SPE-173200-MS

Algebraic Multiscale Solver for Flow in Heterogeneous Fractured PorousMedia

M. Tene, Delft University of Technology; M. S. Al Kobaisi, The Petroleum Institute; H. Hajibeygi, Delft Universityof Technology

Copyright 2015, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Reservoir Simulation Symposium held in Houston, Texas, USA, 2325 February 2015.

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Abstract

A general algebraic multiscale framework is presented for fractured porous media, which enables thetreatment of fractures with multiple length scales and wide ranges of conductivity contrasts. To this end,fully integrated local basis functions for both matrix and fractures are constructed. These basis functionsare employed to construct the multiscale coarse system for both matrix and fractures, and then interpolatethe coarse solutions back to the fine-scale reference system. Combined with a second stage fine-scalesolver, here, ILU(0), our development leads to an iterative multiscale strategy for heterogeneous fracturedmedia, allowing for error reduction to any arbitrary level, while honoring mass conservation after anymultiscale finite volume stage. In order to maintain generality, it is shown that when each fracture networkis modeled using a single coarse grid cell, our formulation automatically reduces to that proposed byHajibeygi et al. (2011). However, in order to facilitate the treatment of general ranges of conductivitycontrasts and deviation term norms, we introduce the ability to refine the coarse grid to more than one cellper network. Our experiments show that this flexibility results in a significant improvement on conver-gence properties. This added degree of flexibility is made possible through an algebraic formulation,which leads to a multi-stage multiscale conservative linear solver for multiphase flow in fractured media.

IntroductionMathematical formulations describing flow in hydrocarbon reservoirs entail highly heterogeneous aniso-tropic static and dynamic coefficients, which span several orders of magnitude over their entire lengthscales (See, e.g., Aziz and Settari (2002)). Accurate treatment of these parameters using state-of-the-artreservoir simulators is not possible due to their grid resolution (i.e., computational capability) limits. Inaddition, many of the worlds hydrocarbon reserves are fractured, either naturally or as a result of artificialfracturing processes. And, while the majority of the hydrocarbons reside in the matrix blocks, the fracturesact as lower dimensional highly conductive channels, with a significant impact on flow and transport offluids. As such, the presence of fractures in heterogeneous media is considered as an additional challengeto the simulators, especially due to their scale and physical characteristics, which are significantlycontrasting with those of the matrix. As a result, efficient and accurate simulation of multiphase flow inheterogeneous fractured porous media is a challenging task, yet crucial for several important applications,

including those of Enhanced and Improved Oil Recovery (EOR and IOR, respectively) and CarbonCapture and Storage (CCS). Also note that fractures are typically of a wide range of length scales, andtheir structure exhibits complex patterns: they can become tangential or cross each other at differentpoints, forming networks. And, their shapes can change dramatically between different reservoir layers.Therefore, the explicit treatment of all fractures present in the entire large-scale reservoir (order of km)is computationally intractable, as for the computational and grid generator limits.

In order to resolve these limitations, Lee et al. (2000) proposed a hierarchical modeling approach. Intheir approach fractures were classified based on their length scale. The small and medium fractures(typically, the size of one matrix grid cell) are homogenized within the matrix, altering its effectivepermeability. The large-scale fractures, however, are explicitly treated in the discretized system. Thisapproach allows for the separation of matrix and fracture fluxes, which in turn, enables the independentspecification of computational grids for the two domains.

Even though the homogenization of small-scale fracture leads to a significant reduction in the overallDegrees Of Freedom (DOF), the flow problem remains still challenging, even for the most efficientnumerical solvers. This makes multiscale methods (See, e.g., Cortinovis and Jenny (2014); Efendiev andHou (2009); Hajibeygi and Tchelepi (2014); Hajibeygi et al. (2012); Hou and Wu (1997); Hou et al.(1999); Jenny et al. (2003); Kippe et al. (2008); Lee et al. (2008); Zhou and Tchelepi (2012); Zhou et al.(2011) and referenced therein) quite attractive options for next-generation reservoir simulation. Theapplication of multiscale methods to fractured porous media was first proposed by Hajibeygi et al. (2011).They developed a sequential strategy on the basis of expressing the pressure inside fracture networks interms of an average value and fine-scale deviation terms. As such, the matrix grid block pressure couldbe coupled to the average pressure for each fracture network and solved for implicitly using the MultiscaleFinite Volume (MSFV) method. Then, the deviation terms were treated in a separate second stage.Therefore, as shown in their numerical experiments, their method was efficient for media with highlyconductive fracture networks which span short spatial length scales (relative to that of the domain).However, for test cases where the average network pressure is not a good zero-order guess for the wholefracture pressure distribution, the convergence rate of their method would degrade.

In this work, we propose a general and novel multiscale method for hierarchical fracture modeling inlarge-scales heterogeneous hydrocarbon reservoirs. Given a fine-scale discrete system, which describesflow in the matrix (prescribed with effective parameters, after the homogenization of small scale fractures)and fractures (with lengths larger than a fine grid cell size), our method constructs an appropriatecoarse-scale solution, on the basis of coarsening strategies (i.e., multiscale superposing expressions) forboth matrix and fracture domains.

In order to recover the fine-scale solution, a new prolongation operator is assembled, which containsmodified multiscale basis functions to capture heterogeneities, as well as fully-integrated fracture and wellbasis functions, which account for external mass transfers, in a conservative fashion. The basis functionsare solved locally over dual-coarse grid blocks, similar to the original MSFE of, e.g., Hou and Wu (1997)and MSFV of Jenny et al. (2003), having the fracture and well coarse nodes set to 0. Well functions arecomputed by setting the well pressure to 1, while all other coarse nodes to 0, over perforated dual-coarseblocks (similar as Jenny and Lunati (2009), but here algebraically formulated and implemented for 3Dproblems). The novel development of this work, i.e., general fracture functions (interpolators insidefracture domains), on the other hand, are first localized within their respective fracture networks,neglecting matrix-fracture couplings, and setting the pressure in the corresponding fracture coarse nodeto 1. These solutions are used as Dirichlet condition to calculate fracture basis functions inside perforatedmatrix dual-coarse grid blocks.

Our novel multiscale method for fractured media reduces in the limit to that of Hajibeygi et al. (2011)if only one fracture coarse node is considered. In this case, the basis function for fracture domain reducesto constant 1 along the fracture network, thus it employs only one additional DOF per fracture network

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implicitly coupled within the coarse-scale matrix equation. More importantly, when more fracture coarsenodes are prescribed, the fracture pressure is shown to be captured more accurately, also leading tosignificant improvements for the solvers convergence rate. Our numerical results show that only a fewDOF per network are necessary to obtain a reasonable tradeoff between convergence rate and computa-tional complexity.

Following the traditional algebraic multiscale formulation, our method is general, in the sense that itcan be easily extended to account for complex physics, such as gravity, as in Wang et al. (2014), orcompressibility (Tene et al. 2014). Furthermore, it allows for the direct implementation as a back-boxprocedure in existing reservoir simulators.

The paper is structured as follows. First we study the fine-scale discrete system describing flow infractured media using the hierarchical fracture modeling approach. Then, in Section 3, our generalmultiscale method for fractured media is presented. Section 4 consists of numerical results on both 2D and3D test cases, and, finally, conclusions and plans for future research are drawn in Section 5.

Governing Equations and Discretization on Fine ScaleThe single-phase flow equations for fractured media can be written as

(1)

(2)

where superscripts m and f denote quantities and operators for matrix and fracture media, respectively.Therefore, km stands for effective matrix permeability (matrix rock and homogenized small-scale frac-tures), and kf for fractures permeability. Also, and are the fluid density (single-phase flow) and rockporosity, respectively. These equations are to be solved for matrix and fracture pressures pm and pf, onmatrix m and fracture f domains. Note that fractures are in a lower dimensional space, i.e. f c Rn-1,than matrix (reservoir rock) n c Rn. Also, qmw and qfw are the matrix and fracture external source termsdue to injection/production wells, respectively, described by Peaceman (1977) as, e.g., for matrix

(3)

where productivity index is PI. Similarly, the fracture-matrix coupling term, i.e., qfm is formulated as

(4)

which is based on a linear pressure distribution assumption between the two overlapping fracture andmatrix grid cells. The Qm and Qf terms describe other external source terms for matrix and fractures (e.g.,gravity terms). The CI is the connectivity index, defined on a discrete system (grid cells) and bearssimilarity to PI, which is the productivity index in the Peaceman well model. As described by Hajibeygiet al. (2011), the CIij is defined as

(5)

where Aij is the area fraction of fracture element i overlapping with the matrix element j, and is theaverage distance of the two elements. The k* is the effective permeability (weighted harmonic average)between the overlapping fracture and matrix nodes. For highly conductive fractures, it becomes nearly km,while for impermeable fractures it becomes close to kf.

The main advantage of this type of formulation is that the matrix and fracture grids are totallyindependent. Their formulations also can be described easily by considering appropriate physics for eachmedium, with high applicability to highly fractured reservoirs or dynamically generated (and closed)

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models, e.g., geothermal and tight-gas/oil formations modeling and simulations, (i.e., when geo-mechanics effects are considered).

Equations (1) and (2) are discretized over fine-grid cells based on the resolution of the heterogeneouspermeability and small-scale fractures present in the domain. When nonlinearities are present (e.g.,compressible flows, where the density and porosity terms are pressure-dependent), the discrete nonlinearequations need to be linearized, leading to,

(6)

and then iteratively solved until the converged solution is achieved. Note that system in (6) shows animplicit treatment of the coupling between fracture and matrix through the Afm entries, and that the matrixsystem A can be non-symmetric, due to the compressibility effects, as described by Tene et al. (2014) andHajibeygi and Jenny (2009).

Developing an efficient solution strategy for the linear system (6) is quite challenging for severalreasons. On the one hand, the size of this system can exceed 107 for realistic test cases. On the other hand,the value of the condition numbers for the system matrix is worsened by high contrasts between reservoirproperties (permeability is highly heterogeneous over large scales).

Clearly a classical upscaling method cannot be employed here due to the highly resolved fractures,which could have important effect on the mass transport. Multiscale methods have the unique advantageof being conservative (MSFV), while honoring fine-scale data. They operate by solving coarse-scalesystems, the solution of which will be interpolated (prolonged) back to the original resolution. This wouldallow one to calculate the error (residual) in fulfilment of the fine-scale system (6) and employ a fewiterations adaptively, when necessary (see, e.g., Hajibeygi et al. (2012) and Hajibeygi and Jenny (2011)).

Algebraic Multiscale Solver for Fractured Heterogeneous Porous Mediawith Complex WellsIn this paragraph, we describe how the system (6) is solved using a multiscale finite volume (MSFV),multiscale finite element (MSFE), or a hybrid multiscale finite element and volume (MSMIX) methods.

Given a reservoir with Nf fracture networks and Nw wells, the Algebraic Multiscale Solver for flowswith Fractures and Wells (Frac-Well-AMS) approximates the matrix and fracture pressure fields as

(7)

and

(8)

respectively, where Ncm and Ncfi are the number of coarse cells for matrix and fracture network i,respectively (Figures 1(b) and 1(c)). Here, basis functions associated with matrix coarse cells ,matrix-fracture effects and matrix-well effects are employed in order to capture the effects ofall the important factors in the matrix pressure field pm. Important, and the novel aspect of this work, isthat the pressure field inside fracture field, i.e., pf, is also described with a multiscale formulation (8). Thismeans that the fracture grid cells are also decomposed into coarse blocks, similar to the matrix, i.e.,fractures are also mapped to the coarse scale and back to the original resolution. More specifically, eachfracture network f is decomposed into Ncfi coarse grid blocks (Figure 1(c)), for which a basis function iscalculated. Also, one can also use the same formulation for wells, which are typically assigned a singleDOF, due to their homogeneous properties.

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Algebraically formulating, one can re-state Eq. 7 as

(9)

where

(10)

is the matrix prolongation operator, having three sub-columns of matrix-matrix, matrix-fracture, andmatrix-wells. Also,

(11)

Similarly, for fracture medium, according to Eq. (8), one can define:

(12)

with

Figure 1Multiscale grids for matrix and fractured media.

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(13)

and

(14)

Starting from the given fine-grid cells (Figure 1(a)), multiscale methods impose a coarse (Figures 1(b))and dual-coarse grid (Figures 1(d)) (See, e.g., Jenny et al. (2003, 2006)). The former is anon-overlappingdecomposition (with the total number of Ncm), each having a coarse node inside (typically, but not strictly,the center cell). By connecting the coarse nodes, the dual-coarse grid cells are obtained (with the totalnumber of Ndm), an overlapping partition of the domain with the overlap width of one sets of fine cells(boundary cells), which serves as support for the basis functions.

The same strategy can be now proposed for fractures, where for their given fine grid cells, anon-overlapping coarse partitioning is considered with Ncfi coarse grids for fracture network fi.

Basis functions for matrix domain are computed over matrix dual-coarse grid cells as

(15)

(16)

(17)

For all above equations, zero Drichlet condition is set to at all coarse nodes except the one for whichits basis function is to be calculated. Moreover, all of these basis functions are subject to reduced-dimensional boundary conditions. Note that the basis functions are calculated locally, hence will beupdated adaptively if the fine-scale properties change beyond a threshold value as described, e.g., byHajibeygi and Jenny (2011).

For fracture medium, similarly, basis functions are obtained as

(18)

subject to Dirichlet condition at fracture coarse nodes xk. It is important to note that, forthe basis functions of fractured medium, the matrix-fracture and well-fracture coupling is not considered,in order to ensure localization. Though the coupling terms can be easily considered, and will positivelyimpact the solution quality, their consideration would lead to additional computational complexity. Thisformulation for mapping the fracture pressure from fine to coarse scale results in an efficient and accurate,yet flexible, multiscale strategy. The detailed coupling between the two media will be obtained as ahigh-frequency error in a separate second stage smoothing procedure.

Figure 2 depicts an example of a basis and a fracture function in a dual-coarse block. Notice that thesum of the basis and fracture basis functions form a partition of unity.

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Once the prolongation operator is computed, the restriction operator (i.e., map from coarse to finescale) needs to be defined. As for having three distinct features in the domain, i.e., matrix, fracture, andwells, the restriction operator can be defined in much more general form than those previously publishedworks. We allow for having a full Finite-Volume restriction, as in the classic MSFV, a full Finite Elementmethod, R PT, or a Mixed Finite Element-Volume, where the matrix is restricted according to FiniteElements, while the fractures and wells use Finite-Volumes.

Numerical ResultsIn order to assess the performance of our newly developed multiscale framework for fractured porousmedia, we investigate two test cases. The first is a 2D reservoir with one central fracture network,composed of 60 interconnected segments, and two horizontal wells. In the second scenario, as a proof ofconcept, we extrude the 2D fracture network along the Z axis, in order to present the behavior of a 3D5-spot reservoir in a synthetic manner, i.e., assuming only 1D flow inside fracture plates.

2D test caseFor the 2D case, we consider the heterogeneous permeability field depicted in Figure 3(a), traversed bya rate-constrained injector on the left edge and by a pressure-constrained producer on the right edge,respectively (Figure 3(b)). There is a fracture network located in the center of the reservoir, whosepermeability was considered to be 2 orders of magnitude higher than the matrix average.

In the fine-scale, there are 4096 matrix cells (64 x 64) and the fracture network was discretized into60 segments. Using multiscale, we reduce the number of degrees of freedom (DOF) of the single-phasepressure solution to 64 coarse-scale matrix blocks (each containing 8 x 8 fine cells), accompanied by

Figure 2A basis function (a), a fracture function (b), and summation of all four basis functions (c) with fracture functions (d). Note that sum ofall basis and fracture functions (and well functions, if they perforate the shown dual cell) is 1.

Figure 32D test case setup: permeability field (a) and geometries of fractures and wells (b).

SPE-173200-MS 7

different coarsening strategies for the fracture network. The fracture control volumes are constructed byfirst specifying the distribution of the coarse nodes within the fracture network (depicted as green lineson the left-hand-side plots of Figure 5) and then assigning the remaining segments (red lines) to theirclosest nodes, in terms of number of hops (dots).

Consequently, we vary the number of coarse cells for fractures and investigate the performance of theiterative method. As second stage (smoother), we employ 5 iterations of ILU(0), as an optimum choicesuggested by the systematic studies of Tene et al. (2014) and Wang et al. (2014). We consider subsets of1, 3, 4, 6, 10 and 15 DOF from the original 60. It is important to note that we do not perform the correctionstage of Hajibeygi et al. (2011), where the fine-scale fracture pressure deviations are determined in asecond stage. In our framework, the implicitly computed fracture functions serve this purpose, instead.However, in experiments with only 1 DOF, the single fracture function will, by definition, show constantpressure within its fracture network. As such, these cases must rely on the smoother to account for thefine-scale deviations.

The right plots in Figure 5 shows the pressure solution after the first multiscale iteration for differentnumber of fracture coarse grids. For higher numbers of coarse DOF for fractures, the pressure after thefirst multiscale stage is already a very good approximation to the fully converged solution and, as such,only a small number of multiscale iterations is needed to converge. However for 1 to 4 DOF, somenoticeable features are not captured and, hence, extra effort is required to achieve convergence, which istranslated into an increased number of both multiscale and smoother iterations.

Illustrated also in Figure 4, the case of 1 coarse DOF for fracture entails slow convergence rate,because of its lack of effective interpolation within the fracture network. However, the rate is dramaticallyimproved when only a few extra DOF are added, since the fracture functions are becoming effective asinterpolators within the network, thus offloading the extra effort from the smoother. It is also interestingto note that there is a small difference between the experiments with 6 and 10 DOF. Therefore, the numberof fracture coarse nodes should not be the only criterion in developing an effective strategy. Rather, thepositioning of these nodes also plays an important role. Finally, we see that with 15 DOF, we obtain aconvergence behavior quite similar to the case when all 60 DOF are considered. Therefore, consideringthe efficiency-accuracy tradeoff, the case with 15 seems quite efficient. Of course the CPU time shouldbe measured for a more reliable assessment.

Figure 4Convergence behavior of the algebraic multiscale fracture solver with different fracture coarsening strategies.

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Figure 5Fracture coarsening strategies. The left plots depict the fracture coarse nodes (in green), while the figures on the right show the pressuresolutions after the first multiscale iteration (without smoothing).

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Figure 6Components of the multiscale prolongation operator for a fracture coarsening strategy with 3 DOF: (a-c) three assembled fracturefunctions, (d) sum of the basis functions, (e) injector well function, (f) producer well function. Note that each fracture function (a, b, and c) and wellfunction (e and f) plot is obtained by assembling the locally computed fracture and well functions over matrix dual cells. Also, (a, b, and c) plotsillustrate the fracture functions for matrix medium.

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Figure 6 provides further insights into the way our newly developed multiscale method performsinterpolation for the matrix domain, for the test case at hand. This time, the fracture network is dividedinto 3 coarse blocks. Figures 6 (a)-(c) depict the footprints of the corresponding fracture functions on thematrix, which are assembled after being computed locally over dual-coarse cells. Plot (d) shows the sumof the basis functions, where we can clearly distinguish the outline of the fracture network, and, finally,the two well functions are given in subplots (e), for the rate-constrained injector, and (f) for the pressureconstrained producer. As mentioned before, these functions reside in the columns of the multiscaleprolongation operator and it can be clearly seen that, together, they form a partition of unity, whichguarantees the monotonicity of the pressure solution.

3D 5-spot test caseThe performance of our devised multiscale framework for a 3D reservoir with 262144 (6x 64 64)fine-scale grid cells is tested. The fracture network was obtained by extruding those used in the previous

Figure 73D 5-spot test case setup: geometry of the medium with wells (a) and fracture plates (b) are shown. Also shown are the permeability field(c) and the pressure solution (d). Note that vertically lumped grid cells are considered for fractures, thus, 60 grid cells are used to discretize thefracture plates along their longitudinal direction.

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paragraph along the Z axis (Figure 7), between levels 20 and 44. For the sake of simplicity, we do notaccount for the vertical inter fluxes within the fracture plates, i.e., the number of degrees of freedom withinthe fracture network remains 60. In other words, vertically lumped grid cells are considered for fractures.The reservoir has a heterogeneous permeability field and is driven by 5 wells vertical wells: a rate-constrained injector, placed in the middle, and 4 pressure-constrained producers, lated in the corners.Finally, the steady-state single-phase pressure solution is also depicted in Figure 7.

Figure 8 depicts the convergence history of the multiscale solver using the 3D-equivalent fracturecoarsening strategies to those shown previously, in Figure 5. The same trend as in the previous experimentis noticed, which means that the proposed multiscale method can efficiently capture the fine-scale solutionusing only 15 DOF to represent the flow in the fracture network. For reference, we also show the resultsobtained using the SAMG commercial solver, developed by Fraunhofer SCAI (Stuben 2010), which, justlike the multiscale solver, was run in a Richardson loop. The different behavior can be, most likely,attributed to the matrix heterogeneity which is better captured by SAMGs adaptive coarse grid generationstrategies, whereas for our solver we prescribed a regular coarse grid of 8 8 8 blocks.

ConclusionsIn this paper, we proposed a novel general multiscale framework for efficient and accurate simulation offlow through heterogeneous porous media with complex wells and fractures of various length scales. Weintroduced, for the first time, the possibility to prescribe an arbitrary number of degrees of freedom perfracture network at coarse scale. This was done in order to obtain the desired tradeoff betweenconvergence properties and computational expense. Such a novel formulation was made possible by thecomputation of fully-integrated fracture and well functions, in full agreement with the classical multiscalealgebraic formulations. As such, the algorithm can be directly implemented as a black-box procedureinto the existing and next-generation reservoir simulators, and poses no algorithmic restrictions toaccommodate the inclusion of complex physics such as compressibility, capillarity, gravity and geo-mechanics. These extensions are the subject of our ongoing research.

Figure 8Convergence histories with different fracture coarsening strategies for 3D quarter-five-spot test case. Note that our algebraic multiscalefracture solver is compared with SAMG in terms of number of iterations.

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Our numerical experiments on 2D and 3D fractured reservoirs revealed that the new method is quiteefficient and flexible, i.e., it can effectively reduce the number of required degrees of freedom per fracturenetwork, at the same time maintaining appropriately fast convergence rates.

As an extension of this work, an effective and automatic coarsening strategy for both matrix andfractures are quite important. This is also the topic of our future research.

AcknowledgmentsFinancial supports of PI/ADNOC are greatly acknowledged.

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Algebraic Multiscale Solver for Flow in Heterogeneous Fractured Porous MediaIntroductionGoverning Equations and Discretization on Fine ScaleAlgebraic Multiscale Solver for Fractured Heterogeneous Porous Media with Complex WellsNumerical Results2D test case3D 5-spot test caseConclusionsAcknowledgmentsReferences