A SPECIALIZED UPSCALING METHOD FOR ADAPTIVE GRIDS

A SPECIALIZED UPSCALING METHOD FOR ADAPTIVE GRIDS: TIGHT INTEGRATION OF LOCAL-GLOBAL UPSCALING AND ADAPTIVITY LEADS TO ACCURATE SOLUTION OF FLOW IN HETEROGENEOUS FORMATIONS M. GERRITSEN∗ AND J. V. LAMBERS†

Abstract. We propose a methodology, called multi-level local-global upscaling, for generating accurate upscaled models of permeabilities or transmissibilities for flow simulation in highly heterogeneous subsurface formations. The method generates an initial adapted grid based on the given fine scale reservoir heterogeneity and potential flow paths. It then applies local-global upscaling [1], along with adaptivity, in an iterative manner. For highly heterogeneous (e.g., channelized) systems, this integration of grid adaptivity and upscaling is shown to consistently provide more accurate coarse scale models for global flow, relative to reference fine scale results, than do existing upscaling techniques applied to uniform grids of similar densities. The algorithm allows for rapid updating of the coarse scale model when the grid is adapted dynamically during transport to resolve important features, or as a result of changing boundary conditions. There is no need for downscaling and both refinement and coarsening can take place dynamically without destroying accurate global flow resolution on the adapted grid.

Key words. Scale up; Subsurface; Heterogeneity; Flow simulation; Channelized; Permeability; Transmissibility; Adaptivity; Enhanced Oil Recovery; Gas Injection

1. Introduction. In this paper, we propose an efficient strategy for generating accurate upscaled models of permeabilities or transmissibilities for flow simulation on highly adapted grids in heterogeneous reservoir formations. Adaptive grids are very attractive when an accurate representation of reservoir heterogeneity is desirable in simulation of fluid flow. An example is the simulation of gas injection processes in reservoirs, in which the mobile gas will seek high permeability flow paths, greatly affecting global sweep. One difficulty caused by coarse grid modeling of these processes ∗ Department of Petroleum Engineering, Stanford University, Green Earth Sciences Building, Stanford, CA 94305-2220, USA † Department of Petroleum Engineering, Stanford University, Green Earth Sciences Building, Stanford, CA 94305-2220, USA

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M. GERRITSEN AND J. V. LAMBERS

is that it tends to smooth permeability contrasts that are not aligned with the coarse grid, resulting in unreliable performance predictions. However, fine grid simulations of EOR processes are generally prohibitively expensive. Adaptive grids also allow important flow features with small spatial support, such as frontal regions, to be resolved efficiently without sacrificing computational efficiency, see for example [2, 3, 4]. Grid adaptivity can be used to capture important large scale connected flow paths provided that the upscaling strategy used on the adaptive grids retain the important permeability contrasts and connectivities. The integrated adaptivity and upscaling strategy presented in this paper was developed specifically for this purpose. It computes an upscaled permeability or transmissibility field for an adapted structured grid without use of downscaling. Permeability or transmissibility fields for dynamically adapted grids can be updated efficiently during run time while retaining global flow accuracy. 1.1. Upscaling techniques. The permeability of reservoir rocks is often highly variable and uncertain. It can greatly affect flow and transport in subsurface formations however, and must therefore be adequately modeled. The effects of uncertain reservoir heterogeneity can be included either through stochastic modeling or, wich is the underlying assumption in this paper, by simulating a number of deterministic geostatistical realizations of the reservoir [5]. Because the computational costs of solving for a large number of realizations are high, simulations are generally performed on grids that are coarse compared to the given geological models. These coarsened flow models should adequately represent key behaviors, such as the overall flow rate for given boundary conditions or critical connected flow paths. A large variety of upscaling techniques exist that are used to find representative permeability or transmissibility fields on the coarse grids. In purely local approaches the upscaled parameters are computed over fine scale regions that correspond to the target coarse block under study. In extended local upscaling methods, regions neighboring the coarse block are included. In both approaches, the local flow is driven by generic Dirichlet or periodic boundary conditions imposed on the fine scale

INTEGRATING ADAPTIVITY WITH LOCAL-GLOBAL UPSCALING

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regions, see [6, 7, 8]. Although local methods may give satisfactory results for some reservoirs, they generally do not perform well on reservoirs with important large-scale connected flow paths. For flow-based grids [9] these local upscaling methods may lead to good results provided the grids capture the reservoir heterogeneity well. But, in strongly heterogeneous three-dimensional problems it is very challenging to construct flow-based grids that are well aligned with the flow and have sufficient grid quality. Global methods reduce the connected flow path errors by using global fine scale flow simulations to determine coarse scale parameters, but they are significantly more expensive [10]. Quasi-global approaches reduce the computational requirements of full global procedures by substituting approximate global information in place of global fine scale results. An alternative and attractive method to improve large scale connectivity in upscaled permeabilities is the recently developed local-global method [1]. In local-global upscaling, a standard extended local method is used to obtain an approximate coarse permeability or transmissibility field. Then, a coarse global solve is performed with generic boundary conditions. Interpolation of the coarse solution provides the Dirichlet boundary conditions for the next iteration of local solves. The process is repeated until the coarse field converges, which generally takes only one or two iterations. The local-global method has been shown to lead to much improved upscaled permeability and transmissibility values in reservoirs that have important high permeability flow paths (eg channelized systems). It also generally reduces the process dependency; that is, as compared to other upscaling methods, it reduces errors in the computation of global flow driven by boundary conditions different from the generic boundary conditions used in the upscaling process. 1.2. Integrating upscaling and adaptivity. When a dynamically adapted grid is used in reservoir simulation to refine around important geological, flow or transport features, upscaled parameters must be found for the new grid levels introduced. In most existing strategies this is done using downscaling strategies based on geostatistical methods [5], or by local upscaling methods on the new cells. Using these methods it is however difficult, if not impossible, to recover important permeability contrasts or connected flow path information on the refined grids that were present on

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the fine geological grid. Our first goal is to find an upscaling strategy for adapted grids that preserves connected flow paths and high permeability contrasts present on the fine geological grid. Our second aim is to construct the adapted grids and upscaling method such that the process dependency that is generally introduced by upscaling methods is minimized. This is especially important for the above mentioned injection processes that are strongly boundary condition dependent. The method we propose is an extension of the local-global upscaling method to adapted grids. In this work, we apply the upscaling strategy to a cell-based anisotropic Cartesian refinement strategy, which we find attractive because it allows very aggressive refinement and coarsening. However, the multi-level local-global method developed here is applicable to many types of structured grid adaptivity strategies, such as those used in [3, 11, 12, 13], and we are currently extending it to unstructured grids also. Uniform Cartesian grids have been criticized in terms of their insufficient flexibility to create suitable alignment with flow or geological features. Although such non-alignments can certainly affect flow accuracy for coarse Cartesian grids, the local-global upscaling method has been shown to alleviate these errors [1]. Adaptivity of the grid, when combined with an appropriate local-global method, can further reduce alignment errors, which is supported by the results presented in this paper. In this work, we restrict ourselves to upscaling for diagonal permeability tensors, or the corresponding transmissibilities. We have found that grid refinement combined with accurate upscaling reduces the need for full tensor flow solves, and will report on these findings in a future paper. We start the paper by giving a brief overview of the governing equations (section 2), CCAR grids and discretization on CCAR grids (section 3). In section 4, we review local-global upscaling, which serves as the foundation of the current work, and discuss how it is applied to our CCAR grids. We present an indicator that we use to adapt the grid during the multi-level local-global iteration to reduce flow solver and upscaling errors, and discuss how to efficiently update upscaled parameters in case of dynamic grid adaptation during the simulation process. Although we use permeability upscaling to illustrate the multi-level local-global method in section 4,


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transmissibility upscaling can be implemented efficiently and is in many cases preferred over permeability upscaling. In section 5, we discuss in some more detail how we upscale transmissibilities on CCAR grids. Numerical results are presented in section 6. We apply our algorithm to a two-dimensional channelized domain, with flow in several directions in order to illustrate the reduced process dependency of our approach. Although the numerical results are given in two dimensions only for ease of visualization, the method is fully three dimensional. Discussion and conclusions are presented in the last two sections. 2. Governing equations. The upscaling strategy is based on single phase, steady and incompressible flow in a heterogeneous reservoir. The governing dimensionless pressure equation is obtained by combining the continuity equation ∇ · u = 0 and Darcy’s law given by u = −k · ∇p, which leads to

∇ · (k · ∇p) = 0.

(2.1)

Here p is the pressure, u is the Darcy velocity and k the permeability tensor, all of which are non-dimensionalized by appropriate reference values. This equation is valid on the fine scale. Although homogenization of equations with variable coefficients generally yields coarse scale equations of a different form than the original fine scale equations, it is common practice to keep the same equation as (2.1) for the coarse pressure, with the permeability tensor replaced by the coarse scale permeability tensor k∗ , also referred to as the effective permeability tensor, and p by the coarse scale pressure. This has been found to lead to adequate coarse scale pressure solutions [6, 8, 14]. In the remainder of the paper we will use Eqn. (2.1) indiscriminantly of scale. In this work, we further assume that the permeability tensor is diagonal. As stated in the introduction we have found that our combination of adaptivity and upscaling reduces the need for full tensor upscaling, so we do not consider this to be a severe restriction. 3. Solving the Pressure Equation Using An Adaptive Cartesian Approach.

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3.1. Cartesian Cell-based Anisotropically Refined (CCAR) grids. When designing an efficient adaptive simulation approach a grid topology must ideally facilitate fast transitions from dense to coarse grids without sacrificing grid and solver quality, allow efficient computation despite potentially high levels of refinement, lead to fast grid generation. We have found a Cartesian grid topology with anisotropic cell-based refinement and coarsening quite attractive because it allows aggressive grid adaptation while maintaining grid quality. We are relying on adaptivity to align the grid with important flow and geological features and ensure flow accuracy. If conformity to exterior (or interior) domain boundaries is necessary, these grids can potentially be extended with an Immersed Boundary Method (IBM), such as described in [15]. IBM has certainly passed the proof of concept in Computational Fluid Dynamics applications, but a robust extension to heterogeneous media has not been formulated as yet. The CCAR grid is formed by a number of anisotropic refinements of its cells, i.e. splitting a cell in half along a right-direction, to construct a suitably refined mesh. This is illustrated for three refinement levels in Figure 3.1.

Fig. 3.1. CCAR grid cells with 3 levels of refinement. Left: isotropic refinement. Right: anisotropic refinement.

We impose the restriction that no cell can have more than four neighbors in each direction (or two neighbors in two dimensions) to maintain solution accuracy across interfaces [16]. Figure 3.2 illustrates a CCAR grid for a gas injection process that is dynamically adapted based on saturation gradients. Near solution fronts, the grid


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is refined. The grid is coarsened again once the fronts have passed. Thanks to the anisotropic refinements the grids have a good alignment with the irregular saturation fronts. The CCAR grids are stored using the unstructured data approach first suggested in [17], which leads to very efficient look-ups.

Fig. 3.2. CCAR grids (bottom) following saturation fronts (top) during an injection process at an earlier time (left) and a later time (right).

3.2. Discretization on CCAR grids. We use the new compact finite volume discretization of the pressure equation 2.1 on CCAR grids developed in [18]. For homogeneous media, this discretization is formally second order, with the fluxes are computed by a general three-point finite difference approximation. The anisotropic refinements require the three-point stencil to be combined with a cubic interpolation of the cell centered pressure values in the adjacent cells. Details are provided in [18]. For heterogeneous media, in which the pressure equation has discontinuous coefficients, we assume diagonal permeability tensors. To illustrate the computation of the fluxes on anisotropic CCAR grids in heterogeneous media, consider Figure 3.3. For i = 1, 2, 3, the cells are assumed to have dimensions hix and hiy , and diagonal

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h2 x

2

hy

c

2

c

c

h3 y

1

h1 y

3

3

1

hx

hx

Fig. 3.3. the properties of the cells centered at c1 , c2 and c3 , along with the pressure values at these points, are used to compute the flux across the left face of the cell centered at c1 .

permeability tensors with diagonal elements kxi and kyi . We will construct a flux for the face of cell 1 bordering cells 2 and 3. The area of the face of cell i that borders this face is denoted by Ai . The average pressures in the cells i are denoted by p1 , p2 and p3 . A straightforward computation of the flux is given by f lux = −2h1y

where W =

P

i

h1y p1 − h2y p2 − h3y p3 h1x h1y 1 kx

+

h2x h2y 2 kx

+

h3x h3y 3 kx

= −2A1

A1 p1 − A2 p2 − A3 p3 , W

(3.1)

Vi /kxi , with Vi being the volume of the ith cell. This is a direct

extension to anisotropic grids of the flux for isotropically refined cells suggested in [19]. Unfortunately, this approach can lead to inaccuracies in the global flow. For example, suppose that kx2 = 0. Then, regardless of the values of kx1 and kx3 , the flux across the face computed by 3.1 will be zero. This is a consequence of the coupling of pressure values at the midpoints of the “half-faces” between cell 1 and cell 3, and


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between cell 1 and cell 2, which is induced by the constraints imposed on the flux in [19]. To alleviate this problem we modify the computation of the flux by uncoupling pressure values at these midpoints, indicated by the points m1 and m2 in Figure 3.4 .

q2

c2

m2 q1

q3

c

3

m

3

d2 c1

d3

Fig. 3.4. The points cj , j = 1, 2, 3, dj , j = 2, 3 and mj , j = 2, 3, are used to compute the fluxes q2 and q3 across the half-faces between cell 1 and cell 2, and between cell 2 and cell 3.

When we require continuity of the flux across the abovementioned half-faces, we obtain the equations kxj

(p(mj ) hjx

− p(cj )) =

kx1 (p(dj ) − p(mj )), h1x

j = 2, 3,

(3.2)

and the corresponding fluxes

qj =

2hjy h1x 1 kx

+

hjx j kx

(p(dj ) − p(cj )),

j = 2, 3.

(3.3)

To complete the definition of the stencil, we must represent the pressure at the points d2 and d3 , indicated in Figure 3.4, in terms of the pressure at appropriate

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cell centers. To that end, we use a convex combination of two interpolations, one with a vertical orientation and one with a horizontal orientation. For the verticallyoriented interpolation, we use bilinear interpolation at c1 , and the two closest cell centers besides c2 and c3 , that form a triangle containing d2 , and similarly for d3 . Such triangles are illustrated in Figure 3.5.

c2

d

2

c c

3

1

d3

Fig. 3.5. Triangulations used to represent the pressure at d2 and d3 in terms of cell centers. The values at these points are obtained by bilinear interpolation of the values at the vertices of the triangle containing them. The point c1 , the center of the cell containing the points at which the pressure is interpolated, is always chosen to be a vertex. The centers of the neighboring hanging nodes, c2 and c3 , are never chosen as vertices to represent fluxes across their own faces.

For the horizontally-oriented interpolation of d2 , we use bilinear interpolation at c1 , c2 , and the closest cell center that completes a triangle containing d2 . Similarly, to interpolate the pressure at d3 , we use bilinear interpolation at c1 , c3 , and the closest cell center that completes a triangle containing d3 . Such triangles are illustrated in Figure 3.6. To determine the combination of these two interpolations, we use the angle of the pressure gradient through the point di , i = 2 or 3, at which we are interpolating, as determined by averaging the pressure gradients from the local solves


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over the half-cell containing di and weighing them by the magnitude of the fluxes across the face.

c2

d

2

c c

3

1

d3

Fig. 3.6. Triangulations used to represent the pressure at d2 and d3 in terms of cell centers. The values at these points are obtained by bilinear interpolation of the values at the vertices of the triangle containing them. The point c1 , the center of the cell containing the points at which the pressure is interpolated, is always chosen to be a vertex. The centers of the neighboring hanging nodes, c2 and c3 , are also chosen as vertices to represent fluxes across their own faces. The third vertex is chosen so that no extrapolation is performed.

The anisotropic refined cells increase the bandwidth of the resulting system of linear equations, and perturb the structure of the matrix as compared to uniform Cartesian grids. But the matrix remains sparse with a reasonable condition number thanks to the compact finite volume discretization. We solve it by the parallel algebraic multigrid solver BoomerAMG [20], which shows almost linear convergence properties for our test cases. 4. Effective upscaling for CCAR grids. As mentioned in the introduction, it has been shown that for Cartesian grids, the local-global method leads to improved results as compared to local, or extended local, methods [1]. For each coarse grid cell, an effective permeability or face transmissibilities are computed by solving a local

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flow problem on the fine geo-cellular grid directly around the coarse cell. This local flow problem is driven by boundary conditions interpolated from a coarse global solve. The resulting coupling between the local flow problems and the coarse global solve leads to an iterative scheme that generally converges rapidly. We now review the details of the local-global upscaling algorithm. For the coarse global solves, we solve one flow problem corresponding to each coordinate direction, with generic boundary conditions: a constant pressure difference in the given coordinate direction, and no flow across boundaries corresponding to the other directions. In the following description of the algorithm for a 2D domain [0, lx ] × [0, ly ], we denote by px a pressure field that is constant for x = 0 and x = lx , with no flow across y = 0 and y = ly . The pressure field py satisfies analogous boundary conditions. With each cell Ci in the coarse grid G, we associate an extended region Ei = Ci ∪ Bi , where Bi is a border region of width r coarse cells around Ci . For example, if r = 0.5, then Bi has a width of half of a coarse cell in each direction, whereas if r = 1, then Bi consists of the eight coarse cells that border Ci . Such an extended region is illustrated in Figure 4.1(c). Typical choices for r are 0.5 or 1.5, so that the pressure values at coarse cell centers can conveniently be used to obtain boundary data for local solves. while not converged do for each coarse cell Ci do if first iteration then Use generic boundary conditions: constant pressure in one coordinate direction, no-flow in all other coordinate directions else Use interpolation of global solutions to approximate pressure on Ei Use Dirichlet boundary conditions using these pressure values end Solve pressure equation on Ei with Dirichlet data obtained from interpolated values


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(a)

(b) (c)

Fig. 4.1. Schematic showing (a) fine and coarse scale grids, (b) fine scale local region and (c) fine scale extended local region, with r = 1

Define a Cartesian grid Gi on Ei containing Ci as a cell Compute fluxes between cells of Gi and pressure at cell centers of Gi Use fluxes and pressure values on Gi to compute new permeability tensor Ki for Ci end Solve generic flow problems on G to obtain pressure fields px and py Check for convergence done Local-global upscaling for Cartesian grids serves as the cornerstone of the local-global upscaling algorithm for CCAR grids, which we now present. 4.1. Upscaling to the Base Grid. The base grid is designed with the aim of preserving important high permeability flow paths. By accounting for these paths, we hope to reduce process dependency, reduce the need for full tensor permeabilities,

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and minimize upscaling errors that can occur when such features are lost due to coarsening. Thus, upscaling and base grid generation are coupled processes. The local-global method fits very naturally in this coupled process, since its consistency check between the coarse and fine grids can be used to ensure that the effects of fine-scale features are properly accounted for in the coarse-scale model.

8000

0.9

7000

0.8

6000 5000

0.7 0.6 0.5

4000 3000

0.4 0.3

2000

0.2

1000

0.1

Fig. 4.2. Permeability field (left), and boundaries of connected flow paths detected by leaky connected-set approach (right)

We apply as a first static refinement indicator, the method proposed by Younis and Caers in [21]. The method is based on a “leaky” connected-set approach that delineates clusters of fine cells with similar permeability. Heuristic measures are used to color each connected set, indicating the level of the importance of capturing the set by refinement. Refinement flags are then obtained by explicit boundary detection. An illustration of the process appears in Figure 4.2. In the current implementation, a minimum level of refinement is set based on the resolution of the geocellular grid and efficiency considerations, and a CCAR grid G is generated by iteration until these level constraints are met. Such a grid is illustrated in Figure 4.3. With each cell Ci in the CCAR grid G, we associate an extended region Ei = Ci ∪ Bi , where Bi is a border region of constant width r around Ci . We note that in the local-global upscaling method in [1] the width of each dimension of the border Bi around Ci is chosen to be a multiple of the corresponding dimension of Ci instead. The reason for this difference is two-fold. First, larger cells need less of a border (relative to the cell size) because the fine-scale permeability within such cells tends


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Number of cells = 1363

60

50

40

30

20

10

0

0

10

20

30

40

50

60

Fig. 4.3. base CCAR grid generated from permeability field and connected sets from Figure 4.2

to be smoother. Second, if a cell Ci with corresponding extended region Ei should be bisected during refinement, the resulting new cells will have extended regions whose union coincides with Ei . This allows pressure values on the boundary of Ei to be re-used for the new regions, thus facilitating rapid dynamic adaptation of the grid, and the coarse-scale permeability field, during transport. After the base grid is generated, we apply local-global upscaling, as in the Cartesian case. We now describe the algorithm in further detail, for a two-dimensional domain, and for permeability upscaling. The algorithm computes a diagonal permeability tensor for each cell, whereas the local-global upscaling algorithm in [1] returns a full tensor. In the next section we discuss how the method can be applied to trans-

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missibility upscaling. Construct the base CCAR grid G using static refinement indicators while not converged do for each cell Ci in G do if this is the first iteration then Use generic boundary conditions: constant pressure in one coordinate direction, no-flow in the other coordinate direction else Use bilinear interpolation of global solutions px and py to approximate pressure on the boundary of Ei Use Dirichlet boundary conditions using these pressure values end Solve pressure equation on Ei with selected boundary conditions Define a Cartesian grid Gi on Ei containing Ci as a cell Compute fluxes between cells of Gi and pressure values at cell centers of Gi Use fluxes and pressure values on Gi to compute new diagonal permeability tensor Ki for Ci end Solve generic flow problems on G to obtain pressure fields px and py Check for convergence done

4.2. Boundary Data for Local Solves. For each cell Ci of the CCAR grid G, we must solve local flow problems (in two dimensions) over the corresponding extended region Ei . In order to obtain the Dirichlet boundary data for each such problem, we use bilinear interpolation, with the interpolation points obtained from each of the global coarse-scale solutions computed during the previous iteration of local-global upscaling. While it is possible to use a higher-order interpolation scheme,


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it is not advisable to do so because the interpolant would exhibit oscillatory behavior, as opposed to the monotonicity we would expect from a pressure field. In Figure 4.4, a cell Ci is shown, along with its extended region Ei. For each of the points pij , j = 1, 2, . . . , 8 along the boundary of Ei , we find the three cells of G whose centers are as close to pij as possible, and form a triangle containing pij . We then use barycentric coordinates [22] to approximate the pressure at pij .

p

p

i1

p

i2

i3

E

i

C

i

p

p

i8

i

p

i4

p

i7

pi6

p

i5

Fig. 4.4. points pij , j = 1, 2, . . . , 8 on the boundary of an extended region Ei at which global coarse-scale pressure is interpolated. The region Ei corresponds to the cell Ci of the CCAR grid G whose center is pi .

4.3. Local Flow-based Refinement of the Base Grid. The fluxes across a face, from each global pressure solve, can be used to determine whether cells bordering the face should be refined. Specifically, let Fi be a face with area Ai , and let qi be the flux across the face obtained by a pressure solve on the CCAR grid with flow driven in the direction perpendicular to the face. If Q is the global flow, and A is the area of the face of the domain perpendicular to the direction of the flow, then we determine that the cells

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sharing the face should be refined if

|qi | > α

Ai A

Ai |Q|, A

(4.1)

where α(Ai /A) is a proportionality constant that depends on the ratio of areas, and Ai |Q|/A can be interpreted as the face’s “fair share” of the global flow. To choose α(Ai /A) > 1, we use the following heuristics. First, when Ai is the area of a face of a cell at the coarsest scale in the grid, then α(Ai /A) should be approximately 1, so that refinement will always take place except in areas that are receiving less than their fair share of flow. This low tolerance is necessary to avoid overlooking high-flow paths that occur within large regions that otherwise have very low flow. On the other hand, as Ai approaches the size of a face in the fine grid, then α(Ai /A) should be approximately A/Ai , so that refinement will not take place unless there is extremely high flow; that is, nearly all of the global flow is crossing the face Fi . This progression from low to high tolerance is used to avoid superfluous refinement while not missing any finer-scale regions of high flow. Under the assumption that α(Ai /A) = C(Ai /A)−p for some positive real number p, we obtain δ α(Ai /A) = (A/Ac )p

Ai A

−p

,

where Ac is the area of the largest face in the grid parallel to Fi , δ ≈ 1 is the chosen value of α(Ac /A), and p is given by

p=

A ln δA f Ac ln A f

,

where Af is the area of each fine grid face that is parallel to Fi . If the face Fi borders only two cells, then both are refined. Otherwise, the face borders two hanging cells and one non-hanging cell; only the non-hanging cell is refined (e.g., cell 1 in Figure 3.3). All refinement is in the direction that is orthogonal to the face. We use additional refinement criteria to reduce the number of “hanging” nodes


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that occur in high-flow regions. First, we apply the above flow-based criterion to each “half-face” that is shared by a hanging cell and a non-hanging cell. If the flow across this half-face is more than its fair share, then the non-hanging cell is refined along the direction that is orthogonal to the face. This refinement also takes place if too much of the flow across the entire face (75%, in practice) is through one hanging node. In the case of transmissibility upscaling, we also refine if the computed transmissibility across Fi is negative. This is discussed further in Section 5.1. If we are using extended local upscaling instead of local-global, we can use transmissibility instead of local flow, interpreting transmissibility as the potential flow across a face in order to anticipate changes in boundary conditions that can cause high flow to occur in regions for which high flow does not occur in the generic global flows. In this case, we use a modified version of the test (4.1). Let W be the size of domain along the dimension that is orthogonal to the face, and let Wi be the sum of the sizes of the cells bordering the face, along the same dimension. In the case of hanging nodes, the maximum such size is used. Then we refine the cells bordering the face if the upscaled transmissibility across the face, denoted by Ti , satisfies

|Ti | > α

Ai A

Ai W |Q|. A Wi

Once the grid is refined based on this criteria, we can repeat local-global upscaling on the updated grid. The result is an algorithm with an outer iteration and an inner iteration, where the inner iteration is a single application of local-global upscaling for CCAR grids, as described in Section 4.1, and the outer iteration consists of alternating upscaling and adaptation. The number of outer iterations is bounded above by the number of levels of refinement between the coarsest scale of the base grid, and the scale of the fine grid. Future work will include the development of a grid generation algorithm that combines the preprocessing algorithm of Younis and Caers from [21] with the criteria described in this section. This will require the formulation of criteria for coarsening cells as well as refining them. In addition, these adaptation strategies will have to be

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generalized to the multiphase case, keeping in mind that in gas injection processes, there tend to be paths of strong preferential flow that must be detected. Furthermore, for the multiphase case it is important to refine wherever velocity and saturation vary significantly, in the hope that we will not have to rely as much on pseudo functions. 4.4. Efficient Upscaling During Dynamic Adaptation. As the CCAR grid G is adapted, either as a result of excessive upscaling error or dynamic adaptation during transport, the hierarchy of permeability fields is examined to determine if permeability values have already been computed for cells that have been added to the grid. If so, then the new cells are simply populated with these values. Otherwise, we compute the needed values by upscaling to the new cells, as in the previously described algorithm for upscaling to the base grid. Permeability values in all other cells are fixed, and values in the new cells are computed in such a way as to preserve flow through the cells at the next finer scale that they contain. 5. Transmissibility Upscaling. We now discuss how we can upscale transmissibility to a CCAR grid. For the most part, the algorithm proceeds as in permeability upscaling, except that extended regions are face-centered instead of cell-centered. In this section, we focus on the details of computing the upscaled transmissibilities themselves, and how the stencil for the pressure solve on the CCAR grid is obtained from these transmissibilities. Referring back to Figures 3.3 and 3.4we choose the extended region E centered at the face for which we compute the flux to be large enough to contain the cells centered at the points c1 , c2 and c3 . We solve the pressure equation on E with Dirichlet boundary conditions as in permeability upscaling. The computation of the fluxes given by equation 3.3 from the local solution, along with the approximate pressure differences ∆pj = p(dj ) − p(cj ) obtained by averaging the local pressure field over the appropriate regions, can be used to compute the transmissibilities T2 and T3 corresponding to the fluxes q1 and q2 indicated in Figure 3.4. These transmissibilities are then used in the stencils centered at c2 and c3 . In the stencil centered at c1 , we use the transmissibility T1 = T2 + T3 .


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5.1. Local Flux-based Weighting of Transmissibilities. Since we are using two generic flow problems, we actually obtain two fluxes for each face. To reduce process dependency, we use both fluxes to compute W by solving an overdetermined linear system using the method of weighted least squares, where the weights are determined by the magnitude of the fluxes. This weighting prevents a flow that is nearly parallel to the face from unduly affecting the result. To describe this process in detail, we consider the simple case of a face shared by only two cells. Let q (1) and q (2) denote the fluxes across the face obtained from the two local pressure solves, and let ∆p(1) and ∆p(2) denote the differences in the cell averages of the pressure from these solves. From these values, we obtain two approximations for the transmissibility, T (j) = −

q (j) , j = 1, 2. ∆p(j)

(5.1)

We solve this system of two equations and only one unknown using weighted least squares, with the weights determined by the fluxes q (1) and q (2) . This approach yields the system W AT = W b for the unknown transmissibility T , where 

 q W =

(1)

0

0 q (2)



 ,



(1)



 ∆p  A= , (2) ∆p



(1)



 q  b = − . (2) q

It follows that the solution is

T =

AT W T W b [q (1) ]3 ∆p(1) + [q (2) ]3 ∆p(2) = − . AT W T W A [q (1) ]2 [∆p(1) ]2 + [q (2) ]2 [∆p(2) ]2

In the case of a face adjacent to hanging nodes, we apply this same approach to compute the transmissibilities across the half-faces. If either of the equations in (5.1) would yield a negative transmissibility, then it is not included in the computation of T . If both equations would yield negative transmissibilities, then we set T equal to the sum of the fine-scale transmissibilities along the face, and flag the cells for refinement.

22


F

d’ i

d

i

f

i

ci

c’ i

Fig. 5.1. The cells ci , di , c0i and d0i used to compute the pressure differences ∆pi and ∆p0i across the face fi that is a portion of the face F , the flux across which is used to compute the upscaled transmissibility across F .

We are able to prevent most instances of negative transmissibility from arising in the first place by computing flux across each face as follows. Assume that a face F consists of fine-scale faces fi , i = 1, . . . , n. We compute the flux using

q=

n X

Ti ∆pi ,

i=1

where, for each i, Ti is the transmissibility across fi , and ∆pi is the pressure difference across the face. To compute this pressure difference, we first use the pressure values in the two fine-scale cells ci and di bordering fi . Then, we compute a pressure difference ∆p0i using the pressure values in the two cells c0i and d0i that are one cell removed from fi along the direction orthogonal to fi . If ∆pi and ∆p0i have opposite sign, then we assume that ∆pi is an isolated fluctuation and compute the flux across fi using 31 ∆p0i ; otherwise we use ∆pi . The cells involved in the computation are shown in Figure 5.1.


23

6. Numerical Results. An implementation of the local-global upscaling algorithm for CCAR grids has yielded encouraging results. When upscaling the channelized domain shown in Figure 4.2 from a fine grid of 4096 cells to a CCAR grid containing 1345 cells, we obtain a good match with fine grid global flow for generic flow problems in both the x- and y-directions, whereas local-global upscaling only produced a good match in the x-direction, as indicated in Table 6.1. The global flow in the x- and y-directions are denoted by Qx and Qy , respectively. Algorithm and resolution Local-global on uniform grid, 1024 cells Local-global on CCAR grid, 1345 cells

% error in Qx

% error in Qy

1.78

8.62

2.29

1.02

Table 6.1 Percentage error in generic global flows Qx and Qy due to permeability upscaling of the fine field shown in Figure 4.2

6.1. Process Dependency. We now illustrate the reduced process dependency obtained by combining local-global upscaling with adaptivity. Using a base CCAR grid containing 445 cells, we solve the pressure equation with flow originating from a corner of the domain. We then compare the global flow to that obtained by solving the same problem using a Cartesian grid containing 512 cells using local-global upscaling. The results are listed in Table 6.2. Note that local-global upscaling on the Cartesian grid only yields high accuracy in one case, while the accuracy of local-global upscaling on the CCAR grid yields is much less dependent on the boundary conditions.

Corner Lower left Upper left Lower right Upper right

% error in global flow, 512-cell Cartesian grid 14.68 1.19 11.01 12.49

% error in global flow, 445-cell CCAR grid 6.95 2.55 4.71 6.27

Table 6.2 Percentage error in corner-to-corner flows due to permeability upscaling of the fine field shown in Figure 4.2

The upscaled permeability fields are illustrated in Figure 6.1. As can be seen in the

24


Cartesian grid, 512 cells

CCAR grid, 445 cells 8000

8000

6000

6000

4000

4000

2000

2000

Fig. 6.1. Left, upscaled permeability field generated by local-global upscaling on a 512-cell Cartesian grid. Right, upscaled permeability field generated by local-global upscaling on a 445-cell CCAR grid.

figure, the channels are more accurately resolved by upscaling on the CCAR grid. 6.2. Transmissibility Upscaling. We now demonstrate the accuracy of transmissibility upscaling on CCAR grids using a domain in which the channels from the domain in Figure 4.2 are extended by reflection across boundaries, and then rotated 45◦ . The new domain is shown in the left plot of Figure 6.2. In order to construct the base CCAR grid, we use a different approach from the leaky connected-set method from permeability upscaling. In this case, we flag each fine cell based on the maximum difference between its permeability and the transmissibilities across each of its faces (scaled appropriately if cell’s dimensions are not equal). We examine signed differences so that each contrast is only flagged once. Table 6.3 summarizes the error in global flow for various flow problems. As with permeability scaling, the use of a CCAR grid leads to greater accuracy in most cases, and much more consistent accuracy. Therefore, we have less process dependency than with local-global transmissibility upscaling on Cartesian grids. It should be noted that the results reported in the table for the Cartesian grid were computed using a modified version of local-global upscaling introduced in [1]. In the modified version, the flux-based weighting described at the end of Section 5 is used, which significantly reduces process dependency for Cartesian grids as well as CCAR grids.


6000 5000

25

0.9 0.8 0.7

4000 3000

0.6 0.5 0.4

2000 1000

0.3 0.2 0.1

Fig. 6.2. Rotated and extended permeability field (left), and refinement flags computed from transmissibility variations (right)

Flow direction In x-direction In y-direction From lower left From upper right

% error in global flow, 128-cell Cartesian grid 0.66 2.41 1.94 3.10

% error in global flow, 127-cell CCAR grid 0.85 0.38 1.31 1.93

Table 6.3 Percentage error in global flows due to transmissibility upscaling of the fine field shown in Figure 6.2

In addition to the global flow, we examine the velocity throughout the domain. Figure 6.3 displays the velocity of the fine-scale pressure field, and the pressure computed using upscaled transmissibilities on a 128-cell Cartesian grid. For the case illustrated, we use Dirichlet boundary conditions with flow coming only from the lower left corner. As can be seen in the figure, significant amounts of fluid flow in the wrong direction. This is particularly noticeable near the bottom of the domain, and near the center. Also, flow near the upper right corner of the domain is underestimated. Next, we perform the same comparison between fine-scale and coarse-scale pressure fields, where the coarse grid is a 127-cell CCAR grid. Although there is room for improvement, the flow computed on the CCAR grid mimics the flow computed on the fine grid with much greater accuracy, as can be seen by visual inspection of Figure 6.4. We then rotate the domain and repeat the comparison. The rotated fine-scale

26

M. GERRITSEN AND J. V. LAMBERS Fine scale velocity

Coarse scale velocity, 128−cell Cartesian grid

Fig. 6.3. Velocity vectors on 128-cell Cartesian grid, computed using pressure values computed at fine scale (left) and coarse-scale (right) Fine scale velocity

Coarse scale velocity, 127−cell CCAR grid

Fig. 6.4. Velocity vectors on 127-cell CCAR grid, computed using pressure values computed at fine scale (left) and coarse-scale (right)

permeability field is shown in Figure 6.5. In this case, we choose the initial grid to be a 4 × 4 Cartesian grid and then apply the grid refinement criteria described in Section 4.3 to obtain a CCAR grid with 49 cells, then 93 cells, and then again to obtain a final grid with 131 cells. Table 6.4 summarizes the error in global flow, compared to a 16 × 8 Cartesian grid.


27

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500

Fig. 6.5. Fine permeability field obtained by a 30-degree rotation of the field shown in Figure 6.2

Flow direction In x-direction In y-direction From lower left From upper left From lower right From upper right

% error in global flow, 128-cell Cartesian grid 2.30 6.61 5.93 18.59 44.55 5.23

% error in global flow, 131-cell CCAR grid 0.49 3.17 4.54 2.76 10.50 7.44

Table 6.4 Percentage error in global flows due to transmissibility upscaling of the fine field shown in Figure 6.5

Figures 6.6 and 6.7 show the velocity fields on both grids, for the case of flow in the x-direction. It can easily be seen that the fine-scale flow is captured much more accurately on the CCAR grid, even though it only has a few more cells than the Cartesian grid. We further demonstrate the accuracy of our algorithm on a modified version of SPE Layer 44 (see [23]), for which the fine-scale permeability field is shown in Figure 6.8. Table 6.5 summarizes the results for global flow, compared to a 32 × 16-cell

28



Fig. 6.6. Velocity vectors on 128-cell Cartesian grid, computed using pressure values computed at fine scale (left) and coarse-scale (right)

Fine scale velocity


Fig. 6.7. Velocity vectors on 131-cell CCAR grid, computed using pressure values computed at fine scale (left) and coarse-scale (right)

Cartesian grid. We see that in all cases, we obtain comparable accuracy with a CCAR grid that contains significantly fewer cells due to coarsening in low-flow regions.

In Figures 6.9 and 6.10, we compare the velocity fields obtained using these same grids. Because the CCAR grid depicted in Figure 6.10 provides increased resolution


29

10000 9000 8000 7000 6000 5000 4000 3000 2000 1000

Fig. 6.8. Fine-scale permeability field for modified version of SPE layer 44. The grid contains 256 × 64 cells, with ∆x = ∆y = 1.

Flow direction In x-direction In y-direction From lower left From upper left From lower right From upper right

% error in global flow, 512-cell Cartesian grid 6.24 0.34 0.23 13.82 1.84 1.72

% error in global flow, 389-cell CCAR grid 5.27 0.27 2.72 5.43 1.17 1.60

Table 6.5 Percentage error in global flows due to transmissibility upscaling of the fine field shown in Figure 6.8 with different gridding strategies

around the high-velocity paths, the essential features of the velocity field can be captured much more efficiently with a CCAR grid than with a Cartesian grid containing a greater number of cells. In Figures 6.11 and 6.12, it can be readily seen that the CCAR grid resolves the profile of the magnitude of the velocity field just as accurately as the denser Cartesian grid.

30



Fig. 6.9. Left plot: Velocity field for the fine scale solution of the pressure equation with permeability defined by a modified SPE layer 44, constant pressure on the left and right boundaries, and no flow across the top and bottom boundaries. Right plot: velocity field for the coarse-scale solution of the same problem, on a 32 × 16 uniform grid

6.3. Local Flux-Based Weighting of Transmissibilities. The original localglobal upscaling algorithm in [1] computed upscaled transmissibility across a face by considering only data obtained from a global flow driven by a pressure difference in the direction perpendicular to the face. We compare this approach to weighting the transmissibilities obtained from global flows in all coordinate directions according to the flux across the face induced by each flow. Figures 6.13 through 6.16 depict the velocity field for global flow in the x-direction, with permeability defined by the modified SPE layer 44 shown in Figure 6.8, for the fine grid and a 16 × 8-cell Cartesian grid. In Figures 6.13 and 6.15, the transmissibilities in each direction are obtained from the global flow in the same direction, whereas in Figures 6.14 and 6.16, the transmissibilities from all coordinate directions are weighted. It can be seen that weighting the transmissibilities leads to more accurate resolution of the magnitude of the velocity, whereas the directions of the velocity vectors in the high-velocity paths are very similar in both cases. 6.4. Combination of Local-Global Upscaling and Adaptivity. To demonstrate that both local-global upscaling and adaptivity contribute to accurate resolu-

INTEGRATING ADAPTIVITY WITH LOCAL-GLOBAL UPSCALING Fine scale velocity

31


Fig. 6.10. Left plot: Velocity field for the fine scale solution of the pressure equation with permeability defined by a modified SPE layer 44, constant pressure on the left and right boundaries, and no flow across the top and bottom boundaries. Right plot: velocity field for the coarse-scale solution of the same problem, on a 389-cell CCAR grid

tion of flow, we compare our algorithm to a modification of the algorithm in which extended local upscaling is used for each CCAR grid, as opposed to using extended local only for the first inner iteration with the initial grid. Table 6.6 lists global flow errors for both algorithms. In all cases with a significant amount of flow, local-global upscaling yields significantly more accuracy, even with fewer cells.

Flow direction In x-direction In y-direction From lower right From upper right

% error in global flow, extended local, 422-cell CCAR grid 7.18 1.15 4.19 3.31

% error in global flow, local-global, 389-cell CCAR grid 5.27 0.27 1.17 1.60

Table 6.6 Percentage error in global flows due to transmissibility upscaling of the fine field shown in Figure 6.8 with different upscaling methods

6.5. Convergence of Global Flow. In Figure 6.17, we demonstrate the rapid convergence of the global flow as the grid is refined, using the criteria defined in Section 4.3. We can see that for both generic global flow problems, with flow driven by pressure differences in the x-direction and y-direction respectively, convergence to

32


Fig. 6.11. Left plot: Magnitude of the velocity field for the fine scale solution of the pressure equation with permeability defined by a modified SPE layer 44, constant pressure on the left and right boundaries, and no flow across the top and bottom boundaries. Right plot: magnitude of the velocity field for the coarse-scale solution of the same problem, on a 32 × 16 uniform grid

the fine-scale global flow is obtained after just two passes of refinement. Furthermore, a subsequent local-global iteration on the final base grid converges immediately to the desired accuracy. The grids from the iterative refinement process are shown in Figure 6.18. 7. Discussion. In this paper, we present a novel approach that integrates grid adaptivity and local-global upscaling. The method generates accurate upscaled models of permeabilities or transmissibilities for flow simulation on highly adapted grids in heterogeneous reservoir formations. In the first step, an initial adapted grid is constructed from the given fine scale reservoir heterogeneity and potential flow paths using a method based on the leaky connected set approach by [21]. Then, the local-global upscaling method, for permeability or transmissibility, is applied to the initial grid. We then perform local refinements on the grid and apply local-global upscaling again, continuing this process until no more refinement is necessary. Each application of local-global upscaling consists of an inner iteration that is continued until consistency between the global solve on the adapted grid and the local solves is obtained. Generally, this takes only one or


33

Fig. 6.12. Left plot: Magnitude of the velocity field for the fine scale solution of the pressure equation with permeability defined by a modified SPE layer 44, constant pressure on the left and right boundaries, and no flow across the top and bottom boundaries. Right plot: magnitude of the velocity field for the coarse-scale solution of the same problem, on a 389-cell CCAR grid

two iterations, and hence the computational costs of this approach are low. In this process, the border region is held fixed for all levels of grid refinement to improve computational efficiency. Also, we designed a new flux-based weighting of transmissibilities which improves the local flow accuracy. We use a flow-based indicator to adapt the grid dynamically during the upscaling procedure to reduce flow solver errors. Future work will include the investigation of other possible adaptivity criteria, and alternative methods of handling occurrences of negative transmissibilities. The results are very encouraging, as indicated in the tables included in the previous section. We compared global flow errors for channelized systems, which are generally challenging for upscaling methods, produced by upscaled CCAR grids to errors produced by local-global upscaling on Cartesian grids of similar density. In [1] it was shown that the uniform local-global method on these channelized systems strongly outperformed local or extended local upscaling methods. For the generic boundary conditions used in the upscaling process the upscaled CCAR grids cut these global flow errors again, often as much as 50% or more. Process dependency is also reduced significantly. These results suggest that refinement is an effective means to control

34


Coarse scale velocity, unweighted T*

Fig. 6.13. Left plot: Velocity field for the fine scale solution of the pressure equation with permeability defined by a modified SPE layer 44, constant pressure on the left and right boundaries, and no flow across the top and bottom boundaries. Right plot: velocity field for the coarse-scale solution of the same problem, on a 16 × 8 uniform grid, without weighting of transmissibilities

upscaling errors and reduce process dependency. We are especially encouraged by the accuracy in the local flow. The combination of adaptivity and local-global upscaling can clearly retain important connected flow paths and accurately compute local flow directions and speeds. This is very promising for displacements with high mobility contrasts, such as gas injection processes, where the mobile fluid will seek these high permeability paths. We are implementing the CCAR methods in a compositional solver to further investigate this potential benefit. In the current method, we assume diagonal permeability tensors. The design of full tensor discretization methods on grids involving hanging nodes is very challenging and an area of active research. However, the results of this work suggest that full tensor permeabilities are perhaps not required to obtain accurate flow on non-aligned grids, provided that grid adaptivity is used in high flow areas, and an appropriate upscaling method that preserves connected flow paths, such as the one presented here, is used. We are currently conducting a comparative study to investigate this further. We are also investigating the use of alternative stencils based on multi-point fluxes for capturing full-tensor effects within both hanging and non-hanging nodes.

INTEGRATING ADAPTIVITY WITH LOCAL-GLOBAL UPSCALING Fine scale velocity

35

Coarse scale velocity, weighted T*

Fig. 6.14. Left plot: Velocity field for the fine scale solution of the pressure equation with permeability defined by a modified SPE layer 44, constant pressure on the left and right boundaries, and no flow across the top and bottom boundaries. Right plot: velocity field for the coarse-scale solution of the same problem, on a 16 × 8 uniform grid, with weighted transmissibilities

The upscaling approach presented here, while suitable for CCAR grids, can just as easily be applied to other types of grids. Future work will include generalization of our algorithm to other gridding strategies such as patched refinements, unstructured grids, or curvilinear grids. We feel that the proposed algorithm is a promising and natural approach for obtaining accurate adaptive coarse scale models needed to simulate flow and transport processes in highly heterogeneous reservoirs. 8. Conclusions. The following main conclusions can be drawn from this study: • For highly heterogeneous (e.g., channelized) systems, the proposed integration of grid adaptivity and upscaling is shown to consistently provide more accurate coarse scale models for global flow, relative to reference fine scale results, than do existing upscaling techniques, including local-global upscaling, applied to uniform grids of similar densities. As expected, transmissibility upscaling outperforms permeability upscaling. The upscaling method ensures consistency between the coarse and fine scale models. Adaptation allows the use of a coarse scale model that contains fewer grid points than would be

36


Fig. 6.15. Left plot: Magnitude of the velocity field for the fine scale solution of the pressure equation with permeability defined by a modified SPE layer 44, constant pressure on the left and right boundaries, and no flow across the top and bottom boundaries. Right plot: magnitude of the velocity field for the coarse-scale solution of the same problem, on a 16 × 8 uniform grid, without weighting of transmissibilities

required when using a Cartesian grid to achieve the same accuracy during simulation. • Apart from more accurate global flow results, the proposed method also computes improved local flow results. Connected high permeability flow paths are preserved, and local velocity directions and magnitude in these important paths are computed accurately. • We developed a refinement indicator based on local fluxes that guides the grid adaptation process during upscaling and further reduces solver and upscaling errors. • Process dependency is strongly reduced, that is, the approach computes accurate global flow results also for flows driven by boundary conditions different from the generic boundary conditions used to compute the upscaled parameters. • Process dependency can be reduced further, for both Cartesian and CCAR grids, by considering fluxes and pressure gradients computed from multiple local flow problems. The contribution of each flow problem to the transmis-


37

Fig. 6.16. Left plot: Magnitude of the velocity field for the fine scale solution of the pressure equation with permeability defined by a modified SPE layer 44, constant pressure on the left and right boundaries, and no flow across the top and bottom boundaries. Right plot: magnitude of the velocity field for the coarse-scale solution of the same problem, on a 16 × 8 uniform grid, with weighted transmissibilities

sibility across a cell face is weighted based on the corresponding flux. Acknowledgements. This work was supported by DOE Grant DE-FC26-04NT15530M001, Computer Modeling Group, and our SUPRI-C affiliates. The authors would also like to thank Jonas Nilsson and Rami Younis for their contributions of crucial code and helpful discussion.

REFERENCES [1] Chen Y, Durlofsky LJ, Gerritsen MG, Wen XH. A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Advances in Water Resources, 26:10411060, 2003 [2] Trangenstein J, Bi Z. Large Multi-Scale Iterative Techniques and Adaptive Mesh Refinement for Miscible Displacement Simulation. SPE/DOE Improved Oil Recovery Symposium, 2002, SPE75232. [3] Sammon, P H, ”Dynamic Grid Refinement and Amalgamation for Compositional Simulation”, SPE RSS, 2003, SPE 79683 [4] Gerritsen MG, Jessen K, Mallison BT, Lambers JV. A Fully Adaptive Streamline Framework for the Challenging Simulation of Gas-Injection Processes. SPE ATC, 2005, SPE 97270. [5] Caers J. Petroleum Geostatistics. SPE, 2005

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M. GERRITSEN AND J. V. LAMBERS Convergence of global flow 15

Q

x

10

5 coarse fine 0

1

2

3 iteration

4

5

50

Q

y

40 30 20 10

coarse fine 1

2

3 iteration

4

Fig. 6.17. Convergence of the global flow from the coarse-scale solutions to the pressure equation with permeability defined by a modified SPE layer 44, and generic boundary conditions. In both cases, the five outer iterations correspond to an initial iteration using extended local upscaling on a 32-cell Cartesian grid, three iterations of local-global upscaling on CCAR grids containing 32, 116, and 389 cells, respectively, followed by a final iteration of local-global upscaling on the 389-cell grid. Top plot: global flow in the x-direction. Bottom plot: global flow in the y-direction.

[6] Durlofsky LJ. Numerical Calculation of Equivalent Grid Block Permeability Tensors for Heterogeneous Porous Media. Water Resources Research, 27:699-708, 1991. [7] Wen XH, Durlofsky LJ, Edwards MG. Use of Border Regions for Improved Permeability Upscaling. Mathematical Geology, in press. [8] Pickup GE, Ringrose PS, Jensen JL, Sorbie KS. Permeability Tensors for Sedimentary Structures. Mathematical Geology, 26:227-50, 1994. [9] He C. Structured Flow-Based Gridding and Upscaling for Reservoir Simulation. PhD Thesis, Department of Petroleum Engineering, Stanford University. [10] Holden L, Nielsen BF. Global Upscaling of Permeability in Heterogeneous Reservoirs: the Output Least Squares (OTL) Method. Transport in Porous Media, 40:115-43, 2000. [11] Berger MJ, Oliger JE. Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations. Journal of Computational Physics, 53:484-512, 1984. [12] Gautier Y, Blunt MJ, Christie MA. Nested Gridding and Streamline-Based Simulation for Fast Reservoir Performance Prediction. SPE Reservoir Simulation Symposium, 1999, SPE51931 [13] Hornung R, Trangenstein J. Adaptive Mesh Refinement and Multilevel Iteration for Flow in Porous Media. Journal of Computational Physics, 136:522-545, 1997. [14] Bourgeat A. Homogenized Behavior of Two-Phase Flows in Naturally Fractured Reservoirs with Uniform Fractures Distribution. Computational Methods in Applied Mechanics and Engineering, 47:205-16, 1984

INTEGRATING ADAPTIVITY WITH LOCAL-GLOBAL UPSCALING Number of cells = 32


39


Fig. 6.18. Grids obtained by successive refinement, based on the distribution of the flux obtained from solutions of the pressure equation, with permeability defined by a modified SPE layer 44 and generic boundary conditions.

[15] Mittal R, Iaccarino, G. Immersed Boundary Methods. Annual Review of Fluid Mechanics, 37:239-61, 2005. [16] Berger MJ, Colella P. Local adaptive mesh refinement for shock hydrodynamics. Journal of Computational. Physics, 82:64-84, 1989. [17] Ham FE, Lien FS, Strong, AB. A Cartesian Grid Method with Transient Anisotropic Adaptation. Journal of Computational Physics, 179:469-494, 2002. [18] Nilsson J, Gerritsen MG, Younis RM. A Novel Adaptive Anisotropic Grid Framework for Efficient Reservoir Simulation. Proc. of the SPE Reservoir Simulation Symposium, 2005, SPE 93243 [19] Edwards MG. Elimination of Adaptive Grid Interface Errors in the Discrete Cell Centered Pressure Equation. Journal of Computational Physics, 126:356-372, 1996. [20] Henson VE, Yang UM. BoomerAMG: a Parallel Algebraic Multigrid Solver and Preconditioner. Applied Numerical Mathematics, 41:155-77, 2002. [21] Younis RM, Caers J. A Method for Static-based Upgridding. Proc. of the 8th European Conference on the Mathematics of Oil Recovery, Sept. 2002 [22] Watson DF. Contouring: A guide to the analysis and display of spacial data. Pergamon, 1994. [23] Christie MA, Blunt MJ. Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reservoir Evaluat Eng 2001;4:308-17.