Modeling of Failure along Predefined Planes in Fractured Reservoirs

Transcription

1 PROCEEDINGS, Thirt-Ninth Workshop on Geothermal Reservoir Engineering Stanford Universit, Stanford, California, Februar 24-26, 2014 SGP-TR-202 Modeling of Failure along Predefined Planes in Fractured Reservoirs Rajdeep Deb and Patrick Jenn Institute Of Fluid Dnamics, ETH Zurich, Sonneggstrasse 3, ML H 45, Zurich 8037, Switzerland address, debr@ethz.ch, jenn@ifd.mavt.ethz.ch Kewords: Fracture, Shear Modulus, Slip, Finite Volume, Friction, Failure ABSTRACT A numerical approach to model stress and failure in fractured reservoirs, including slip along predefined planes, is presented. The model is based on linear elasticit theor and uses the static/dnamic friction law. The displacement vector is computed b a finite volume method, and in addition to the discretization of the whole domain, the fractures are discretized b lower dimensional segments. After ever time step it is decided individuall for each of these fracture segments, whether the specified slip criterion based on the static/dnamic friction law is reached. A numerical scheme coupling irreversible slip along the segments with the elastic displacement in the domain is developed. The irreversible slip information, which also matters for the stress calculation, is stored and used in a mathematicall consistent wa at each adjacent finite volume interface, i.e. the actual grid geometr remains unaltered. The coupled sstem for elastic displacement and irreversible slip is solved b an implicit solver. The method is implemented in two dimensions for multiple fractures, and a number of numerical simulation studies have been performed to analze the stress distribution adjacent to single fracture and fracture networks. 1. INTRODUCTION Numerical studies of enhanced geothermal sstems (EGS) requires efficient modeling and simulation of the coupled solid and fluid mechanics in the reservoirs. Geothermal sstems are natural heat echangers, which depend on fluid flow paths through the rock and heat echange between fluid and rock. Rock mechanics obviousl plas a ke role in forming and creating new flow paths, and therefore a first objective is to obtain a phsicall accurate numerical modeling strateg to describe the geo-mechanics. Tpical EGS reservoirs can be approimated b an elastic medium with a high fracture densit. In order to properl describe poroelastic coupling of fluid and solid in such sstems, primar fractures are generall represented as discrete manifolds embedded in the elastic domain, where the sstem cannot bear traction forces beond a certain maimum limit. Failure occurs along the fracture manifolds, once the local shear force eceeds the local sustainable traction force. Then an irreversible rock displacement occurs along these manifolds, which is termed as slip in the rest of the paper. Slip is ver important to determine the hdrodnamic flow radius in these fractures. In this paper an accurate and efficient solution approach for such slip calculations and coupled stress distributions is presented. The paper is divided into four sections. In section 2, the continuum problem of fractured reservoirs is described. In section 3, the numerical method used for the linear elastic problem is described. Section 4 describes the numerical modeling of slip and stress after failure. Finall in section 5, numerical results are presented and studies of failure scenarios with multiple fractures are discussed. 2. CONTINUUM PROBLEM Numerical simulations of fractured reservoirs rel on a continuum domain with a number of embedded fracture manifolds, whereas force or displacement boundar conditions are applied. An eternall imposed displacement or stress induces a shift and thus changes the stress distribution in the domain. Figure 2.1: Representation of a fractured domain in an elastic medium with predefined fracture manifolds. Figure 2.1 depicts a fractured domain, where the blue lines represent predefined fractures. The green area is treated as a continuum, in which force balance is described b the wave propagation equation. (2.1) Equation 2.1 describes the balance between divergence of the stress tensor, the weight force per unit volume and inertial acceleration. Initiall, the fractures are closed and have no effect. Once the local maimum traction limit is reached, a separate mathemati- 1

2 cal treatment is required along these manifolds to capture the slip along them. Numerous constitutive friction laws based on slip distance and rate have been described in the literature (Dieterich, 2007, Segall, 2010). A conclusion thereof is that the sstem acquires a new configuration due to irreversible slip, and fractures eventuall lock in and again static friction becomes the phsical mechanism responsible for force balance, i.e. equation 2.1 again becomes applicable in the entire domain. Obviousl, these irreversible slip events have to be taken into account for the effective elastic strain calculations across fracture manifolds. The strateg regarding this strain calculation is further described in section Linear Elasticit In order to obtain a closure of equation 2.1, a constitutive law needs to be defined for the relation between stress and strain. A simple description is based on a linear elastic description of the rock, i.e. the stress and strain tensors are assumed to be linearl related. For most geo-mechanics applications in the finite stress limit, linear elasticit is found to be a valid assumption and is described b. (2.2) Equation 2.2 defines the stress tensor at an point in the domain as a linear combination of bulk strain and deviatoric strain. The phsical relevance of the Lamé constant l is derived from the Young modulus and Poisson ratio, from which also the shear modulus G can be computed. 2.2 Equilibrium Problem Further, equilibrium is generall assumed for the geo-mechanics in enhanced geothermal sstems (EGS), i.e. it is assumed that an perturbation almost immediatel propagates across the whole domain. In that case, the inertial term in equation 2.1 can be omitted leading to the elliptic equation, (2.3) which describes force balance between stress and weight forces. Note that this assumption is not valid for large slip as during earthquakes, but for EGS simulations it is justified. This assumption can be verified b comparing compressive (P-wave) and shear wave (S-wave) propagation speeds with the ratio of relevant problem length and time scales. 2.3 Failure Criterion and Slip Solution There are two mechanisms, due to which geo-mechanical failure occurs, i.e. either due to shear or tensile forces (shear or tensile failure, respectivel). In the former case, slip occurs when the rock grains in a fracture cannot bear tractional forces applied due to local shear stress, which leads to a new stress distribution. Shear failure leads to an increased shear stress at the tips of a fracture, possibl resulting in secondar failures. In the latter case, tensile failure occurs when tensile stress overcomes the maimum limit, thus leading to fracture openings. Note that especiall shear failure is a complicated phenomenon and requires careful modelling approimations. Net, modelling of shear failure and the resulting stress redistribution is described. A simple model to stud the failure criterion on a predefined fracture surface due to applied traction is based on the static/dnamic friction law. Traction force on a fracture plane is compared with the static friction limit for a given compressive force. If it eceeds the maimum static friction limit, the traction force on the fracture surface drops to the dnamic frictional limit, which also depends on the compressive force. Tensile failure occurs when the normal stress on the fracture surface becomes positive. Therefore, the domain eposed to dnamic shear or tensile stress eperiences failure, once a corresponding condition on a fracture manifold is reached. Compressive and shear forces are described b and the slip criterion is and (2.41), (2.42), (2.43) where S 0 is a cohesive force on the fracture plane and μ s an internal static friction coefficient. Note that modified failure criteria also accounting for local fluid pressure have to be emploed, if also flow is considered. Like for the wave propagation speed it is assumed that the time scale for slipping along fracture manifolds is much smaller than all other relevant time scales. Therefore, it is not required to resolve the slipping events in time, i.e. the irreversible slip can directl be computed together with the elastic displacements within the domain. This is described in more detail in section 4, which is dedicated to numerical failure modeling. The use of such assumptions allows for much faster numerical solution algorithms compared to methods in which such slip failure events are resolved. With this assumption in place, the problem can be solved b finding another equilibrium based on the dnamic friction coefficient. 2

3 Figure 2.2: Cartesian mesh with finite volumes. The red dots represent the nodes at which the displacement solutions are stored. The red line represents an arbitraril oriented fracture. 3. NUMERICAL METHOD FOR STRESS EQUILIBRIUM In this paper, a two dimensional problem is solved using plain strain assumption, i.e. the displacement normal to the the plane of figure 2.2 is zero. A finite volume method similar as the one used b Benjemaa et. al. (2009) is emploed to solve the stress equation 2.3. Here, the computational domain is discretized into rectangular finite volumes and the displacement vectors are stored at their centres; see figure 2.2. In each volume Ω, (3.11) has to be fulfilled and with Gauss theorem one obtains the requirement. (3.12) The stress tensor is assumed to be constant over each face of a finite volume, thus discretization of equation 3.12 leads to ( s -s E W )D + ( s N -s S )D = 0 and (3.21) ( s E -s W )D + ( s N -s S )D - rgdd = 0. (3.22) Equations 3.21 and 3.22 need to be understood with respect to the discretization stencils depicted in figure 3.1. The describe horizontal and vertical force balance, respectivel. Figure 3.1: Discretization stencils. The left figure illustrates the naming convention for the volume interfaces, and the right figure illustrates the naming convention for central nodes. Replacement of the stress components in equations 3.21 and 3.22 using the linear elastic constitutive law of equation 2.2 and numerical approimation of the spatial displacement derivatives at the volume interfaces lead to a set of closed discrete equations for the displacement vectors: æ u ( l + 2G) è E W ø D + l æ u è E W ø D + G æ u è N S ø D + G æ u è N D = 0 and (3.31) S ø æ u ( l + 2G) è N S ø D + l æ u è N S ø D + G æ u è E W ø D + G æ u è E D - rgdd = 0. (3.32) W ø The above set of equations involves derivatives with respect to face tangential and normal directions. With reference to the notation introduced in figure 3.1 the can be epressed (for the eastern face) as 3

4 u /» (u e - / uc / ) D (3.41) E and» 1 æ E 4 è u / u / + u / + u / + u / EN ES N S ö. (3.42) ø B replacing equations 3.31 and 3.32 with the numerical approimations proposed above, one obtains æ - 2 è (l + 2G)D D l 4 (une + usw - unw - use + 2 GD D ö ø uc æ (l + 2G)D ) - 2 è D (l + 2G)D + u e D + 2 GD D ö ø uc (l + 2G)D + u w D + GD D un + GD D us + l 4 (une+ usw- unw (l + 2G)D + u n D - use ) = 0 and (3.51) (l + 2G)D + u s D + GD D ue + GD D uw - rgdd = 0 (3.52) for each volume. For this work, successive over relaation was used to solve the resulting linear sstem for the displacement vectors in each volume. In the following section it is described how failure due to slip is treated in this framework. 4. NUMERICAL METHOD FOR FAILURE MODELING Figure 4: Fracture manifold with fracture segments (represented b black dots) embedded in a conforming mesh (blue lines). The maimum fracture resolution is given b the grid resolution. Each predefined fracture manifold is discretized into finite segments as shown in the figure 4. The maimum possible fracture resolution is limited b the grid resolution and the minimum fracture resolution b the size of the whole manifold. The number of fracture segments is denoted b N s. The slip solution, which is stored on each fracture segment, quantifies the irreversible displacement between adjacent grid points on either side of the fracture. Once the displacement and stress fields are computed as described in the previous section, traction and compressive forces on the fracture manifolds can be obtained from equations 2.42 and 2.41 and the failure criterion 2.43 can be consulted for each fracture segment. Unless the traction force magnitude is less than the maimum limit proposed b equation 2.43, the linear sstem of equations are solved for obtaining the net time step solution based on time dependent boundar conditions. If for a segment the failure limit is reached, i.e. if the local traction force eceeds the sustainable limit, the resulting slip displacement has to be computed along with the displacement field due to linear elasticit. The governing slip dnamics is usuall modelled based on a slip or slip rate dependent friction law. However, since the timescale of this process is etremel small compared to all other relevant time scales, a new static equilibrium is directl calculated. The latter is achieved b seeking slip solutions for each failing segment, such that the corresponding local traction forces subject to dnamic friction are balanced. Note that the dnamic friction coefficient is alwas smaller than the static one. In order to capture the influence of slip and slip rate, the dnamic friction coefficient can be modelled b a function thereof as described b McClure and Horne (2010). For simplicit, in this paper the dnamic friction coefficient is assumed to be constant and suitable values related to EGS are selected. Therefore, slip for all failing fracture segments are solved together with the linear sstem for the displacement field due to linear elasticit b adding t (s,s ) = t ( S 0 + m d s c (s,s )) or (4.11) t (s,s ) = t ( S 0 + m d s c (s, s )), (4.12) where s = t s t. (4.13) Equation 4.11 and 4.12 describe the new equilibrium obtained after failure and successive slip. The constant μ d is the dnamic friction coefficient. Note that traction and compressive forces depend on the stress along the fracture segments, whereas for the stress calculation the accumulated slip has to be subtracted from the elastic displacement between adjacent nodes on opposite sides of the corresponding segment, i.e. the numerical approimation 3.41 has to be replaced b 4

5 u /» ( u n / - uc - s / / ) D (4.2) N for central difference approimations of all the derivatives along the normal direction to the fracture line. Finall, an etended linear sstem has to be solved for the elastic displacement vectors in each finite volume and the irreversible slip along the fracture segments. An important numerical parameter is the number N s of segments per fracture line, which can var from one to the number of intersecting finite volumes. The decisive traction and compressive forces for a segment are computed as averages with contributions from all the intersecting finite volumes. This wa, a coarse slip solution can be coupled with a fine scale displacement solution. Especiall for scenarios with huge number of fractures this is useful to effectivel reduce computational cost, if numerical accurac can be achieved b interpolating the slip solution between fracture segments. In the following section, numerical studies to validate the solution algorithm discussed in sections 3 and 4 are presented. 5. PROBLEM SETUP AND RESULTS To assess the accurac of the algorithm outlined above, three 2D test cases are considered. All of them emplo a domain of size 1,000m 1,000m consisting of homogeneous isotropic rock with predefined fracture manifolds. Periodic boundar conditions are applied on the left and right sides (at = 0m and = 1000m), and constant vertical and time dependent horizontal forces are imposed at the top and bottom boundaries (at = 0m and = 1,000m) in such a wa that the same shear rate applies everwhere in the domain until first failure. To obtain displacement vectors for each finite volume and the accumulated slip for each fracture segment, equations 2.3 and 4.11 are solved. Therefore, a shear modulus of G=15GPa and a Lamé constants of l =15GPa are chosen, and the densit is ρ = 2300 kg/m 3. The choice of these properties results in S-wave and P-wave speeds of 2,550m/s and 4,420m/s, respectivel, leading to a time scale of less than half a second, which is much smaller than the one imposed b the boundar conditions (the forces per unit area at the top and bottom boundaries are varied at a rate in the order of a few MPa/s). Therefore, it is justified to assume quasi-equilibrium based on the static and dnamic friction coefficient at the fracture manifold, which are set to 0.6 and 0.55, respectivel. For all simulations, a time step size of 1s is emploed. Further, the cohesion parameter is set to 0 and no weight force is considered. 5.1 Shear Test with Single Fracture Figure 5.1: Illustration of test case with a single horizontal fracture manifold. Shear is induced at the top and bottom boundaries and periodic boundar conditions are applied on the left and right sides. Here, a horizontal fracture manifold of size 500m located at the domain center is considered, which is depicted b the blue line in figure 5.1. Before first failure occurs, traction force per unit area on the fracture manifold increases at a rate of 1MPa/s. The domain is discretized into finite volumes and the number of fracture segments is 80, i.e. each fracture segment is adjacent to one finite volume on each side. The compressive force per unit area in vertical direction is kept at 50MPa and the shear rate at 1MPa/s. Under the prescribed compression, the segments can sustain a shear force per unit area of up to 30 MPa as calculated b equation With these specifications, first failure occurs after 31 time steps, i.e. after 31s. The stress distribution after the first failure event is shown b the plots of figure 5.2 and on the left of figure 5.3. The right plot in figure 5.3 depicts the slip along the horizontal fracture line. Figure 5.2: Contour plots for normal stress components s (left) and s (right) after 31s. 5

6 Qualitativel, normal stress results are in good agreement with those b Bourne and Willemse (2001). Shear stress decreases near the fracture line and increases sharpl near the fracture tips. This is due to the fact that after failure shear forces decrease to the dnamic friction limit. Therefore, shear stress near the fracture manifold is reduced to a level, which allows for a new equilibrium. Figure 5.3: Contour plot for shear stress component s (left) after 31s and slip solution along the horizontal fracture line. A volume near a fracture tip, which is not adjacent to the fracture, eperiences a much higher shear force than those neighbouring volumes, which share an interface with the fracture manifold. Similarl, a sharp rise in normal stress (both in and direction) is apparent near the fracture tips, which can be eplained b the force balance. The decrease of compressive stress and sharp rise of shear stress at the fracture tips leads to regions prone for secondar fracture generation. 5.2 Shear Test with Multiple Fractures Here, shearing of the same rock as in the previous subsection is considered, but with multiple fractures of length range from m. Nine of them have horizontal orientation with their left tips located at (250m,250m), (425m,250m), (625m,250m), (250m,500m), (425m,500m), (625m,500m), (250m,750m), (425m,750m) and (625m,750m), and eight have vertical orientation with their lower tips located at (200m,300m), (400,300m), (600m,300m), (800m,300m), (200m,550m), (400m,550m), (600m,550m) and (800m,550m). As in the previous test case, shearing is imposed at the top and bottom boundaries, which first leads to failure of the vertical fractures. The resulting slip at the corresponding fracture segments changes the stress distribution. It can be observed that the local stress solution around individual vertical fractures is in qualitative agreement with the solution for a single fracture presented above. However, the stress perturbation near the vertical fractures increases the tendenc for failure of the neighbouring horizontal fracture segment. Results after 15s, 30s and 70s are shown in the upper, middle and lower plots of figure 5.4, respectivel; on the left the shear stress is depicted and on the right the failed segments are indicated b blue dots. The initial boundar condition eerts a compressive stress σ of 50MPa in vertical direction, which results in a compressive stress σ of 16.67MPa in horizontal direction. At 30s it can be observed that the horizontal fractures in the bottom row start to fail near their left tips, while those in the top row start failing at their right tips. This is due to the stress field solution resulting from failure of the vertical fractures. The left and right plots of figure 5.5 depict the normal stress components σ and σ (after 70s), respectivel, and the slip along the fracture lines is shown in figure 5.6, whereas for visualization purpose the fractures are assembled in one line ordered according to the list of respective tip coordinates mentioned above. Along each individual fracture, the resulting slip distribution closel follows a parabolic profile similar as in the test case with one fracture. 5.3 Shear Test with Oblique Fracture Here, the same setup as for the previous two test cases is emploed, but with a single oblique fracture at the center, which has an orientation of 45 0 and a length of 354m; see right plot of figure 5.7. To trigger failure at an earlier stage, here a vertical compressive force per unit area of 20MPa and the same constant shear rate of 1MPa/s as before, but in opposite direction, have been applied. The dnamic friction coefficient is set to 0.5. The oblique fracture is approimated b a staircase representation as illustrated in the left plot of figure of 5.7. Each fracture segment consists of one horizontal and one vertical volume face; note that the whole fracture etends over 40 finite volumes, both in horizontal and vertical directions. The stress components σ, σ and σ are shown in the three contour plots of figure 5.8 after first failure occurred, i.e. after 5s. 6 CONCLUSION A numerical algorithm is devised, which allows to simulate failure in fractured reservoirs due to shearing. Stress and displacement fields are computed based on the linear elasticit assumption, while for failure along prescribed fracture manifolds static and dnamic friction laws are emploed. A further assumption is that inertia effects can be ignored. The domain is discretized b a Cartesian finite volume grid, in which the fracture manifolds are embedded and approimated b staircase representations. Failure is computed individuall for each fracture segment, which ma be coarser than the grid volumes. High efficienc is achieved, since displacement and slip can be computed b solving a linear sstem. The time steps have to be small enough in order to capture individual failure events, but wave propagation and the dnamics during slip do not have to be resolved. Thus, b emploing coarse fracture segments, the intervals between individual failure events can be increased, which allows for larger average time steps and at the same time reduces the number of unknowns. For demonstration, the framework was applied for cases with single and multiple fractures and the simulation results are discussed. In the future, this geo-mechanics framework will be etended b including models for tensile failure and where necessar for nonlinear elastic behavior. Further, it will be coupled with flow and transport. Therefore, constitutive relations between local stress, slip, fluid pressure and fracture aperture are required, and the fluid pressure has to be taken into account b the failure criteria. 6

7 Deb and Jenn Figure 5.4: Contour plots of shear stress component (left) and failed fracture segments in blue (right) at 3 different times. Figure 5.5: Contour plots of normal stress component after 70s. 7

8 Figure 5.6: Slip solution after 70s along all fracture lines, which are aligned here for visualization purpose. Figure 5.7: Problem setup for the oblique fracture test case (right), with an illustration of the stair case fracture representation (left). Figure 5.8: Contour plots of all stress components after 5s. 7 ACKNOWLEDGEMENTS The work was performed as part of the GEOTHERM project that is funded b the Competence Center for Environment and Sustainabilit of the ETH-Domain. The authors are grateful to Keith Evans and Dimitrios Karvounis for the constructive remarks and suggestions during the course of the project. 8

9 8 NOMENCLEATURE = Stress tensor at an point of the given domain. = Volume densit of force vector ( weight force ) at an point of the given domain. = Displacement vector at an point given domain. = Unit tensor. ˆn = Normal direction to a particular plane. t = -directional component of unit vector along shear traction force for a plane. t = -directional component of unit vector along shear traction force for a plane. s c = Normal force on a plane. Ñ = Gradient operator. Ñ = Divergence operator. V p = P-wave (compressive wave) speed. V s = S-wave (shear wave) speed. G = Shear modulus. l = Lamé constant. r = Densit. dw= Differential volume segment over which the equilibrium equation is averaged. g = Acceleration due to gravit. s = Normal stress tensor component along -direction which means force applied along -direction applied on a unit area normal to -direction. s = Shear stress tensor component along /-direction which means force applied along /-direction applied on a unit area normal to /-direction. s = Normal stress tensor component along -direction which means force applied along -direction applied on a unit area normal to -direction. u = The displacement component along -direction. u = The displacement component along -direction. D = Grid discretisation along -direction. D = Grid discretisation along -direction. 9

10 REFERENCES Bourne, Stephen. J., and Willemse, Emanuel J.M.: Elastic stress control on the pattern of tensile fracturing around a small fault network at Nash Point, UK, Journal of Structural Geolog, 23, (2001), McClure, Mark W., and Horne, Roland N.: Investigation of Injection-Induced Seismicit Using a Coupled Fluid Flow and Rate/State Friction Model, Proceedings, 36th Workshop on Geothermal Reservoir Engineering, Stanford Universit, Stanford, California, Januar 31 - Februar 2, 2011 SGP-TR-191. Dieterich, J. H, (2007), Applications of Rate- and State-Dependent Friction to Models of Fault Slip and Earthquake Occurrence, Treatise on Geophsics, vol. 4, chapter 4, ed. Kanamori, H., Elsevier, Amsterdam and Boston, Segall, P., Rubin, Allan M., Bradle, Andrew M., and Rice, James R.:Dilatant Strengthening as a Mechanism of Slow Slip Events., Journal of Geophsical Research, 115, B12305, (2010). Segall, P., and Pollard, D. D.: Mechanics of Discontinuous Faults. Journal of Geophsical Research, 85, B8, (1980), McClure, M., and Horne, R. N., (2010), Numerical and Analtical Modeling of the Mechanisms of Induced Seismicit During Fluid Injection, GRC Transactions, 34, Benjemaa, M., Glinsk-Olivier, N., Cruz-Atienza, V.M., Virieu, J., and Piperno, S.:Dnamic non-planer crack rupture b finite volume method, Geophs. J. Int., 171, (2007). Da, Steven M., Dalguer, Luis A., Lapusta, N., and Liu Yi:Comparison of Finite Difference and Boundar Integral Solutions to Three-Dimensional Sponteneous Rupture, Journal of Geophsical Research, 110, B12307, (2005). 10