Summary

Imbibition in water-wet matrix blocks of fractured porous media is commonly considered to be countercurrent. The modeling studies of this paper indicate that when a matrix block is partially covered by water, oil recovery is dominated by cocurrent imbibition, not countercurrent. It is also found that the time for a specified recovery by the former can be much smaller than that by countercurrent imbibition. Consequently, use of the imbibition data by immersing a single block in water and its scale-up may provide pessimistic recovery information. Moreover, it is shown that the application of the diffusion equation for modeling of oil recovery by cocurrent imbibition leads to a large error. Through a detailed study of the governing equations and boundary conditions, significant insight is provided into the mathematical and physical differences between co- and countercurrent imbibition.

Introduction

Countercurrent imbibition, in which water and oil flow through the same face in opposite directions, has received considerable attention in the literature. The mathematical formulation of countercurrent imbibition is of the form of a nonlinear diffusion equation.1 Analytical and semianalytical solutions of countercurrent imbibition have been recently emphasized.2–4 These solutions assume a semi-infinite domain and are valid before the saturation front reaches the far boundary.

Much experimental work on countercurrent imbibition has been reported in the literature. In these experiments, the oil-saturated cores are either immersed in water, or sealed such that water in-flow and oil out-flow occur through the same faces.5–10 Some imbibition studies have reported oil production from a face not covered with water (i.e., cocurrent imbibition).6,8,11,12 The majority of these studies were inconclusive regarding the differences between cocurrent and countercurrent imbibition. In a detailed experimental study, Bourbiaux and Kalaydjian13 examined the cocurrent and countercurrent imbibition processes on a laterally coated core. When the two opposing faces of a cylindrical core were open to flow, and water was in contact with one face, oil was mostly produced by cocurrent imbibition from the face in contact with oil; oil production from the water-contacted face was very small—about 3%. The authors expressed uncertainty about the source of this oil. It was not clear if this small amount was produced from the rock, or was the oil from the dead volume. The countercurrent experiment had a slower recovery than the cocurrent experiment; the half-recovery time for cocurrent imbibition was 7.1 hours and that for countercurrent imbibition was 22.2 hours, for one set of experiments. The measured saturation profiles for the two processes were also different.

In a recent experimental study, a stack of Berea and chalk matrix blocks were used to study water injection in fractured porous media.14 Visual observation and recovery data from a number of tests revealed that (1) the injected water did not fill the fracture space surrounding the matrix blocks rapidly, and (2) the recovery mechanism was mainly cocurrent imbibition before a block was fully covered by water. Once the rock was covered by water, the oil recovery rate decreased due to low efficiency of countercurrent imbibition.

The above experimental studies13,14 show that there are significant recovery differences between co- and countercurrent imbibition. To our knowledge, there is currently no theoretical study comparing the two imbibition processes. The major objective of this paper is to study the similarities and differences of co- and countercurrent imbibition and point out the consequences for practical applications. In the following, we first investigate the equations and boundary conditions for countercurrent and cocurrent imbibition. A numerical model is then developed to compare the behavior and recovery efficiency of the two processes. Scaling studies are used to draw general conclusions.

Mathematical Investigation

The one-dimensional (1D) countercurrent imbibition process can be described by a nonlinear diffusion equation1 of the form

$∂∂x(D(Sw)∂Sw∂x)=∂Sw∂t,(1)$

where

$D(Sw)=−kϕkroμof(Sw)dPcdSw(2)$

and

$f(Sw)=11+krokrwμwμo.(3)$

The initial and boundary conditions are

$Sw=Swi,t=0,0≤x≤L,(4)$
$Sw=1−Sor,t=0+,x=0,(5)$
$qw=0,t=0+,x=L.(6)$

The symbols are defined in the nomenclature. Eq. 1 assumes that the fluids are incompressible, and the effect of gravity is neglected. The diffusion coefficient of Eq. 2 is bell shaped with respect to water saturation, attaining a value of zero at Swi and Sor (see Ref. 2). Eq. 5 expresses the continuity of capillary pressure at the inlet face (Pc=0) and Eq. 6 is the no-flow boundary condition at the outlet.

Formulation of cocurrent imbibition includes an additional convective term 2

$∂∂x(D(Sw)∂Sw∂x−qtf(Sw))=∂Sw∂t,(7)$

where the functions D and f are given by Eqs. 2 and 3, respectively. In Eq. 7, qt=qo+qw is unknown, and an additional equation, i.e., the pressure equation, with initial and boundary conditions is required to complete the formulation.

Some studies of the imbibition process have assumed that the pressure gradient in the displaced oil phase may be neglected.15–17 This assumption is based on the common practice in hydrology, where the mathematical formulation of unsaturated water flow ignores the air pressure gradient (see Ref. 18 for an account of this assumption). Under this assumption, the problem is formulated19 as in Eq. 1 with initial and boundary conditions 4 to 6. However, the corresponding diffusion coefficient is

$D(Sw)=−kϕkrwμwdPcdSw.(8)$
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