Analytical Model for the Waterflood Honouring Capillary Pressure (With applications to laboratory studies)
Authors: Bedrikovetsky P. , Rodrigues J.R.P. , Britto P.R.F. 
Published: December 2007

Abstract
An analytical model for the coreflood honouring the capillary pressure is developed. The analytical solution for the problem of the displacement of oil by water taking into account the capillary pressure is obtained using the method of matched asymptotic expansions.
The model is based on the matching of the capillarity-free Buckley-Leverett solution with two asymptotic expansions in regions of strong capillary pressure effects: with the continuous stabilized-zone solution in the neighbourhood of the displacement front, and with the end-effect solution near to the core outlet. Numerical simulation shows the high accuracy of the analytical solution proposed.
The analytical solution is used for regularization of the inverse problem for determination the relative permeabilities from the coreflood data. Analytical solution allows to transform the inverse problem, which is the ill-posed one in the general formulation, to the system of two integral equations. The system can be solved asymptotically using the regular small parameter method.

Introduction
Relative phase permeabilities are the basic functions which determine the efficiency of the waterflooding So the accurate laboratory determination of these functions from the routine coreflood procedure is an important problem of applied reservoir engineering.
The most widely spread procedure is the non-steady-state method which consists on the displacement of oil with connate water by water. Measurements of the water cut during the displacement allow to calculate the fractional flow function (Welge's method), while the pressure drop on the core after the breakthrough allows to determine the total mobility function. The fractional flow and the total mobility functions determine relative permeabilities for both phases.
The above mentioned methods are derived from the Buckley-Leverett equation and are based on the assumption that the capillary forces are negligible if compared with the viscous ones. Actually, they do not use the solution of the direct problem by itself, being based only on the self-similarity of the solution.
In short low permeable cores or for low displacement velocities the capillary pressure is of the same order of magnitude as the pressure drop on the core during the displacement. So the above mentioned assumption which allows to neglect the capillary pressure in the governing equation is not valid any more.
To overcome this difficulty, numerous papers have reported simplified approximate solutions of the inverse problem of determination of relative permeabilities honouring capillary pressure from the coreflood tests.
Another possible approach is to couple optimization methods with the numerical solution of flow equations including the capillary effects, where the solution of the inverse problem is found from minimization of the difference between the data measured and the ones predicted by the numerical model. This inverse problem is an ill-posed one. Therefore, this approach can give stable results only for fixed analytical expressions for relative permeability. Nevertheless, very different shapes of relative permeability have been reported in the literature, thus the more-or-less general formulae are to contain a lot of adjusting parameters, which reduces the stability of the optimization method.
Another possibility for the solution of the inverse problem could be the use of an analytical solution for the direct problem. The self-similar solution of the 1-D displacement problem without the capillary pressure is well known.
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