Analytical Well Models for Reservoir Simulation

Abstract

Abstract The computation of flowing-well bottomhole pressure from the pressure of the block containing the well or of well now rate when the flowing bottomhole pressure is specified are important considerations in reservoir simulation. While this problem has been addressed by several authors, some important aspects of the problem are not treated adequately in the literature. We present an analytical method for computing the wellblock factors (constants of the PI) for a well located anywhere in a square or rectangular block (aspect ratio between 1/2 and 2). Equations for well geometric factors and well fraction constants are given for gridblocks of various types, containing a single well, encountered in reservoir simulation studies. The equations given in this paper can be used for both block-centered and paper can be used for both block-centered and point-distributed grids in five- and nine-point two-dimensional point-distributed grids in five- and nine-point two-dimensional (2D), finite-difference formulations. The radial flow assumption used in deriving the equations in this paper is not always strictly valid; however, for most practical situations it provides an adequate approximation for near-well flow. Introduction Handling of wells in reservoir simulators presents several difficulties that require special considerations. These difficulties generally can be divided into two classes.Problems arise because the block size usually is large compared to the size of the well, and hence the pressure of the block computed by the reservoir simulator is not a good approximation for the well pressure.Problems can be caused by the complex interaction (coupling) between the reservoir and the wellbore in both injection and production wells. Some aspects of this second problem are discussed by Settari and Aziz and Williamson and Chappelear, and other important aspects remain unresolved. This paper, however, deals with only the first problem-the problem of relating well-block pressure in the finite-difference model to the well pressure. The discussion is further restricted to single-phase 2D areal models, without any direct consideration of three-dimensional (3D) and cross-sectional flow problems. In die absence of more accurate model, well factors derived from single-phase flow considerations may be used even when two- or three-phase flow exists near the well. Well-Block Equations Peaceman has defined an equivalent well-block radius, Peaceman has defined an equivalent well-block radius, ro, as the radius at which the steady-state flowing pressure in the reservoir is equal to the numerically pressure in the reservoir is equal to the numerically calculated pressure, po, of the block containing the well This definition of ro can be used to relate the well pressure, pw, to the flow rate, q, through po: pressure, pw, to the flow rate, q, through po:Peaceman has obtained an approximate value of ro for Peaceman has obtained an approximate value of ro for an interior well in a uniform square grid by assuming radial steady-state flow between the well block and the blocks adjacent to this block:where i=1, 2, 3, 4 for the four surrounding blocks in the five-point finite-difference scheme. Combining this equation with the steady-state difference equation for the well block,Peaceman obtained the value Peaceman obtained the valuewhich is close to the more precise numerically computed value of 0. 1982 ( – 0.2). Peaceman obtained this more precise value by use of the difference in pressure between precise value by use of the difference in pressure between injection and production wells in a repeated five-spot as derived by Muskat, who used potential theory. Peaceman applied this solution to the difference in Peaceman applied this solution to the difference in pressure between the injection and production blocks and pressure between the injection and production blocks and obtainedwhere Delta pm is the numerically computed pressure difference between injection and production blocks for an M × M grid. The right side of Eq. 5 approaches an approximately correct value of 0.194 for M=3. This implies that the assumption of radial flow used to obtain Eq. 4 is reasonable even for a very coarse 3 ⨯ 3 grid. SPEJ P. 573