Pseudofunctions for Water Coning in a Three-Dimensional Reservoir Simulator
Authors: Woods E.G. , Khurana A.K. 
Published: December 2007

Abstract
Three-dimensional numerical models of bottom-water-drive reservoirs show delayed water breakthrough into individual wells when compared with observed well performance and individual-well coning models. This reservoir-model behavior results from masking of the well coning effect by volume-averaging pressure and saturation profiles around a well over a grid block with a large volume. The reservoir-simulator prediction of well performance can be improved by mathematically performance can be improved by mathematically transforming the production performance of a detailed well-coning model into a set of time-independent pseudorelative-permeability and capillary-pressure curves that then can be used in the reservoir model. A procedure for obtaining the required pseudofunctions is described and the results of their application in simple models and in a large reservoir-simulator model are shown.

Introduction
The prohibitive cost of numerical reservoir simulation with fine-grid definition models of large reservoirs has led to development of techniques whereby vertical saturation distribution and/or localized flow conditions in the vicinity of individual wells can be approximately accounted for in relatively coarse-grid models at an acceptable incremental cost. In particular, vertical cross-section models under capillary and gravity equilibrium have been used to derive pseudorelative permeabilities and capillary pressures for use in two-dimensional, areal models to simulate the average vertical distribution of flow without having to pay the computing price of a full three-dimensional model. Coats et al. described the use of the vertical equilibrium concept for developing pseudorelative-permeability and capillary-pressure pseudorelative-permeability and capillary-pressure functions for simulating the vertical dimension in a two-dimensional, areal simulator model This method assumes gravity-capillary equilibrium in the vertical direction. Also, Coats et al. developed a dimensionless parameter for estimating when these conditions are valid.
Martin formed a mathematical basis for pseudofunctions by reducing the equations for pseudofunctions by reducing the equations for three-phase, three-dimensional, compressible flow to two-dimensional relations by partial integration of the equations of flow. Hearn extended the pseudorelative-permeability concept by adapting it pseudorelative-permeability concept by adapting it to stratified reservoirs where viscous rather than gravity and capillary forces dominate the vertical sweep efficiency. Hawthorne studied the effects of capillary pressure on pseudorelative permeability derived from the Hearn stratified model. Jacks et al. further enlarged thepseudorelative-perrneability concept by developing dynamic pseudorelative permeabilities. (Dynamic pseudos, denoting pseudos permeabilities. (Dynamic pseudos, denoting pseudos determined under flowing rather than static conditions, allow one to account for the interaction between viscous and gravity forces resulting from rate variation in the vertical plane.) Kyte and Berry generalized the work of Jacks et al. by introducing the concept of pseudocapillary pressures and improving dynamic pseudofunction calculations to include varying flow potential gradients.
Emanual and Cook expanded the concept of vertical cross-section, pseudorelative permeabilities to include the vertical performance of individual wells. Their procedure combines the effect of coning and well pseudorelative permeabilities for use in a two-dimensional, areal model. Chappelear and Hirasaki used a different approach to including of coning effects in a two-dimensional, areal simulator by developing a functional relationship among water cut, average oil-column thickness, and total rate based on an analytical incompressible, steady-state model. The most sophisticated approach to representing well-coning effects in a reservoir simulator has been taken by Mrosovsky and Ridings and Akbar et al. They incorporated detailed numerical well models into the reservoir-model grid.
SPEJ
P. 251