An Integrated Discrete Fracture Model for Description of Dynamic Behavior in Fractured Reservoirs

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1 PROCEEDINGS, Thirty-Ninth Workshop on Geothermal Reservoir Engineering Stanord University, Stanord, Caliornia, February 24-26, 2014 SGP-TR-202 An Integrated Discrete Fracture Model or Description o Dynamic Behavior in Fractured Reservoirs Jack Norbeck 1, Hai Huang 2, Robert Podgorney 2, and Roland Horne 1 1 Department o Energy Resources Engineering, Stanord University, Stanord, CA, 94305, USA 2 Idaho National Laboratory, Idaho Falls, ID, 83415, USA jnorbeck@stanord.edu Keywords: dynamic racture model, racture mechanics, racture propagation, leako ABSTRACT We present the ramework or a numerical model that is capable o calculating the coupled interaction o mass transer between ractures and surrounding matrix rock, racture deormation, and racture propagation. We call the ramework an integrated discrete racture model (idfm) approach, because it is necessary to combine several numerical modeling strategies that were each developed originally to solve particular types o problems in order to capture the dynamic behavior o ractured systems appropriately. In this work, we extended the coupled luid low, geomechanics, and racture propagation model introduced by McClure (2012) to incorporate mass exchange between ractures and surrounding matrix rock. We adopted a technique called hierarchical racture modeling or the matrix-racture mass transer component o the model to ensure that the racture propagation problem remained tractable in terms o numerical eiciency. In this paper, we irst present the ormulation or the idfm approach. We veriied the accuracy o the model against an analytical solution to a ractured reservoir problem, and subsequently characterized several o the model s numerical properties. We ound that the matrix-racture mass transer model was able to yield reasonably accurate solutions with only a modest increase in the total number o degrees o reedom beyond what is required to solve the racture low and deormation problem. The convergence rate o the matrixracture mass transer model improved or low matrix permeability settings. Finally, we applied the model to a synthetic example o a minirac analysis in a geothermal well in order to demonstrate the model s ability to calculate complex behavior in dynamic racture systems. 1. INTRODUCTION The development o geothermal and unconventional hydrocarbon resources requires reservoir characterization and reservoir management strategies that are tailored to systems in which the low behavior is dominated by the engineering properties o connected racture networks. It is important to recognize that the luid low characteristics o ractured reservoirs will remain dynamic throughout the entire liecycle o a resource. In these settings, it is crucial that a strong understanding o the undamentals o ractured reservoir geomechanics be leveraged in order to optimize stimulation treatments and long-term production strategies. The purpose o this research was to develop the ramework or a numerical model that is able to capture the dynamic behavior o ractured reservoir systems in which both the properties o individual ractures and the connectivity o racture networks are expected to evolve over time. We reer to this modeling ramework as an integrated discrete racture model (idfm) approach because several dierent numerical modeling strategies that are each customized or particular types o problems must be combined eectively in order to capture the broad range o physics that is expected to have irst-order impacts on reservoir behavior. The local state o stress controls the permeability o individual ractures and governs racture propagation behavior, both o which contribute to reservoir-scale low patterns. Poroelastic eects due to luid transer between ractures and surrounding matrix rock as well as stress redistribution resulting rom racture deormation can have signiicant impacts on the local state o stress throughout the reservoir. In the present work, we ocused on a ull-physics coupling o three distinct processes: mass exchange between ractures and surrounding matrix rock, racture deormation and mechanical interaction between ractures, and racture propagation. The model described in this paper merges two approaches that were developed previously into a uniied ramework. The ramework is lexible and could be extended to incorporate additional physics, such as poroelastic and thermoelastic eects that arise rom pore pressure and temperature gradients in the matrix rock, while maintaining the ability to treat racture networks as dynamic systems. The model has direct applications or reservoir stimulation modeling, well test interpretation, and production optimization in ractured reservoir settings. The racture mechanics and racture propagation calculations are perormed using the strategy introduced by McClure (2012). The twodimensional displacement discontinuity method (DDM) and a discrete racture network inite volume method are used to couple luid low in ractures to mechanical deormation o ractures. The approach described in McClure (2012) assumed that matrix permeability was negligible and required that only the racture domain be discretized. In order to incorporate matrix-racture mass exchange into the geomechanical model, we adopted the hierarchical racture modeling (HFM) approach proposed originally by Lee et al. (2000, 2001) and Li and Lee (2008). Moinar et al. (2013) implemented the HFM approach with a relatively simple treatment o geomechanics that related racture permeability to eective stress using a unctional relationship and did not incorporate racture propagation. Hajibeygi et 1

2 al. (2011) recognized that the HFM method is particularly well-suited or problems in which the racture system grows over time, but did not incorporate a geomechanical component in their model. In our model, we calculate racture deormation with a rigorous treatment o racture mechanics, allow new tensile ractures to nucleate and propagate, and apply the HFM method to capture luid exchange behavior between ractures and surrounding rock. This paper is organized as ollows. In Section 2, we present the numerical ormulation or the idfm ramework. The matrix-racture mass transer approach used in the idfm approach was veriied against an analytical solution to a ractured reservoir problem as shown in Section 3. Several o the method s numerical properties were investigated and characterized in Section 4. As an example o the utility o the model, a practical application o the model or interpretation o minirac tests in geothermal wells is presented in Section 5. Finally, several concluding remarks are discussed in Section FORMULATION OF THE INTEGRATED DISCRETE FRACTURE MODEL In this section, the ormulation or the integrated discrete racture model (idfm) is presented. In Section 2.1, we describe the hierarchical racture modeling approach or matrix-racture mass transer and emphasize its utility or racture propagation problems. In Section 2.2, the racture mechanics and racture propagation model is described. In Section 2.3, we illustrate the overall iterative coupling strategy o the algorithm. 2.1 Matrix-Fracture Mass Transer In this work, we made use o the hierarchical racture modeling (HFM) approach introduced originally by Lee et al. (2000). In the HFM approach, it is recognized that small scale ractures aect low locally and can be homogenized (i.e., upscaled) to orm an equivalent matrix permeability. On the other hand, large scale ractures aect low on a reservoir-scale and must be treated explicitly through the use o a discrete racture network approach. The HFM approach is unique rom other discrete racture modeling (DFM) strategies (e.g., see Karimi-Fard and Firoozabadi, 2003; Karimi-Fard et al., 2004) in that two completely independent computational domains are used or the matrix and large-scale racture systems (see Fig. 1). Mass conservation is strictly enorced through a coupling term that is similar in concept to well source terms in conventional reservoir simulation. The mass conservation equations are written separately or the racture and matrix domains (modiied rom Aziz and Settari, 1979) and are discretized separately so that an unstructured, conorming mesh is no longer required. The HFM approach provides an elegant ramework or racture propagation problems, because new racture elements appear simply as source terms in the matrix system o equations. For a single-phase luid, the mass conservation equations can be written, or low in the racture domain, as: ek p q w m E, (1) t where e is racture hydraulic aperture, k is racture permeability, is luid mobility, p is luid pressure in the racture, is luid w density, q is a normalized volumetric source term rom a well, and E is racture void aperture. For low in the matrix domain, the mass conservation equation is: where k m m is matrix permeability, p is luid pressure in the matrix, q k m p m q wm m, (2) t wm is a normalized volumetric source term rom a well, and is m m matrix porosity. In Eqs. 1 and 2, and represent the mass transer terms between the racture and matrix domains. These mass transer terms are necessary to guarantee mass conservation between the racture network and surrounding matrix rock and take the ollowing orm: and p p A, m m (3) p p V. m m (4) In Eqs. 3 and 4, the parameter is called the racture index and is analogous to the Peaceman well index (Peaceman, 1978). The terms are normalized by the racture control volume surace area, A, and the matrix control volume, V, to ensure continuity upon integration over the respective control volumes (Hajibeygi et al., 2011). Similar to the treatment o wells in traditional reservoir simulators, the racture index serves to capture subgrid behavior o the pressure gradient near ractures. In this work, we ollowed the derivation o Li and Lee (2008) to calculate the racture index. The assumptions in the derivation are: a) low in the vicinity o the racture is linear, b) the racture ully penetrates the matrix control volume in the vertical direction, and c) the matrix pressure represents the average pressure over the control volume (Li and Lee, 2008). 2

3 The mass lux rom a racture control volume into a matrix control volume is deined as: m p p m. (5) This term has units o mass per time. Using Darcy s law and assuming that low is linear in the local region near the racture, we can alternatively describe the mass lux term as: where m m A k p n, (6) A is the racture surace area, n is the unit normal vector to the racture ace, and the pressure gradient term is: m p p. p n (7) d Here, d represents the average normal distance rom the racture surace in the matrix control volume. Equating the right hand side expressions in Eqs. 5 and 6 allows or the determination o the racture index: where I is a grid dependent property with units o length that can be calculated as: m Ik, (8) A I. (9) d The quantity d can be calculated numerically or complex racture and matrix control volume geometries (Hajibeygi et al., 2011). It is important to recognize the utility o the racture index or problems that involve complex networks o preexisting ractures. Traditional DFM approaches that use unstructured, conorming grids can quickly become computationally expensive due to the detailed grid reinement that is necessary near racture intersections. Furthermore, problems that involve racture propagation are intractable with traditional DFM because they would require the domain be continually rediscretized as each new racture propagates, resulting in a huge amount o computational overhead. Treating the ractures as source terms through the use o racture indices completely eliminates this issue. The HFM approach becomes more attractive as the level o racture network complexity increases. Figure 1: Illustration o the hierarchical racture modeling conceptual approach or using separate computational domains or the racture and matrix systems. Only the matrix control volumes that contain at least one racture element (shaded in gray) pick up a matrix-racture coupling term. Subgrid scale ractures can be homogenized to orm an equivalent matrix permeability as discussed in Li and Lee (2008) and Haijibeygi et al. (2011). 2.2 Fracture Mechanics and Fracture Propagation In this work, we adopted the approach introduced by McClure (2012) to couple luid low in ractures and racture mechanics. Friction evolution, racture deormation, and the mechanical interaction between ractures were modeled directly. In addition, tensile ractures were allowed to propagate along prespeciied planes according to racture propagation criteria based on stress intensity actors at racture tips. A detailed description o the assumptions and equations used in the geomechanical model are given in McClure (2012) and McClure and Horne (2013). In this section, we will present the major components o the model. The model assumes a two-dimensional, linear elastic ractured medium with homogeneous mechanical properties in the matrix rock. The domain is initially saturated with a single-phase luid and is in mechanical equilibrium. Fractures are able to deorm as luid pressure changes, giving rise to a discontinuous displacement ield. A boundary element method called the displacement discontinuity method (DDM) was used to calculate the opening and sliding displacements along the ractures and the displacements in the matrix rock 3

4 that result rom changes in traction boundary conditions along the racture suraces. Fractures were discretized into discrete elements, and the approach described by Crouch and Starield (1983) was used to arrive at a system o equations that relate opening and shear displacements to changes in normal and shear traction boundary conditions: t = Au, (10) where t is a vector o changes in normal and shear traction boundary conditions along the racture suraces, A is a matrix o DDM interaction coeicients, and u is a vector o unknown displacement discontinuities (i.e., opening and shear displacements). The interaction coeicients were calculated using the higher order DDM approach o Shou and Crouch (1995). As with all boundary element methods, the normal and shear displacements o each element aect every other element, leading to a ully dense system o equations. An algorithm called HMMVP is used to solve the system o equations eiciently (Bradley, 2012). A sequential approach is used to solve or the mechanical displacements and luid pressure o the racture elements. The basic strategy is to irst solve or shear displacement while holding opening displacement and luid pressure constant. Upon convergence, the normal stress and luid pressure equations are solved while holding shear displacement constant. This process is repeated iteratively until the changes in all the primary variables (i.e., shear displacement, opening displacement, and racture luid pressure) all below a prescribed tolerance. This approach is not guaranteed to converge, but has been applied successully by other researchers or coupled low and geomechanics problems and has proven to work well in practice (e.g., see McClure and Horne, 2010; Kim et al., 2011; McClure, 2012). During the iterative process, the state o stress is continually updated and evaluated or all racture elements. The eective normal stress acting on a racture element is:, (11) ' n n p where n is the resolved normal stress on the racture plane, and compressive stresses are taken to be positive. I the eective normal stress is positive, the racture is bearing compression and is considered closed. The opening displacements or closed ractures are zero and can be removed rom the set o primary variables in Eq. 10. The shear stress acting on closed ractures is compared against the rictional resistance to slip based on the Coulomb ailure criterion: ' crit s n S, (12) where crit is the rictional resistance to slip, s is the static coeicient o riction, and S is the cohesion o the racture surace. I the resolved shear stress acting on a racture is less than the rictional resistance to slip, the racture is considered stuck. The shear displacements are zero or stuck ractures can be removed rom the set o primary variables in Eq. 10. The aperture o closed ractures is a unction o the eective normal stress acting on the racture and the amount o dilation due to shearing (Willis-Richards et al., 1996): e0 e Dtan, ' ' n e, re n re (13) wheree 0 is aperture at zero eective stress, e, re is a laboratory derived constant, D is cumulative shear slip, and is shear dilation angle. I the eective normal stress acting on a racture element is negative, the walls o the racture element are in tension and the element is considered open. For open ractures, the opening and shear displacements are determined rom the solution o Eq. 10, and the apertures are calculated as: e e0 e Dtan, (14) where e is the opening displacement. The transmissivity o a racture is calculated according to the cubic law (Snow, 1965): T e 3 ek. (15) 12 It should be noted that in Eq. 1, a distinction was made between the hydraulic aperture, e, that is related to the lux term and the void aperture, E, that is related to the storage term. For ractures that could be considered parallel plates, the hydraulic aperture is equal to the void aperture. For ractures or aults with a damage zone, these parameters could dier signiicantly. Eqs. 13 and 14 are used or void aperture as well, and the constants are allowed to be dierent as necessary. Hydraulic racture propagation is governed by the state o stress near racture tips. In the current version o the model, potentially orming racture planes are prespeciied as a numerical convenience or discretization purposes. Initially, the potentially orming ractures are considered inactive and are not included in any o the systems o equations. During the simulation, the state o stress is continually evaluated at the potentially orming racture elements, and i the eective normal stress acting on an element goes into 4

5 tension that element is activated. Upon activation, the mode one stress intensity actor or racture tip elements is calculated based on the opening displacement discontinuity (Shultz, 1988): G 2 KI 41 a 1/2 e, (16) where G is shear modulus, is Poisson s ratio, and a is the racture tip element hal-length. I the stress intensity actor reaches a critical threshold, then the racture will propagate. 2.3 Overall Coupling Strategy In this work, we have extended the McClure (2012) model to allow or luid leako into the surrounding matrix rock. Mass transer between the racture and matrix domains is computed using the HFM approach described in Section 2.1 through an iterative strategy, and the mechanics equations are solved using a sequential iterative strategy as described in Section 2.2. Here, we illustrate the overall coupling strategy. In the outer loop, matrix pressures, racture pressures, and opening displacements are held constant while the shear stress equations are solved or shear displacements (Eq. 10). In the inner loop, the algorithm irst calculates matrix pressures while holding racture pressures and opening displacements constant (Eq. 1). Upon solving or the matrix pressure distribution, the matrix pressures are held constant while solving or racture pressures and opening displacements (Eqs. 2 and 10). The matrix pressure residual equations are then evaluated or convergence. This inner loop is repeated until matrix pressures, racture pressures, and opening displacements have converged. At this point, the shear stress residual equations are evaluated or convergence, and the outer loop is repeated as necessary. A low chart o the algorithm is shown in Fig. 2. Figure 2: Overall coupling strategy or idfm ramework. 3. VERIFICATION OF NUMERICAL MODEL In this section, we show the veriication that the HFM matrix-racture mass transer term (see Eq. 5) is capable o capturing leako behavior o ractured systems accurately. The numerical model was compared against the analytical solution presented by Ghassemi et al. (2008) or a reservoir that contains one production well and one injection well connected by a single vertical racture. The problem coniguration is illustrated in Fig. 3. Fluid is injected at a constant volumetric rate and the production well is maintained at a constant pressure equal to the initial reservoir pressure. Poroelastic eects were neglected so that the racture aperture remains constant. The luid in the racture is incompressible. In order to obtain an analytical expression or pressure distribution in the racture, the luid leako rate was assumed to be constant along the racture and also in time. The resulting racture pressure distribution was then used as a boundary condition to solve the slightly compressible diusivity equation or the transient pressure distribution in the surrounding matrix rock. Fluid leako was assumed to be one-dimensional low in the direction perpendicular to the racture. Fluid pressure in the matrix is then given by the ollowing expression (Ghassemi et al., 2008): m y p x, y, t x LC 1x LC2 erc p0, 2 t 5 (17)

6 where L is the length o the racture, is hydraulic diusivity, and p 0 is the initial reservoir pressure. The two constants are: and 12q L 1 3 where q L is the constant leako rate and q 0 is the constant injection rate. C, (18) e 12q0 C2, (19) 3 e All relevant model parameters are listed in Table 1. Several levels o matrix discretization reinement were tested. In Fig. 4, the analytical solution is compared to the numerical solution or the times o 100 days and 1000 days ater initiation o injection and production. It is clear that the numerical solution with the highest level o grid reinement was able to capture the leako behavior very well at both early times and late times. The late time solution appears to have some minor dierences near the boundaries o the domain. This dierence is most likely due to the act that the analytical solution assumes one-dimensional low while the numerical solution is calculating the more realistic case o two-dimensional low. Boundary eects may also be introducing additional error because the analytical solution assumes an ininite domain. Visual inspection o Figs. 4e and 4 might seem to indicate that the pressure is underestimated signiicantly or the lowest level o grid reinement, especially in the region near the injection well. O course, the reservoir pressures yielded rom inite volume schemes represent an average pressure over the control volume and so the accuracy must be quantiied with respect to volume averages. To quantiy the error introduced by the numerical scheme, the root mean square error relative to the analytical solution was calculated across the domain and normalized by the pressure drop between the injection and production well. The error is presented in Fig. 5 or ive dierent levels o grid reinement at dierent solution times. The error or the lowest level o grid reinement ranged between 1% and 6% over the duration o the simulation. The error or the highest level o grid reinement ranged between roughly 1% and 2%. The goal o this numerical experiment was to veriy that the HFM matrix-racture mass transer approach is capable o accurately calculating leako behavior by treating ractures essentially like wells, in contrast to more conventional DFM approaches. These results indicate that the assumptions involved in deriving the HFM racture index are well ounded, at least or this relatively simple model o a single vertical racture. This experiment was perormed rom the point o view o the matrix rock, because the racture pressure distribution was held constant throughout time. This was done purely or the sake o having the ability to compare against an analytical solution. In Section 4, we will show more realistic simulations in which both racture pressure and matrix pressure had transient eects. Nonetheless, we have demonstrated that the conceptual approach o using two separate computational domains or the racture and matrix system can provide reasonably accurate solutions without the need o an unstructured, conorming grid. Figure 3: Schematic or the one-dimensional leako model (modiied rom Nygren and Ghassemi, 2006). Table 1: Model parameters or model veriication study. 6

7 (a) (b) (c) (d) (e) () Figure 4: Comparison o the analytical solution to a one-dimensional leako model or a vertical racture to the HFM numerical solutions at very high reinement (111 x 111 grid blocks) and very low reinement (5 x 5 grid blocks). The let side (a, c, e) shows the solutions ater 100 days o injection and production. The right side (b, d, ) shows the solution ater 1000 days o injection and production. Figure 5: Normalized mean square error or dierent levels o grid reinement or the matrix domain. The dierent symbols represent the error at various simulation times. Errors are calculated with respect to control volume averages o the analytical solution and normalized by the maximum pressure drop over the system. 7

8 4. CHARACTERIZATION OF NUMERICAL PROPERTIES In hydraulic racture applications, the leako eect has two main physical implications. The irst is that luid pressure inside the racture is reduced, which acts to limit the growth and aperture o the hydraulic racture. The second is that the surrounding matrix rock expands, acting to reduce the hydraulic racture aperture. The latter is termed the poroelastic eect, and is ignored in this paper. In this section, we show the ability o the HFM matrix-racture mass transer approach to calculate the pressure distribution within the racture accurately, because the racture pressure will be the key component governing the mechanical behavior o racture deormation. The McClure (2012) model assumed that matrix permeability was negligible so that no luid was allowed to leak o rom the ractures. The motivation or this assumption was to reduce numerical complexity in order to ocus on a rigorous treatment o racture mechanics and racture propagation. One o the main goals o this research was to extend the capabilities o that model to incorporate leako eects without increasing the computational burden signiicantly. Thereore, it was important to understand the numerical properties o the matrix-racture mass transer model and leverage that understanding to achieve a balance between physical accuracy and computational eiciency. To test some numerical properties o the method, we used a model that is similar to the one described in Section 3. A single vertical racture connects two wells. Fluid is injected at a constant mass low rate and the production well is held at constant pressure. The racture aperture is constant. Fluid in the racture is now taken to be slightly compressible. The leako rate is no longer assumed to be constant in space or time, but is now ully coupled to both racture and matrix pressures and is determined iteratively using the HFM racture index approach. In addition to testing the sensitivity to matrix discretization reinement, we also tested the sensitivity to contrast between racture permeability and matrix permeability. One o the main numerical advantages o the HFM approach is that two separate systems o equations are solved or the racture and matrix domains, whereas traditional DFM strategies solve a single system o equations. The coeicient matrices or DFM models thereore involve terms related to both racture and matrix transmissibilities that can dier by many orders o magnitude, especially or settings with very tight matrix rock. These coeicient matrices can become ill-conditioned, which can have negative consequences on convergence rates. Fluid exchange between the racture and matrix system is expected to decrease as matrix permeability becomes tighter, and so, conversely, the convergence rate o the HFM iterative strategy is expected to improve in these settings. The relevant model parameters or the base case simulations are given in Table 2. The matrix permeability was then modiied as necessary to observe the eect o contrast in racture permeability to matrix permeability. For this model, an analytical solution is not easily obtainable, and so errors are calculated with reerence to a simulation with high level o grid reinement ( grid blocks) or each permeability contrast. The racture pressure distribution along the length o the racture is illustrated in Fig. 6 or the base case permeability contrast. For reerence, the pressure distribution or the case o no leako is also shown (black line). It is clear that the solutions are convergent upon grid reinement. A more interesting observation is that the solutions tend towards the zero leako case as the level o reinement becomes coarser. This result indicates that the matrix-racture mass transer model tends to underestimates the amount o leako or coarser matrix grids. Thereore, the solution can be considered to yield a conservative estimate o racture pressure with respect to the original McClure (2012) model assumption o negligible leako. The implication is that solutions with improved physical accuracy can be obtained by paying only a small price in terms o additional computational burden. The error in racture pressure due to matrix discretization reinement is presented in Fig. 7 or an early time and late time solution. In this problem, the maximum error occurs in the racture control volume that contains the injection well. The reported errors are deined as the dierence in racture pressure rom the reerence case normalized by the pressure drop between the injection and production well or the impermeable matrix case. As expected, the error is observed to decrease as the level o grid reinement increases. In addition, the solution generally becomes more accurate as matrix permeability decreases, especially or lower levels o discretization reinement. For the relatively high matrix permeability cases, the error never exceeded 6.5%. The error tended to drop o rapidly as the permeability contrast increased, which can be attributed mainly to the act that leako becomes negligible or low matrix permeability. In Fig. 7, the positive errors indicate that racture pressure was overestimated or all levels o grid reinement and permeability contrast, and that the magnitude o the overestimate was larger or coarser grids. We reiterate that this result indicates that the matrix-racture mass transer model tends to underestimate the amount o luid leako, yielding a conservative estimate o racture pressure. From a modeling perspective, this result is encouraging because the modeler is able to saely ramp up the level o grid reinement as necessary to achieve the desired level o accuracy. The average number o matrix-racture mass transer coupling iterations over the duration o the simulation is shown in Fig. 8. It is observed that the coupling process typically converged ater 0 2 iterations. I the amount o mass transer over a timestep is insigniicant, then the algorithm does not require any iteration. In timesteps that did require iteration, it was observed that convergence usually occurred ater one or two iterations. Occasionally, convergence was not observed ater ive coupling iterations at which point the timestep was discarded or a smaller timestep length. These wasted iterations are not relected in Fig. 8. As expected, the number o coupling iterations decreased or the cases with relatively low matrix permeability. It is also interesting to note that the coarse grids tended to require ewer coupling iterations. 8

9 Table 2: Model parameters or study to evaluate numerical properties o the matrix-racture mass transer model. (a) (b) Figure 6: Fracture pressure distribution ater a) 7 days and b) 1000 days or the base case permeability contrast. As the level o grid reinement increases, the solutions appear convergent. More interestingly, the solution is tending towards the zero leako case (black line) or coarser grids. This result is encouraging, because it shows that the amount o leako is underestimated or coarser grids. This can be considered a conservative estimate with respect to the McClure (2012) assumption o zero leako, and implies that improved solutions can be obtained with only a relatively small number o additional degrees o reedom. (a) (b) Figure 7: Error in the racture pressure o the control volume that contains the injection well with respect to the high grid reinement case (101 x 101 grid blocks) or dierent racture-matrix permeability contrasts. Error is normalized by the pressure drop between the injection and production well or the zero leako model. a) error ater 7 days o and b) error ater 1000 days o injection and production. 9

10 Figure 8: Average number o matrix-racture mass transer coupling iterations over 1000 days o simulation time. Reducing matrix permeability improves the convergence rate o the algorithm. 5. PRACTICAL APPLICATION OF MODEL TO MINIFRAC ANALYSIS In this work, we extended a tool used to model hydraulic stimulation to incorporate luid leako eects. Here, we present an application o the newly developed model to a practical geothermal reservoir engineering problem. Minirac tests or extended leako tests are commonly perormed in the geothermal and oil and gas industries to measure reservoir parameters that are required to design hydraulic stimulation treatments. In a minirac test, a relatively small volume o luid is injected, typically at constant rate, or a short duration o several minutes in order to propagate a small hydraulic racture. In some cases, the well is then shut in or a period o time and luid is allowed to bleed o into the matrix rock. The pressure response at the well can be used to iner inormation about hydraulic racture initiation, propagation, and closure (Nolte, 1979; Zoback, 2007). One o the most important measurements gained rom these tests is an estimate o the in-situ minimum principal stress. There are ive distinct measurements that can be used to iner minimum principal stress: leako pressure (LOP), ormation breakdown pressure (FBP), racture propagation pressure (FPP), initial shut-in pressure (ISIP), and racture closure pressure (FCP). Whether the target reservoir stimulation mechanism is hydraulic racturing or shear stimulation, accurate knowledge o the minimum principal stress is key to optimizing the treatment design because the magnitude o this stress controls the luid pressure necessary to drive hydraulic racture propagation. In oil and gas settings, minirac tests are typically perormed under relatively controlled conditions. The wells are usually cased, and a target location (ree o preexisting ractures or aults) is identiied, isolated with packers, and perorated. In geothermal settings, the tests may be perormed in less ideal conditions. For example, wells are usually open-hole and intersect several conductive natural ractures. Regardless, the results rom geothermal ield tests are oten analyzed using principals that were developed based on the more ideal assumptions. In this section, we illustrate the extent to which natural ractures interere with the interpretation o minirac ield tests. The problem setup was motivated by a recent ield test at a geothermal injection well targeted or stimulation. It was clear rom image log and temperature log data that this well was accepting luid at a single natural racture. Borehole breakouts and drilling-induced tensile ractures identiied in the image log allowed or the determination o the orientation o the horizontal principal stresses, and the conductive natural racture was oriented roughly 45 degrees rom the principal stress directions. One o the unknowns at this site was the length that the natural racture extended away rom the wellbore. In this synthetic example, we modeled a minirac test perormed in a strike-slip environment. In the model, luid is injected at a rate o 2.0 kg/s or a period o our minutes, ater which the well is shut-in or 48 hours. The matrix rock permeability is 0.1 md. As a base case, we simulated a conventional minirac test or a well that did not intersect a natural racture. Six additional cases were designed to simulate minrac tests or wells that intersected a single natural racture oriented at 45 degrees rom the minimum horizontal stress direction with lengths ranging rom 1.4 m to 70.7 m. We hypothesized that the stress perturbation due to slip on the natural racture would inluence the estimate o minimum principal stress inerred rom the wellbore pressure measurements (see Fig. 9). To test or this eect, two simulations were perormed or each racture length. In one simulation, the racture was given an artiicially high coeicient o riction so that slip was prevented rom occurring. In the other simulation, a normal coeicient o riction was assigned so that slip could occur. The model parameters are listed in Table 3. The wellbore bottomhole pressure observations or the ull duration o the test are shown in Fig. 10. The solid lines represent the simulations where no shear slip was allowed and the dashed lines represent the simulations where slip occurred. A magniied illustration o the pumping phase is shown in Fig. 11. In all simulations, hydraulic ractures did initiate and propagate rom the tips o the natural racture with the exception o the two cases with the longest preexisting racture (70.7 m). Upon shut-in, luid leaked o into the surrounding ormation and the wellbore pressure eventually decayed to the initial reservoir pressure. Interpretations o the LOP, FBP, FPP, ISIP, and FCP or all cases are summarized in Fig. 12. As a reerence, the case where no preexisting racture intersected the wellbore behaved as expected. The LOP was exactly equal to the minimum principal stress ( 3 ), a clear FBP was observed, and the FPP reached a steady value slightly larger than 3. The cases with preexisting natural ractures that were not allowed to slip (solid lines) each displayed behavior that is qualitatively similar to the 10

11 reerence case. The LOP, FBP, and FPP are higher than the reerence case due to the injectivity o the well, rictional losses along the length o the racture, storage eects related to the length o the racture, and storage eects related to changes in racture aperture resulting rom changes in eective stress. The cases in which the preexisting ractures were allowed to slip (dashed lines) display markedly dierent behavior. The LOP consistently occurred signiicantly below 3 (see Fig. 12a). This was the expected result, because shear slip induces local antisymmetric zones o tension and compression near the racture tip (see Fig. 9). Hydraulic ractures are then able to propagate more easily within the zones o induced tension. Thereore, using the LOP as a proxy or 3 resulted in an underestimate o the minimum principal stress o up to nearly 8% in this example. For the ractures longer than 5.7 m, a clear FBP was not observed. Our interpretation is that the unsteady propagation behavior was masked by storage eects. Once the racture begins to propagate stably, it is typically assumed that the FPP will reach a steady value slightly above 3 because it does not take much additional luid pressure to extend a racture once it has reached a length greater than about 1.0 m. In these examples, we see a steady increase in the FPP as racture length increases (see Fig. 12c). This is most likely due to rictional eects. Additionally, an interesting behavior is observed or the smaller length ractures that display a clear FBP. Ater the FBP is reached, the pressure drops quickly, but then begins to rise again, ultimately reaching a FPP value that is higher than the FBP. Our interpretation is that the racture initiates and propagates relatively easily in the zone o induced tension near the racture tip, but once the racture has extended beyond the perturbed stress zone it takes a relatively higher pressure to continue to extend the racture. This interpretation is also guided by the observation that the FPP values or the ractures that were allowed to slip appear to be converging towards the FPP values or the counterparts that were not allowed to slip. Upon shut-in, the pressure decline behavior can also provide inormation about the in-situ stress state. In act, the ISIP and FCP can be more accurate measures o 3 because they should not include the eects o riction that are present when the well is lowing (Zoback, 2007). Because we have extended our model to incorporate leako eects, we are able to interpret the allo data. The results o this numerical experiment indicate that the measurements o ISIP and FCP are not greatly aected by the shear slip-induced stress perturbation. O the ive dierent minirac pressure measurements, ISIP and FCP are able to predict 3 most accurately over the range o cases studied (see Figs. 12d and 12e). Note that clear ISIP and FCP could not be observed or the cases with the 70.7 m natural racture because a hydraulic racture did not propagate a signiicant distance in either o these two simulations. Figure 9: Schematic o conceptual model or minirac analysis. The injection well (black dot) intersects a natural racture that is well-oriented or shear slip in the current stress ield. As the racture slips, the near-tip stress ield is perturbed. The cool colors indicate areas where tensile stresses are induced. The red dashed lines indicate potentially orming hydraulic ractures. Table 3: Model parameters or minirac numerical experiment. 11

12 Figure 10: Wellbore bottomhole pressure measurements over the entire duration o the minirac test. The dierent colors represent simulations with dierent preexisting natural racture length. Solid lines are or simulations where shear slip was not allowed to occur. Dashed lines are or simulations where shear slip did occur. Figure 11: Wellbore bottomhole pressure measurements during the pumping phase o the minirac test. 12

13 (a) (b) (c) (d) (e) Figure 12: Summary o the results o the estimate o minimum principal stress inerred rom the minirac test. Minimum principal stress was estimated using ive dierent parameters available rom the minirac pressure data: a) leako pressure (LOP), b) ormation breakdown pressure (FPP), c) racture propagation pressure (FPP), d) initial shut-in pressure (ISIP), and e) racture closure pressure (FCP). The LOP was observed to signiicantly underestimate the minimum principal stress when shear slip o preexisting ractures occurred. In contrast, both ISIP and FCP were relatively unaected by the shear slip eect. 6. CONCLUDING REMARKS In this paper, we have presented the ramework or an integrated discrete racture modeling (idfm) approach. The concept o idfm is to combine various numerical modeling strategies in a uniied ramework in order to capture the multitude o coupled physical processes that are expected to have irst-order eects on the luid low behavior in ractured reservoirs. These physical processes include: mass exchange between ractures and surrounding matrix rock, racture deormation and riction evolution, mechanical interaction between ractures, poroelasticity, and racture propagation. In the current version o the model, we have included each o these eects with the exception o poroelasticity in the matrix rock. We have extended the coupled luid low and geomechanical model introduced originally by McClure (2012) in order to incorporate luid exchange between racture and matrix rock domains though the use o a hierarchical racture modeling (HFM) technique. We demonstrated that the HFM conceptual model is particularly well-suited or problems that involve racture propagation due to the unique implementation o matrix-racture mass transer coupling terms that involve a parameter called the racture index. We evaluated 13

14 several numerical properties o the matrix-racture mass transer model, and ound that reasonably accurate solutions can be obtained with relatively ew additional degrees o reedom beyond what is needed to capture the racture low and deormation problem. In addition, we ound that the numerical eiciency o the method improved or increasing contrast in racture to matrix permeability (i.e., tight ormations), an advantage over traditional DFM strategies. A practical application to the analysis o minirac tests in geothermal wells showcased the utility o the model. A numerical experiment was perormed in order to investigate the inluence o shear slip o preexisting natural ractures that intersect the well on the interpretation o the minimum principal stress inerred rom minirac data. It was observed that using the leako pressure as a proxy or the minimum principal stress could result in signiicant underestimates o the stress magnitude. The result is a direct eect o the stress perturbation near the racture tip due to racture deormation. However, initial shut-in pressure and racture closure pressure were not aected signiicantly by shear slip, and were ound to provide reliable estimates o the minimum principal stress. These results are not meant to be used to make deinitive arguments about well test interpretation strategies, but rather to indicate that incorporating a rigorous treatment o the mechanical deormation o ractures and associated stress perturbations into a numerical model is necessary to accurately model behavior in ractured reservoirs. In order to optimize the exploitation o geothermal and unconventional hydrocarbon resources, engineers must leverage an understanding o the dynamic behavior o ractured reservoir systems. Fluid low behavior in these systems can be highly nonlinear, largely due to complex geomechanical processes that occur during all phases o a reservoir lietime, including both stimulation and production. The model presented in this work can be used as an eective engineering tool in both research and practical applications in order to enhance understanding o dynamic behavior in ractured reservoirs. NOTATION a racture element hal-length A surace area o racture control volume A racture surace area A matrix o DDM interaction coeicients C irst constant in analytical pressure solution C 1 2 d second constant in analytical pressure solution average normal distance rom racture surace D shear displacement discontinuity e racture hydraulic aperture e racture aperture at zero eective stress 0 e E G I k m k K I L n p p q q L 0 m normal displacement discontinuity racture void aperture shear modulus connectivity index racture permeability matrix permeability mode one stress intensity actor racture length unit normal vector to racture surace racture pressure matrix pressure luid leako rate luid injection rate w q normalized well-racture volumetric source wm q normalized well-matrix volumetric source S racture cohesion t vector o shear and normal tractions T racture transmissivity u vector o shear and normal displacements V volume o matrix control volume Greek Symbols hydraulic diusivity porosity racture index luid mobility luid viscosity s static coeicient o riction Poisson s ratio shear dilation angle luid density reerence eective stress e, re n normal stress ' n eective normal stress resistance to shear slip crit m racture-matrix mass lux m normalized racture-matrix mass lux m normalized matrix-racture mass lux ACKNOWLEDGEMENTS This work was supported by the Stanord Center or Induced and Triggered Seismicity. The authors would like to thank Dr. Mark McClure or the use o the code, CFRAC, and or the many helpul conversations. REFERENCES Aziz, K. and Settari, A Petroleum Reservoir Simulation. Khalid Aziz and Antonin Settari. Bradley, A.M H-matrix and Block Error Tolerances. arxiv: v2, source available at paper available at Crouch, S.L., and Starield, A.M Boundary Element Methods in Solid Mechanics. London: Allen and Unwin. 14

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