Reservoir Simulation With a Control-Volume Finite-Element Method

Abstract

Summary This paper describes a control-volume finite-element (CVFE) method incorporating linear triangular elements for the simulation of thermal multiphase flow in porous media. The technique adopts the usual finite-element shape functions to evaluate flow potentials at the control-volume boundaries and uses the conservation equations for each control volume. The main advantage of the CVFE method over the finite-difference method is in the representation of complex reservoir geometries. In addition, desirable features, such as local grid refinement for near-well resolution, can be achieved simply and consistently. The control-volume approach enforces local mass conservation and permits a direct physical interpretation of the resulting discrete equations. These are significant advantages over the classical Petrov-Galerkin or variational finite-element methods. The method was implemented in a general-purpose thermal simulator. Numerical examples compare the proposed method with five-point and nine-point finite-difference schemes in terms of grid-orientation effects and run time. The CVFE method was found to reduce grid-orientation effects significantly. At the same time, computational cost was much lower than for the nine-point scheme. The geometric flexibility of the method also is demonstrated. Introduction In many reservoir simulation problems, a flexible discretization method is extremely useful in the definition of complex reservoir geometries and discontinuities (such as faults) and in enhancing the resolution near the wells. The use of Cartesian grids with finite-difference methods has created difficulties and/or complexities in the definition of complex geometries or grid refinements.1-6 It is desirable to adopt the intrinsic grid flexibility of the finite-element method. However, combining upstream weighting with the usual finite-element method for the multiphase multidimensional flow problem presents difficulties. Although asymmetric weighting procedures like the Petrov-Galerkin method7 have been introduced to deal with the convective terms in the mixed convective-diffusive flow problems, such methods are in general not mass-conservative in the local sense. On the other hand, local mass conservation is a specific requirement of the control-volume methods. In addition, reservoir simulation problems can be very complex, involving multiphase mass and heat flow with interphase transfers and chemical reactions. The mass-conservative aspect of the control-volume methods is a distinct advantage in the programming and testing of these simulators. The CVFE method was proposed in computational fluid dynamics for solving the Navier-Stokes equations,8,9 where flexible gridding and local mass, momentum, and energy conservation are achieved. In this paper, a CVFE procedure for the reservoir flow equations is developed where flexible grid geometry is obtained without sacrificing the advantageous attributes of the control-volume finite-difference method. The derivation shows that the use of the perpendicular-bisection10 grid and the seven-point finite-difference method11 are special cases of this discretization method. Recently, Forsyth12 applied a CVFE method to the local-mesh-refinement problem by providing a smooth transition between the coarse and fine grids. As discussed in detail later, a proper choice of the triangular finite-element mesh is crucial to the reduction of grid-orientation effects. The construction of a CVFE grid by triangulation with one of the diagonals of each rectangle in a Cartesian grid (as in Ref. 12) will result in a five-point discretization scheme because the diagonal flow terms for this grid are identically equal to zero. The method results in a set of discretized conservative equations where the Jacobian construction for Newton's method and the upstream weighting of mobilities can be done in the usual way. For the incompressible single-phase flow problem, the method gives the same stiffness matrix as the Petrov-Galerkin weighted-residual finite-element method when linear shape functions are used. The criterion for maintaining positive transmissibility coefficients of a general anisotropic system also is derived. A number of examples are included to demonstrate the geometric flexibility, non-grid-orientation characteristics, and efficiency of the proposed method. p. 349-357