A Computer Model for Two-Phase Coning Simulation
Authors: Settari Antonin , Aziz Khalid 
Published: December 2007

Abstract
Published coning studies do not treat the wellbore/reservoir interaction propers. A coning model is developed here with a new finite-difference formulation of this boundary condition. It is shown that there are two factors, often ignored, that must be considered:the outlet effect which requites that the capillary pressure approach zero at the sand lace, andthe compatibility condition, which requires that the vertical pressure gradient in the well be the same as the pressure gradient at the reservoir/wellbore boundary.
We show here that when both these factors, along with point-distributed grid, are incorporated in a model, a highly stable scheme with a low discretization error is obtained. Results of this study are compared with those of several studies reported in the literature and also with some currently available commercial models.

Introduction
In mathematical modeling of multiphase flow in reservoirs, boundary conditions representing production and injection are usually accounted for by production and injection are usually accounted for by introducing source and sink terms in the appropriate differential equations. These terms are actually singular (Dirac functions), as they are zero everywhere except at the source or sink points (wells). One can also introduce source terms on difference rather than differential level, with the same result. Such a concept, first used by Douglas et al., gives good results in many situations and A has been naturally carried over to the modeling of production of a well in coning simulations. Mathematical simulation of coning is very difficult and it has not been possible to evaluate the importance of the boundary conditions among all other important factors in mathematical modeling. Much of the recent research effort has been focused on improving the stability of the difference schemes, resulting in models that use various approximations to the fully implicit treatment of transmissibilities. Only minor attention has been paid to the boundary conditions. Although it was recognized some time ago by Douglas et al. that the "source" boundary conditions are not physically correct, there has been no proof in the literature that it affects the performance of coning models. performance of coning models. A conventional finite-difference model uses source terms at the well that are distributed among the production points according to mobilities or potentials (more precisely according to the product potentials (more precisely according to the product of mobility and the potential gradient at the well). Such a model does not take into account two physical aspects of the flow: the outlet effect due physical aspects of the flow: the outlet effect due to capillary forces and the condition of continuity of pressure in the wellbore with pressure in the reservoir (compatibility condition). Two recent works have attempted to incorporate a more accurate description of the boundary conditions into a model. Nolen and Berry give only an empirical method of satisfying the compatibility condition at the well and do not deal with the outlet effect at all. Sonier et al. consider both effects; however, they neglect the pressure drop due to friction in the wellbore, and their calculation of the outlet effect is not a logical part of their finite-difference scheme. The present treatment of the boundary conditions ensures that the proper physical conditions are satisfied, and it forms a logical part of the finite-difference model. Although, as will be shown later, the outlet effect and the pressure drop due to friction do not seem to be important at the field scale when viewed separately (the outlet effect cannot be ignored for small-scale laboratory experiments), it is necessary to consider them in order to arrive at the proper formulation of the finite-difference model of the coning phenomenon. We will demonstrate by an example that much of the time-discretization error currently reported in the literature for coning models must be actually attributed to incorrect treatment of the boundary conditions. The time-step sensitivity of our model is two times better than the sensitivity of the semi-implicit model of Nolen and Berry and vastly better than the sensitivity of the Letkeman and Ridings model.
SPEJ
P. 221