Development and Validation of a Modified Well Model Equation for Nonuniform Grids With Application to Horizontal Well and Coning Problems

Abstract

Abstract In this paper we demonstrate the important limitations of the standard Peaceman well model equation when applied to non-uniform areal Cartesian grids and when used in low problems with a significant vertical flow component near the well. We present and validate simple modifications to the Peaceman well model which restore its validity for non-uniform areal grids and for problems which model vertical flow processes like gas and water coning. Introduction In reservoir simulation, a well model equation is required to properly relate well rates and flowing bottom hole pressures when the grid blocks which contain the wells are much larger than the wells themselves. Using quasi steady-state, radial flow theory, Peaceman developed a well model equation which has been the standard for reservoir simulation. The main attractions of this well model are (a) that it effectively removes sensitivity of the predicted well rates to the well block size and (b) that it is written entirely in terms of well block variables. Although the Peaceman well model equation was written for radial flow problems, on uniform cartesian grids, it is routinely used on non-uniform gas (X not constant in the x-direction and/or Y not constant in the y-direction) and for modelling vertical flow phenomena, such as gas and water coning. For the latter problem, the argument is usually made that, at least at the perforations, the flow is locally radial. Also, the need to resolve these local, small scale, but physically important, near-well flow features, necessitates the use of fine grids around wells. Practical limitations on computer resources prevents the use of such refined grids throughout the entire reservoir model. Hence simulation models often run with highly non-uniform grids. In this paper, we investigate the consequences of using the standard Peaceman well model equation in problems formulated on non-uniform Cartesian grids and for problems in which the vertical flow transmissivity is not insignificant compared with that of horizontal, radial flow. We show in Section 2.0 that, for non-uniform grids, it is no longer possible to specify an equivalent well block radius at which to apply the computed well block pressure. We then propose and validate, a simple modification of the standard well model equation to correct this problem. We also present a methodology for performing non-uniform grid refinement, which ensures increased numerical accuracy. In Section 3.0, we demonstrate that the radial well model is not adequate for properly including vertical flow effects. We then propose and validate a cylindrical well model equation, for uniform and non-uniform grids, which may be simply implemented in existing simulation models. 2.0 LIMITATIONS OF PEACEMAN'S WELL MODEL FOR AREAL PROBLEMS 2.1 INHERENT ASSUMPTIONS This section investigates the development of the Peaceman well model equation so that an understanding of its practical limitations may be obtained. The starting point is the analytical steady-state, radial pressure distribution about a well (single phase; no gravity): (1) Peaceman assumes that Eq.(1) applies to the numerical finite difference model and associates the numerically calculated well grid block pressure, p, with an equivalent radius, r : (2) P. 397^