Water Resources Research

Robust streamline tracing using inter-cell fluxes in locally refined and unstructured grids

Authors


Corresponding author: Y. Zhang, Department of Petroleum Engineering, Texas A&M University, Richardson Bldg. 401G, College Station, TX 77843, USA. (yanbinzhang@tamu.edu)

Abstract

[1] We present a comprehensive study of velocity interpolation methods in polygons. These methods are often used as postprocessing procedures for numerical schemes that do not directly calculate the velocity field but only provide cell boundary flux conditions, such as the finite volume schemes. These methods extend the widely used velocity interpolation algorithms, such as the Pollock's algorithm, to more complex geometries such as perpendicular bisection (PEBI) grids, unstructured triangular grids and grids with local refinement. Once the velocity field is interpolated, streamline trajectories and time of flight along the streamlines can be calculated for reservoir simulation, model calibration and waterflood management, for instance. These velocity interpolation methods assume known lower-order or higher-order cell boundary fluxes, which satisfy global mass conservation and normal flux continuity. However, they differ in the interpolation of velocities within the interior of the cells. The interpolating velocity may be locally conservative or nonconservative, continuous or discontinuous, lower order or higher order. Results show that the interpolated velocity field has to be locally conservative in order to guarantee the correct volumetric transformation for the calculated streamlines and the time of flight. Velocity continuity is not as important as local conservation for the purpose of streamline applications. Compared to higher-order interpolation for the streamline trajectories, lower-order interpolation has the advantage of an analytic solution and an efficient implementation. Based on our analysis, we recommend a lower-order locally conservative method for the most robust and numerically efficient calculation of streamline trajectories on unstructured grids.

1. Introduction

[2] Streamline based methods are widely used for various subsurface flow modeling problems, especially for advection-dominated displacements. To apply these methods it is necessary to construct streamlines and calculate the time of flight (TOF) along each streamline in an accurate and robust manner. This is usually achieved in three steps. First, a simulator or a numerical method is used to obtain the cell boundary face fluxes. Second, a suitable velocity field is interpolated throughout the entire computational domain. Third, streamlines are integrated using the velocity field, and the time of flight calculated along those trajectories. Of these three steps, the third step is the most straight-forward and utilizes either analytic solutions or Runge-Kutta techniques for the trajectory calculation. The first step is outside the scope of the current study. However, we will discuss enough of the flux calculation to lay a solid foundation for the discussion of the velocity interpolation models, which is the main subject of this paper.

[3] Unstructured grids are an important topic in reservoir simulation and three-dimensional (3-D) geologic modeling due to their flexibility especially for representing highly complex geologic structures and fluid flow near multilateral and fractured wellbore trajectories. In recent years, this research topic has received even more attention in the context of the emerging next generation reservoir simulators as well as the new data exchange standard for reservoir characterization, earth and reservoir models [King et al., 2012]. It is both theoretically important and practically necessary to develop velocity interpolation models which are applicable in unstructured grids. Some of the frequently encountered unstructured grids in field applications are shown in Figure 1 and will be discussed in detail later.

Figure 1.

Velocity interpolation problem description in (a) PEBI cells, (b) LGR cells, (c) nonmatching (faulted) cells, (d) point-distributed control volumes, (e) control volumes under CVFE scheme. Shaded areas in Figures 1d and 1e have constant rock properties.

[4] This paper is organized as follows. We begin with a literature review of the numerical calculation of flux and of velocity models. The next section is dedicated to a brief description of the problem to be solved, emphasizing the cell boundary conditions which then constrain the velocity interpolation discussions. The fourth section begins with an investigation of various velocity interpolation spaces in the unit square, and then discusses possible extensions of those velocity spaces to polygons. In the fifth section, several alternative methods based on subcell refinement are proposed. The rest of the paper evaluates the characteristics of the different velocity interpolation schemes and then demonstrates their application to several 2-D and 3-D reservoir modeling situations, including 2.5D PEBI grids, faulted grids, and grids with local refinement.

2. Literature Review

[5] We start with a review of the numerical methods which have been proposed to solve the flow equations for various kinds of unstructured grids. These calculations determine the flux on the boundaries of our unstructured polygonal elements. The numerical methods themselves also offer insight into simpler velocity interpolation methods previously developed on triangular and quadrilateral grids. We have considered three different numerical approaches: (1) Finite Volume (FV), (2) Galerkin Finite Element (FE), and (3) Mixed Finite Element (MFE).

[6] The FV schemes can be further categorized into the cell-centered, the point-distributed, and the control volume finite element (CVFE) schemes. The common features of these schemes are as follows.

[7] 1. The discretized system of equations is setup by applying mass conservation over the cells (called “mass conserving cells” in this paper) of the grid or the dual grid; and

[8] 2. The face fluxes of these mass conserving cells are approximated in various ways from the pressures at the center of these mass conserving cells. We consider globally mass conservative schemes for which the flux is continuous across the face.

[9] The distinction between the cell-centered and the point-distributed FV schemes is not essential, and they can be treated in a uniform manner as shown by Aavatsmark et al. [1998]. Both the cell-centered and the point-distributed FV schemes define the rock properties as piecewise constant over the mass conserving cells, which may be polygons. In contrast, the CVFE schemes define the rock properties as piecewise constant over the triangular or quadrilateral cells, where pressure is interpolated by piecewise linear or bilinear functions [Prakash, 1987; Forsyth, 1990]. For the CVFE schemes, the rock properties within a mass conserving cell (control volume) need not be homogeneous. The flux approximation for the cell-centered and the point-distributed FV schemes can be either two-point or multipoint. Usually the two-point flux approximation (TPFA) is used for structured corner point grids or perpendicular bisection (PEBI) grids (K-orthognal PEBI grids in general). For rigorous treatment of full tensor permeability and/or highly distorted grids, the multipoint flux approximation (MPFA) is necessary [Aavatsmark et al., 1998; Edwards and Rogers, 1998; Lee et al., 2002]. For each interface between two mass conserving cells, TPFA calculates one flux on the interface while MPFA calculates two half face fluxes. For the CVFE schemes, the flux approximation is obtained from the piecewise linear or bilinear pressure interpolation in each triangular or quadrilateral cell.

[10] The Galerkin FE schemes solve for pressure on the grid using a linear or bi-linear interpolation over the triangular or quadrilateral cells. The rock properties are assumed to be piecewise constant over the same cells. The MFE schemes solve simultaneously for both pressure and velocity.

[11] Of these numerical schemes, only the MFE schemes contain explicit velocity interpolation which may be used for streamline tracing. We will not discuss this particular application since it has already been extensively studied by other authors, e.g., Darlow et al. [1984] and Matringe et al. [2007]. However, the velocity interpolation methods developed in the MFE schemes provide a foundation for the current paper. We will focus on the other numerical schemes, for which the velocity field is not directly available but needs to be interpolated afterward.

[12] Pollock [1988] proposed a velocity interpolation method for rectangular cells constrained by the cell face fluxes. This was later extended to corner point cells by other authors [Cordes and Kinzelbach, 1992; Prévost et al., 2001; Jimenez et al., 2007] on the basis of the lowest-order Raviat-Thomas (RT0) space and an isoparametric mapping. We will call this the Extended Pollock Method (EPM). Hægland et al. [2007] developed a Corner Velocity Interpolation (CVI) method for corner point cells constrained also by the cell face fluxes. The CVI method addressed the issue of reconstructing uniform flow, removing a grid orientation effect for corner point grids from the previous methods. Matringe et al. [2006] introduced the Brezzi-Douglas-Marini space of order one (BDM1) for streamline tracing in order to improve accuracy. This velocity interpolation is of a higher order than the RT0 space and has been shown to work with MPFA FV schemes which calculate half face fluxes for triangles or quadrilaterals [Matringe et al., 2008].

[13] All the above velocity interpolation methods have been developed for triangular or quadrilateral cells in 2-D and tetrahedral or hexahedral cells in 3-D. When the mass conserving cells are of these geometries just mentioned, they can be applied directly to interpolate the velocity field within each cell. However, the following two cases make things more complicated.

[14] 1. The face fluxes originally obtained for triangular or quadrilateral cells are not continuous across the cell faces;

[15] 2. The mass conserving cells, with which the face fluxes are associated, are not triangles or quadrilaterals but n-polygons ( inline image).

[16] In the first case, postprocessing is necessary in order to recover global mass conservation and flux continuity. For example, using the Galerkin FE numerical scheme the velocity obtained by directly taking the gradient of the pressure field will not satisfy flux continuity. This velocity field, if unprocessed, will give poor streamline tracing results due to spurious sources and sinks introduced on the cell faces. Cordes and Kinzelbach [1992] discussed this problem and proposed a postprocessing procedure. The idea is to first introduce more degrees of freedom by dividing cells into subcells and then using mass conservation, flux continuity, and irrotationality in a local “patch” to constrain the newly introduced degrees of freedom. For triangular based grids, the velocity interpolation they used was piecewise constant over subtriangles. Such a velocity model does not have enough degrees of freedom to represent compressible flow, which is required in general. For quadrilateral based grids, subquadrilateral face fluxes are reconstructed first and then the velocity can be interpolated by the EPM. Later Prévost et al. [2001] pointed out that the subtriangular construction cannot be applied to triangular based grids under the point-distributed FV schemes since the rock properties may no longer be constant within a triangular cell. They proposed to use subquadrilaterals rather than subtriangles for triangular based grids, which leads to a construction similar to the CVFE scheme. They also demonstrated that for divergence free velocity fields their subquadrilateral construction will reproduce exactly the same exit point as the subtriangular construction. However, they did not discuss the case of compressible flow. Sun et al. [2005] proposed a general velocity postprocessing approach based on the Gauss-Seidel method to recover local mass conservation and flux continuity simultaneously. The method can work directly on the original grid of the Galerkin FE scheme instead of the dual grid, but it requires solving a global optimization problem by iteration. It also requires a priori knowledge of the source terms, which may be difficult to obtain accurately for compressible, multiphase flow.

[17] The second case (i.e., mass conserving and flux continuous n-polygons) is the focus of this paper. It occurs when using cell-centered FV scheme for PEBI grids or grids with faults and/or local grid refinement (LGR). It also occurs when using point-distributed FV scheme or CVFE schemes for unstructured triangular or quadrilateral grids. There are basically two strategies to solve this problem:

[18] 1. Finding direct extensions of the existing velocity interpolation spaces to n-polygons;

[19] 2. Dividing the n-polygons into subtriangles or subquadrilaterals.

[20] Recently Rasmussen [2010] attempted the first strategy by extending CVI to polygonal cells. The previous work of Cordes and Kinzelbach [1992] and Prévost et al. [2001] have already shed some light on the second strategy. The current paper will systematically investigate both strategies and propose the most robust and efficient methods to be used under different circumstances based on both theoretical analysis and numerical experiments.

3. Problem Description

[21] Let us consider a polygonal shaped mass conserving cell which arises in the discretization of some FV scheme. The Galerkin FE scheme can be treated in the same way as CVFE scheme just by introducing control volumes. The numerical solution of the flow equations will provide fluxes on the cell interfaces. This flux information is global because the cell interface fluxes satisfy conservation for each mass conserving cell and continuity across neighboring cells. The velocity interpolation methods, on the other hand, only work locally within each mass conserving cell, consistent with these fluxes. This means that the global cell face flux information serves as known boundary conditions for the velocity interpolation of a particular mass conserving cell.

[22] As just mentioned, the boundary conditions provide knowledge of the face fluxes. When we utilize lower-order boundary conditions, we mean that flux along the boundary face is uniform, i.e., the normal component of the velocity is characterized by a single average value. When we utilize higher-order boundary conditions, the normal component of the velocity varies linearly, i.e., it is characterized by two variables: the average and the slope of variation. (Note that in 3-D there is a distinction between the same boundary conditions in the physical cell and in the isoparametric reference cell, but in 2-D they are equivalent [Hægland et al., 2007].) In particular, a lower-order boundary condition can be viewed as a constrained higher-order boundary condition with zero slope of the linear variation. Thus a general discussion can be made assuming higher-order boundary conditions; the results for lower-order boundary conditions arise as a special case.

[23] The boundary conditions can be equivalently expressed by corner velocities instead of fluxes for the nondegenerate cases. As shown in Figure 2, if the angle between two adjacent faces is different from inline image, the face fluxes can be used to uniquely define the corner velocities. When the angle between two adjacent faces is close to inline image (degenerate case), the corner velocity will not be well defined. The degenerate case may occur for grids with local grid refinement and faults, as well as CVFE grids. The unrobustness caused by the degeneracy of the physical cell will be addressed in section 4.2 and 4.3.

Figure 2.

The degrees of freedom for a polygonal cell. The degrees of freedom may be associated with the average flux and the slope of linear variation of flux for each face (blue). Or equivalently they may be associated with the components of corner velocities (red).

[24] It is worth mentioning that Matringe and Gerritsen [2004] have proposed a local refinement tracing method where the boundary conditions on the subgrid are determined using the fluxes of the neighboring coarse blocks with a “slope limiter.” This will generate a higher-order boundary condition for the coarse blocks without using a higher-order numerical scheme for the fluxes. If a similar method was developed for unstructured grids, it would be immediately applicable to our analysis since we start with the known boundary conditions.

[25] The various situations that are frequently encountered in practice are depicted in Figure 1 and explained as follows. As mentioned earlier, TPFA calculates one average flux for each interface between two cells. This is most often seen in simulations using PEBI grids, grids with LGRs, and grids with faults as shown in Figures 1a, 1b, and 1c. The point-distributed FV schemes for unstructured grids usually require MPFA, which is able to calculate two half face fluxes. As shown in Figure 1d, the interface between two mass conserving cells is part of a nonconvex polygon and consists of two straight lines. Thus for the polygonal cell, only one average flux is known for each edge of the polygon. The situation of the CVFE schemes are similar to that of the point-distributed FV schemes as shown in Figure 1e. The only difference is that the CVFE scheme is based on heterogeneous rock properties within the polygonal cell. Note that all the situations depicted in Figure 1 involve only lower-order boundary conditions. Thus for practical purposes and simplicity, the test cases and examples in this paper assume lower-order boundary conditions. However, the theory and the methods developed in this paper are not restricted to lower-order boundary conditions.

[26] Now that we have discussed the boundary conditions, the problem we are trying to solve becomes easier to state. The local velocity field within a cell is not determined uniquely by the boundary conditions. It also depends on the local velocity interpolation method used, which will make assumptions of the functional form of the local velocity field as well as certain properties that it should satisfy. As a basic requirement, the velocity interpolation method should provide sufficient degrees of freedom in order to be able to honor the boundary flux conditions; on the other hand, it should not introduce too many additional degrees of freedom without physically meaningful constraints on the velocity field to obtain a robust solution.

[27] The following discussion will focus on 2-D situations. The extension to 3-D is straightforward in some cases but may require additional study for some classes of 3-D unstructured grids.

4. Velocity Interpolation Spaces

[28] In this section, we will discuss the first strategy mentioned in the literature review, namely finding direct extensions of the existing velocity interpolation spaces to n-polygons. We will begin with a discussion in the unit square.

4.1. Velocity Interpolation Spaces in the Unit Square

[29] As shown by Figure 3, in the unit square the eight degrees of freedom can be equivalently associated with the eight half face fluxes or the eight corner velocity components. In order to treat the RT0, CVI, and BDM1 spaces in a uniform manner, we will use the eight corner velocity components as degrees of freedom in the following discussions.

Figure 3.

The degrees of freedom for the unit square.

4.1.1. RT0 Space

[30] The lowest-order shape function associated with inline image and inline image in Figure 3 is

display math

It is uniform on the face. The normal component of the velocity vanishes on all other faces (Figure 4a). The divergence of this shape function is simply 1. Such a velocity field is locally conservative, but is also sufficiently general to represent compressible fluid flow. The other three shape functions may be obtained by rotation and reflection symmetry. For example, by rotation symmetry, the shape function associated with inline image and inline image in Figure 3 is inline image. Then the other two shape functions can be obtained by reflection symmetry, specifically substituting inline image with inline image and inline image with inline image to obtain inline image and inline image. The velocity field that can be represented in this basis can be written as the linear combination of the four shape functions:

display math

For the RT0 space, we have inline image and similar relations on the other three boundary faces. The above velocity field can be simplified as

display math
Figure 4.

(a) A shape function of the RT0 space in the unit square; (b) a higher-order shape function of the CVI space in the unit square; (c) a higher-order shape function of the BDM1 space in the unit square; (d) a lower-order shape function of the CVI space in the unit pentagon; (e) a higher-order shape function of the CVI space in the unit pentagon; (f) the higher-order shape function of the BDM1 space in the unit pentagon is not known.

4.1.2. CVI Space

[31] The CVI space consists of the RT0 space with additional higher-order shape functions. For instance, the higher-order shape function, orthogonal to the RT0 basis but associated with inline image and inline image in Figure 3, is

display math

Figure 4b shows this shape function. The total flux through the face vanishes but the local velocity does not. The divergence of this shape function is calculated to be inline image, which is not constant in the cell. This means that this interpolation space is not locally conservative. The upper half of the cell acts as a source while the lower half of the cell is a sink. The other three higher-order shape functions can be obtained easily by reflection and rotation symmetry. The velocity field interpolated is the linear combination of all the shape functions:

display math

This can be rearranged into the following with only linear and bilinear terms:

display math

4.1.3. BDM1 Space

[32] Similar to the CVI basis, the higher-order shape function associated with inline image and inline image in Figure 3 is:

display math

Figure 4c shows this shape function. The divergence of this shape function is identically zero; this interpolation space is locally conservative. Compared with CVI space, the addition of the quadratic term inline image, which vanishes at the four corners, makes the interior velocity field locally conservative. The other three higher-order shape functions can be obtained easily by reflection and rotation symmetry. The velocity field interpolated is the linear combination of all the shape functions:

display math

This can be rearranged into the following with linear, bilinear, and quadratic terms:

display math
display math

The BDM1 space can be viewed as the CVI space plus an extra term which has zero contribution at the corners but cancels out the variation of divergence in the interior.

4.2. Isoparametric Mapping

[33] An isoparametric mapping is used to define a corner point simulation grid as a mapping from a unit square (or cube in 3-D) to a general cell defined by the location of its corners in 2-D or 3-D. The following discussion is quite general. It can be applied to n-polygons and is not restricted to the unit square. Suppose that the corners of the reference cell are inline image, and the isoparametric mapping maps the reference cell corner inline image to the corner inline image of the physical cell. A point inline image in the reference cell can be written as

display math

where inline image are generalized barycentric coordinates [Wachspress, 1975; Floater et al., 2006]. Then the point image of inline image in the physical cell can be written as

display math

The above equation can be seen as the coordinate transformation from inline image to x (see Figure 5 as an example for pentagons). The velocity transformation usually adopted is called the Piola transformation, which has the property of preserving boundary flux [Brezzi and Fortin, 1991]. According to the Piola transformation, the velocity inline image at point inline image in the reference cell can be transformed to the velocity u at point x in the physical cell by

display math

where the Jacobian matrix J is defined as

display math

and is calculated from equation (12).

Figure 5.

Isoparametric mapping for a pentagon.

[34] The equations of motion for streamline trajectories in the physical cell are

display math

Converting into the reference cell we have

display math

or

display math

We can introduce the time variable in the reference cell T, related to the physical time of flight using

display math

In the reference cell, the trajectory is now easy to calculate

display math

The Jacobian is a known function in the reference space. Once the trajectory in reference space is calculated as a function of T, then equation (18) can be integrated to determine the actual time of flight across the polygonal cell. Using equations (12), (18), and (19) is more robust than using equations (13) and (15) for streamline tracing. The reason is that if the physical cell is degenerate or nonconvex, the Jacobian may become zero or negative, and consequently equation (13) may not be well defined. However, equations (12), (18), and (19) will still remain applicable. The interpolation of velocity in the reference space, together with the technique shown here, makes the RT0 and BDM1 velocity interpolation methods robust for both nonconvex and degenerate quadrilaterals.

4.3. Velocity Interpolation in Polygons

[35] In this section we will seek direct extensions of the previously discussed continuous velocity spaces, i.e., the CVI space and the BDM1 space, to n-polygons inline image. Such extensions should produce continuous velocity field interpolations in n-polygons just as they do for triangles and quadrilaterals. This is desirable for cells with homogeneous rock properties, which is almost always the case for triangles and quadrilaterals but only true for polygons under the cell-centered or the point-distributed FV schemes.

4.3.1. CVI in Polygons

[36] Rasmussen [2010] recently proposed an extension of the CVI method to convex polygons. The interpolation can be written as

display math

where the weights inline image are given by generalized barycentric coordinates in the convex n-polygon, e.g., the Wachspress coordinates [Wachspress, 1975]. The interpolation is directly carried out in the physical cell without the application of an isoparametric mapping. The limitation of this extension is that it cannot be applied to degenerate or nonconvex polygons. First, in degenerate or nonconvex polygons the generalized barycentric coordinates inline image are not well defined. Second, in degenerate cells, some corner velocities may not be well defined, as shown in section 3. We must point out that Rasmussen's CVI method does not reduce to Hægland's CVI method [Hægland et al., 2007] for the case of n = 4.

[37] The original CVI method proposed by Hægland et al. [2007] for quadrilaterals can be expressed as

display math

A direct extension of this equation to n-polygons should be

display math

Note that the weighting functions inline image are the generalized barycentric coordinates in the reference cell rather than the physical cell. Please compare equation (20) with equation (22) to see why Rasmussen's extension does not reduce to Hægland's CVI method. Because the reference cell is always a convex polygon, the weighting functions inline image are always well defined even if the physical cells are degenerate or nonconvex. This is why Hægland's CVI method does not require the cell to be convex. However, this method still fails when the polygon has degenerate adjacent faces because of the inability to calculate the corner velocity from the face flux just as with Rasmussen's method. In addition, to apply equation (22), it is necessary to map points from the physical cell back to the reference cell, i.e., solve for inline image from equation (12). This requires numerical iterations which is computationally expensive.

[38] We propose a new robust CVI method which is applicable for both degenerate and nonconvex polygons. This new method interpolates the velocity field in the reference cell rather than in the physical cell:

display math

Using the isoparametric mapping technique developed in section 4.2, we will trace streamline trajectories in the reference cell and then map them to the physical cell. Because the velocity interpolation and integration are both carried out in the reference cell, this method will be robust regardless of the geometry of the physical cell. Moreover, unlike Hægland's CVI method, this method does not require the expensive back-mapping from the physical cell to the reference cell.

[39] We have presented three different CVI methods for polygons. Rassmussen's method has the simplest form without any notion of isoparametric mapping, but it cannot be applied to degenerate and nonconvex polygons. Hægland's method interpolates velocity in the physical cell but calculates the weights in the reference cell. It is not robust for degenerate polygons and requires expensive back-mapping from the physical cell to the reference cell. We propose a new robust method applicable for both degenerate and nonconvex polygons which avoids the expensive back-mapping. Both the Rasmussen and the Hægland methods will preserve a uniform velocity, which is the intent of their constructions. The isoparametric extension of the CVI method, equation (23), need not have this property. Uniform velocities will be preserved if the Jacobian of the isoparametric mapping belongs to the space spanned by the nodal basis functions, inline image. Otherwise the direction of uniform velocities is preserved, but not the magnitude.

[40] There are two major drawbacks of all three CVI methods. First, they are not locally conservative, the consequences of which will be discussed in section 6.1. Second, in general, the numerical computation of the trajectory cannot be obtained analytically and requires a more expensive numerical integration.

4.3.2. BDM1 in Polygons

[41] As we have already shown in the unit square (section 4.1), the BDM1 space can be seen as the CVI space with an additional velocity term which has zero contribution at the corners but cancels out the variation of divergence in the interior. Figures 4a and 4b show the shape functions of the CVI space and BDM1 space in the unit square. The mass conserving term provides a circulation in the BDM1 space which is not present in the CVI space. As an example, Figure 4d and 4e show the lower-order and the higher-order shape functions of the CVI space in the unit pentagon. They are similar to Figure 4b in that the velocity direction is maintained but the magnitude of the velocity reduces away from the corners. For n-polygons, the basis functions are rational functions of polynomials. The divergence of such a velocity field is not constant. By analogy, the shape function of the BDM1 space in the unit pentagon should have an extra term which has zero contribution at all five corners but cancels out the variation of divergence in the interior of the pentagon. For pentagon and n-polygons inline image in general, such a velocity field has not been constructed and is unlikely to have a simple analytical expression. In other words, a direct extension of BDM1 to arbitrary polygons is not known.

5. Subcell Refinement Methods

[42] The CVI schemes provide a method for smooth interpolation of properties within polygons. However, if we wish to work with conservative velocity fields, then no solution is known for n-polygons once inline image. In such a case, we may refine the polygon into either triangles or quadrilaterals, and then use conservative schemes on the subcells. The problem then becomes one of flux reconstruction on the subcells, knowing the flux on the boundary of the polygon. This is the approach that we will study in this section.

[43] In section 3 we discussed the various situations where polygonal cells arise. For the situations shown in Figure 1d and 1e, the n-polygon has already been naturally refined into quadrilaterals by the faces of the triangular grid cells, although a triangular refinement may also be constructed. For the situations shown in Figure 1a, 1b, and 1c, there is no natural refinement and the n-polygonal cell can be either refined into n quadrilaterals or n triangles (see Figure 6 as an example). We will focus our discussion on a 2-D PEBI cell to illustrate both quadrilateral refinement (QR) and triangular refinement (TR) (Figure 6). Instead of seeking a direct interpolation of velocity from the known boundary conditions to every point inside the polygon, the strategy is to divide and conquer by interpolating in each subcell separately. The boundary conditions for each subcell then have to be constructed from the boundary conditions on the polygon.

Figure 6.

Subcell refinement with (a) quadrilaterals and (b) triangles. Arrows show the positive flux directions.

[44] Let us consider quadrilateral refinement first. As in our discussion of shape functions, let us suppose each quadrilateral is associated with 8 degrees of freedom in general. Thus there are 8n degrees of freedom in the representation. The normal projection of the velocities on the boundary of the polygon are known. This means that for the 2 faces of each quadrilateral which are on the boundary of the polygon, 4 degrees of freedom are specified. Flux continuity across the newly introduced n internal faces will constrain another 2n degrees of freedom. Therefore, there are 2n degrees of freedom remaining. If we choose to constrain the velocity model to the lowest order, then the variation of the normal velocity on each internal face vanishes, providing n additional constraints and reducing the problem to n degrees of freedom.

[45] For triangular refinement, each triangle has 6 degrees of freedom, with 6n degrees of freedom in total. There are 4 degrees specified on the polygon boundary for each triangle. Total flux continuity across the newly introduced internal faces will constrain another n degrees of freedom, reducing the problem to n degrees of freedom.

[46] According to this analysis, only the case of quadrilateral refinement with higher-order (BDM1) internal face fluxes provides 2n degrees of freedom. Otherwise, only n degrees of freedom are available. For the schemes with n degrees of freedom, we will show that there is a competition between continuity of the solution and whether the velocity field is locally conservative. The higher-order quadrilateral refinement, with 2n degrees of freedom, is able to satisfy both requirements. This is explained in more detail in the remainder of this section.

5.1. Local Conservation (LC)

[47] First we will discuss in general the constraints which are imposed by the requirement of local conservation on the remaining degrees of freedom. There is no restriction of our construction to incompressible flow. The total source of the polygonal cell is given by

display math

where the inline image are the known polygon boundary face fluxes. The positive directions are outwardly directed, as shown in Figure 6. Assuming a constant velocity divergence, i.e., local conservation, then the source term for subcell i should be

display math

where inline image, inline image, and Ai are the total compressibility, the porosity and the area of subcell i. Therefore, the conservation residual for each subcell is

display math

for quadrilateral refinement, and

display math

for triangular refinement. The inline image and inline image are the half face fluxes for the ith polygonal boundary face (we have inline image); the Qi are the fluxes of the internal faces whose positive directions are counterclockwise as shown in Figure 6. Because inline image, identically, local conservation provides only n – 1 independent constraints:

display math

5.2. Closure Constraint

[48] Local conservation only provides n – 1 constraints (equation (28)). An additional closure constraint is needed in order to complete the solution of the Qi. There are three existing options to formulate the closure constraint, which will be summarized as follows.

5.2.1. Option 1

[49] This option assumes irrotationality of the velocity field:

display math

The above equation can be formulated in a weak sense

display math

The closed loop for the integration is usually chosen to run through the cell and face centers. This option is simple to implement because it does not refer to any rock properties.

5.2.2. Option 2

[50] This option assumes the existence of a local potential function, the pressure P, which means

display math

In combination with Darcy's Law, this equation can be formulated in a weak sense as

display math

with the same closed loop as before. This was first used by Cordes and Kinzelbach [1992] and later by Prévost et al. [2001] and often referred to as the “irrotationality” condition in the literature. It reduces to the first option for constant scalar rock permeability. If the heterogeneity or the anisotropy of the permeability is significant, equation (32) should be applied in preference to equation (30).

5.2.3. Option 3

[51] This option, proposed by Datta-Gupta and King [2007], is equivalent to a finite difference form of equation (32). It utilizes a local pressure solution, which provides easy extension to 3-D and to compressible flow. A similar construction has recently been applied by Kachuma [2010] to trace streamlines in nonmatching grids across faults. If the principal axes of the permeability tensor are aligned with the subgrid axes, then a two point flux approximation is adequate. Otherwise a multipoint flux approximation should be considered. Let Pi be the pressure at the center of each subcell, and let Ti be the two-point transmissibility between subcell i and i + 1. Then the inner face flux from subcell i to i + 1 is related to the pressure difference according to

display math

Divide by Ti and sum over i, and the total pressure drop must vanish, which gives

display math

Equation (34) is used as the final constraint to supplement equation (28).

5.3. Higher-Order Quadrilateral Refinement, BDM1(QR)

[52] As we have already mentioned, higher-order quadrilateral refinement, BDM1(QR), with 2n degrees of freedom, is able to satisfy local conservation and produce smooth velocity fields at the same time. We will first impose the n – 1 local conservation constraints discussed in section 5.1 and one closure constraint discussed in section 5.2. The remaining n degrees of freedom can be further reduced to only 2 if we impose a single velocity at the center point, shared by all the subquadrilaterals. We calculate this center velocity using one of the CVI schemes discussed in section 4.3, usually equation (23). Like the CVI methods, this scheme provides a smooth interpolation of properties within polygons. Unlike the CVI methods, it produces locally conservative velocity fields, which are advantageous in many applications, as we will show.

[53] This scheme is the best technical solution as it satisfies all of our stated requirements. However, it has two disadvantages. First, it requires numerical calculation of the streamline trajectories. This is in contrast to lower-order schemes where the trajectories may be integrated analytically. Second, it imposes a continuous solution when this may not be appropriate. If either the subgrid boundary conditions or the subgrid properties are heterogeneous, then the lower-order scheme may provide a more physically reasonable solution. For both these reasons, we continue our analysis by examining lower-order velocity interpolation schemes.

5.4. Lower-Order Quadrilateral Refinement, RT0(QR)

[54] As discussed earlier, the lower-order schemes, based on the RT0 basis, imposes a uniform normal velocity across each internal face and thus reduces the problem from 2n to n degrees of freedom. If we choose to satisfy the n – 1 local conservation constraints first, the remaining degree of freedom is not sufficient to provide a smooth velocity field. Otherwise, if we choose to satisfy continuity of the solution first by imposing a single velocity at the center point shared by all the subquadrilaterals, the n degrees of freedom will be reduced to only 2, which are not sufficient to satisfy the n – 1 local conservation constraints. The competition between the two requirements leads to either a local conservation method or a continuous velocity method as follows.

5.4.1. Local Conservation (LC)

[55] If the polygonal cell properties are only piecewise constant on the subquadrilaterals, or if the flux boundary conditions are also heterogeneous, then the requirement of smooth velocities becomes unnecessary. In this case, a lower-order scheme is adequate to satisfy local conservation, without constraining the solution to be too smooth. We may also choose to work with a lower-order scheme in order to simplify the implementation and increase numerical efficiency. Specifically, if the polygonal boundary flux conditions are themselves of lower order, then utilizing a high-order velocity for interpolation shows no advantage in the tests we have performed. Also, use of the RT0 interpolation in each subquadrilateral provides an analytic solution for both streamline trajectory and the time of flight calculation.

[56] We first apply the n – 1 local conservation constraints. The remaining degree of freedom may be constrained by any of the closure constraints discussed previously in section 5.2. In addition to these three options, we propose an additional option (option 4). The last degree of freedom (e.g., Q1) can be resolved by minimizing the velocity variance at the center node, i.e., maximizing velocity continuity.

display math

where vci is the velocity at the center point in the ith subcell and inline image is the average of these velocities. As with option 1, this is expected to work best on problems with no subcell heterogeneity. For higher-order quadrilateral refinement, this residual vanishes, although not for the lower-order scheme.

5.4.2. Continuous Velocity (CV)

[57] We impose a single velocity at the center point shared by all the subquadrilaterals. This reduces the n degrees of freedom to only 2. The velocity at the center point can then be specified by applying one of the CVI methods (option 1), as was done for higher-order quadrilateral refinement. Alternatively, one can construct the center velocity by minimizing the conservation error (option 2), which is expressed as

display math

5.5. Triangular Refinement, BDM1(TR)

[58] Triangular refinement is based on the BDM1 basis, but is more tightly constrained by the boundary velocities than quadrilateral refinement, providing only n degrees of freedom. Hence, the analysis of triangular refinement is similar to that of lower-order quadrilateral refinement. Therefore, either local conservation or the continuous velocity constraint can be applied with triangular refinement, but the satisfaction of both requirements is not possible with triangular refinement.

5.6. Summary

[59] At this point we have described a large number of possible interpolation schemes, as summarized in Table 1. The four major categories are corner velocity interpolation (CVI), higher-order quadrilateral refinement (BDM1(QR)), lower-order quadrilateral refinement (RT0(QR)), and triangular refinement (BDM1(TR)). Additional categories distinguish between local conservation (LC) and continuous velocity (CV), and as discussed, each scheme may require specific closure options. We will now test the performance of these schemes against an increasingly complex series of test cases, in section 6.

Table 1. Descriptions of the Velocity Interpolation Schemes
SchemesCVIBDM1(QR)RT0(QR,LC)RT0(QR,CV)BDM1(TR,LC)BDM1(TR,CV)
Cell RefinementNoneQuadrilateralQuadrilateralQuadrilateralTriangularTriangular
BasisCVIBDM1RT0RT0BDM1BDM1
Continuous VelocityYesYesNoYesNoYes
Local ConservationNoYesYesNoYesNo
Number of Options334242

6. Results and Discussions

6.1. Single Polygonal Cell

[60] The single polygonal cell used for the test is a regular convex octagon, with incompressible flow coming in from three adjacent boundary faces and going out of the remaining five boundary faces. We will impose uniform (lower order) flux on each face. The flux is 5 for each inflow boundary face, and 3 for each outflow boundary face. The launching points of the streamlines are at the three inflow boundary faces and distributed uniformly.

[61] The three different CVI schemes (see section 4.3) are in this case identical because of the regularity of the test cell. They will generally give slightly different results for irregular convex polygons, but their major difference is in the robustness for degenerate and nonconvex polygons, which has already been discussed.

[62] Different options of the closure constraint should produce almost identical results if the cell permeability is a scalar constant. However, if the cell permeability is anisotropic ( inline image), the anisotropy will contribute to the formulation of options 2 and 3, but not to options 1 and 4. As an example, Figure 7 shows the streamline trajectories using the RT0(QR,LC) scheme with option 3 with inline image (Figure 7a), option 3 with inline image (Figure 7b), and option 4 (Figure 7c). Even with strong anisotropy, we see negligible differences between the different closure constraints for these conservative methods. Because of the existence of an underlying stream function, which is identical on the boundary of the polygon for all of these three cases, the exit point of each trajectory is uniquely specified by its entry point. However, if the cell permeability is not constant, i.e., the subcells have strong contrasts in permeability, then the face fluxes will not be uniformly distributed and either option 2 or 3 should be used for velocity reconstruction as then neither velocity irrotationality nor a requirement of velocity continuity would have a physical motivation.

Figure 7.

The streamline trajectories of the RT0(QR,LC) with (a) option 3 with inline image, (b) option 3 with inline image, (c) option 4.

[63] In Figure 8 we examine the slightly more complicated case of interpolation on an irregular convex polygon. For the RT0(QR,CV) method, the two options of interpolating the center point velocity produce only slightly different results. The regular octagon example reveals no difference at all due to its symmetry. Here a nonregular pentagon is used instead to show that these is only a minor difference between the closure options. However, unlike the conservative schemes, the exit points of the trajectories will now vary for different options of the RT0(QR,CV) method, and will also differ from the RT0(QR,LC) exit points. As this deviation will accumulate, we expect less accurate streamline trajectories as we trace across more cells.

Figure 8.

The streamline trajectories using the RT0(QR,CV) scheme in a nonregular pentagon with (a) option 1 and (b) option 2. The arrows and numbers associated indicate the lower-order boundary fluxes.

[64] In Figure 9 we now contrast a wider range of methods. For the locally conservative methods we close the equations using minimum velocity variance (option 4). For the continuous velocity schemes we utilize CVI to determine the center point velocity (option 1). Figure 9 shows the streamline trajectories for the methods discussed in this paper. The CVI method requires no cell refinement; the continuous velocity and the locally conservative schemes may use either lower-order quadrilateral refinement or triangular refinement. The CVI, the BDM1(QR), the RT0(QR,CV), and the BDM1(TR,CV) methods produce smooth streamlines. The RT0(QR,LC) and the BDM1(TR,LC) produces streamlines with kinks or refractions at the inner faces due to discontinuous tangential velocities. For the BDM1(QR), the RT0(QR,LC), and the BDM1(TR,LC) methods, which are locally conservative, the streamlines are distributed uniformly on the outflow boundary faces, and their exit points are uniquely determined by the entry points. For the CVI, the RT0(QR,CV), and the BDM1(TR,CV) methods, which do not satisfy local conservation, streamlines are distributed erratically and their exit points are not uniquely determined by the entry points.

Figure 9.

Single cell streamline tracing results for (a) CVI, (b) RT0(QR,CV), (c) BDM1(TR,CV), (d) BDM1(QR), (e) RT0(QR,LC), and (f) BDM1(TR,LC).

[65] The consequences of not satisfying local conservation can be illustrated by the example shown in Figure 10. The directions of flux on the boundary faces of the quadrilateral are indicated by the thick arrows and the flow rates across the boundary faces are the same Q as shown in Figure 10a. Therefore, there is no net flow coming in or going out of the cell. However, the velocity field calculated by the CVI method shows negative divergence in the upper region and positive divergence in the lower region separated by the dotted line. Although the integral of the divergence in the entire cell is still zero, locally there are volume sources introduced in the lower region and volume sinks introduced in the upper region. Let us consider a streamtube shown in Figure 10b in the lower region of the cell starting between points A and B. Because the flux on each boundary face is constant, the streamtube carries a flowrate of inline image when it enters the cell. But because artificial sources are introduced in the lower region, the streamtube will carry more and more fluid as it passes through the cell, and delivers more than inline image at the exit boundary between points inline image and inline image. For incompressible fluid, such a result violates mass balance. Under the same situation, the locally conservative RT0 velocity interpolation scheme will generate a divergence free velocity field. As shown in Figure 10c, the streamtube starting between the same points A and B will always carry a flowrate of inline image and the distance between the exit points is exactly one 10th of the boundary length. When tracing streamlines in multiple cells, the error created in a single cell as shown in Figure 10b may accumulate or cancel in the succession of cells that are penetrated by the streamlines, depending on the specific geometries of those cells. In contrast, in a locally conservative scheme, the flow carried by a streamtube will always be preserved regardless of cell geometries. Looking from the other side of the same problem, if we assume each streamline carries the same amount of flow, then for a nonconservative scheme, the fluxes obtained by counting the number of streamlines passing through each outflow face will be inconsistent with the value specified as the boundary condition.

Figure 10.

(a) The divergence of the velocity field obtained by the CVI method in a quadrilateral is negative in the upper region and positive in the lower region divided by the dotted line. (b) Using the CVI method, a streamtube between streamlines inline image and inline image is shown. The flowrate between AB is inline image, while the flowrate between inline image is larger than inline image. (c) With the locally conservative RT0 velocity interpolation, the streamtube starting between AB is shown. The flowrate between the points inline image and inline image is inline image, which is same as the flowrate entering AB.

[66] Another important observation from Figure 9 is that quadrilateral refinement and triangular refinement produce quite different results under the same method. Quadrilateral refinement provides systematically more regular results. We believe that these results are due to the more continuous boundary conditions with quadrilateral refinement, as each adjacent boundary quadrilateral face shares a common boundary flux. In contrast, triangular refinement has no such continuity in boundary conditions, with each triangle having an independent boundary condition with polygonal boundary face. This difference is also reflected in the count of degrees of freedom, as discussed in the beginning of section 5. The triangular refinement constructions have only n degrees of freedom in the interior of the cell, even with higher-order boundary conditions on the inner faces. This is why triangular refinement does not have sufficient degrees of freedom to satisfy both local conservation and velocity continuity. BDM1(QR) is the only method that satisfies both local conservation and velocity continuity. The result of BDM1(QR) surpasses the other methods in the combined regularity and smoothness of the streamline trajectories.

6.2. Quarter 5-Spot Pattern Flood

[67] The 2-D quarter 5-spot pattern flood is one of the most frequently used test cases for streamline tracing. We use a PEBI grid of 24 polygonal cells in the unit square with homogeneous rock properties. The injector and the producer are placed at the lower left corner and the upper right corner of the unit square, respectively. For this particular case, the face fluxes of all cells can be calculated using the analytic stream function for the quarter 5-spot [Datta-Gupta and King, 2007]. Lower-order flux boundary conditions are assumed on each cell for all the velocity interpolation methods to be applied.

[68] Figure 11 shows the streamline trajectories initiated uniformly from the injector cell with methods corresponding to those used in Figure 9. The dashed streamlines are the references obtained from the analytic streamfunction. The results of time of flight are summarized in Table 2 (top). According to Table 2 (top), the CVI and BDM1(TR,CV) methods have better performance. However, the continuous velocity schemes tend to avoid the extremes in velocity, and do not adequately sample the lower velocity portions of the flood pattern, especially in the two stagnation corners. In contrast, the conservation schemes are constrained in such a way as to better sample these regions. To avoid such bias, another set of test results are calculated with the streamlines initiated uniformly from the line connecting the upper left and lower right corners in the domain. This new set of initiation points provides a better match between the calculated and the reference trajectories for all the methods but not to the same degree. The time of flight error analysis is shown in Table 2 (bottom). In this case, the RT0(QR,LC), BDM1(TR,LC), and CVI methods have better performance.

Figure 11.

Streamline tracing results of the homogeneous quarter 5-spot case. Solid red lines show the streamlines obtained by corresponding methods; dotted black lines show the reference streamlines for comparison. (a) CVI, (b) RT0(QR,CV), (c) BDM1(TR,CV), (d) BDM1(QR), (e) RT0(QR,LC), and (f) BDM1(TR,LC).

Table 2. Time of Flight Results in the Homogeneous Quarter 5-Spot Casea
SchemesRef.CVIBDM1(QR)RT0(QR,LC)RT0(QR,CV)BDM1(TR,LC)BDM1(TR,CV)
  • a

    Velocity interpolation schemes are compared based on particular streamlines launched from the same points.

Streamlines Originated at the Injector
Min TOF (a.u.)0.60310.60150.60310.60540.61740.60990.6019
Max TOF (a.u.)1.52881.47801.72391.68281.29881.66981.5428
Avg TOF (a.u.)0.84800.82890.90040.89920.80310.90280.8408
Avg. TOF Err. %2.266.186.035.306.450.85
 
Streamlines Originated on a Line From Upper Left to Lower Right in the Domain
Min TOF (a.u.)0.61840.61140.61050.61310.61690.61710.6133
Max TOF (a.u.)2.16622.19852.27972.22822.33962.20152.2045
Avg TOF (a.u.)1.13271.14331.15021.13971.18271.14381.1588
Avg. TOF Err. %0.931.540.614.420.982.30
Table 3. Performance of the Velocity Interpolation Schemes
SchemesCVIBDM1(QR)RT0(QR,LC)RT0(QR,CV)BDM1(TR,LC)BDM1(TR,CV)
Analytical streamline integrationNoNoYesYesNoNo
Acceptable single cell resultsYesYesYesYesNoNo
Acceptable quarter 5-spot resultsNoYesYesNoYesNo

[69] From these two numerical experiments, we find it difficult to provide a quantitative comparison for the different schemes by sampling specific streamlines. The results are quite variable depending on the specific points in the domain where the streamlines are initiated. In the comparison of Table 2 (top) and 2 (bottom), we see that the magnitudes of the errors and the ranking between schemes varies depending upon the selection of launching points. Qualitatively, we realize that continuous velocity schemes systematically under-sample the stagnation region compared to the locally conservative schemes, but quantitatively it is difficult to draw conclusions in this fashion. Instead, in order to avoid the bias generated by sampling specific streamlines, we will seek a quantitative error analysis based upon a convergence study of the various schemes.

[70] A less biased “global” error analysis may be obtained by recognizing the relationship between volume, flux and time of flight. This transformation is important in correctly developing the equations for streamline simulation, as without them, mass (or volume) is not conserved. As discussed in the work of Datta-Gupta and King [2007], there is a transformation in volume elements between physical space and streamline coordinates

display math

where inline image is the time of flight, and inline image and inline image are the bi-streamfunctions. This relationship holds for incompressible flow, and may be readily generalized to compressible flow. However, we will use an incompressible test case in the convergence analysis. The volume integral gives

display math

Thus we may evaluate the global volumetric error using

display math

Figure 12 shows the global volumetric error plotted against the number of streamlines for various methods for the flow patterns shown in Figure 11. It is clear to see that the three locally conservative methods converge while the other three methods do not, with an error of approximately 10%. This lack of convergence is a major pitfall of the CVI method and the continuous velocity methods. The global error indicates that the conservative velocity fields are all first-order convergent. This is the same order of convergence demonstrated on regular Cartesian grids [Datta-Gupta and King, 2007]. The order of the convergence is dominated by the low-velocity stagnation regions. A local error analysis indicated that the convergence is second-order away from such regions, and we would expect the same convergence here on polygons.

Figure 12.

Global volumetric error of the homogeneous quarter 5-spot case. The locally conservative schemes demonstrate convergence while the nonconservative schemes do not.

[71] The CVI method and the continuous velocity methods produce visually appealing streamline trajectories in Figure 11. However, it should be recognized that this is a homogeneous case, for which the underlying analytic solution is smooth. We believe that a high degree of heterogeneity, which is often the case in practical applications, favors the methods which honor local conservation. To illustrate this point, we use the same PEBI grid as in Figure 11 but assign a heterogeneous permeability field of 1 md in two cells near the center and 1000 md otherwise (Figure 13). Again, we place an injector at the lower left cell and a producer at the upper right cell. Flow equations are solved using the cell-centered finite volume scheme with a two point flux approximation assuming single phase incompressible fluid. Lower-order cell boundary fluxes are identical for each calculation, while the streamlines are traced using various velocity interpolation methods. The launching points are uniformly distributed along the two boundary faces of the injector. The calculated streamline trajectories are shown in Figure 13. In this case, we encounter unphysical termination of some streamlines for the CVI method. In addition, the streamline trajectories obtained by CVI and the continuous velocity methods show spurious high concentrations along some streaks as if they were channels, which is purely artificial. In contrast, there is a region near the producer where almost no streamlines are found. This is verified by using as many as 2048 streamlines in the domain. In the discussion of the homogeneous case (Figure 11) we pointed out that the continuous velocity schemes tend to under-sample the low-velocity stagnation region. In the heterogeneous case, this effect is more apparent since the stagnation regions are on the interior of the domain, as would be the case with all real full field or multiwell applications. Similar failure cases were also shown by Sun et al. [2005] for some other nonconservative situations. In contrast, the BDM1(QR) and the locally conservative methods are constrained in such a way as to better sample the stagnation regions and produce much more reasonable streamline trajectories. Due to local conservation, these methods share the same exit point at the cell boundaries. Their primary difference is in the smoothness of streamline trajectories inside each cell: BDM1(QR) is the smoothest while BDM1(TR,LC) is the most tortuous.

Figure 13.

Streamline tracing results of the heterogeneous quarter 5-spot case. The permeability of the two cells shaded in gray is 1 md. All other cells have 1000 md permeability. Solid red lines show the streamlines obtained by corresponding methods: (a) CVI, (b) RT0(QR,CV), (c) BDM1(TR,CV), (d) BDM1(QR), (e) RT0(QR,LC), and (f) BDM1(TR,LC).

[72] Up to now there are no obvious failures displayed for the BDM1(QR) and the locally conservative methods. Although BDM1(TR,LC) may produce very tortuous streamlines within a cell, it is still acceptable from a field-level point of view because usually the details of the streamline trajectories inside a cell are not of much concern for field applications. For these methods, we argue that the major error appears to come from the grid discretization rather than from the velocity interpolation methods. As we have already discussed in section 3, the cell boundary flux conditions are obtained by solving the flow equations according to some numerical scheme. A better numerical scheme and/or a better/finer grid will produce a more accurate representation of the boundary fluxes. For the conservative methods, this is the primary factor concerning the accuracy of the interpolated velocity field.

6.3. Robustness, Simplicity and Efficiency

[73] Robustness, simplicity, and efficiency should not be overlooked for evaluating various velocity interpolation methods especially for large-scale practical applications. Of all the methods presented in this paper, the nonconservative methods show the most issues with robustness, especially as heterogeneity increases. CVI is the most complicated method to implement and the most expensive in term of numerical calculation. The other methods, based on subcell refinement, ultimately reduce to simpler tracing problems in quadrilaterals or triangles using either the lower-order RT0 space or the higher-order BDM1 space. The RT0 space can be analytically integrated leading to simple implementation and numerical efficiency. The BDM1 space requires the numerical solution of an ordinary differential equation usually by Runge-Kutta integration. It is also necessary to devise a robust and efficient calculation of when a streamline exits from each subcell. In contrast, for the RT0 space, these calculations may be solved analytically. Only if the polygonal cell boundary flux conditions are of higher order and if the subcell problem has no heterogeneity, should the BDM1 space be considered. Even then, using RT0 with half-cell boundary face fluxes, is expected to give a comparable solution, with the efficiency of an analytic solution. In summary, the performance of the various velocity interpolation schemes are shown in Table 3. The RT0(QR,LC) method is our recommended approach. Since it is based on the RT0 basis it has the efficiency of an analytic treatment. Because it is conservative, we have none of the issues of robustness experienced by the nonconservative schemes. We will examine a number of applications of the RT0(QR,LC) scheme in section 7.

7. Applications

7.1. Two-Dimensional LGR Grids

[74] The 2-D cell as shown in Figure 1b can be simply considered as a degenerate polygonal cell. Therefore, the methods discussed in this paper can be easily applied. As an example, we show a simple quarter 5-spot case with nested LGRs (Figure 14). The injector and the producer are located at the lower left corner and the upper right corner, respectively. The streamline trajectories shown in the figure are calculated using the RT0(QR,LC) method.

Figure 14.

Streamline tracing example using the RT0(QR,LC) method on a grid with local refinement.

7.2. Three-Dimensional LGR Grids

[75] It is not trivial to apply the unstructured streamline tracing methods developed in 2-D scenarios to 3-D LGR grids. Here we follow the approach of Jimenez et al. [2010] using a boundary layer approximation, which reduces the 3-D problem to a sequence of 2-D calculations. As an example, consider a coarse cell with four LGR cells adjacent to one of its faces. The boundary layer for that face is depicted in Figure 15, where the thickness of the boundary layer is inline image. The boundary layer is within the coarse cell and provides a region in which the uniform flux on the coarse cell face is redistributed to match the individual fluxes coming from each of the refined cell local grid faces. The flux from the coarse cell on the –x side of the boundary layer is allocated proportionally to the area of each fine cell, while the fluxes on the +x side of the boundary layer are known from the fine fluxes obtained for the LGR cells. The only problem still unsolved is how to trace streamlines across this boundary layer. This problem will be simplified if we allow the boundary layer thickness to approach zero, in which case the boundary fluxes in directions other than –x and +x will vanish. The local problem will appear to be that of two dimensional incompressible flow, with heterogeneous source and sink terms, irrespective of the actual fluid properties.

Figure 15.

A global cell with four LGR cells adjacent to one of its face. The boundary layer is outlined in red.

[76] As illustrated in Figure 16a, suppose the total flux through the boundary layer on the coarse cell side is 1, while on the other side, three out of four local fine cell faces have zero flux and the last one has flux 1. If the boundary layer thickness inline image, the situation becomes equivalent to a closed boundary 2-D problem as shown in Figure 16b. The differences of fluxes on the –x side and the +x side can be treated equivalently as sources or sinks in the 2-D problem for each of the four cells as shown by the red numbers. The next step is to solve this 2-D local flow problem.

Figure 16.

(a) Flow rates are shown for the boundary layer. On the –x side the total flux is 1. On the +x side, the fluxes across three LGR faces are zero, and only one LGR face has a flux of 1. All other boundary faces have zero flux. (b) As inline image, the boundary layer reduces to a 2-D closed boundary problem. The red number is the equivalent source (positive) or sink (negative) term for each fine 2-D cell. The black arrows show the calculated inner face fluxes using the RT0(QR,LC) method.

[77] The situation of this particular application favors the requirement of local conservation rather than velocity continuity. The reasons are that the boundary flux distribution should be honored rigorously and, the heterogeneity of the refined cell rock properties makes the velocity continuity requirement unnecessary. For tracing within the boundary layer, we require that the integral of the flux associated with the streamlines honor the boundary flux and the exit point of a streamline. Because the detailed path of the streamline is not required, the locally conservative methods with either option 1 or 4 may be applied for simplicity, as these two options do not require information about rock properties. The values and directions of the inner face fluxes are calculated and shown in Figure 16b with black numbers and arrows. Once the inner face fluxes are determined, streamlines can be traced in the boundary layer by ordinary Pollock interpolation in the unit cube, as shown in Figure 17. As the thickness of the boundary layer decreases and finally approaches zero, the traced streamlines ultimately become only slips on the 2-D boundary face, which conserve flux on both the coarse cell and the local cell faces. The incremental time of flight in the boundary layer will vanish. The example shows tracing from the coarse side to the fine side of the boundary layer. The reversed trajectories are identical. Once the boundary layer is crossed, tracing can be continued within either a coarse or a fine cell.

Figure 17.

Streamline tracing for 3-D LGR boundary layer. The thickness of the boundary layer decreases from Figure 17a to 17c. In Figure 17c inline image, and thus the streamlines simply slip on the boundary face.

7.3. The 2.5D PEBI Grids

[78] The 2.5D polygonal example shown in Figure 18 has one injector and one producer in a three layered reservoir. The grid is refined near the wells. In this example the flow in the k direction is simply treated separately from the i and j directions and the traced streamlines are shown in Figure 18. The rock properties within each PEBI cell is a scalar constant and thus the RT0(QR,LC) method with minimum velocity variance is applied for this application. This pattern flood demonstrates that there is no difficulty in extending the 2-D few cell results to larger reservoir models.

Figure 18.

Streamline tracing example using the RT0(QR,LC) method in a 3 layer 2.5D PEBI grid.

8. Conclusions

[79] We have presented a comprehensive study of the velocity interpolation methods in n-polygons. These methods are often used as postprocessing after the cell boundary fluxes are calculated for numerical schemes that do not directly calculate the velocity field, such as the FV schemes.

[80] We have discussed the relationship between the CVI and the BDM1 velocity interpolation spaces. Both provide 2n degrees of freedom in general n-polygons, but the BDM1 scheme includes additional higher-order terms to honor mass conservation. We have described three versions of the CVI method in n-polygons. Their major difference concerns robustness when dealing with nonconvex and degenerate polygons. Based on the technique of isoparametric mapping, the version we propose is the only one that is robust for both nonconvex and degenerate polygons. Although it is rather straight-forward to extend the CVI space to n-polygons, a direct extension of the locally conservative BDM1 space to n-polygons seems difficult, and has not been demonstrated.

[81] Subcell refinement, using either quadrilaterals or triangles, provides an effective way to simplify the problem. Local conservation or velocity continuity or both may be used as constraints on the degrees of freedom introduced inside the n-polygonal cell. Only the higher-order BDM1(QR) method satisfies both requirements. The BDM1(QR) method can be viewed as an indirect extension of the BDM1 space to n-polygons.

[82] Numerical experiments have provided strong evidence of the necessity of local conservation for a velocity interpolation method to produce robust physical results. An error analysis of comparing particular streamlines with the “true” streamlines are shown to be ambiguous in nature. Instead, a global volumeric error analysis shows that only the locally conservative schemes are convergent. Moreover, the heterogeneous quarter 5-spot case demonstrates failure modes for both the CVI and the continuous velocity methods.

[83] The primary difference between the locally conservative methods is the smoothness of streamline trajectories inside each cell, and in their numerical efficiency. BDM1(QR) is the smoothest while BDM1(TR,LC) with triangular refinement is the most tortuous. RT0(QR,LC) utilizes the RT0 basis and has an analytic solution for the streamline trajectories while both BDM1(QR) and BDM1(TR,LC) utilize the more expensive BDM1 basis. Therefore, robustness, simplicity and numerical efficiency seem to be decisive factors in term of looking for the best method to be applied in large-scale field applications. In summary, we would only consider the BDM1(QR) method for constant subcell permeability and high-order cell boundary flux conditions, but even then, a piecewise implementation of RT0(QR,LC) may be more than adequate; otherwise we recommend the RT0(QR,LC) method, which is robust, easy to implement, and highly efficient. The option for closing the locally conservative methods may be selected based on how much subcell constitutive information is available and how much effort is deemed to be worthwhile in incorporating subcell heterogeneity information.

Acknowledgments

[84] We acknowledge the support of the Texas A&M MCERI JIP and its member companies. We acknowledge the support from the Department of Energy, Basic Energy Sciences Division.

Ancillary