Modeling Reservoir Geometry With Irregular Grids
- Z.E. Heinemann (Mining U. of Leoben) | C.W. Brand (Mining U. of Leoben) | Margit Munka (Mining U. of Leoben) | Y.M. Chen (Mining U. of Leoben)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- May 1991
- Document Type
- Journal Paper
- 225 - 232
- 1991. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 5.5 Reservoir Simulation, 5.4.6 Thermal Methods, 5.3.2 Multiphase Flow, 5.1.5 Geologic Modeling
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This paper describes a practical method in which irregular or locallyirregular grids are used in reservoir simulation with the advantages offlexible approximation of reservoir geometry and reduced grid-orientationeffects. Finite-difference equations are derived from an integral formulationof the reservoir model equations equivalent to the commonly used differentialequations. Integrating over gridblocks results in material-balance equationsfor each block. This leads to a finite-volume method that combines theadvantages of finite-element methods (flexible grids) with those offinite-difference methods (intuitive interpretation of flow terms).Grid-orientation effects are investigated. For grids based on triangularelements, the more isotropic distribution of gridpoints diminishes theorientation effect significantly. Numerical examples show that the regions ofinterest in a reservoir can be simulated efficiently and that well flow can berepresented accurately.
The accurate and efficient simulation of complex reservoirs depends highlyon proper grid selection. Grids based on a Cartesian coordinate system havebeen widely used, but have some disadvantages: (1) inflexibility in descriptionof faults, pinchouts, and discontinuities of reservoir parameters; (2)inflexibility in representation of well locations; and (3) the influence ofgrid orientation on the results.
Local grid refinement and nine-point difference schemes were introduced toimprove the performance of the Cartesian grid technique.
To diminish grid-orientation effects, Pruess and Bodvarsson presented auniform hexagonal gridblock pattern. Grids based on presented a uniformhexagonal gridblock pattern. Grids based on orthogonal curvilinear coordinatesystems were proposed by Hirasaki and O'Dell and investigated by Sonier andChaumet, Robertson and Woo, and Fleming. Curvilinear coordinate systems,defined by streamlines and equipotential lines, can be more appropriate for theflow pattern under consideration. If the coordinate system is not strictlyorthogonal, however, mixed derivative terms have to be introduced that distortthe structure of the linear equation. For this reason, the mixed derivativeswere usually neglected. The finite-difference approximation based on suchnonor-thogonal grids, sometimes called corner-point geometry, can result inconsiderable errors and is not applicable for simulation.
In this paper, irregular gridblock systems for simulation of complicatedreservoirs are proposed. The multiphase flow equations are discretized with afinite-volume approach. Anisotropic permeability is taken into account.Gridblock systems based on permeability is taken into account. Gridblocksystems based on perpendicular bisectors are described, and grid-orientationeffects are perpendicular bisectors are described, and grid-orientation effectsare examined. The performance of these irregular grids is demonstrated bynumerical examples (a single-phase, a black-oil, and a steamdrivesimulation).
An integral approach similar to that described by Nghiem and Pedrosa andAziz is applied to discretize the multiphase flow Pedrosa and Aziz is appliedto discretize the multiphase flow equations. The conservation of mass within acontrol volume V is expressed for the component k as a balance between theaccumulation within V and the flow through its surface A: (1)
The phase velocity is given by the multiphase formulation of Darcy's law,(2)
Discretization of the conservation Eq. 1 involves two main steps:construction of an appropriate gridblock system and setup of proper differenceequations for each gridblock.
Because Eq. 1 holds for volumes of arbitrary shape, some flexibility inconstructing gridblocks is possible. The derivation of differ-ence equationsfor blocks such as those presented in this paper lies, in some sense, midwaybetween the finite-difference and finiteelement approaches. This method offersthe advantage of flexible approximation of the reservoir geometry by regular orirregular grids. In particular, the blocks used in the examples in this paperare based on triangular elements commonly used in finite-element methods. Onthe other hand, the method preserves the simple derivation and handling offinite-difference approximations, the local and illustrative modeling of flowin the neighborhood of a gridpoint, and the universality of finite differenceswith respect to varous types of differential equations.
The method is called a finite-volume or -balance method or also acontrol-volume or integrated-finite-difference method. As these names indicate,such methods have been used in many formulations (see Refs. 16 through 18 andthe references cited therein). In fact, such an approach is used (at leastimplicitly) when the conventional conservative finite-difference equations fora Cartesian gridblock system are derived. Nevertheless, the freedom andflexibility in choosing the shape of the gridblocks apparently have not beenexploited extensively in practical reservoir simulation examples thus far.
A gridblock system, as described here, consists of gridpoints and blockssurrounding each point. The blocks cover the whole reservoir region withoutleaving gaps or overlapping each other. For a given gridblock, the spatialdiscretization of the accumulation term is obtained by approximating the volumeintegral in Eq. 1 by a simple one-point quadrature formula: (3)
where ( ), is the average value of S over VI, the volume of Gridblock I Forthe time derivative, a first-order finite difference is used. The particularshape of the control volume VI does not enter the difference expressions butessentially determines the form of the flow terms, which we explain in greaterdetail now.
Flow Terms. Surface A of a gridblock is partitioned into subsurfaces.Subsurface A is the part of A that is shared by Block I and neighboring BlockJ. Substituting Darcy's law (Eq. 2) into Eq. 1 and splitting the integral intoa sum of integrals over subsurfaces allows the flow of the k component fromBlock I to J to be written as (4)
To derive a finite-different expression, one has to approximate the surfaceA, the value of the scalar product ( ) , and the component mobility p on thesurface A.
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