 is the last step). The orientation of the fracture is ~50 with respect to the x axis and shear stress over the middle element is ~0.98 MPa. In time step 3, 2, and 1 total normal stress over the fracture is ~1.54, 1.46, and 1.38 MPa and friction angle of 30, 26.82, and 26.27, respectively. Solid lines in Fig. 8a show MC envelops for different friction angle, which in this case, start from zero point (0,0) since cohesion is zero. Thus, as we can see, the middle element of the fracture fails in accordance with MC criterion and yields which leads to lower friction angle as the plastic deformation increases. Solid lines in Fig. 8a shows the MC envelop which is the highest amount of shear stress that an element can tolerate without yielding at a particular normal stress. Figure 8b shows the seismic moment of elements along the indicated sample fracture in Fig 7a during the last two time steps. Radius of each circle expresses the magnitude of seismic moment based on the shear displacement (see Eq. (6)). The larger circles have higher seismic moments. The element at position of x=0 m has the highest shear displacement and consequently the highest seismic moment. As we go forward on time steps (#2 to #1 in Fig. 8a) shear displacement of elements increases and leads to higher seismic moment. The summation of seismic moment over slip area helps us to estimate the amount of micro-earthquake. 6

Figure 8: (a) Stress conditions of the middle element of indicated sample fracture during the last three time steps; (b) Seismic moment of elements a sample fracture. Fractures in red might have lower or higher orientation angle (in this sample case we have only lower orientation) with respect to x axis and consequently they have lower shear stress over the surface which prevents them from meeting the MC criterion. The magnitude of the micro earthquake produced after the simulation with slip-weakening model has magnitude of -4.40 for the indicated sample fracture in Fig. 7a at the last time step. Figure 9: Field stress distribution at the end of the simulation. A contour of field stress distribution is shown in Fig. 9. Pressure inside the fracture network is almost 7.5 MPa which leads to increase in aperture (see Fig. 6b) but the pressure is still low that prevents any elements to change its state to separation. Thus all elements are still in state of stick and have stiffness. 4.2.2. A larger simulation Consider a scale fracture network consist of 25 fractures with irregular orientation and different length. This fracture network has 3144 elements (9432 DOFs) and pressure and fracture aperture change is simulated under injection through FMDDM. The geometry of the fracture network is shown in Fig. 10. In this sample case, the network is in equilibrium under in-situ stress of and. The initial pore pressure is 15 MPa and injection rate in all injection wells is 5 10-6 m 3 /s. Friction angle changes from 30 (intact) to 26 (residual) and 2 of dilation affect normal displacement as a result of shear displacement. Other parameters are similar to those in the verification cases. Compressibility of water and its viscosity are 4.2x10-10 Pa -1 and 2.03x10-4 Pa.s, respectively. 7

Figure 10: Geometry of the large scale fractures network and injection wells. Pressure changes by 0.5 MPa throughout the simulation and pressure is almost equal to 15.35 MPa everywhere inside the fracture. However, each fracture faces different effective normal stress based on its orientation (see Fig. 11a). The middle fracture connects both sides of the network and fluid flow and consequently aperture change should be higher in the middle of the fracture network (see Fig. 11b). Aperture of fractures on sides of the network has not changed significantly, except some places which might be because of inaccuracy in calculation. But, aperture of fractures in the middle has increased as it was expected. Figure 11: (a) Normal effective stress; (b) aperture change. Note that fracture permeability (Fig. 11b) is higher around the middle injection wells as it was expected according to Eq. (1). Changes of permeability depend upon various factors such as the position and orientation of the facture and induced normal and shear stress. The stress distribution within the fracture network is presented in Fig. 11a. Since the aperture of fractures has not changed significantly throughout the fracture network, field stress is almost uniform except at fractures tips. Figure 12 shows field stress distribution around the fracture network. Tension at the tip of fractures and compression between fractures can be seen clearly. 8

Figure 12: (a) Field stress contour of the large scale fracture network; (b)(c)(d) zoom-in contour of selected area in (a). Figures 12b and 12c show that at the tip of fractures a kidney-like stress distribution is formed. Selected area in Fig. 11b shows that aperture size in the middle fractures has increased leading to compression stress around and the area in-between fractures. The result of these induced compression on the field stress is shown in Figs. 12d and 12c. CONCLUSIONS A combination of fast multipole method and Displacement discontinuity method has been used to estimate the behavior of fracture in natural fracture networks. Change of fracture aperture, pressure inside the network, friction angle were investigated. According to slip-weakening model, with increasing slip, state friction angle decays towards a residual friction angle value. The Mohr- Coulomb criterion is used to check the yield stress and calculate slip over the surface of fractures. The latter is used to estimate potential MEQ magnitudes. Total shear stress acting upon the fracture may vary depending on the orientation of fractures. As would be expected, a fracture with higher shear displacement over its surface can potentially yields a higher seismic moment over its area. REFERENCES Liu, Yijun. Fast Multipole Boundary Element Method, Cambridge University Press, (2009). Liu, Y.J., and Nishimura, N.: The fast multipole boundary element method for potential problems: A tutorial. Engineering Analysis with Boundary Elements, 30 (5), (2006), 371-381. 9

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