Admissible Parameters for Two-Phase Coreflood and Welge–JBN Method

Abstract

The Welge–JBN method for determining relative permeability from unsteady-state waterflood test is commonly used for two-phase flows in porous media. We discuss the theoretical criteria that limits application of the basic Buckley–Leverett model and Welge–JBN method and the operational criteria of the accuracy of measurements during core waterflood tests. The objective is determination of the waterflood test parameters (core length, flow velocity and effluent sampling frequency) that fulfil the theoretical and operational criteria. The overall set of criteria results in five inequalities in three-dimensional Euclidian space of these parameters. For known rock and fluid properties, a formula for minimum core length to fulfil Welge–JBN criteria is derived. For cases where the core length is given, formulae for test’s flow velocity and sampling period are provided to satisfy the test admissibility conditions. The application of the proposed methodology is illustrated by two coreflood tests.

Appendices

Appendix A: Mathematical Model for Two-Phase Immiscible Displacement

Following Rapoport and Leas (1953), Barenblatt et al. (1991), Lake et al. (2014), here we present the mathematical model for two-phase flow of immiscible incompressible fluids in porous media. The modified Darcy’s law expresses the momentum balance for each phase

$$u_{\text{w}} = - \frac{{kk_{\text{rw}} (s)}}{{\mu_{\text{w}} }}\frac{{\partial P_{\text{w}} }}{\partial x}\quad u_{\text{o}} = - \frac{{kk_{\text{ro}} (s)}}{{\mu_{\text{o}} }}\frac{{\partial P_{\text{o}} }}{\partial x}$$

(A.1)

where k is the permeability, u_w and u_o are the water and oil velocities, respectively, K_rw and K_ro are the relative permeability for water and oil, s is the saturation, μ_w and μ_o are the viscosity for water and oil and P_w and P_o are the phase pressures in water and oil.

Volumetric balance for water is:

$$\phi \frac{\partial s}{\partial t} + \frac{{\partial u_{\text{w}} }}{\partial x} = 0$$

(A.2)

Here ϕ is the porosity.

The total flux conservation follows from incompressibility of both phases:

$$u_{\text{w}} + u_{\text{o}} = U(t)$$

(A.3)

where U is the total velocity.

The difference between phase pressures is equal to capillary pressure

$$P_{\text{o}} - P_{\text{w}} = P_{\text{c}} (s) = \frac{\sigma \cos \theta }{{\sqrt {(k /\phi } )}}J(s)$$

(A.4)

Here P_c is the capillary pressure, σ is the interfacial tension, θ is the contact angle, and J is the capillary function.

Substituting Darcy’s law for both phases (1, 2) into the expression (4) for the total flux, expressing pressure in oil from Eq. (5) and also substituting it into Eq. (4) yield

$$U = - k\left( {\frac{{k_{\text{rw}} (s)}}{{\mu_{\text{w}} }} + \frac{{k_{\text{ro}} (s)}}{{\mu_{\text{o}} }}} \right)\frac{{\partial P_{\text{w}} }}{\partial x} - \frac{{kk_{\text{ro}} (s)}}{{\mu_{\text{o}} }}\frac{\sigma \cos \theta }{{\sqrt {\left( {k /\phi } \right)} }}\frac{\partial J(s)}{\partial x}$$

(A.5)

Expressing pressure gradient in water and substituting it into Eq. (1) yields

$$u_{\text{w}} = Uf(s) + \frac{{kk_{\text{ro}} (s)}}{{\mu_{\text{o}} }}\frac{\sigma \cos \theta }{{\sqrt {\left( {k /\phi } \right)} }}\left( {\frac{\partial J(s)}{\partial x}} \right)f(s),\quad f(s) = \left( {1 + \frac{{k_{\text{ro}} (s)\mu_{\text{w}} }}{{k_{\text{rw}} (s)\mu_{\text{o}} }}} \right)^{ - 1}$$

(A.6)

where f is the fractional flow for water.

Substituting expression for water flux (A.6) into volume balance equation for water (A.2) results in one equations for unknown saturation s(x, t):

$$\phi \frac{\partial s}{\partial t} + U\frac{\partial f(s)}{\partial x} = - \frac{\partial }{\partial x}\left[ {\frac{{kk_{\text{ro}} (s)}}{{\mu_{\text{o}} }}\frac{\sigma \cos \theta }{{\sqrt {\left( {k /\phi } \right)} }}\left( {\frac{\partial J(s)}{\partial x}} \right)} \right]$$

(A.7)

Introduce the following dimensionless variables and parameters:

$$x_{\text{D}} = \frac{x}{L},\quad t_{\text{D}} = \frac{1}{\phi L}\int_{0}^{t} {U(y){\text{d}}y,\quad P = \frac{kp}{{\mu_{\text{o}} UL}},\quad \varepsilon_{\text{c}} = \frac{{\sigma \cos \theta \sqrt {k\phi } }}{{\mu_{\text{o}} UL}}}$$

(A.8)

Here x_D is the dimensionless distance, L is the core length, t_D is the dimensionless time, P is the dimensionless pressure, and ε_c is the capillary-viscous ratio.

Equations (A.5, A.7) become

$$\frac{\partial s}{{\partial t_{\text{D}} }} + \frac{\partial f(s)}{{\partial x_{\text{D}} }} = \varepsilon_{\text{c}} \frac{\partial }{{\partial x_{\text{D}} }}\left( { - k_{\text{ro}} (s)f(s)J^{{\prime }} (s)\frac{\partial s}{{\partial x_{\text{D}} }}} \right),\quad f(s) = \left( {1 + \frac{{k_{\text{ro}} (s)\mu_{\text{w}} }}{{k_{\text{rw}} (s)\mu_{\text{o}} }}} \right)^{ - 1}$$

(A.9)

$$1 = - \frac{k\lambda (s)}{LU}\frac{\partial P}{{\partial x_{\text{D}} }} - \varepsilon_{\text{c}} k_{\text{ro}} (s)\frac{\partial J(s)}{{\partial x_{\text{D}} }},\quad \lambda (s) = \frac{{k_{\text{rw}} (s)\mu_{\text{o}} }}{{\mu_{\text{w}} }} + k_{\text{ro}} (s)$$

(A.10)

Here λ(s) is the total mobility of two phases.

Equations (A.1) and (A.3) are decoupled, which means that the saturation distribution during the displacement s(x_D, t_D) is determined from Eq. (A.1). Afterwards, the pressure distribution p(x_D, t_D) is determined from Eq. (A.2) for the obtained saturation distribution.

Water flux in continuity Eq. (A.1) consists on the advective and capillary components, defined by Eq. (A.9).

The core is initially saturated with oil and connate water, i.e. the initial condition for Eq. (A.1) is:

$$t_{\text{D}} = 0{:}\quad s = S_{\text{wi}}$$

(A.11)

Here S_wi is the initial water saturation.

Only water flows through the inlet cross section, so the boundary condition at the inlet of the core is:

$$x_{\text{D}} = 0{:}\quad f - \varepsilon_{\text{c}} k_{\text{ro}} \,f\,J^{{\prime }} (s)\frac{\partial s}{{\partial x_{\text{D}} }} = 1$$

(A.12)

The boundary condition at the core outlet after the breakthrough is given by the condition of continuity of pressures in the both phases across the outlet interface, from which follows the continuity of the capillary pressure also. At the right-hand side of the core outlet, we assume a segregated flow regime, so the capillary pressure is zero. Therefore, the capillary pressure is zero behind the core outlet too. Thus, the boundary condition at the outlet of the core corresponds to zero capillary pressure:

$$x_{\text{D}} = 1{:}\quad J(s) = 0$$

(A.13)

and the outlet saturation after the breakthrough is equal to its maximum value s⁰ = 1 − S_or. Here S_or is the residual oil saturation.

Appendix B: Large-Scale Approximation of the Buckley–Leverett Equations

For small values of ε_c, one can neglect terms with ε_c in right-hand sides of Eqs. (A.9, A.10):

$$\frac{\partial s}{{\partial t_{\text{D}} }} + \frac{\partial f(s)}{{\partial x_{\text{D}} }} = 0$$

(B.1)

$$1 = - \frac{k\lambda (s)}{LU}\frac{\partial P}{{\partial x_{\text{D}} }}$$

(B.2)

For this case, fractional flow function f(s) is the ratio between the water flux and the total flux.

Approaching ε_c > 0 in the boundary condition (A.6), we obtain:

$$x_{\text{D}} = 0{:}\quad f = 1$$

(B.3)

The solution s(x_D, t_D) of the 1 − D capillary–pressure–free displacement problem (B.1), (A.5)–(A.7) is self-similar, depending only on the group ξ = x_D/t_D, i.e. s(x_D, t_D) = s(ξ):

$$s\left( {x_{\text{D}} ,t_{\text{D}} } \right) = \left\{ {\begin{array}{*{20}l} {1 - S_{\text{or}} ,} \hfill & {0 < \frac{{x_{\text{D}} }}{{t_{\text{D}} }} < D_{\text{or}} = f^{{\prime }} \left( {1 - S_{\text{or}} } \right)} \hfill \\ {\frac{{x_{\text{D}} }}{{t_{\text{D}} }} = f^{{\prime }} \left( s \right),} \hfill & {D_{\text{or}} < \frac{{x_{\text{D}} }}{{t_{\text{D}} }} < D = f^{{\prime }} \left( {S_{\text{f}} } \right) = \frac{{\left( {S_{\text{f}} } \right)}}{{S_{\text{f}} - S_{\text{wi}} }}} \hfill \\ {S_{\text{wi}} ,} \hfill & {D < \frac{{x_{\text{D}} }}{{t_{\text{D}} }} < \infty } \hfill \\ \end{array} } \right.$$

(B.4)

The self-similarity of the solution is the only information which is required for inverse problem to have exact solution (see). In two following sections exact shape of s(x_D, t_D) will not be used, and only the fact of self-similarity will be exploited.

Appendix C: Welge’s Method for Determination of Fractional Flow Function

Let us discuss how to determine the fractional flow function f(s) from the water-cut history f(1, t_D) measured during the coreflood.

Let us integrate Eq. (B.1) over the region Δ on the plane (x_D, t_D) which is limited by the triangle ∂Δ: (0, 0) → (1, 0) → (1, t_D) → (0, 0) and apply the Green’s formula (Bedrikovetsky 2013):

$$0 = \iint\limits_{\Delta } {\left( {\frac{\partial s}{{\partial t_{\text{D}} }} + \frac{\partial f}{{\partial x_{\text{D}} }}} \right)}{\text{d}}x_{\text{D}} {\text{d}}t_{\text{D}} = \oint\limits_{{\delta\Delta }} {f{\text{d}}t_{\text{D}} - s{\text{d}}x_{\text{D}} }$$

(C.1)

Let us calculate the contour integral in (C.1) over the sides of triangle ∂Δ:

$$(0,0) \to (1,t_{\text{D}} ){:}\quad 0 \times 0 + ( - s_{i} \times 1) = - s_{i}$$

(C.2)

$$(1,0) \to (1,t_{\text{D}} ){:}\quad \int_{0}^{{t_{\text{D}} }} {f(1,\tau ){\text{d}}\tau } - s \times 0$$

(C.3)

$$(0,0) \to (1,t_{\text{D}} ){:}\quad f\left( {s\left( {1 /t_{\text{D}} } \right)} \right) - s\left( {1 /t_{\text{D}} } \right)f_{\text{s}}^{{\prime }} (s)$$

(C.4)

Substituting the expressions for the integrals over the sides of the triangle (C.2–C.4) into Eq. (C.1), we obtain the expression for the saturation on the core outlet:

$$s\,\left( {1,\,t_{\text{D}} } \right) = s_{i} + f\,\left( {1,t_{\text{D}} } \right)\,t_{\text{D}} - \int\limits_{0}^{{t_{\text{D}} }} {f\,\left( {1,t} \right)\,} {\text{d}}t$$

(C.5)

Corresponding the saturation values, calculated by (C.5), to the water-cut values, which have been measured at the same moments, we obtain the dependence f = f(s).

Appendix D: JBN Method for Determination of Relative Permeability

Following the works by Johnson et al. (1959) and Jones and Roszelle (1978), let us calculate the pressure drop on the core during the waterflood:

$$\Delta \,p\,\left( {t_{\text{D}} } \right) = \int\limits_{0}^{1} {\left( { - \,\frac{\partial p}{{\partial x_{\text{D}} }}} \right)\,{\text{d}}x_{\text{D}} = \frac{U\,L}{k}\,\int\limits_{0}^{1} {\frac{{{\text{d}}x_{\text{D}} }}{\lambda \,\left( s \right)}} } = \frac{{U\,L\,t_{\text{D}} }}{k}\,\,\int\limits_{0}^{1 /T} {\frac{{{\text{d}}\xi }}{\lambda \,\left( s \right)}}$$

(D.1)

Expressing the integral from (D.1) and taking its derivative with respect to the upper limit, we obtain the explicit expression for the total mobility

$$\lambda = \frac{\text{d}}{{{\text{d}}\xi }}\,\left( {\frac{{k\;\Delta \,p\,\left( \xi \right)\;\xi }}{U\;L}} \right),\quad \xi = \frac{1}{{t_{\text{D}} }}$$

(D.2)

Corresponding the values of total mobility calculated by (D.2) to the values of saturations calculated by (C.5) for the coherent moments, we obtain the dependence λ = λ(s).

From the expression for the fractional flow function (A.9), the formulae for determination of relative permeabilities for both phases are:

$$k_{\text{rw}} = f\,\lambda \;\mu_{\text{W}} ;\quad k_{\text{ro}} = \,\left( {1 - f} \right)\,\lambda \;\mu_{\text{o}}$$

(D.3)

It is important to emphasise that formulae (C.5) and (D.2), (D.3) deliver the solution f = f(s) and λ = λ(s) only for saturations which are realised during the displacement process. In the case of the displacement of oil with the connate water by water, the method provides with the fractional flow function and total mobility only for saturations higher than the frontal saturation s_f and below the s⁰ and also in the initial point S_wi.

Appendix E: Determination of the Frontal Saturation and Displacement Velocity from the Waterless Period Data

Let us calculate the pressure drop on the core, as in (D.1), but for the moment before the breakthrough. Acting by analogy to Eq. (D.1), we obtain

$$\Delta \;p\,\left( {t_{\text{D}} } \right) = \frac{U\,L}{k}\,\left( {t_{\text{D}} \;\int\limits_{0}^{D} {\frac{{{\text{d}}\xi }}{\lambda \,\left( s \right)} + \frac{{1 - D\,t_{\text{D}} }}{{\lambda \,\left( {s_{\text{i}} } \right)}}} } \right)$$

(E.1)

where initial saturation s_i can exceed connate water saturation S_wi.

Let us transform Eq. (E.1) to the form

$$\frac{{k\,\Delta \;p\,\left( {t_{\text{D}} } \right)}}{U\,L} = T\,\left( {\;\int\limits_{0}^{D} {\frac{{{\text{d}}\xi }}{\lambda \,\left( s \right)} - \frac{D}{{\lambda \,\left( {s_{i} } \right)}}} } \right) + \frac{1}{{\lambda \,\left( {s_{i} } \right)}}$$

(E.2)

Equation (E.2) gives the straight line versus t_D. The free constant in it is determined by the total mobility before the flood, which is known from the initial saturation process. The slope of the line together with the condition that the velocity D is the tangent of the fractional flow curve in the frontal saturation point makes it possible to determine the slope D and the frontal saturation s_f. The slope of the pressure drop as defined by Eq. (E.2) and calculated from the measured data is an additional information for relative permeability tuning from the coreflood data.

Appendix F: Determination of Pore Throat Radius from Permeability and Porosity

Following Barenblatt et al. (1991), here we derive permeability and porosity for cubic lattice with the tube radius r and the bond length l. One cube is adjacent to 12 bonds and one bond belongs to four cubes, so each cube includes three bonds. The porosity is equal to

$$\phi = 3\frac{{\pi r^{2} l}}{{l^{3} }}$$

(F.1)

Flow through a cube side corresponds to flow through a single tube. Comparing Darcy and Poiseuille laws

$$U = \frac{k}{\mu }\frac{\Delta p}{l} = \frac{1}{{l^{2} }}\frac{{\pi r^{4} }}{8\mu }\frac{{\Delta p}}{l},$$

(F.2)

we obtain the formula for permeability

$$k = \frac{{\pi r^{4} }}{{8l^{2} }}$$

(F.3)

From Eqs. (F.1) and (F.3) follows the expression for pore throat radius r

$$r = \sqrt {\frac{24k}{\phi }} .$$

(F.4)