An Analytical Solution for Capillary Gravity Drainage with Dominant Viscous Forces

Abstract

The development of the capillary fringe during gravity drainage has a significant influence on saturation and pressure distributions in porous formations (Sarkarfarshi et al. in Int J Greenh Gas Control 23:61–71, 2014). This paper introduces an analytical solution for gravity drainage in an axisymmetric geometry with significant capillary pressure. The drainage process results from the injection of a lighter and less viscous injectant into a porous medium saturated with a heavier and more viscous pore fluid. If the viscous force dominates the capillary and the buoyancy forces, then the flow regime is approximated by differential equations and the admissible solution comprises a front shock wave and a trailing simple wave. In contrast to existing analytical solutions for capillary gravity drainage problems (e.g., Nordbotten and Dahle in 47(2) 2011; Golding et al. in J Fluid Mech 678:248–270 2011), this solution targets the saturation distribution during injection at an earlier point in time. Another contribution of this analytical solution is the incorporation of a completely drained flow regime close to the injection well. The analytical solution demonstrates the strong dependency of the saturation distribution upon relative permeability functions, gas entry capillary pressure, and residual saturation. The analytical results are compared to results from a commercial reservoir engineering software package ($\hbox {CMG } \hbox {STARS}^{\mathrm{TM}}$).

Appendices

Appendix 1: Conservative Equation Derivation

By substituting Eq. (9) in Eq. (6), the conservation equation is reduced to

$$\begin{aligned} \begin{array}{lll} &{}&{}\displaystyle \frac{Q_\mathrm{o} B}{2\pi r}=-\left( {\int \limits _{0}^{Bb} {\left( {\frac{K}{\mu _\mathrm{n} }} \right) } {\hbox {d}}z+\displaystyle \int \limits _{Bb}^{B} {K\left( {\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }+\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }} \right) } {\hbox {d}}z} \right) \partial _\mathrm{r} p-\displaystyle \int \limits _{Bb}^{B} {\left( {\frac{Kk_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }\partial _\mathrm{r} P_\mathrm{c} } \right) } {\hbox {d}}z \\ &{}&{}\qquad \qquad -B\displaystyle \int \limits _{Bb}^{B} {\left( {\frac{Kk_\mathrm{rn} \left( S \right) \Delta \gamma _\mathrm{w} }{\mu _\mathrm{n} }{\hbox {d}}z} \right) } \partial _\mathrm{r} b \\ \end{array} \end{aligned}$$

(39)

The velocity of water and the injectant in the drained zone can therefore be expressed as follows. Radial derivative of pressure is derived as a function of injection rate and capillary pressure.

$$\begin{aligned} \partial _\mathrm{r} p=-\frac{\frac{Q_\mathrm{o} B}{2\pi r}+\displaystyle \int \limits _{Bb}^{B} {\left( {K\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }\partial _\mathrm{r} P_\mathrm{c} } \right) } {\hbox {d}}z+B\partial _\mathrm{r} b\displaystyle \int \limits _{Bb}^{B} {\left( {K\frac{k_\mathrm{rn} \left( S \right) \Delta \gamma _\mathrm{w} }{\mu _\mathrm{n} }{\hbox {d}}z} \right) } }{\frac{BbK}{\mu _\mathrm{n} }+\displaystyle \int \limits _{Bb}^{B} {K\left( {\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }+\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }} \right) } {\hbox {d}}z} \end{aligned}$$

(40)

Velocity of the wetting phase becomes:

$$\begin{aligned} \mathbf{u}_\mathrm{w} \cdot \mathbf{e}_\mathrm{r} =\frac{\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }\left( {\frac{Q_\mathrm{o} B}{2\pi r}+\displaystyle \int \limits _{Bb}^{B} {K\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }\partial _\mathrm{r} P_\mathrm{c} } {\hbox {d}}z+B\partial _\mathrm{r} b\displaystyle \int \limits _{Bb}^{B} {K\frac{k_\mathrm{rn} \left( S \right) \Delta \gamma _\mathrm{w} }{\mu _\mathrm{n} }{\hbox {d}}z} } \right) }{\left( {\frac{Bb}{\mu _\mathrm{n} }+\displaystyle \int \limits _{Bb}^{B} {\left( {\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }+\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }} \right) } {\hbox {d}}z} \right) } \end{aligned}$$

(41)

The mass conservation equation becomes:

$$\begin{aligned} 2\frac{\partial }{\partial r^{2}}\frac{\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }\left( {\frac{Q_\mathrm{o} B}{2\pi }+r\displaystyle \mathop {\int }\limits _{Bb}^{B} {K\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }\partial _\mathrm{r} P_\mathrm{c} } {\hbox {d}}z+Br\partial _\mathrm{r} b\displaystyle \mathop {\int }\limits _{Bb}^{B} {\frac{Kk_\mathrm{rn} \left( S \right) \Delta \gamma }{\mu _\mathrm{n} }{\hbox {d}}z} } \right) }{\left( {\frac{Bb}{\mu _\mathrm{n} }+\displaystyle \mathop {\int }\limits _{Bb}^{B} {\left( {\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }+\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }} \right) } {\hbox {d}}z} \right) }=\varphi \frac{\partial S}{\partial t} \end{aligned}$$

(42)

By a simple vertical averaging, Eq. (9) becomes:

$$\begin{aligned}&2\frac{\partial }{\partial r^{2}}\left( {\begin{array}{l} \frac{\frac{Q_\mathrm{o} B}{2\pi }\displaystyle \mathop {\int }\limits _{Bb}^{B} {\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }{\hbox {d}}z} }{\left( {\frac{Bb}{\mu _\mathrm{n} }+\displaystyle \mathop {\int }\limits _{Bb}^{B} {\left( {\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }+\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }} \right) } {\hbox {d}}z} \right) }+\displaystyle \frac{r\mathop {\int }\limits _{Bb}^{B} {\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }{\hbox {d}}z} \displaystyle \mathop {\int }\limits _{Bb}^{B} {K\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }\partial _\mathrm{r} P_\mathrm{c} } {\hbox {d}}z}{\left( {\frac{Bb}{\mu _\mathrm{n} }+\displaystyle \mathop {\int }\limits _{Bb}^{B} {\left( {\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }+\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }} \right) } {\hbox {d}}z} \right) }+ \\ \frac{rB\Delta \gamma \partial _\mathrm{r} b\displaystyle \mathop {\int }\limits _{Bb}^{B} {\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }{\hbox {d}}z} \displaystyle \mathop {\int }\limits _{Bb}^{B} {K\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }} {\hbox {d}}z}{\left( {\frac{Bb}{\mu _\mathrm{n} }+\displaystyle \mathop {\int }\limits _{Bb}^{B} {\left( {\frac{k_\mathrm{rw} \left( S \right) }{\mu _\mathrm{w} }+\frac{k_\mathrm{rn} \left( S \right) }{\mu _\mathrm{n} }} \right) } {\hbox {d}}z} \right) } \\ \end{array}} \right) \nonumber \\&\quad =\varphi \frac{\partial \left( {S_\mathrm{r} Bb+\displaystyle \mathop {\int }\limits _{Bb}^{B} {S{\hbox {d}}z} } \right) }{\partial t} \end{aligned}$$

(43)

Equation (43) sums the wetting phase conservation equations and incorporates contributions of viscous, capillary and buoyancy forces.

Appendix 2: First-Order Solution

Equation (37) is the first order of the conservation equation. After solving Eq. (36), the zeroth-order solution of the equation is known and Eq. (37) becomes linear.

$$\begin{aligned} U_{1}^{i} \frac{\partial o_{1} }{\partial \xi }+U_{2}^{i} o_{1} +U_{3}^{i} =\frac{\partial o_{1} }{\partial \tau } \end{aligned}$$

(44)

$U_{1}^{i}$, $U_{2}^{i}$, and $U_{3}^{i}$ are functions of $o_{0}$ and Eq. (44) is a linear differential equation. By using the method of characteristics and introducing a new variable as $\omega =\xi -U_{1}^{i} \tau$, Eq. (44) is simplified to

$$\begin{aligned} \frac{\partial o_{1} }{\partial \omega }+U_{2}^{i} o_{1} +U_{3}^{i} =0 \end{aligned}$$

(45)

Analytical solution of Eq. (45) is straightforward.

$$\begin{aligned} o_{1} =-\frac{\displaystyle \mathop {\int }\limits _{\omega _{0} }^{\omega } {e^{\mathop {\int }\limits _{\omega _{0}}^{{\omega }^{\prime }}{U_{2}^{i} {\hbox {d}}{\omega }^{\prime \prime }} }U_{3}^{i} d{\omega }^{\prime }} }{\displaystyle \mathop {\int }\limits _{\omega _{0}}^{\omega } {e^{\mathop {\int }\limits _{\omega _{0}}^{{\omega }^{\prime }} {U_{2}^{i} {\hbox {d}}{\omega }^{\prime \prime }} }d{\omega }^{\prime }} } \end{aligned}$$

(46)

$o_{1}$ is the first-order correction of saturation. It is possible to evaluate higher-order solutions of the equations resulting from the perturbation method.