Use of a Hybrid Grid in Reservoir Simulation

Abstract

Summary A technique is presented to improve the treatment of wells in reservoir simulators by the use of an orthogonal curvilinear grid (cylindrical or elliptical) in well regions and a rectangular grid elsewhere in the reservoir. Special methods are developed to handle irregular blocks connecting the two types of regions. The nonlinear equations for the two regions are solved with different levels of implicitness; the simultaneous solution (implicit) method is applied to the well regions, while an implicit pressure, explicit saturation (IMPES) technique is used for the reservoir region. An efficient approach, requiring no outer iterations, between well and reservoir regions is used to couple different regions. The scheme is validated by the simulation of the flow of a slightly compressible fluid in a square domain with one boundary open to flow. For this case, the numerical solution with the hybrid grid approach matches the analytical solution exactly. A computer model based on the approach was developed for simulating two-phase flow in multiwell horizontal reservoirs. We show that the proposed technique provides much better prediction of the phase split in the producers than provides much better prediction of the phase split in the producers than the conventional approach. Introduction The treatment of wells in reservoir simulators can have a strong influence on computed results. The common approach is to relate the wellbore pressure and the reservoir block pressure through single-phase flow models in the well region. Peaceman showed that for a square block, the pressure obtained from the steady-state radial pressure pressure obtained from the steady-state radial pressure distribution equals the wellblock pressure at an equivalent radius of r 0.2 x. With this concept of equivalent radius, a relationship between wellbore flowing pressure and wellblock pressure can be derived easily. This approach has been extended to wells in the center of rectangular blocks and to off-centered wells. Strictly speaking, these well models are invalid when high rates of saturation change occur in the neighborhood of the wellbore. Improper handling of phase mobility leads to errors in the calculation of WOR and GOR. Furthermore, the use of Cartesian coordinates everywhere in the reservoir does not allow the simulation of real geometry of the nearwell flow. While multiphase flow near the well can be properly modeled through a coning simulator, it is difficult to do this in large simulation studies that involve many wells. On the other hand, in many situations, the usual approach of using pseudofunctions provides only a crude approximation of the real problem. One method of handling wells rigorously is the coupling of well coning models with a reservoir model. Akbar et al. incorporated a one-dimensional (1D), three-phase, radial coordinate well simulator within a twodimensional (2D), three-phase, Cartesian coordinate reservoir simulation model. The radial model simulated a rectangular gridblock in the areal model. The equality of the volumetrically weighted average pressure within the radial model and the corresponding reservoir block pressure was established. In addition, the summation of pressure was established. In addition, the summation of the fluxes into the four vertical faces of the well gridblock was taken as the influx into the radial system. Alternate calculations for the two models were performed until both models predicted the same production rates. However, material balance between the two models was not exactly maintained. Mrosovsky and Ridings extended the Akbar et al. approach by coupling a'2D cylindrical model with a three-dimensional (3D), rectangular-grid reservoir simulator. To obtain good resolution in the vicinity of the wellbore, a fine grid spacing around the well is often used in reservoir simulation. Rosenberg developed a method of localized mesh refinement for finite-difference techniques. Recent developments in the use of local grid refinement and adaptive implicit methods are major steps toward the improvement of well treatment in reservoir simulators. These approaches require the use of the same coordinate system everywhere in the reservoir, however, and cannot take advantage of the almost radial nature of flow near the wells. The need for an accurate and simple way to represent wells in reservoir simulators has led to the development of a hybrid grid approach. In this technique, a Cartesian grid is used for the entire reservoir with arbitrarily fine curvilinear orthogonal grids in well regions that may span one or more blocks of the Cartesian grid (see Fig. 1). An integral approach is applied to derive the discretized flow equations. The solutions for the various regions can be decoupled in such a way that different levels of implicitness in the treatment of transmissibilities can be considered. Furthermore, the well region problem can be solved in 1, 2, or 3D, as appropriate. SPERE p. 611