Intergrid Operations - Dynamic Gridding Techniques in Nonlinear Domain Decomposition Methods

Abstract

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractDomain decomposition methods provide major advantages to large-scale and high-resolution reservoir simulation studies in terms of computer resource requirements. In this paper, a directional, intergrid mapping technique is introduced together with a nonlinear domain decomposition algorithm -the Fast Adaptive Composite Grid (FAC). Domain decomposition techniques have often been applied to solve the linear system of equations resulting from reservoir numerical models. However, a stronger, more efficient approach is to apply the domain decomposition techniques directly to the nonlinear problem. As a result, mapping dynamic reservoir properties from one grid scale to another one, a process also known as intergrid mapping, becomes a key operation in dealing with composite grid systems.Normally, upscaling and downscaling techniques focus on geological structures and formation properties. The proposed intergrid mapping operations in the FAC method honor the static reservoir properties while preserving the properties of fluid flow dynamics between grids with different scales. Most importantly, the intergrid mapping operator in FAC can accurately represent local phenomena in much coarser domains, so that the composite grid residual, which can be viewed as the relative overall solution error, can be effectively minimized. Application of the FAC method brings significant reduction in CPU time and memory requirements when compared with simulations having the same level of global grid and timestep refinement, making it possible to attain large simulation scales and resolution with limited computer resources.In this paper, the approximation of the differential conservation equations is based on the control volume approach and uses cell-centered finite difference methods. The composite grid solution is obtained through a defect-correction process of refinements. Unlike many domain decomposition algorithms, the proposed method provides full approximation storage (FAS) by computing and storing the whole coarse grid solution, including the grids for which a refinement exists.Computational results from a model featuring local spacetime refinements are presented to illustrate that a rigorous intergrid mapping operation can significantly improve the efficiency of nonlinear domain decomposition algorithms, especially in cases of multiphase flow with irregular local timestepping. The presented numerical examples demonstrate the applicability of the proposed numerical method. t J J IJ c t J rJ J J J J = − ∇Φ , represents the superficial flow associated with phase J, where the potential gradient is given by J J J , , c c c X Y Z ∆ ∆ ∆ , over which certain cells are further divided into subdomains with local grid cells, fc Ω , fc = 1, 2… N fc , of dimension ( ) p c N J J I J J