Unstructured-Coarse-Grid Generation Using Background-Grid Approach M. Evazi, SPE, Sharif University of Technology, and H. Mahani, SPE, Shell International Exploration and Production
Summary Reservoir flow simulation involves subdivision of the physical domain into a number of gridblocks. This is best accomplished with optimized gridpoint density and a minimized number of gridblocks, especially for coarse-grid generation from a fine-grid geological model. In any coarse-grid generation, proper distribution of gridpoints, which form the basis of numerical gridblocks, is a challenging task. We show that this can be achieved effectively by a novel grid-generation approach based on a background grid that stores gridpoint spacing parameters. Spacing parameter (L) can be described by Poisson’s equation (∇ 2 L = G), where the local density of gridpoints is controlled by a variable source term (G); see Eq. 1. This source term can be based on different gridpoint density indicators, such as permeability variations, fluid velocity, or their combination (e.g., vorticity) where they can be extracted from the reference fine grid. Once a background grid is generated, advancing-front triangulation (AFT) and then Delaunay tessellation are invoked to form the final (coarse) gridblocks. The algorithm produces grids varying smoothly from high- to low-density gridpoints, thus minimizing use of grid-smoothing and -optimization techniques. This algorithm is quite flexible, allowing choice of the gridding indicator, hence providing the possibility of comparing the grids generated with different indicators and selecting the best. In this paper, the capabilities of approach in generation of unstructured coarse grids from fine geological models are illustrated using 2D highly heterogeneous test cases. Flexibility of algorithm to gridding indicator is demonstrated using vorticity, permeability variation, and velocity. Quality of the coarse grids is evaluated by comparing their two-phase-flow simulation results to those of fine grid and uniform coarse grid. Results demonstrate the robustness and attractiveness of the approach, as well as relative quality/performance of grids generated by using different indicators. Introduction Practical handling of detailed geological models has long been a serious challenge for reservoir simulation and management. Efficient coarsening of the fine-scale model is a potential choice to alleviate the problem and has been addressed by several researchers. Original grids may be coarsened to generate structured or unstructured grids, each having pros and cons. As long as a structured grid works well, it is preferable to an unstructured grid. However, when the physical domain is highly heterogeneous and geometrically complex, structured grids may fail to capture the complex features well, unless a very fine grid is used. This contributes to the need for use of unstructured grid. Unstructured-grid-generation (UGG) techniques generally entail three steps: gridpoint insertion, triangulation, and construction of gridblocks. Insertion of gridpoints is completely arbitrary and is the main advantage of UGG. It is well understood that computational gridpoints must be distributed in the physical domain so that they are denser where flow and rock properties vary more or their magnitudes are large. Thus regions such as the near-well
Copyright © 2010 Society of Petroleum Engineers This paper (SPE 120170) was accepted for presentation at the SPE Middle East Oil and Gas Show and Conference, Bahrain, 15–18 March 2009, and revised for publication. Original manuscript received for review 11 December 2008. Revised manuscript received for review 7 July 2009. Paper peer approved 4 September 2009.
326
region, high-flow regions, fractures, and faults are potential regions to be gridded finer. Several methods for gridpoint insertion have been used in reservoir simulation. Local grid refinement (LGR) has been developed with this idea. Cartesian LGR (Cimet and Sweet 1973; Nacul 1991) and hybrid LGR (Pedrosa and Aziz 1986) are two well-known types of LGR. Because these grids are based on a Cartesian grid, both have difficulty aligning the grid lines with complex features and boundaries. Modular gridding, proposed by Palagi and Aziz (1994), uniformly distributes the gridpoints in the domain, resulting in a coarse, uniform Voronoi grid as the base grid. Then, several local grid systems suitable for specific geological and geometrical features are constructed independently and placed in the appropriate location of that feature in the base grid. This procedure is able to generate flexible grids. However, it is best suited for capturing notable features such as faults, large fractures, and vertical and horizontal wells, not necessarily for key flow regions and heterogeneities. An unstructured perpendicular-bisector (PEBI) -type grid was developed by Mlacnik et al. (2004) on the basis of the success of structured flow-based gridding (Durlofsky et al. 1996, 1997; Verma and Aziz 1997; Portella and Hewett 2000) that aimed at preserving fine-grid flow information in an unstructured coarse grid. Prévost (2003) and Prévost et al. (2005) proposed a 3D unstructuredcoarse-gridding framework that conforms to the large-scale geometries by using a topological model (soft-frame model). This model is used as the basis for incorporation of flow information from streamline simulations to adapt blocks for the small-/medium-scale heterogeneities. This provided an improved unstructured grid with capability of capturing both large-scale geometrical constraints, such as fault units, and grid-resolution objectives. Nonetheless, the literature on unstructured grids in geosciences focuses mainly on the numerical aspects of flow simulation, such as development of efficient and robust flow solvers and numerical techniques in control-volume and finite-element schemes, handling full-tensor permeability, and accurate permeability/transmissibility upscaling on the unstructured grid [see, for example, Verma and Aziz (1997), Edwards and Rogers (1998), and Edwards (2000)]. Less attention is paid to the creation of gridblocks, more specifically to the gridpoint-insertion algorithms by which the initial triangular/tetrahedral mesh is guided and constructed. In this work, we present a new approach for the insertion of gridpoints on the basis of static or dynamic information from the fine-grid geological (geocellular) model. The technique is very flexible to the choice of gridding indicator and allows sufficient control on the distribution and density of gridpoints in the domain. AFT serves as the main algorithm for gridpoint insertion, after a structured background grid has been generated from the fine-grid information. Highly heterogeneous fine-grid models are synthesized to test the method using different well configurations. Local permeability variation, velocity, and vorticity are employed as different measures of heterogeneity to generate computational coarse grids. Two-phase, incompressible flow simulations are performed to validate the new-grid-generation algorithm and assess various gridding indicators. Background-Grid Approach The background-grid approach (BGA) was first used in the area of aerodynamics (nonporous media) by Pirzadeh (1993) with introduction of only limited source elements to serve as areas where computation of flow dynamics required finer gridblocks. Our work extends its promising application in geosciences, particularly in June 2010 SPE Journal
equation on the structured (here Cartesian) grid with specified sources and Dirichlet-type boundary conditions: ∇ 2 L = G , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
(i,j+1)
(i–1,j)
(i+1, j)
(i,j)
rn
n
(i,j–1)
where L is the spacing parameter and G is sum of source terms, each with a specified spacing parameter. In absence of any source term [e.g., when porous media are (physically) homogeneous], the resulting mesh would be of uniform structure. In the presence of heterogeneities or well in- and out-flow, source terms must be prescribed. In this situation, source elements are allocated spacing and intensity (strength) values by which the density of gridpoints is controlled. Eq. 1, with the external boundary condition of L = Lb, where Lb is the prescribed value of the spacing parameter at the outer boundaries, and internal boundary condition describing the spacing at source points, is numerically descretized. The spacing parameter at each gridpoint is assumed to be inversely proportional to the square of distance from the source points. The principle of linear superposition is used to calculate the resulting source term at each gridpoint (interpolation point) for distributed discrete source points, as follows:
n Li , j − Ln , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2) 2 n =1 rn N
Gi , j = ∑
Spacing value at each source (circle)
Source point Gridblock center
Fig. 1—Source elements in Cartesian geocellular model (fine grid). All sources are placed at the gridblock centers. Sources are measures of heterogeneity impact on flow. Wells can be treated as linear discrete sources.
the area of coarse-grid generation by using (prescribed) number of source elements acting as key measures of heterogeneity and criticality of flow. BGA entails insertion of gridpoints in the domain of interest, using the information stored in a “background grid.” This information specifies the local density of gridpoints. Although, in principle, any parameter related to flow and rock properties can be used as the criterion for the distribution of spacing parameter in the background grid, the criterion is not meant to be arbitrary, in view of the challenges and complexities in creation of an optimum coarse grid. As such, in this work, permeability variation, singlephase-flow average velocity, and vorticity on a geocellular model are examined as the criteria. A background grid can be structured or unstructured. A structured background grid is used here, because it is efficient and simple to construct. In BGA, we determine the spatial variation of grid parameters (spacing and density) in a field by a process similar to that of computing the diffusion of heat from discrete heat sources in a conducting medium. Gridpoint spacing parameter distribution is obtained by the solution of a steady-state Poisson’s
)
where n, rn, and N refer to intensity factor of the nth source, distance between interpolation point (i, j) and nth source, and number of the source elements respectively. Li,j and Ln correspond to the spacing parameters at interpolation point (i, j) and at the nth source point (see Fig. 1). n, which is dimensionless, determines the extent by which the effect of a source element diffuses in the domain. The larger n is, the more extended the impacted area will be. n has a relative and unknown value and can be used as a tuning parameter in the algorithm to adjust spacing parameters and achieve desired coarse-grid structure. The effect of n on the distribution of the grid is demonstrated in Fig. 2 using a rectangular homogeneous domain by generating two grids with different values of n. In both cases, eight source points are placed at the corners and midpoint of the boundary edges with equal intensity factor of 10, and one extra source element is positioned in the center of the domain. Intensity factors of 1 and 10 are assigned to this source, and two grids are generated (see Figs. 2a and 2b). As seen, with an increase of intensity factor, gridblocks are distributed more smoothly toward the boundary (i.e., radius of effect becomes larger). Once distribution of n and all other source elements’ specifics are prescribed, one can determine spacing distribution in the rest of the domain by combining Eqs. 1 and 2 and solving the linearized sets of equations iteratively: for example, by implementing a Gauss-Seidel scheme with successive overrelaxation (Pirzadeh 1993):
Intensity Factor = 1
( a)
(
Intensity Factor = 10
( b)
Fig. 2—Effect of source intensity on grid distribution: (a) intensity factor = 1; (b) intensity factor = 10. June 2010 SPE Journal
327
N ⎛ r +1 n Ln ⎞ r 2 r +1 r ⎜⎝ Li , j −1 + Li , j +1 + Li −1, j + Li +1, j + h ∑ r 2 ⎟⎠ n =1 n Lri ,+j1 = (1 − ) Lri , j + . N ⎛ n ⎞ 2 4 + h ∑ 2 ⎟ ⎜⎝ n =1 rn ⎠
. . . . . . . . . . . . . . . . . . . . . . . . (3) Here, stands for relaxation factor, h is gridblock size, and superscript r refers to iteration level. The unknowns are spacing parameters at the gridpoints except source points. Initial values of the unknown spacing parameters can be obtained simply by averaging the spacing parameter of source points with the inverse of the square of their distance to the gridpoint: N
Ln rn2 Li , j = nN=1 . 1 ∑ n r 2 n =1 n
∑
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)
Because of the diagonal dominance of the coefficient matrix, this iterative solution is stable. The convergence rate of this process can be accelerated with an optimum value of . For some cases of special geometries, theoretical optimum values are available, which can be found in Lapidus and Pinder (1982). In BGA, as opposed to elliptic structured grid-generation techniques, no mapping from physical space to logical space is required; therefore, this technique has no complication of transformation and back transformation between spaces. In addition, the resulting grid distribution is smooth, which may not be the case for elliptic grid-generation techniques. The preceding discussion dealt with discrete nodal source elements, which mainly represent, for instance, points of high permeability variation or high-flow regions. However, subsurface flow is often driven by wells, and because of the difference between flow regimes in the vicinity of a well and away from a well, wells’ effect on flow must be accounted for in the gridding procedure. Injectors and producers can be assumed as point-source terms in 2D areal maps and can be easily included in the procedure. It is also possible to handle horizontal, deviated, or vertically penetrating wells in 3D. In this case, one can simply treat wells as (linear) line sources and use the integral form of Eq. 2 (equivalent to infinite discrete sources) or simply divide the well into a finite number of intervals, each acting as a single source element. Therefore, one separate term can be added to Eq. 2 to account for the contribution of each well. In the following, the equation is derived only for a single well, but it can be extended to any number of wells similarly:
n Li , j − Ln + w 2 Iw n =1 rn N
Gi , j = ∑
( a)
(
)
∫
Iw
Li , j − fw ( l ) rw ( l ) 2
dl . . . . . . . . . . . . . (5)
Here, w, Iw, rw2 ( l ), and fw(l) refer to the intensity factor of the well, length of the well (perforated intervals only), distance between well segment and the interpolation point (i, j), and spacing variation function along the well, respectively. The line integral in Eq. 5 is evaluated along the well. fw(l) can be assumed to be constant for simplicity or can be a function of expected flow rate at each perforated interval. Once the background grid is constructed, gridpoints can be inserted in the domain. This is accomplished by applying the AFT. In this technique, gridpoints are inserted in the domain according to the spacing parameter stored in the background grid. First, the boundary of the domain is divided into edges by insertion of gridpoints. Usually, the length of these edges is equal to the spacing value of the sources at the boundaries (external sources). The smallest edge is selected, and the spacing parameter of its midpoint is interpolated from the background grid. A new point is inserted in the domain according to this value, and two new edges are formed with the new point and the vertices of the selected edge. The selected edge is removed from the edge array, and this procedure is repeated on the smallest edge, and the front is advanced in the domain until no edge remains [see Fig. 3; more relevant information can be found in Farrashkhalvat and Miles (2003) and Mavriplis (1997)]. This technique provides gridpoints and the triangles connecting these points. These points can be used to generate any kind of numerical grid, depending on the type of formulation to be used for the solution of flow equations. In this work, we use control-volume finite-difference (CVFD) formulation suitable for Voronoi grid. In order to generate a Voronoi grid, gridpoints are triangulated with Delaunay tessellation, which is the dual of a Voronoi grid. Delaunay tessellation forms triangles from the points in a way that circles passing through each triangle vertices contain no other point. Connecting the centers of these circles will lead to the construction of the desired Voronoi grid. Before attempting to simulate flow on the generated Voronoi grid, proper rock properties must be assigned to the gridblocks and connections of gridpoints. In this work, we simply use an analytical (arithmetic/harmonic) permeability-upscaling technique similar to the one used by Palagi (1992). This procedure is simple and easy to apply. However, it is not as realistic as flow-based upscaling methods and may affect the flow results adversely. However, because our work focuses on the BGA, its effect is not investigated and is supposed to be of equal effect in all coarse grids. It is expected that more-accurate, flow-based upscaling techniques would improve the flow results. However, flow-based upscaling is complicated on unstructured mesh and is not yet readily available. Application of BGA in Coarse-Grid Generation For coarse-grid generation in reservoir simulation, we may authenticate the candidate points for the source points according to the static or dynamic information of the field. Permeability or porosity variation may be chosen to determine the source points. In fact,
( b)
Fig. 3—Illustration of AFT: generation of a new triangle using new point (a) or existing front point (b). Dashed-line edges represent current front (Mavriplis 1997). 328
June 2010 SPE Journal
regions where permeability or porosity variation is considerable are marked for refinement. Velocity as indication of flow and density of streamlines can be chosen also because regions of higher velocity are more sensitive to the variation of properties in the gridblocks and require finer gridblocks to capture accurately fluidfront movement and displacing-fluid-breakthrough prediction. It is even possible to use the combination of permeability and velocity by using vorticity. For simplicity, the criterion for the construction of the background grid is normalized by dividing values into the maximum found in the domain. In order to determine the internal source elements, a cutoff (threshold) value between 0 and 1 is specified for each criterion. Blocks with a normalized value above the cutoff are selected as internal source elements, to which a spacing parameter and an intensity factor are assigned. Values of these two parameters are dependent on the magnitude of the normalized value, such that, in the areas where the normalized criterion is greater than the threshold, finer and denser gridding is introduced. For simplicity, we assign equal spacing parameter and intensity factor to the qualified points, resulting in equal block size (also representing minimum block size in the coarse grid) at each source element. In the following, computation of permeability variation, velocity, and vorticity is briefly discussed. In this paper, we focus on 2D systems. Permeability-Based BGA. Permeability is an important property of porous media; its distribution directly affects the flow dynamics. When the permeability is homogeneous or has small overall variations, it does not cause difficulty to the flow field and to the grid-generation process. However, subsurface formations are heterogeneous where permeability varies throughout the domain at different length scales (e.g., between layers or within layers), which complicates upscaling and efficient coarse-grid generation. In fact, for large heterogeneous models, a computational grid should be constructed by addressing the heterogeneities. Permeability-based grid generation has been investigated before (Garcia et al. 1992; Farmer et al. 1991; Li et al. 1995; Qi et al. 2001), where the underlying idea is to preserve key fine-grid heterogeneities in the coarse grid by refining areas of high permeability variation. The technique is commonly used because it does not entail any flow simulation. However, it is static and independent of actual flow pattern, thus resulting in identical grid regardless of well configuration in the domain. This, in practice, does not fulfill reservoir-simulation/various-flow-scenarios objectives. In our work, we define the permeability gradient as k ( i , j ) − k ( i + 1, j ) + k ( i , j ) − k ( i − 1, j ) k ( i , j ) = 4h k ( i , j ) − k ( i , j + 1) + k ( i , j ) − k ( i , j − 1) + , . . . . . . . (6) 4h where k ( i , j ) stands for average permeability gradient at point (i, j) and h = x = y designates gridblock length. Obviously, this definition weighs permeability variation in both x and y directions equally. However, this may limit its application when permeability variation in the vertical direction dominates the equation (e.g., in layered systems). In such cases, a weighting average of the gradients would be more appropriate. Nevertheless, for the moment, we use the simple definition of Eq. 6, which assumes that the fine grid is regular and the underlying permeability is isotropic. Once the average-permeability gradients are computed for each fine block, then they are normalized as described earlier. Next, a cutoff value is specified for the selection of source points corresponding to areas with high permeability variations. The cutoff value is casedependent and is adjusted by the user according to the required upscaling level. Velocity-Based BGA. Flow is another important dynamic property that indicates where fluid is actually moving in porous media. Understanding flow movement is not possible simply by looking June 2010 SPE Journal
only at permeability map. It is well understood that a computational grid must capture high-flow regions corresponding to high-permeability/connected areas, as typically used in flow-based gridding techniques (Durlofsky et al. 1996, 1997; Darman et al. 2000; Castellini 2001). To construct flow-based grid, a singlephase-flow experiment on a fine grid is performed. This may be a disadvantage for flow-based griddings, compared to permeabilitybased approaches. However, this can be alleviated thanks to recent advancements in streamline simulation technology that is significantly faster than the conventional finite-difference, finite-volume methods. Once the flow and velocity information is extracted from this experiment, one can generate grids on the basis of the mean velocity at each gridblock using the following equations: u ( i , j ) = ux ( i , j ) + u y ( i , j ) , . . . . . . . . . . . . . . . . . . . . . . . (7) 2
2
where ux ( i , j ) =
(
)
(
),
. . . . . . . . . . . . . . . . . . . (8)
) (
).
. . . . . . . . . . . . . . . . . . . (9)
ux i + 12 , j + ux i − 12 , j 2
and uy (i, j ) =
(
1 uy i, j + 1 2 + uy i, j − 2 2
The indices i±½ and j±½ refer to faces of gridblocks in the x and y direction, respectively, where velocity components are calculated. The procedure of incorporation of velocity in the BGA is similar to that for permeability. Note that the velocity-based gridding applied here is slightly different from the common flow-based gridding approach because we preserve local velocity distribution by looking at velocity magnitude at each point; whereas, in the flow-based algorithm, time of flight along each streamline or total flux in stream tubes is selected as a measure for refinements or coarsening. Both techniques resolve high-permeability streaks and flow channels, but they could produce different grids, particularly when the correlation length of permeability is small compared to system size (no dominance of high-flow channels). In our velocity-based approach, only velocity magnitude is preserved, while velocity direction should be considered also. Vorticity-Based BGA. Combination of velocity and permeability variation is another choice to generate a grid. Vorticity as a convenient measure of heterogeneity effect on flow has been examined recently by Mahani and Muggeridge (2005), Ashjari et al. (2007, 2008), Evazi et al. (2008), Evazi and Mahani (2009), and Mahani et al. (2009). In fact, flow in porous media is characterized by vorticity resulting from the presence of rock heterogeneities on different length scales, permeability anisotropy, mobility differences, and gravity effects. It is believed that the vorticity caused by heterogeneities is dominant because it exists even in single-phase flow (Mahani et al. 2009). Conventional upscaling approaches do not capture vorticity because a full-tensor upscaled permeability distribution is required to describe the rotational character of flow in a coarse grid properly. Instead, conventional upscaling techniques preserve translational velocity, which is different from actual velocity distribution in a coarse grid (Mahani et al. 2009). A vorticity-based technique alleviates this problem by adjusting coarse gridblock sizes according to the underlying fine-grid vorticity intensity. The main difference/advantage of this technique over the permeability-based technique is that the direction of the permeability gradient with respect to local velocity direction is considered to be as important as gradient magnitude. This means that existence of high permeability variation at a location does not necessarily imply a finer grid unless there is considerable flow in that region. This leads to an improved permeability-based grid that honors flow information, as does a flow-based grid [see Mahani et al. (2009) and Mostaghimi et al. (2009) for further details on the technique]. 329
Vorticity measure can also be incorporated easily in BGA. Given that we have the flow information from single-phase-flow simulation on fine grid, vorticity can be computed on the same grid according to the gradient of velocity profile, as follows (Mahani et al. 2009; Mahani 2005):
1,200 Log Permeability (md) 17 15 14 13 11 10 8 7 6 4 3 2 0 –1 –3
z (i , j ) =
∂u y ∂x
i, j
−
∂ux ∂y
i, j
⎡ ⎛ 1 ⎞ ⎛ 1 ⎞⎤ ⎢u y ⎜⎝ i + 2 , j ⎟⎠ − u y ⎜⎝ i − 2 , j ⎟⎠ ⎥ ⎣ ⎦ ≅ Δx 1⎞ ⎤ 1⎞ ⎡ ⎛ ⎛ ⎢ux ⎜⎝ i , j + 2 ⎟⎠ − ux ⎜⎝ i , j − 2 ⎟⎠ ⎥ ⎦ , . . . . . . . . . . . . . . . (10) −⎣ Δy where uy(i±½, j) and ux(i, j±½) are tangential velocities at gridblock boundaries. Note that, in 2D flow in an x–y plane, vorticity is only created in the z direction and the x and y components of the vorticity vector are zero. In 3D applications, all vorticity components are generally nonzero and they all should be used to adjust coarse-gridblock size and location. Results and Discussion Test Model 1. In order to illustrate unstructured gridding using background grid and examine the quality and efficiency of the generated grids with different indicators, a synthetic channelized model is first tested. The original 2D fine-scale model contains 14,400 (120×120) gridblocks, as shown in Fig. 4. Two different well configurations are considered. The quality of coarse grids generated on the basis of different indicators is judged by comparing the two-phase incompressible flow results predicted by these grids and the results of the fine grid. For more evaluation of the grid, results simulated by a uniform coarse grid of equal upscaling level are presented also. To test the proposed gridding algorithm further and provide more insights about quality/performance of different grids generated from different gridding measures, a second complex model will be used. Permeability-Based Grid. The first well configuration for the model is obtained by placing injection and production wells as shown in Fig. 5b. Normalized permeability-gradient map is depicted in Fig. 5a. Because of the high variation in permeability values, permeability gradients greater than 0.001 are made equal to 0.001. On the basis of a permeability-variation map, a background grid is generated that shows a smooth distribution of spacing parameter throughout the domain. Irrespective of well location, spacing parameter is derived by magnitude of permeability gradient such that the lowest values are found around boundaries of facies (source elements) and the largest in the rest of the system and domain boundaries. A triangular grid (obtained from advancing front) and a Delaunay grid are shown in Figs. 5c and 5d, respectively. Parameters used for generation of the grids are listed in Table 1. The Voronoi grid, created from a Delaunay grid and used for flow simulation, is shown in Fig. 5e. As seen, gridblocks are finer in the high-permeability-variation zones and are coarser in the more-homogeneous zones. This grid preserves the high permeability variations very well but has difficulty in capturing connectivity of the channel between injector and producer. Two-phase-flow simulation is performed with the well configuration shown in Fig. 5b. Water is injected from the injector well, and a water/oil mixture is produced from the producer. Well rates and constraints along with fluid mobilities and relative permeability curves are summarized in Table 2. Fine grid, uniformly gridded coarse grid, and permeability-based unstructured grid are prompted to mimic this condition. Fig. 5f represents the oil-recovery factor predicted by each of these grids. The permeability-based grid generated by BGA shows an improved accuracy over uniform coarse grid. However, it is not that much more persuasive compared with the fine grid. 330
Y (ft)
800
400
0
0
400
X (ft)
800
1,200
Fig. 4—Distribution of logarithm of permeability of the Test Model 1.
The grid-generation procedure was not repeated for the second well configuration (Fig. 6a). Note that, in principle, we can retain the coarse-grid structure of Fig. 5e and just add new well locations for flow simulations, provided that the upscaling level does not change. Therefore, only two-phase-flow results are presented for this case, while coarse-grid-generation steps are skipped. As seen in Fig. 6b, a permeability-based grid performs worse and even fails to compete with the uniform grid results because the grid has not been adjusted for the new well location. In fact, this is a drawback of permeability-based griddings. Velocity-Based BGA Grid Generation. The procedure described for the first well configuration is applied here again. The average velocity, as described in the Velocity-Based BGA subsection, is computed and mapped onto a fine grid to be used as the basis for the construction of a background grid. Comparatively, as shown in Fig. 7a, the velocity profile has less variation throughout the system than the permeability gradient in Fig. 5a. This makes handling of the velocity map easier than permeability gradient in BGA. High-velocity-channel regions, candidates for finer gridding, appear as areas with small spacing parameter in Fig. 7b. The generated triangular and Voronoi grids also capture this; therefore, it is expected to observe a good match of the fine-grid with coarse-grid model (Fig. 7f). This process is repeated for the second well configuration, and the results are shown in Figs. 8a through 8f. Basically, the background grid has to be regenerated because the well location was altered. Again, the velocity-based grid works well in reproducing the fine-scale behavior. Vorticity-Based BGA Grid Generation. A similar type of exercise is repeated by using a vorticity map in the BGA.I Vorticity, I I according to its definition (Mahani et al. 2009) ( = − u × ∇ ln k ), is highest at the boundaries of significant permeability gradient perpendicular to flow direction. In this test case, the highest permeability gradient is observed around the boundaries of the channel where the high velocities are expected. This can be seen clearly in Fig. 9a for the first well configuration. A background grid generated on the basis of the vorticity map is shown in Fig. 9b. The background grid is of acceptable quality because the smallest spacing parameters are assigned to the high-vorticity source elements and the largest to the rest of the model. The ratio of highest to lowest spacing is approximately 5 (see Table 1), which allows construction of a suitable coarse grid by applying AFT and Voronoi tessellation. The produced grid from BGA is rather smooth and does not require any post-processing effort. The grid resembles the flow-based grid for this example (although this is not generally the case), in that they both resolve the high-permeability/high-flow channel. The simulated two-phase-flow results on the unstructured coarse grid are close to those of the fine grid and better than a uniformly gridded coarse grid that does not honor heterogeneities. June 2010 SPE Journal
1,200
1,200 Spacing Parameter (ft)
Normalized Permeability Variation 0.0009 0.0009 0.0008 0.0008 0.0007 0.0006 0.0006 0.0005 0.0004 0.0004 0.0003 0.0003 0.0002 0.0001 0.0001
400
0
0
400
( a)
X (ft)
800
400
0
1,200
1,200
1,000
1,000
800
800
600
600
400
400
200
200
200
400
600
0
400
X (ft)
( b)
1,200
0 0
85 80 76 72 68 63 59 55 50 46 42 38 33 29 25
800
Y (ft)
Y (ft)
800
800
1,000
0 0
1,200
(c)
200
400
600
800
800
1,200
1,000
1,200
( d)
1,200
0.9 0.8
1,000
Oil Recovery Factor
0.7 800 600 400
0.6 0.5
Fine grid, 14,400 gridblocks Uniform grid, 515 gridblocks Permeability-based grid, 513 gridblocks
0.4 0.3 0.2
200
0.1 0 0
200
400
600
800
1,000
( e)
1,200
0 0
0.1
(f)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pore Volume Injected
Fig. 5—(a) Normalized permeability-variation map (Configuration 1); (b) background grid based on permeability variation; (c) triangular grid generated by AFT; (d) triangular grid generated by Delaunay tessellation; (e) Voronoi grid, dual of Delaunay grid; (f) oil-recovery factor by the permeability-based grid, uniform coarse grid, and fine grid. TABLE 1—PARAMETERS USED IN THE BGA FOR THE GENERATION OF GRIDS Permeability-Based Grid, Configuration 1
Permeability-Based Grid, Configuration 2
Velocity-Based Grid, Configuration 1
Velocity-Based Grid, Configuration 2
Vorticity-Based Grid, Configuration 1
Vorticity-Based Grid, Configuration 2
cutoff
0.0001
0.0001
0.011
0.01
0.011
0.08
internal
0.0033
0.0033
0.1
0.05
0.1
0.05
external
100
100
100
120
100
120
Lmin(ft)
20
20
20
20
20
20
Lmax(ft)
100
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TABLE 2—RELEVANT DATA USED IN TWO-PHASE INCOMPRESSIBLE FLOW SIMULATION Porosity
µo
µw
kro
0.17
1 cp
1 cp
Qinj
Pwf at Production Well
500 STB/D
3,000 psi
krw
The results for the second well configuration are presented in Fig. 10. The well location is such that only a small portion of the channel truly contributes to flow between wells where highest vorticity values/smaller spacing parameters are also observed in Figs. 10a and 10b. Therefore, only this part of the channel requires finer gridding and the remainder of the model can be coarsened. Therefore, a smaller number of coarse gridblocks (430) is required compared to the situation in Well Configuration 1, and this produces a very good match to the fine-grid-flow results (Fig. 10f). Test Model 2. The second test model is a fluvial highly channelized system with wide permeability range (10−2 to 1012 md), as shown in Fig. 11a. The 2D fine-grid model is uniformly gridded and is 130×100 ft, containing 13,000 blocks. As the permeability variation suggests, it is a complex, highly heterogeneous model and is expected to be a challenging test case for upscaling and coarsening procedures. Usually, if correlation length of permeability is of system size, then upscaling and coarse-grid generation become challenging tasks. Injector/producer location for the simulation of single-phase and two-phase displacement is shown in Fig. 11a. To make the flow path more complex and nontrivial, injector and producer are positioned in the high-permeability channels, separated by lowpermeability sand, resulting in poor connectivity between injector and producer. This would force the injection water to displace oil mostly in the connected high-permeability area; therefore, most of the oil remains unswept in the low-permeability background and disconnected channels. As demonstrated before, depending on the choice of gridding measure, the structure of the coarse grid would be different. Targeting the permeability variation, velocity, and vorticity distribution, the technique described earlier is repeated to generate coarse nonuniform unstructured grids. We first assess permeability-variation criterion by upscaling the fine grid into a coarse grid by an upscaling factor of approximately 25. The results are depicted in Figs. 11b through 11e. It is clear that the procedure refines the grid at regions of high permeability variation mainly around boundaries of channels, irrespective of the actual flow behavior. In this case, because of the specific configuration of the wells, high-permeability channels at the left side of the model play a minor role in the
flow. However, the permeability-based grid also refines this area. Given the fixed total number of coarse gridblocks (513) available to distribute/create, intuitively one could coarsen inactive channel areas and further refine active branches of channels. As will be mentioned in the next paragraphs, finer grids are indeed generated in active high-permeability channels by flow- and vorticity-based techniques compared to a permeability-based technique. This potentially improves accuracy of flow simulation on these grids. Two-phase-flow simulations performed on two types of coarse grid, uniform and nonuniform, are compared in Fig. 11f. For reference, the fine-grid result is shown also. As evident, the performance of permeability-based grid (nonuniform) is as poor as the uniform coarse grid of same size. The velocity map from single-phase-flow simulation on the fine grid shows that fluid moves mostly in the middle part of the domain and most of the oil is bypassed in other areas (Fig. 12a). Flow-based gridding suggests that one needs to focus more on high-flow regions and tries to capture those areas in the coarse grid. A background grid produced on the basis of velocity is shown in Fig. 12b. Triangular grids created by AFT and Delauny tessellation are shown in Figs. 12c and 12d, respectively. The Voronoi grid resulting from the Delauny triangular grid is used for two-phase-flow simulation. Analysis of fine grid, velocity-based coarse grid, and uniform coarse grid results (Fig. 12f) reveals a close match of flow-based grid oil recovery factor to that of fine grid compared with uniform coarse grid, which does not honor the fast-flow area. Likewise, a fine-grid model is coarsened using vorticity distribution. Inclusion of both velocity and permeability-variation information results in the vorticity map shown in Fig. 13a. This map is comparable to the velocity map in Fig.12a, thus generation of a similar grid (although not exactly the same) is expected. Fig. 13b shows the output background grid that is fairly similar to Fig. 12b. The main difference (compare the left half of Figs. 13b and 12b) is linked to the difference between vorticity and velocity definitions. For instance, in the left half of Fig. 12b, where velocity is near zero, the spacing parameter is largest, while, in Fig. 13b, in the same area, the permeability gradient is still high (e.g., around channel boundaries) no matter if velocity is small. This results in spacing parameters smaller than the maximum, thus 0.7
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0 0
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Recovery Factor
86 81 77 73 68 64 60 55 51 47 43 38 34 30 25
800
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0.6
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( b)
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Pore Volume Injected
Fig. 6—(a) Background grid based on permeability variation; (b) oil-recovery factor by the permeability-based grid, uniform coarse grid, and fine grid (Well Configuration 2). 332
June 2010 SPE Journal
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Normalized Velocity 0.94 0.88 0.81 0.75 0.69 0.63 0.56 0.50 0.44 0.38 0.31 0.25 0.19 0.13 0.06
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Fig. 7—(a) Normalized velocity map (Configuration 1); (b) background grid based on velocity; (c) triangular grid generated by AFT; (d) triangular grid generated by Delaunay tessellation; (e) Voronoi grid, dual of Delaunay grid; (f) oil-recovery factor by the velocity-based grid, uniform coarse grid, and fine grid.
requiring some refinements in that area. This implies that, although both velocity- and vorticity-based grids capture high-flow zones, vorticity-based grids preserve better boundaries between highand low-permeability units in low-flow areas. Two-phase-flow results for vorticity-based grids in Fig. 13f confirm the success of June 2010 SPE Journal
vorticity-based grids. Nonetheless, for this test case, velocity- and vorticity-based-grid results are not that different, as they can be in general (e.g., in heterogeneous models with small correlation length compared to system size or when high-permeability channels are not the predominant flow path). 333
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Pore Volume Injected
Fig. 8—(a) Normalized velocity map (Configuration 2); (b) background grid based on velocity; (c) triangular grid generated by AFT; (d) triangular grid generated by Delaunay tessellation; (e) Voronoi grid, dual of Delaunay grid; (f) oil-recovery factor by the velocity-based grid, uniform coarse grid, and fine grid.
Summary and Concluding Remarks In this paper, a novel unstructured (coarse) grid-generation approach was introduced and applied using a structured background grid. Being designed for application in reservoir simulation, the technique aims at guiding gridpoint insertion by an advancing-front method. A structured/Cartesian fine-grid distribution of properties (typically fine 334
grids have this structure) is used to create a background grid or a spacing-parameter map. The technique exploits the advantages of elliptic gridpoint distribution, while it does not require complex transformation and back transformation between physical and logical domains. Various fine-grid information, either static or dynamic, can easily serve as input to the BGA. Using two highly heterogeneous June 2010 SPE Journal
1,200
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Pore Volume Injected
Fig. 9—(a) Normalized vorticity map (Configuration 1); (b) background grid based on vorticity; (c) triangular grid generated by AFT; (d) triangular grid generated by Delaunay tessellation; (e) Voronoi grid, dual of Delaunay grid; (f) oil-recovery factor by the vorticity-based, uniform coarse, and fine grids.
2D test cases, we demonstrated that the technique is powerful and flexible in handling three main gridding indicators—permeability variation, velocity, and vorticity—and provides sufficient control for distribution of gridpoints. The produced coarse grids had smooth distribution (an essential feature for an unstructured grid) and were successful for two-phase-flow simulation. June 2010 SPE Journal
The technique presented aims primarily at effectively distributing the gridpoints and has no major limitation on the choice of gridding measure, although the intention is to use meaningful static or dynamic reservoir properties (as used in this paper) rather than making an arbitrary choice of gridding measure. The technique works locally to preserve the determining measure, while it does 335
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Pore Volume Injected
Fig. 10—(a) Normalized vorticity map (Configuration 2); (b) background grid based on vorticity; (c) triangular grid generated by AFT; (d) triangular grid generated by Delaunay tessellation; (e) Voronoi grid, dual of Delaunay grid; (f) oil-recovery factor by the vorticity-based grid, uniform coarse grid, and fine grid.
not guarantee that variation of the gridding measure is also preserved. If this is as important, then the gridding measure has to be redefined to meet the purpose. It is worthwhile to mention that results presented in this paper assumed incompressible fluids. This implies that fluid displacement is the chief recovery mechanism and the effect 336
of fluid expansion/contraction is negligible on fluid dynamics and streamline distribution. Normally flow- and vorticity-based grids are constructed from single-phase-flow information that implicitly assumes that streamlines are mainly controlled by heterogeneity and flow scenario (well type and configuration), and not by compressibility- and saturation-dependent properties. In June 2010 SPE Journal
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Fig. 11—(a) Permeability map (Model 2); (b) background grid based on permeability; (c) triangular grid generated by AFT; (d) triangular grid generated by Delaunay tessellation; (e) Voronoi grid, dual of Delaunay grid; (f) oil-recovery factor by the permeability-based grid, uniform coarse grid, and fine grid.
many cases, this is valid (e.g., in waterflood experiments). Even if this is not the case, the grids can be updated dynamically between steps of major changes in flow paths, or streamline spacing and distribution. Note that this is only applicable for flow- and vorticity-based grids. A permeability-based grid does not vary with simulation scenario. Therefore, the results and conclusions June 2010 SPE Journal
are not biased by this assumption and remain valid, although grid structure in presence of (significant) fluid compressibility and in its absence can differ, as do velocity and vorticity distributions. Last but not least, when extended to 3D space, calculation of permeability gradient, velocity, and vorticity is straightforward and 337
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Fig. 12—(a) Normalized velocity map (Model 2); (b) background grid based on velocity; (c) triangular grid generated by AFT; (d) triangular grid generated by Delaunay tessellation; (e) Voronoi grid, dual of Delaunay grid; (f) oil-recovery factor by the velocitybased grid, uniform coarse grid, and fine grid.
normally comes out of the flow simulator. The BGA requires only minor modification for 3D extension and can be used to effectively distribute the gridpoints and subsequently create Voronoi or other types of unstructured coarse grids. However, there are challenges. A Voronoi grid has its own limitations in 3D, while other types of grids such as control-volume finite-element or its extension, are 338
more suited for 3D applications. To address limitations of Voronoi grid in 3D, a layer-cake approach or 2.5D unstructured grid can be applied for the vertical direction, where Voronoi grids are generated only in 2D areal layers by ensuring proper connection of gridpoints vertically between layers. This can be a starting point to extend the technique to 3D. June 2010 SPE Journal
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Fig. 13—(a) Normalized vorticity map (Model 2); (b) background grid based on vorticity; (c) triangular grid generated by AFT; (d) triangular grid generated by Delaunay tessellation; (e) Voronoi grid, dual of Delaunay grid; (f) oil-recovery factor by the vorticity-based grid, uniform coarse grid, and fine grid.
Nomenclature G = sum of source terms h = gridblock length Iw = Length of well k = permeability kr = relative permeability L = spacing parameter Lmax = maximum spacing parameter June 2010 SPE Journal
Lmin N Pwf Qinj r S I u
= = = = = = = =
minimum spacing parameter number of source points well flowing pressure at producer water injection rate distance between gridpoint and source element phase saturation I I I velocity vector (u = ux i + u y j ) relaxation factor 339
= fluid viscosity = intensity of source element I I I I I = vorticity vector ( = x i + y j + z k ) Superscripts r = iteration number Subscripts b = system boundary i, j = gridblock indices in x, y direction i±½ = gridblock face indices; left (−), right (+) in x direction j±½ = gridblock face indices; left (−), right (+) in y direction n = nth source w = well x, y = x, y coordinate References Ashjari, M.A., Firoozabadi B., Mahani, H., and Khoosan, D. 2008. Using Vorticity as an Indicator for the Generation of Optimal Coarse Grid Distribution. Transport in Porous Media 75 (2): 167–201. Ashjari, M.A., Firoozabadi, B., Mahani, H., and Khoozan, D. 2007. Vorticity-based coarse grid generation for upscaling two-phase displacements in porous media. J. Pet. Sci. Eng. 59 (3–4): 271–288. doi: 10.1016/ j.petrol.2007.04.006. Castellini, A. 2001. Flow based grids for reservoir simulation. MS thesis, Stanford University, Stanford, California. Cimet, M. and Sweet, R.A. 1973. Mesh refinements for parabolic equations. Journal of Computation Physics 12 (4): 513–525. doi: 10.1016/00219991(73)90102-2. Darman, N.H., Durlofsky, L.J., Sorbie, K.S., and Pickup, G.E. 2000. Upscaling Immiscible Gas Displacements: Quantitative Use of Fine Grid Flow Data in Grid Coarsening Scheme. Paper SPE 59452 presented at the SPE Asia Pacific Conference on Integrated Modelling for Asset Management, Yokohama, Japan, 25–26 April. doi: 10.2118/59452-MS. Durlofsky, L.J., Behrens, R.A., Jones, R.C., and Bernath, A. 1996. Scale Up of Heterogeneous Three Dimensional Reservoir Descriptions. SPE J. 1 (3): 313–326. SPE-30709-PA. doi: 10.2118/30709-PA. Durlofsky, L.J., Jones, R.C., and Milliken, W.J. 1997. A nonuniform coarsening approach for the scale-up of displacement process in heterogeneous porous media. Advances in Water Resources 20 (5–6): 335–347. doi: 10.1016/S0309-1708(96)00053-X. Edwards, M.G. 2000. M-matrix flux splitting for general full tensor discretization operators on structured and unstructured grids. J. Comp. Phys. 160 (1): 1–28. doi: 10.1006/jcph.2000.6418. Edwards, M.G. and Rogers, C.F. 1998. Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Computational Geosciences 2 (4): 259–290. doi: 10.1023/A:1011510505406. Evazi, M. and Mahani, H. 2009. Generation of Voronoi Grid Based on Vorticity for Coarse-Scale Modeling of Flow in Heterogeneous Formations. Transport in Porous Media (in press; posted 09 October 2009). doi: 10.1007/s11242-009-9458-2. Evazi, M., Mahani, H., Hejranfar, K., and Masihi, M. 2008. VorticityBased PEBI Grids for Improved Upscaling of Two Phase Flow. Paper SPE 113703 presented at the Europec/EAGE Annual Conference and Exhibition, Rome, 9–12 June. doi: 10.2118/113703-MS. Farmer, C.L., Heath, D.E., and Moody, R.O. 1991. A Global Optimization Approach to Grid Generation. Paper SPE 21236 presented at the SPE Symposium on Reservoir Simulation, Anaheim, California, USA, 17–20 February. doi: 10.2118/21236-MS. Farrashkhalvat, M. and Miles, J.P. 2003. Basic Structured Grid Generation: With an Introduction to Unstructured Grid Generation. Oxford, UK: Butterworth-Heinemann. Garcia, M.H., Journel, A.G., and Aziz. K. 1992. Automatic Grid Generation for Modeling Reservoir Heterogeneities. SPE Res Eng 7 (2): 278–284; Trans., AIME, 293. SPE-21471-PA. doi: 10.2118/21471-PA. Lapidus, L. and Pinder, G.F. 1982. Numerical Solution of Partial Differential Equations in Science and Engineering, Chap. 5, 405–417. New York: John Wiley & Sons. Li, D., Cullick, A.S., and Lake, L.W. 1995. Global scale-up of reservoir model permeability with local grid refinement. J. Pet. Sci. Eng. 14 (1–2): 1–13. doi: 10.1016/0920-4105(95)00023-2. 340
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Mohammad Evazi Yadecuri is a PhD student at the University of Southern California. e-mail:
[email protected]. He holds a bachelor’s degree in petroleum engineering from the Petroleum University of Technology and an MS degree in reservoir engineering from Sharif University of Technology. His research interest includes flow modeling in porous media and upscaling. Hassan Mahani is a research reservoir engineer with Shell International Exploration and Production in Rijswijk, The Netherlands. e-mail:
[email protected]. Before joining Shell, Mahani was Assistant Professor of Petroleum Engineering at Sharif University of Technology. His main research interests include dynamic induced fractures modeling, enhanced oil recovery including low-salinity waterflooding, upscaling, and optimal coarse grid generation. He holds a PhD degree in petroleum engineering from Imperial College London, and an MS (honors) degree from Sharif University of Technology and a BS (honors) degree from Isfahan University of Technology, both in chemical engineering. He was the second runner-up of the student contest in the Canadian International Petroleum Conference in 2004. Mahani is a technical editor for the SPE Journal. June 2010 SPE Journal
