Impact Of Inlet Boundary Conditions On The Numerical Simulation Of One Dimensional Coreflooding
Authors: Shen C. , Ruth D.W. 
Published: March 2010

Abstract
Numerical simulators developed to match coreflooding performance provide a powerful means for the determination of relative permeabilities of a core sample. However, the application of proper capillary boundary conditions at the ends of a sample is not straightforward. In this paper, the impact of inlet boundary conditions on the numerical simulation of one-dimensional coreflooding is studied. It is found that (i) the widely used inlet saturation boundary condition - the inlet saturation rises instantly to its maximum value - is not correct, and instead the inlet saturation increases gradually depending on the capillary number; (ii) if the maximum inlet saturation condition is applied, numerical solutions would not satisfy both the frontal advance equation and the fractional flow equation; and (iii) the pressure drop history is directly related to the inlet saturation. Hence, a variable inlet saturation boundary condition must be used in the simulation to account for inlet capillary end effects. Thereafter, the pressure drop history (including initial responses) can be used to make performance matches.

Introduction
Numerical simulations of immiscible corefloods have been widely used to study two-phase flows in a core sample including capillary and gravity force effects(1,2,3,4.5.6.7), and to interpret the results of unsteady state coreflooding experiments-history matching to determine relative permeabilities and capillary pressure(8,9,10). In coreflooding experiments, capillary end effects always exist and can greatly affect displacement performance(11,12,13,14,15.16,17). Although some experimental techniques have been developed to eliminate these effects(18,19), they may still exist and affect displacement performance, especially when reservoir fluids and flow parameters are utilized. To account for the capillary end effects, proper boundary or initial conditions must be applied at the core ends in numerical simulations. The outlet end capillary effect has been incorporated into the simulations performed by Fassihi(10). Recently, Shen and Ruth(7) derived a set of initial and boundary conditions which can properly describe the capillary effect on the variation of the inlet saturation and applied them in their simulations.
The application of proper capillary boundary conditions at the ends of a sample in simulations is not straight forward. In spite of long awareness of the inlet end effect on coreflooding flow (11,12,13,15,17), previous numerical simulations(1,2.3,4,5,6) usually could not account for it, with the result of errors in the solutions. It can be anticipated that the solution with proper initial and boundary conditions will be self-consistent and the solution with improper ones will not be. Therefore, the theme of the present study is to investigate the impact of the inlet end capillary effect on the results of numerical simulations. In the subsequent part of the paper, the notion of an inlet end effect indicates the inlet end capillary effect on the variation of the inlet saturation.
The numerical simulation technique employed in this study is the same one as used in the previous study(7). The finite element method was employed to obtain dynamic solutions of a Lagrangian equation derived by Bentsen(6) (Bentsen's equation). This paper consists of the following parts: Firstly, a brief summary of the derivation of Bentsen's equation is presented.