Fully Coupled Multiblock Wells in Oil Simulation

A fully coupled treatment of oil wells that are completed in more than one zone results in a bordered matrix. This paper develops solution algorithms that incorporate paper develops solution algorithms that incorporate existing direct and iterative (incomplete LU) solutions in a straightforward manner. Timings in scalar and vector modes on the Cray for a typical reservoir simulation problem are presented. problem are presented. Introduction Numerical simulation of oil reservoirs requires the solution of coupled sets of highly nonlinear partial differential equations. These equations represent the conservation of oil, gas, water, and energy. It usually is necessary to solve from 3 to 10 coupled equations per finite-difference cell. The equations usually are discretized by use of a nearest-neighbor coupling in space and a fully implicit timestep scheme. The resulting set of nonlinear algebraic equations then is solved by Newtonian iteration., Clearly, simulation of large systems requires effective solution of the Jacobian matrix. Many practical reservoir simulation problems involve multiblock wells or fractures. These situations arise when a well is completed in several layers, and consequently the wellbore penetrates several finite-difference cells. Each conservation equation in a cell penetrated by a well will have a source term of the form .....................................(1) where qjt is the mass influx of component k (resulting from the well), Xk is the mobility of component k, 1 pi is the pressure in cell i, and pi, is the unknown wellbore pressure in well j. pressure in well j. To specify the wellbore pressure, pi, an additional equation is required. This extra equation as generally a constraint op the total flow into the well - This constraint is of the form .....................................(2) where qJt. is the total specified fluid flow into well j, Nc, is the total number of components, and is the set of cell numbers penetrated by well j. Because several cells are connected to the same well, there is now an extra degree of coupling between these cells through the well-bore pressure. This coupling generally will not be consistent with the coupling produced by the usual finite-difference molecule. If the well pressures, pjw, are treated explicitly, or are lagged one iteration, convergence difficulties or stability limitations often result. 7 Fully coupled treatment of multiblock wells gives rise to a bordered matrix. We develop various methods to solve these systems. These methods are specifically designed for the block-banded systems arising from fully implicit thermal problems, although similar methods can be used for single-component systems The iterative methods are extensions of the incomplete factorization techniques (ILU), and a direct method is presented for comparison. Existing solution routines can be modified easily to solve the bordered system. Solution of the Bordered Matrix The standard approach to solving fully implicit, fully coupled multiblock wells (or fractures) is to order the unknowns so that those connected with flow in the reservoir (cell pressures, saturations, etc.) appear first in the solution vector. The unknowns connected with the well (well pressures) are placed last in the solution vector. This produces a bordered Jacobian matrix (see Fig. 1). The upper left portion of the matrix has the usual incidence matrix for the Jacobian of nearest-neighbor finite-difference discretization. The incidence matrix for the Jacobian is a matrix with entries zero if the Jacobian elements are zero, and with entries one if the Jacobian elements are nonzero. The border of columns on the upper right of Fig. 1 contains derivatives of the source terms (Eq. 1) with respect to the wellbore pressure The border of rows on the lower left contains derivatives of the constraint equations (Eq. 2) with respect to reservoir variables (i.e., cell pressures). The block on the lower right contains derivatives of the constraint equations with respect to the wellbore pressures and is diagonal. The number of extra columns and rows is proportional to the number of fully coupled wells (or fractures). Although the incidence matrix of the reservoir flow portion of the matrix is symmetric, the incidence matrix of portion of the matrix is symmetric, the incidence matrix of the borders is not necessarily symmetric. George discusses three possible block factorizations of sparse, linear systems. The algorithm used here is based on his second factorization. SPEJ P. 535