b006 multi-scale finite-volume method for highly

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B006 MULTI-SCALE FINITE-VOLUME METHOD FOR HIGHLY HETEROGENEOUS POROUS MEDIA WITH SHALE LAYERS IVAN LUNATI and PATRICK JENNY Institute of Fluid Dynamics –– ETH Zurich, ETH-Zentrum, Sonneggstrasse 3, CH 8092 Zurich, Switzerland

Abstract A multi-scale finite-volume (MSFV) method for solving elliptic problem in highly heterogeneous media was recently developed. The goal of the MSFV method is not simply to capture the large-scale effects of the fine-scale heterogeneity, but to fully describe the fine-scale velocity field with the original resolution. Thus, the MSFV method differs fundamentally from upscaling, since it provides an efficient tool for solving large flow problems with fine-scale resolution. The first step in the MSFV method is to compute the effective parameters that are used to solve the global flow problem on a coarse grid. This is done by means of a first set of numerical basis functions. After that the fine-scale velocity field is reconstructed by means of a second set of basis functions. Employing this second set of basis functions guarantees that the reconstructed fine-scale velocity is conservative. MSFV solutions prove to be in excellent agreement with the corresponding fine-scale solutions even for highly heterogeneous media. However, local deviations of the MSFV solution from the fine-scale solution can be observed in media characterized by very strong permeability contrasts, like in the presence of shale layers or meander like structures. We demonstrate that in media with virtually impermeable shale layers these deviations are caused by a reconstructed fine-scale velocity field that can locally contradict the global flow conditions in order to honor mass conservation in a coarse cell. These errors are only observed if the shale layer subdivides a coarse cell into poorly connected regions. If the shale layer coincides with coarse-cell boundaries, instead, the MSFV method handles such cases correctly. This suggests that the performance of the MSFV methods can be improved by extending it for unstructured grids. In that case the coarse grid can be adequately defined according to the small-scale permeability field. Introduction The hydraulic permeability of a real reservoir exhibits strong spatial variations that span a wide range of scales. A full description of flow and transport that includes all these scales exceed the current computational capabilities: the numerical grid required to describe the heterogeneity with full resolution would be too large to be used for solving the flow problem. To overcome this difficulty the numerical grid is usually coarsened by replacing groups of fine-scale blocks with coarser homogeneous blocks. A number of techniques have been developed to upscale the fine scale permeability field to a coarser equivalent permeability field that adequately describes the large-scale effects of the small-scale heterogeneity (we refer to Renard and de Marsily [8] for a comprehensive review). The price paid by these approaches for reducing the computational cost is the loss of the fine-scale structure in the solution. Whereas that might be acceptable for 9th

European Conference on the Mathematics of Oil Recovery —— Cannes, France, 30 August - 2 September 2004

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simulations aimed to characterize the pressure field of single-phase flow, there is evidence that in many problems connected with contaminant transport or multi-phase flow knowledge of the fine-scale velocity field is essential to correctly capture the evolution of the quantities of practical interest such as contaminant dilution or oil cut. In order to recover the information lost during the upscaling procedure, several multi-scale finite-element methods have been employed [1,3,4]. Recently, also a multi-scale finite-volume (MSFV) method for solving elliptic problems was developed [5,6]. Analogous to several upscaling techniques the MSFV method employs an imposed coarse grid to solve the global flow problem. To compute the effective parameters, a set of local flow problems is solved. In contrast to upscaling, however, the MSFV method targets not only the solution of the coarse-grid problem, but also the solution of the full problem with the original resolution. This is achieved by local reconstruction of the fine-scale velocity field from the coarse-grid pressure solution. The goal is to be as accurate as fine scale simulations with a computational cost comparable to traditional upscaling. To extract the equivalent coarse-grid permeabilities and to reconstruct the small-scale velocity field, two sets of numerical basis functions are employed. To compute the basis functions local flow problems are solved on each cell of the coarse grid (or, as clarified later, on each cell of a dual coarse grid). Difficulties in calculating the basis functions may arise, if coarse cells are subdivided by impermeable layers. The multi-scale finite-volume method The MSFV method applies to nonlinear multiphase flow and transport in highly heterogeneous media and it is based on a pressure-saturation formulation of multiphase flow equations. Here we restrict ourselves to a description of the method that is functional for the purposes of the paper. We refer to the existing MSFV literature [5,6] for a more detailed description of the algorithm and to classical multiphase-flow textbooks (e.g. [2,7]) for a comprehensive formulation of multiphase flow. Here we consider two incompressible fluids and suppose that capillarity and gravity are negligible compared to viscous forces. Under these assumptions the flow can be described by the pressure-saturation formulation, ’ ˜ >utot (S)@ q tot ,

I

wS  ’ ˜ >f (S) utot (S)@ q , wt

(1) (2)

where I is the porosity, S the saturation of one phase, and f and q the fractional flow and the source term of the same phase, respectively. In equations (1) and (2) we used the total source term, q tot q  qc, (positive when extracted) and the total velocity, utot u  uc, where primed quantities refer to the second phase. Assuming Darcy’’s law for phase velocities yields

utot (S) Otot (S)’p ,

(3)

where p is the pressure. The total mobility, Otot, is given by

Otot (S) k >kr (S) P  krc(S) Pc@ ,

(4)

where k is the intrinsic permeability, kr and krc the relative permeabilities of the two phases, and P and Pc their viscosities.

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The system of equations (1)-(2) completely determines the flow. The main ideas of the MSFV method deal with the solution of the elliptic equation (1). Indeed, once the total velocity has been obtained from equations (1) and (3), determination of the saturation distribution from equation (2) is straightforward.

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8 C

4

D 5

A 1

9

6 B

2

3

Figure 1. The fine grid (dotted line), the coarse grid (solid line) and the dual coarse grid (dashed line) for a part of the domain containing a two-dimensional nine-point stencil. The striped rectangle represents the possible location of an impervious shale layer that divides coarse cell : 5 and dual cells

˜ B and : ˜D :

To solve equation (1) an auxiliary coarse grid and a dual coarse grid are imposed on the original fine grid. The nodes of the dual coarse grid are located inside the cells of the imposed coarse grid (see figure 1). To compute the effective transmissibilities between the nodes of the coarse grid a first set of basis functions (dual basis functions, DBF for short) is used. The DBFs are numerical solutions of local flow problems solved on dual coarse cells. Once the effective transmissibilities are computed, the discretized flow equation (1) is solved globally on the coarse grid to obtain the pressures at the nodes of the dual coarse grid. At this point, starting from the DBFs, a second set of basis functions (velocity basis functions, VBF for short) is constructed. The VBFs are used together with the coarse-pressure solution to reconstruct the fine-scale total velocity field. The reason for employing a second set of basis functions is that a velocity field reconstructed directly from the DBFs would be locally non-conservative, whereas our final velocity field satisfies mass conservation at the fine scale. In the following, we focus on the description of how these two sets of basis functions are constructed and on the evaluation of the assumptions that are made. Without loss of generality we consider a two-dimensional problem. Referring to figure 1, we will show how the basis functions related to the coarse-grid block :5 (which has corners A, B, C and ˜ D) are constructed. For each dual cell, : i{A ,B,C ,D} , overlapping with :5 a set of four dual basis ˜ ij , is constructed by solving four local problems on : ˜ i . The the subscript i refers to functions, I the i-th dual block on which the solution is computed and the superscript j refers to the j-th corner. These problems have the form

˜ i. ˜ ij @ 0 on : ’ ˜ >Otot ’I

(5)

Appropriate boundary conditions that approximate the true flow conditions experienced by the sub-domain embedded in the whole domain are required. Hou and Wu [3] suggest imposing the solution of a reduced problem as Dirichlet boundary condition for the local problem. Following 9th

European Conference on the Mathematics of Oil Recovery —— Cannes, France, 30 August - 2 September 2004

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˜j this approach, we obtain the boundary conditions for equation (5) by imposing I i ˜ i and solving the reduced problem th corner of : ª tot w j º w ˜ » 0 on w: ˜i I ¦ «Otk wx t k ¬ wx k i ¼

G jk at the k-

(6)

on the face between two corners. The index t denotes the component tangent to the boundary. Jenny et al. [5] proved that equation (6) provides a good approximation of the boundary conditions in most cases. A physical interpretation of boundary condition (6) can be immediately obtained by decomposition of the total velocity in its tangent, t, and normal, n, components with tot tot respect to the boundary, i.e., utot utot 0 on the t  u n . As equation (6) is equivalent to w x t ut tot boundary, it follows immediately that wx n un 0 since the divergence of the velocity is zero. Thus, assuming condition (6), we assume that uttot uttot ( x n ) and untot untot ( x t ) at the boundary.

˜ ij . From Equation (5) with the boundary condition (6) allows computing all dual basis functions I the DBFs, the fluxes across the boundaries of :5 are computed to extract the effective transmissibilities. Note that if we want to compute, for instance, the flux across the segment AB, ˜ A and : ˜ B , i.e. I ˜ Aj {1,2,4,5} and we have to consider all basis functions related to the dual cells : ˜ Bj {2,3,5,6} . Therefore six coarse cell pressures contribute to the flux across the segment AB. This I results in a nine-point stencil for two-dimensional problems (a twenty-seven-point stencil in three dimensions). The extracted transmissibilities are used to solve the global flow problem on the coarse-grid and to obtain the coarse-cell pressure pj in the coarse cell :j. So far our method could be classified as an upscaling technique. In particular, it resembles the multi-scale finiteelement method by Hou and Wu [3,4]. However, their finite-element scheme does not ensure mass conservation, whereas our finite-volume scheme does. Moreover, our goal is to reconstruct a conservative fine-scale velocity field. For this purpose, we introduce a second set of basis functions. Considering the coarse element :5 (figure 1), a set of nine velocity basis functions I 5j 1,...,9 is built. Each VBF is obtained by solving the local flow problem ’ ˜ >Otot ’I 5j @  f on :5

(7)

with Neuman boundary conditions. The specified fluxes at the boundary are linear combinations of the DBFs related to :5, e.g., to compute the VBF in equation (7) the fluxes are extracted from the pressure field 9

¦ ¦G

p

kj

˜k I i

(8)

i {A ,B,C ,D} k 1

The term on the right hand side of equation (7) is a uniformly distributed source term that is introduced to ensure solvability. From the global pressure solution, pj, and the VBFs obtained from equations (7) and (8) we can compute the fine-scale pressure field in :5 as 9

p

¦p I j

j 5

j 1

and from that the fine-scale velocity field using equation (3).

(9)

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(a)

(b)

(c) Figure 2. a) The log-permeability of field 1 (mean value 2.4, variance 5.5). Darker means lower permeability. b,c) Saturation distribution after 0.5 pore-volume injected in a quarter five-spot problem. The white fluid injected in the bottom-left corner displaces the black fluid. Fine-scale simulation (b) and MSFV simulation (c) are shown.

Numerical results To assess the accuracy of the MSFV solution we compare the fine-scale velocity field obtained with the MSFV method with the velocity field obtained from a direct fine-scale simulation. Even if the MSFV method is implemented for three-dimensional three-phase flow, here only twodimensional two-phase flow problems are considered. Moreover, we assume the same viscosity for the two fluids and linear relation between relative permeability and saturation. This simplifies the problem to solute transport in a single-phase flow and allows to focus on the effects of high permeability contrasts. First, two highly heterogeneous permeability fields, the top layer (Field 1) and the bottom layer (Field 2) of the SPE10 test case, are considered. The logical dimension of the domain is 220u60. The natural logarithm of these two permeability fields is shown in figures 2.a and 3.a, respectively. The log-permeability of field 1 has a mean value of 2.4, a variance of 5.5 and a finite correlation length (figure 2.a). The log-permeability of field 2 has a mean value of 3.6 and a variance of 12.2. Field 2 is characterized by strong spatial contrasts and a meander-like correlation structure with long spatial correlation, which is representative for many natural 9th

European Conference on the Mathematics of Oil Recovery —— Cannes, France, 30 August - 2 September 2004

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formations (figure 3.a). For both fields a quarter five-spot problem is solved. A white fluid is uniformly injected in the coarse cell at the left-bottom corner of the domain and displaces a black fluid that is initially present in the formation. The production term is uniformly distributed over the coarse cell at the right-top corner of the domain. Figures 2 and 3 show the saturation distribution after 0.5 pore volume injected (PVI) obtained by direct fine-scale simulation and with the MSFV method. The MSFV simulations are performed with a uniformly spaced 11u3 coarse grid. For field 1 the two solutions are virtually indistinguishable (figure 2). Overall, also the MSFV simulation performed with field 2 shows excellent agreement with the fine-scale solutions. However, significant deviations can be locally observed (figure 3).

(a)

(b)

(c) Figure 3. a) The log-permeability of field 2 (mean value 3.6, variance 12.2). Darker means lower permeability. b,c) Saturation distribution after 0.5 pore-volume injected in a quarter five-spot problem. The white fluid injected in the bottom-left corner displaces the black fluid. Fine-scale simulation (b) and MSFV simulation (c)

We suggest that these deviations are manly due to the fact that the reconstructed pressure field from which the Neuman boundary conditions for the VBFs are computed yield a velocity field which is not divergence free at the boundary between two dual cells. It is worth to stress that even if the pressure distribution reconstructed from the coarse-grid pressure and the DBFs is not divergence free, the final fine-scale velocity field reconstructed from the coarse scale pressure and the VBFs is divergence free. However, a contrast between the reconstructed divergence-free

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velocity field and the global flow conditions may arise due to the local Neuman boundary conditions.

(a)

(b)

Figure 4. Saturation distribution around the shale layer after 0.5 PVI. The white fluid is injected at the left side and displaces the black fluid that was initially present in the formation. The logical size of the domain is 220u60 and the logical size of the MSFV coarse grid is 11u3. A region of 3u3 coarse-cells around the shale layer is represented. (a) The shale layer consists of 40 fine cells, i.e. from cell (120,11) to cell (120,50). The right border of the central coarse cell coincides with the shale layer. (b) The shale layer consists of 40 fine cells, i.e. from cell (112,11) to cell (112,50), and subdivides the central coarse cell into poorly connected regions.

As an illustrative example of this phenomenon we consider the extreme case of flow through a homogenous medium with a single almost impermeable shale layer (permeability ratio 10-10). As in the previous two examples, a domain with logical dimensions 220u60 is considered. Logical length and thickness of the shale layer, which is parallel to the y-axis, are 50 and 1, respectively. Different positions of the shale layer along the x-axis are considered. A white fluid is uniformly injected in the coarse cells at the left boundary and a black fluid, which initially occupies the formation, is displaced. The MSFV simulations are performed with a uniformly spaced 11u3 coarse grid. The saturation map in figure 4.a shows that the MSFV solution is physically correct, if the shale layer coincides with the boundary between two coarse cells. If the shale layer subdivides a coarse cell, instead, the solution obtained is unphysical as can be observed in figure 4.b. In this case, a counter flow through the shale layer is observed. This can be understood by referring to figure 1. Let us consider the shale layer represented by the striped rectangle, which ˜ B and : ˜ D and the coarse cell :5 into poorly connected regions. Thus, subdivides the dual cells : only the DBFs I 53 , I 56 and I 59 contribute to the pressure on the right side of the layer. The DBFs satisfy the no-flow boundary condition at the surface of the impermeable layer, but they also yield an unphysical no-flow boundary condition across the segment BD, which is used to compute the VBFs. The effects of this unphysical boundary are well visible in the saturation distribution of figure 4.b. Therefore, in the case considered only fluxes into the coarse cell :5 are present in the part of the domain on the right of the layer. This results in a flow through the layer in order to achieve mass conservation. Flow through shale layers and unphysical no-flow boundaries are also observed in problems with heterogeneous fields embedding shale layers. Like in the previous test case, the errors disappear if the shale layers coincide with the coarsecell boundaries. 9th

European Conference on the Mathematics of Oil Recovery —— Cannes, France, 30 August - 2 September 2004

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Conclusions The MSFV method provides an efficient tool for solving large flow problem with fine-scale resolution. MSFV solutions prove in excellent agreement with the corresponding fine-scale solutions, even for heterogeneous media with high variance and long spatial correlation. However, local deviations may occur if the medium is characterized by strong permeability contrasts and contains low-permeability regions. We demonstrate that in media with virtually impermeable shale layers these deviations are due to a reconstructed fine-scale velocity field that may locally contradict the global flow conditions, in order to honor mass conservation. These errors are only observed if shale layers subdivide coarse cells into poorly connected regions. If the shale layers coincide with coarse-cell boundaries, instead, the MSFV method handles the problem correctly. This suggests that the MSFV method can be improved if it is extended for unstructured grids. In that case, the coarse grid can be adequately defined according to the small scale-heterogeneity field. References [1] Arbogast, T., Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow, Computational Geosciences, 6(3-4), 453-481, 2002 [2] Aziz, K., and A. Settari, Petroleum Reservoir Simulation, Elsevier Applied Science Publishers, London, 1979 [3] Hou, T.Y., and X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of Computational Physics, 134, 169-189, 1997 [4] Hou, T.Y., X. Wu, and Z. Cai, Convergence of multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Mathematics of Computation, 68(227), 913943,1999 [5] Jenny, P., S.H. Lee, and H.A. Tchelepi, Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, Journal of Computational Physics, 187, 47-67, 2003 [6] Jenny, P., S.H. Lee, and H.A. Tchelepi, Adaptive multiscale finite-volume method for multiphase flow in porous media, submitted for publication to SIAM Multiscale Modeling and Simulation, 2004 [7] Marle, C. M., Multiphase flow in Porous Media, Institut Français du Pétrole Pubblications, Éditions Technip, Paris, 1981 [8] Renard, P., and G. de Marsily, Calculating equivalent permeability: a review, Advances in Water Resources, 20(5-6), 253-278, 1997