Hybrid-CVFE Method for Flexible-Grid Reservoir Simulation
- Larry S.K. Fung (Computer Modelling Group) | Lloyd Buchanan (Computer Modelling Group) | Sharma Ravi (Computer Modelling Group)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- August 1994
- Document Type
- Journal Paper
- 188 - 194
- 1994. Society of Petroleum Engineers
- 5.1.5 Geologic Modeling, 4.1.5 Processing Equipment, 5.8.5 Oil Sand, Oil Shale, Bitumen, 5.3.2 Multiphase Flow, 5.1.1 Exploration, Development, Structural Geology, 4.3.4 Scale, 5.8.6 Naturally Fractured Reservoir, 4.1.2 Separation and Treating, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 5.4.6 Thermal Methods, 5.5 Reservoir Simulation, 5.6.4 Drillstem/Well Testing
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The flexible control-volume finite-element (CVFE) method is used to construct hybrid grids. The method involves use of a local cylindrical or elliptical grid to represent near-well flow accurately while honoring complex reservoir boundaries. The grid transition is smooth without any special discretization approximation, which eliminates the grid transition problem experienced with Cartesian local grid refinement and hybrid Cartesian gridding techniques.
Well flows and pressures are the most important boundary conditions in reservoir simulation. In a typical simulation, rapid changes and large pressure, temperature, saturation, and composition gradients occur in near-well regions. Treatment of these near-well phenomena significantly affects the accuracy of reservoir simulation results; therefore, extensive efforts have been devoted to the numerical treatment of wells and near-well flows.
To perform practical field-scale simulation, use of coarse grid cells is often necessary where the near-well regions are not represented adequately. Local grid refinement is an efficient method to improve the accuracy of near-well modeling. Previously, Cartesian grid refinement was proposed by several authors.1-3 However, special treatment is required at the fine- to coarse-grid boundary where one big grid cell can be connected to several small grid cells. The other problem is that Cartesian grids do not follow the natural flow geometry around the wells. A hybrid Cartesian grid method was proposed that uses a cylindrical or elliptical grid in the well regions and a rectangular grid elsewhere in the reservoir.4 This method accurately models near-well regions by coupling several coning models with a reservoir model in a multiwell simulation scenario.
Recently, the CVFE method was developed for reservoir simulation.5-8 This method offers considerable geometric flexibility to represent the reservoir region. A properly constructed CVFE grid has low grid-orientation sensitivity, similar to that of nine-point schemes.9,10 Because of the geometric flexibility, CVFE local grid refinement can be obtained easily and consistently without the discretization errors at the coarse- to fine-grid boundary seen in Cartesian refinement approaches.11 The full permeability tensor for transmissibility calculations is included easily on the CVFE surface, which is an important consideration in simulating heterogeneous and anisotropic reservoirs.
The hybrid-CVFE method combines the benefits of the CVFE method with the cylindrical or elliptical refinement around the well. The size and location of the cylindrical regions can be chosen freely without the confines of rectangular grid cells. This approach also eliminates the limitations of the previously proposed hybrid-Cartesian method, which requires (1) that the well to be at the center of the Cartesian grid cells and (2) an approximate method for handling the transition between the rectangular and cylindrical grids.
In this work, the CVFE formulation is derived in the general context of the method of weighted residuals (MWR).12 From this derivation, the related integral-finite-difference (IFD),13 the Petrov-Galerkin finite-element (PGFE), and the CVFE methods can be viewed in the same general perspective. This discussion also shows that the CVFE method includes the perpendicular bisection (PEBI) method, 14 Yoronoi grid,15 and Cartesian finite difference as special cases. The treatment of hybrid-CVFE grids is described, and we show that this hybrid discretization is consistent and does not incur additional discretization error at the grid boundaries.
A description of CVFE and hybrid-CVFE grid generation methods follow. An interactive graphic grid generator was developed to triangulate over any 2D planar region by locally optimizing on the triangle geometry. CVFE local grid refinement and hybrid grids are also interactively generated. Delaunay triangulation is validated by a swap-test algorithm on the generated grid. Generally, swapping is rarely required when the proposed triangulation algorithm is used.
Then, the CVFE methods are validated against analytical solutions for the case of a single well in a bounded cylindrical reservoir and for the case of three wells in a bounded rectangular reservoir. We show that for low-permeability reservoirs, the equivalent-/well-radius ratio, re/rw, constrains the size of the wellblocks for noncylindrical grids so that the desired accuracy cannot be achieved. The cylindrical or hybrid-CVFE grids do not have such limitations.
Numerical examples comparing the accuracy and efficiency of the CVFE locally refined grid and the hybrid-CVFE grid are included. Test cases include a multi well black-oil coning and a multiwell cyclic steam stimulation problem. The gridding requirements for a desired level of accuracy are compared for each test problem. The flexibility of CVFE and hybrid-CVFE methods for field-scale simulation is demonstrated.
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