INTERNATIONAL JOURNAL OF c  2009 Institute for Scientific NUMERICAL ANALYSIS AND MODELING Computing and Information Volume 6, Number 3, Pages 375–388 WELL FLOW MODELS FOR VARIOUS NUMERICAL METHODS ZHANGXIN CHEN AND YOUQIAN ZHANG This paper is dedicated to the special occasion of Professor Roland Glowinski’s 70th birthdate. Abstract. Numerical simulation of fluid flow and transport processes in the subsurface must account for the presence of wells. The pressure at a gridblock that contains a well is different from the average pressure in that block and different from the flowing bottom hole pressure for the well [17]. Various finite difference well models have been developed to account for the difference. This paper presents a systematical derivation of well models for other numerical methods such as standard finite element, control volume finite element, and mixed finite element methods. Numerical results for a simple well example illustrating local grid refinement effects are given to validate these well models. The well models have particular applications to groundwater hydrology and petroleum reservoirs. Key words. Well models, petroleum reservoirs, aquifer remediation, finite difference, finite element, control volume finite element, mixed finite element, fluid flow, numerical experiments 1. Introduction Numerical simulation of fluid flow and transport processes in the subsurface must account for the presence of wells. The pressure at a gridblock that contains a well is different from the average pressure in that block and different from the flowing bottom hole pressure for the well [17]. The difficulty in modeling wells in a field scale numerical simulation is that the region where pressure gradients are the largest is closest to a well and is far smaller than the spatial size of gridblocks. Using local grid refinement around the well can alleviate this problem but can lead to an impratical restriction on time step sizes in the numerical simulation [5]. The fundamental task in modeling wells is to model flows into the wellbore accurately and to develop accurate well equations that allow the computation of the bottom hole pressure when a production or injection rate is given, or the computation of the rate when this pressure is known. The first theoretical study of well equations was given by Peaceman [17] for cell- centered finite difference methods on square grids for single phase flow. Peaceman’s study gave a proper interpretation of a well-block pressure, and indicated how it relates to the flowing bottom hole pressure. The importance of his study is that the computed block pressure is associated with the steady-sate pressure for the actual well at an equivalent radius r e . For a square grid with a grid size h, Peaceman derived a formula for r e by three different approaches: (1) analytically by assuming that the pressure in the blocks adjacent to the well block is computed exactly by the radial flow model, obtaining r e =0.208h, (2) numerically by solving the pressure equation on a sequence of grids, deriving r e =0.2h, and (3) by solving exactly the system of difference equations and using the equation for the pressure drop between the injector and producer in a repeated five-spot pattern problem, finding r e =0.1987h. From these approaches, he concluded that r e ≈ 0.2h. Received by the editors August 30, 2008. 2000 Mathematics Subject Classification. 65N30, 65N10, 76S05, 76T05. 375