2.2. Relative Permeability
The relative permeabilities of water–oil two-phase flow in horizontal capillary channels are expressed as Equations (4) and (5) [
32], which are
where
Jw and
Jo are the seepage velocities for water and oil, respectively, m/s;
krw is the relative permeability of water phase;
kro is the relative permeability of the oil phase;
μw is the dynamic viscosity of water phase, Pa·s;
μo is the dynamic viscosity of the oil phase, Pa·s;
L is the length of the capillary fracture, m. Δ
p is the pressure drop of water–oil flow, Pa.
From Equations (4) and (5), if the properties of fluids and a fracture are fixed, the relative permeabilities can be obtained for water–oil flow as long as the pressure drop is known at a certain flow rate. The pressure drop of water–oil flow in a capillary fracture is the key to study the seepage law of water–oil flow. When water and oil pass together in a capillary fracture, capillary force exists and is related to wettability [
33]. The influence of wettability on the relative permeabilities of stratified and dispersed water–oil flow regimes in capillary fractures is analyzed in detail below.
For oil–water two-phase flow in capillary fractures, it can be assumed as one-dimensional steady flow. Assuming the phase 1 is a wetting fluid and the phase 2 is a non-wetting fluid, considering wall shear stresses, interfacial shear stress and capillary force, the momentum equations can be obtained for laminar two-phase flow in a horizontal fracture according to the two-fluid model developed by Lahey [
34]. The results are shown by the following Equations (6) and (7). The term
in Equation (7), which is related to capillary force, can be expressed by Equation (8) following the approach proposed for dispersed bubbles [
34]. The sum of those terms associated with capillary force is zero by adding Equation (6) to Equation (7), which is expressed by Equation (9). Accordingly, the term
is obtained to be Equation (10). The main radius of curved surface for fluid 1 and 2 are denoted by
r1 and
r2. The interfacial tension between fluid 1 and 2 is denoted by
σ. The term
pc in Equations (8) to (10) can be expressed by Equation (11).
The expressions of the variables in Equations (6) and (7) are relevant to flow regime. For stratified flow,
Table 1 shows the basic relations for the calculations of those variables in Equations (6) and (7) [
35].
In
Table 1,
J1 and
J2 are the seepage velocities for fluids 1 and 2, respectively, m/s;
u1 and
u2 are the average velocities for fluids 1 and 2, respectively, m/s;
Q1 is the volumetric flow rate of fluid 1, m
3/s;
Q2 is the volumetric flow rate of fluid 2, m
3/s;
Q is the total volumetric flow rate of fluid 1 and fluid 2, m
3/s;
ϕ1 and
ϕ2 are the volumetric contents for fluid 1 and fluid 2, respectively.
S1 and
S2 are the saturations for fluid 1 and fluid 2, respectively.
ρ1 and
ρ2 are the densities for fluid 1 and fluid 2, respectively, kg/m
3;
τ1 and
τ2 are the wall shear stresses for fluid 1 and fluid 2, respectively, Pa;
τi is the shear stress of two-phase interface, Pa;
λ1 and
λ2 are the Darcy friction factors for fluid 1 and fluid 2 in laminar case, respectively;
λi and
ρi are the friction factor and the density of faster phase, respectively;
Re1 and
Re2 are the Reynolds numbers of fluid 1 and fluid 2, respectively.
A1 and
A2 are the cross-sectional areas for fluid 1 and fluid 2, respectively, m
2;
B1 and
B2 are the wetted perimeters for fluid 1 and fluid 2, respectively, m;
Bi is the wetted perimeter of two-phase interface, m;
d1 and
d2 are the hydraulic diameters of fluid 1 and fluid 2, respectively, m;
μ1 and
μ2 are the dynamic viscosities for fluid 1 and fluid 2, respectively, Pa·s;
pc is capillary pressure, Pa;
and
are the capillary forces for fluid 1 and fluid 2, respectively, Pa/m.
For stratified two-phase flow, the basic relations in Equations (6) and (7) are shown in
Table 1. Equation (8), Equation (10) and the physical quantities in
Table 1 are taken into Equations (6) and (7), Equations (12) and (13) are obtained as below
where
a and
b are interfacial slip coefficients of relative permeabilities for stratified flow, as shown in
Table 2. These two interfacial slip coefficients were proposed to effectively reflect the influence of two-phase interfacial slip on the relative permeabilities of stratified flow in previous study [
30].
From comparing Equations (4) and (5) with Equations (12) and (13), the relative permeabilities for stratified flow with the effect of wettability are obtained as follows
where
kr,1 is the relative permeability of wetting fluid 1,
kr,2 is the relative permeability of non-wetting fluid 2.
c1 is the wettability coefficient of the relative permeability of fluid 1.
c2 is the wettability coefficient of the relative permeability of fluid 2. These two proposed wettability coefficients are used to study the effect of wettability on the relative permeabilities.
From Equations (14) and (15), the relative permeabilities of stratified water–oil flow are relevant to wetted perimeter, saturation, interfacial slip coefficient and wettability coefficient. The relative permeabilities of stratified water–oil flow in a capillary fracture with the effect of wettability are shown in following Equations (16)–(19) in terms of the wettability coefficients. If oil is the wetting fluid and the water is non-wetting fluid, the expressions for stratified flow are the followings, which are
where
kr,1,o is the relative permeability of oil in the case that oil is the wetting fluid,
c1,o is the wettability coefficient of the relative permeability of oil in the case that oil is the wetting fluid,
kr,2,w is the relative permeability of water in the case that water is the non-wetting fluid,
c2,w is the wettability coefficient of the relative permeability of water in the case that water is the non-wetting fluid.
However, if water is the wetting fluid and oil is the non-wetting fluid, the equations are expressed for stratified flow as below
where
kr,2,o is the relative permeability of oil in the case that oil is the non-wetting fluid,
c2,o is the wettability coefficient of the relative permeability of oil in the case that oil is the non-wetting fluid,
kr,1,w is the relative permeability of water in the case that water is the wetting fluid,
c1,w is the wettability coefficient of the relative permeability of water in the case that water is the wetting fluid.
If wettability is not considered, Equations (20) and (21) are obtained. In this instance, the relative permeabilities of stratified water–oil flow are relevant to wetted perimeter, saturation and interfacial slip coefficient.
For dispersed flow, wall shear stress of non-wetting fluid 2 is zero, the wall shear stress of wetting fluid 1 is equal to the wall shear stress of dispersed flow, and the wetted perimeter for wetting fluid 1 is the total wetted perimeter. Equation (22) can be obtained from Equations (6) and (7) without considering wettability. The basic relations for dispersed flow are shown in
Table 3 as below [
36].
In
Table 3,
τm is the wall shear stress for dispersed flow, Pa;
λm is the Darcy friction factor for dispersed flow in laminar flow case;
ρm is the density for dispersed flow, kg/m
3;
Rem is the Reynolds number of dispersed flow;
μm is the dynamic viscosity for dispersed flow, Pa·s;
um is the average velocity of dispersed flow, m/s.
Equation (22) and the physical quantities in
Table 3 are taken into Equations (4) and (5), the relative permeabilities of dispersed water–oil flow in horizontal capillary fractures are obtained, as shown in the following Equations (23) and (24). It can be seen that the relative permeabilities of dispersed water–oil flow are relevant to saturation and viscosity.
If wettability is considered, Equation (25) is obtained from Equations (6), (7) and (22). Equation (25) and the physical quantities in
Table 3 are taken into Equations (4) and (5), the relative permeabilities of dispersed liquid–liquid flow in a capillary fracture with the effect of wettability are obtained as follows.
From Equations (26) and (27), the relative permeability model for dispersed water–oil flow is related to saturation, viscosity and wettability coefficient. Similarly, the relative permeabilities of dispersed water–oil flow in a capillary fracture with the effect of wettability are shown in following Equations (28)–(31) in terms of the wettability coefficients. If oil is the wetting fluid and the water is non-wetting fluid, the expressions for dispersed water–oil flow are the followings.
However, if water is the wetting fluid and oil is the non-wetting fluid, the equations are expressed for dispersed water–oil flow as below.