Building PEBI Grids Conforming To 3D Geological Features Using Centroidal Voronoi Tessellations

1

Building PEBI Grids Conforming To 3D Geological Features Using Centroidal Voronoi Tessellations

Romain MERLAND1, Bruno LÉVY2, Guillaume CAUMON3 1 CRPG-INPL, France [email protected]

2 INRIA, France [email protected]

3 CRPG-INPL, France [email protected]

Peer-reviewed IAMG 2011 publication

doi:10.5242/iamg.2011.0064

Abstract

For numerical reservoir flow simulation, grids that are conformal to the geological features are needed in order to reduce the homogenization error (in particular between horizons) and to re-trieve the major flow features (such as faults). In this paper, Voronoi Tessellations are obtained by an optimization method where the minimized function is modified from the classical Cen-troidal Voronoi function. The geological features are considered as inner surfaces, dividing the reservoir into closed subdomains. These methodologies are applied successfully to 3D synthetic reservoirs with internal features such as horizons, faults, partly cutting faults and pinch-outs.

Merland et al.

2

1 Introduction

Unstructured grids offer great flexibility for flow simulation, allowing to adapt the cells to vari-ous flow constraints such as local refinement (Durlofsky [2005], Prevost et al. [2005], Evazi and Mahani [2010]), and cell orientation (Souche [2004], Mlacnik et al. [2006], Mlacnik and Dur-lofsky [2006], Merland et al. [2011]). In particular, Voronoi diagrams (also called PErpendicular BIsector) benefit from recent developments in the field of geometry processing to generate au-tomatically well shaped cells and grids suitable for flow discretization.

In the geological model from which the flow grid is generated, several geological features have great impact on flow:

Layers are often fairly homogeneous formations from the permeability and porosity point of view. For upscaling consistency, one cell of the flow grid should not be partly in one layer and partly in another.

Faults can either be barriers or drains, hence are often modeled with transmissibility mul-tipliers. Consequently, one cell should not go across a fault.

Pinch-outs are frequent and should also be recovered as accurately as possible.

Verma [1996], Verma and Aziz [1997], Courrioux et al. [2001] propose two ways of conforming grids to geological features: on Voronoi grids, by tracing the normals to the feature, and placing Voronoi seeds equidistant on each side of the feature; on CVFE grids, by modifying the cell bor-ders in order to conform them locally to the feature. It entails in a CVFE-BAG (Boundary Adapted Grid) and CVFE-Voronoi-BAG. The main drawback is it cannot deal with complex features that are frequent in geological models such as pinch-outs, and for CVFE-Voronoi-BAG, the PEBI property is partly lost.

More recently, Branets et al. [2009] propose to describe in 2D the features as Planar Straight Line Graphs (PSLG) and to define protection areas around them by placing Voronoi seeds in mirror images. The pinch-outs are treated specifically in a similar way but with circular mirror placement. The remaining of the domain is then meshed using constrained Delaunay Triangula-tion. The method gives good results in 2.5D but has not been extended to 3D to our knowledge.

In this article, we choose to optimize the coordinates of the Voronoi seeds in 3D according to the geological features. The geological features are 3D Piecewise Linear complexes (PLC), for in-stance triangulated surfaces, and divide the domain into a finite number of subdomains. Once the seeds are optimized, the final grid is obtained by clipping the 3D Voronoi diagram by the do-main. In this way, the features are not exactly recovered (because it is an iterative optimization) but the error is expected to be small enough to meet discretization requirements.

The optimization process relies on Centroidal Voronoi Tessellations (CVT) algorithms (Sec-tion 1) that are natively designed in 3D and can take in parallel a wide range of constraints into account. The CVT algorithms aim at minimizing a CVT function measuring the error between the Voronoi seeds and the relevant Voronoi cell barycenter. The first idea developed in this ar-ticle is to alter the location of the Voronoi barycenter in such a way that the Voronoi seeds go away from the features (Section 2). As the seeds are constrained to remain in their respective subdomain, they are naturally placed symmetrically on either side of the features. The second idea is to define and minimize a function similar to the CVT function, but measuring the error of conformity, i.e. the part of the cell that is on the wrong side of the feature (Section 3). The CVT function and the conformity function are then weighted by the user and minimized at the same time. For now, 3D synthetic cases have been tested (Section 4) with the different ideas and prove

Building PEBI Grids Conforming To 3D Geological Features Using Centroidal Voronoi Tessellations

3

to give good results, robustness and efficiency. The cases have layers, faults, partly cutting faults and pinch-outs.

2 Centroidal Voronoi Tessellation

A Centroidal Voronoi Tessellation (CVT) is of particular interest for unstructured gridding as it is an optimal Voronoi Diagram, providing the PEBI (PErpendicular BIsector) property, guaran-teeing an optimized compactness of the cells and an optimal sampling of the control volume cen-ters (Voronoi seeds). The discretization errors due to badly shaped cells are thus minimized.

Moreover, the cell size can be controlled using a density function known in each point of the domain, and the orientation and aspect ratio of the cells can be constrained using a background tensor field. It is now also possible to constrain the tessellation to conform to geological features as developed in this article. A brief recall of the main Voronoi definitions is performed in this section. The reader is referred to Merland et al. [2011] for more details.

As described by Du et al. [1999], the Voronoi diagram relevant to a set of seeds is a set of Voronoi cells defined by:

│‖ ‖ ‖ ‖,for where ||.|| denotes the Euclidean norm.

For each Voronoi cell, one can define the mass centroid ∗ . A CVT is a Voro-

noi tessellation where all seeds are placed at their cell mass centroids, i.e. ∗ (Figure 1).

Du et al. [1999] give the definition of a CVT function :

∑ ‖ ‖

and prove that this function is minimal for Voronoi tessellations that are centroidal. Given an initial random set of seeds , computing a CVT is then a minimization problem. The gradient of the CVT function for the seed is given by:

| 2 ∗ where is the mass of the cell.

The CVT function is minimal when the gradient is null, i.e. when the seed is at its cell barycen-ter. This leads to a popular iterative algorithm: (1) starting with a random set of seeds , (2) compute the barycenter of the Voronoi cells, (3) move the seeds to the barycenters, (4) repeat steps 2 and 3 until some convergence criterion is reached.

Liu et al. [2009] propose an efficient minimization algorithm based on the Gauss-Newton's algo-rithm and coupled with a fast approximation of the gradient and the Hessian of the CVT func-tion.

Merland et al.

4

a) b)

Figure 1: Voronoi tessellation and centroidal Voronoi tessellation for ten seeds in two dimensions. (a) Seeds are black dots, mass centroids are small circles. (b) Seeds are located at the mass centro-

ids of their Voronoi regions (from Du et al. [1999]).

3 Modification of the barycenter

In this section, the computation of the barycenter is modified to take into account the effect of the inner boundaries on the Voronoi cells. Actually, only the cut cells have a special treatment to compute their barycenter. This needs to define a consistent input as a boundary representation.

3.1 Definition of input

In this article, we consider a closed surface defining a volumetric domain and an initial random set of seeds inside the domain. The geological features considered are faults and horizons, i.e., inner boundaries inside the domain. These inner boundaries either cut the entire domain, i.e. de-fining closed subdomains, either cut partly the domain, in which case we continue them arbitrari-ly to form closed subdomains (Figure 2a). Each subdomain is treated independently. Further-more, we tag the inner boundaries to indicate that a special treatment needs to be performed on the cut cells, as described in the following.

3.2 CVT weighted by parts

The first step is to determine which seed is inside which subdomain, in order to define an inner part and an outer part for each cell. Note that a subdomain can include the inner part of some cells and the outer part of others, depending on the position of their seed. Actually, it is possible to distinguish 3 parts in each Voronoi cell as indicated on Figure 2b: (in) inner part, (in-b) inner part connected to the boundary, (out) outer part.

The second step is to apply a particular weight to each part:

| | | |

| 2 ∗ ∗ ∗ 2

| 2 ̅ ∗

Where , , are weights chosen by the user. The introduced ̅ ∗ is relevant to a modified ba-rycenter taking into account the effect of the boundary. Experiments seem to indicate that the best results are obtained for 1, 1, 0 : the outer part is simply ignored and the inner part connected to the boundary has a “repulsive effect” on the barycenter.

Building PEBI Grids Conforming To 3D Geological Features Using Centroidal Voronoi Tessellations

5

a) b)

Figure 2: (a) A square 2D domain with outer boundaries (blue) and inner boundaries made of hori-zons and faults (red). Note the fault partly cutting the domain (subvertical) and continued to reach

the upper horizon and form closed subdomains. The virtual continuation is tagged as an outer boundary. (b) Voronoi cells are divided in parts by the boundary: inner part (yellow), inner part connected to the boundary (red), outer part (green). The white cells are not cut by the boundary.

a) b) c)

Figure 3: Parameters taken into account for the computation of the objective function (a) outer volume (b) outer/inner volume ratio (c) variance of the scalar field f with constant values in each

part fin and fout respectively.

4 Definition of objective functions

In this section, rather than modifying the CVT function , we define an objective function computed and minimized at the same time as . The minimized function is the

weighted sum of these two functions: where , are weights chosen by the user.

The objective function is a measure of the error to the expected result. We have defined three objective functions: ( ) sum of the outer volumes , ( ) sum of the volume ratio outer/inner ⁄ , ( ) sum of the variance of a scalar field over the Voronoi cell (Fig-ure 3).

As we try to minimize , we also compute its gradient, i.e. the sum of the gradient of the CVT function and of the objective function. In the following, we only give the objective function and its gradient. The CVT function and its gradient are given in Section 1.

4.1 Outer volume

This objective function and its gradient are simply defined by:

Merland et al.

6

As described in Lévy and Liu [2010], the gradient of the volume can be computed from the sym-bolic information stored in each vertex of the Voronoi cell. is the gradient of the volume

computed only for the vertices in the outer part.

4.2 Volume ratio

This objective function and its gradient are defined by:

, forverticesinouterpart, forverticesininnerpart

4.3 Variance of a scalar field

Given a scalar field defined over the entire domain, the mean value Φ of over a Voronoi cell and the objective function to minimize are given by:

Φ ∑ Φ

Let's define the scalar field as piecewise constant:

, ifybelongstotheinnerpartofthecurrentVoronoicell, ifybelongstotheouterpartofthecurrentVoronoicell

For this scalar field, the objective function and its gradient are given by:

∑

∑

∗ , forverticesinouterpart , forverticesininnerpart

These equations are dependent of and only in a constant multiplier. As we minimize the objective function, it can be simply ignored in the computation of the function and its gradient.

5 Applications to 3D reservoirs

The previously described algorithms have been applied to generate grids of 3D reservoirs. The reservoirs are defined by closed surfaces, one for each subdomain, and the inner boundaries are tagged.

The first reservoir is a synthetic cube with three horizontal horizons defining four subdomains. The resulting meshes for 3,000 seeds are presented on Figure 4.

Building PEBI Grids Conforming To 3D Geological Features Using Centroidal Voronoi Tessellations

7

a)

b)

c)

d)

e)

Figure 4: Synthetic reservoir. For each optimization, an outer view, a vertical slice colorized with volume property (red=high, blue=low), and an horizontal slice at the level of an inner boundary are provided. (a) optimization using standard CVT algorithm. (b) optimization with CVT weighted by

parts α=1, β=-1, γ=0. (c) optimization using an objective function measuring the outer volume (α=0.1, β=0.9), (d) the outer/inner volume ratio (α=0.001, β=0.999), (e) the variance of the scalar

field (α=0.01, β=0.99).

Merland et al.

8

All meshes succeed in recovering the inner boundaries. The following remarks can be made:

The optimization using CVT function weighted by parts leads to a repulsion of the seeds on either parts of the inner boundaries. The volume of the Voronoi cells near the bounda-ries is therefore increased.

For the optimizations using an objective function, no repulsive effect is seen. The volume of the Voronoi cells does not have any significant pattern.

As shown on the horizontal slices, there are still some facets not conforming locally to the boundaries. However, the measured error is small as described in the following. These badly shaped remaining facets may be due to early reach of the convergence crite-rion of the optimization process, or to local minima, as the number of seeds on each side of the boundary is not exactly the same.

An error can be measured as ∑ ⁄ where is the volume of the reservoir. The values for an isotropic CVT and for the above meshes are given in Table 1. The best results are obtained for the optimization using an objective function measuring the variance of the scalar field (Fig-ure 4e).

Table 1: Values of the error for the synthetic case with three horizontal horizons. For optimization using CVT weighted by parts, α, β, γ are relevant to the inner part, the inner part connected to

boundary and the outer part respectively. For optimization using objective function, α, β are rele-vant to the weight of CVT function and objective function respectively.

mesh isotropic CVT weighted

by parts objective function

out volume out volume volume ratio variance f α - 1 0.1 0 0.001 0.01 β - -1 0.9 1 0.999 0.99 γ - 0 - - - -

%error 4.53216 0.414914 0.228797 0.193068 0.324805 0.13933

The second reservoir is a synthetic cube with three layers partly faulted. The fault is virtually continued (in all directions) to form closed subdomains and the virtual part is not tagged as an inner boundary. The resulting mesh for 3,000 seeds with an objective function measuring the outer volume is presented on Figure 5. The errors for all the generated meshes are presented in Table 2a. The best results are obtained for the optimization using an objective function measur-ing the outer volume.

Table 2: Values of the error. (a) Synthetic case with partly cutting fault. (b) Duplex synthetic case.

Mesh isotropic

CVT weighted by parts

objective function out volume volume ratio variance f

(a)

α - 1 0.1 0.001 0 β - -1 0.9 0.999 1 γ - 0 - - -

%error 4.89975 1.1171 0.776342 0.958247 1.00572

(b)

Α - 1 0 0 0 β - -1 1 1 1 γ - 0 - - -

%error 5.57255 0.943777 0.861054 0.917948 1.2233

Building PEBI Grids Conforming To 3D Geological Features Using Centroidal Voronoi Tessellations

9

a) b) c)

d) e) f)

Figure 5: Synthetic case with a partly cutting fault. (a) outer view. (b) inner view showing the fault continued to the top layer. (c-f) optimization using an objective function measuring the outer vo-lume (α=0.1, β=0.9). (c) slice along y axis showing Voronoi seeds. (d) resulting grid along the same

slice. (e) slice along the x axis. (f) zoom of the slice along the x axis.

The third reservoir is a synthetic duplex with six subdomains, branching intersections and pinch-outs. The resulting mesh for 10,000 seeds with an objective function measuring the outer volume is presented on Figure 6. The error for all the generated meshes are presented in Table 2b. The best results are obtained for the optimization using an objective function measuring the outer volume.

Conclusion

The main advantage of all the presented techniques is that it is fully automatic and deals with various issues such as pinch-outs, multiple intersections and 3D meshing. It produces PEBI un-structured grids and can be combined simultaneously with flow based Gridding techniques such as local refinement, around well radial patterns, streamlines-equipotentials cell orientation, cell aspect ratio control, in an integrated Centroidal Voronoi Tessellations approach (Merland et al. [2011]).

Also, it is important to note that the inner boundaries are not exactly recovered, but that it is an optimization process leading to the best possible sampling of the grid cell centers. The badly shaped remaining facets can be removed by several ways:

By changing the geometry of the final grid without modifying the coordinates of the seeds. It is a geometrical modification of the facets. The strict PEBI property is therefore lost, but the error for the discretization is expected to be small, at least when the inner boundary is a barrier from the flow simulation point of view, and the homogenization er-ror is reduced to zero.

Merland et al.

10

a)

b)

c)

d)

e)

Figure 6: Duplex synthetic case. (a) outer view. (b) inner view. (c-e) optimization using an objective function measuring the outer volume (α=0, β=1). (c) outer view of the grid. (d) slice along the y axis.

(e) zoom of the slice along the y axis.

By using a steepest gradient descent instead of the actual Newton like method for the op-timization. Thus, the convergence criterion will not be reached early.

By getting out of the local minima, either by removing or adding seeds on one side of the boundary, either by detecting the wrong configurations and moving the needed seeds away from the boundary.

By finding other objective functions measuring the error to the expected result.

Some other limitations have to be mentioned: (a) if the number of seeds is too small, the inner boundaries are not well recovered. The minimal number of seeds depends on the accuracy needed by the user. Accuracy can be measured by the error described previously. (b) the number

Building PEBI Grids Conforming To 3D Geological Features Using Centroidal Voronoi Tessellations

11

of seeds in each subdomain is fixed at the beginning of the process, i.e. seeds cannot go across a boundary once the optimization process has started. To overcome this problem, a consistent ini-tial sampling is required, for instance using a density field.

As the advantage of such techniques is to provide better control on cell homogenization and flow across inner boundaries, it could be interesting to provide flow simulation results and compari-sons with classical grids. For instance, it could be combined with automatic detection of flow boundary in channelized system to get cell boundaries perpendicular to geobody boundaries.

Acknowledgements

We thank Paradigm for the GoCAD developer licenses and the members of the GoCAD Consor-tium for funding this research. We also thank the ALICE team of LORIA in Nancy for the CVT algorithms and the CGAL team of INRIA Sophia Antipolis for the Delaunay algorithms.

Merland et al.

12

References

L. V. BRANETS, S. S. GHAI, S. L. LYONS. AND X.-H. WU. (2009): Efficient and accurate reservoir modeling using adaptative gridding with global scale up. SPE International Journal for Numerical Methods in Engineering.

G. COURRIOUX, S. NULLANS, A. GUILLEN, J. D. BOISSONNAT, P. REPUSSEAU, X. RENAUD, AND M. THIBAUT. (2001): 3d volumetric modelling of cadomian terranes (northern brittany, france): an automatic method using voronoi diagrams. Tectonophysics, 331(1-2):181-196. ISSN 0040-1951. doi: 10.1016/S0040-1951(00)00242-0.

Q. DU, V. FABER, AND M. GUNZBURGER. (1999): Centroidal voronoi tessellations: Ap-plications and algorithms. SIAM Review, 41:637-676.

L. J. DURLOFSKY. (2005): Upscaling and gridding of fine scale geological models for flow simulation. In 8th International Forum on Reservoir Simulation, Iles Borromees, Stresa, Italy.

M. EVAZI AND H. MAHANI. (2010): Unstructured coarse grid generation using back-ground-grid approach. SPE Journal, 15(2):326-340. ISSN 1086055X.

B. LÉVY AND Y. LIU. (2010): Lp centroidal voronoi tessellation and its application. ACM Transactions on Graphics, 29(4). doi: 10.1145/1778765.1778856.

Y. LIU, W. WANG, B. LÉVY, F. SUN, D.-M. YAN, L. LU, AND C. YANG. (2009): On centroidal voronoi tessellation - energy smoothness and fast computation. ACM Transactions on Graphics, 28(4). doi: 10.1145/1559755.1559758

R. MERLAND, B. LÉVY, G. CAUMON, AND P. COLLON-DROUAILLET. (2011): Build-ing centroidal voronoi tessellations for flow simulation in reservoirs using flow information. In Proceedings of SPE Reservoir Simulation Symposium. doi: 10.2118/141018-MS.

M. MLACNIK AND L. DURLOFSKY. (2006): Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients. Journal of Computational Physics, 216(1):337-361. ISSN 00219991. doi: 10.1016/j.jcp.2005.12.007.

M. MLACNIK, L. DURLOFSKY, AND Z. HEINEMANN. (2006): Sequentially adapted flow-based pebi grids for reservoir simulation. SPE International Journal for Numerical Me-thods in Engineering, pages 317-327.

J. PELLERIN, B. LÉVY, AND G. CAUMON. (2010): Advanced geometry processing in go-cad using graphite. In 30th Gocad Meeting.

M. PREVOST, F. LEPAGE, L. DURLOFSKY, AND J.-L. MALLET. (2005): Unstructured 3d gridding and upscaling for coarse modelling of geometrically complex reservoirs. Petro-leum Geoscience, 11(4):339-345.

L. SOUCHE. (2004): Generation of unstructured 3d streamline pressure-potential-based k-orthogonal grids. In 9th European Conference on the Mathematics of Oil Recovery.

S. VERMA. (1996): Flexible grids for reservoir simulation. PhD thesis, Standford University.

S. VERMA AND K. AZIZ. (1997): A control volume scheme for flexible grids in reservoir simulation. In Proceedings of SPE Reservoir Simulation Symposium. doi: 10.2118/37999-MS.

VITEL, S. (2007): Méthodes de discrétisation et de changement d'échelle pour les réservoirs fracturés 3D. PhD thesis, Institut National Polytechnique de Lorraine.