Cocurrent and Countercurrent Imbibition in a Water-Wet Matrix Block

Abstract

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 ( S w ) ∂ S w ∂ x ) = ∂ S w ∂ t , ( 1 ) where D ( S w ) = − k ϕ k r o μ o f ( S w ) d P c d S w ( 2 ) and f ( S w ) = 1 1 + k r o k r w μ w μ o . ( 3 ) The initial and boundary conditions are S w = S w i , t = 0 , 0 ≤ x ≤ L , ( 4 ) S w = 1 − S o r , t = 0 + , x = 0 , ( 5 ) q w = 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 ( S w ) ∂ S w ∂ x − q t f ( S w ) ) = ∂ S w ∂ 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 ( S w ) = − k ϕ k r w μ w d P c d S w . ( 8 )

A critical review on fundamental mechanisms of spontaneous imbibition and the impact of boundary condition, fluid viscosity and wettability

“…However, many matrix blocks are partly covered by water in fractured reservoirs (Bourblaux, 1990;Pooladi-Darvish and Firoozabadi, 2000). Haugen et al (2014Haugen et al ( , 2015 designed a special experimental setup to study SI in the core partly covered by water, in which one end of the core was in contact with water and the other end of the core was in contact with oil.…”

“…expelled by combination of co-and counter-current imbibition (Bourblaux, 1990;Pooladi-Darvish and Firoozabadi, 2000).…”

“…For the occurrence of SI, the water-rock interfacial energy must be lower than the oil-rock interfacial energy, which embodied that the rock must be water-wet. According to the flow direction of wetting and non-wetting phase, SI generally is divided into two types: co-current imbibition and counter-current imbibition (Pooladi-Darvish and Firoozabadi, 2000;Karpyn et al, 2009;Cai et al 2010;Mirzaei-Paiaman et al, 2011). For co-current imbibition, the wetting and non-wetting phase flow through the same end face in opposite directions.…”

Solute Transport With Chemical Reaction in Singleand Multi-Phase Flow in Porous Media

“…Correlation of the imbibition capillary pressure data depends on the type of application. For example, for water-oil system, see for example, (Pooladi-Darvish & Firoozabadi, 2000), the capillary pressure and the normalized wetting phase saturation are correlated as,…”

Upscaling of fractured oil reservoirs using homogenization including non-equilibrium capillary pressure and relative permeability

“…In fractured reservoirs, capillary forces, leading to counter-current or co-current imbibition [11,22,25,36,40], are the main drivers for waterdrive recovery from the matrix blocks [16]. Reservoir wettability and its effect on oil recovery have been the subject of numerous studies [4,18,30,49,56].…”