A Model for Subsurface Oil Pollutant Migration

Abstract

The subsurface spill of hydrocarbon fluids (from an underground leaking tank or from pipelines) is a widespread problem in the oil industry and hydrology. Under the assumption of low oil saturation, the migration of oil pollutant may be modeled by a nonlinear convection–diffusion equation with power law nonlinearities. The saturation-dependent diffusion coefficient is proportional to a power of saturation for which the exponent may be negative or positive depending on the porous medium and the pollutant characteristics. This paper presents general analytical solutions for the one-dimensional transient saturation distribution in immiscible zone. The solutions describe the longtime behavior of a contaminant oil pollutant caused by a continuous source of contaminant (e.g., leakage from pipelines, disposal dumps or storage tanks). Two general forms corresponding to positive and negative power exponents of the diffusion coefficient are presented. The general forms express implicitly the position as function of oil saturation and time. An alternative explicit closed-form solution near the front position is also proposed. An exact explicit solution is obtained for the particular case of constant diffusion coefficient. The proposed solutions are fully analytical in the sense that they do not require any numerical implementation. The solutions may be used to study the porous medium and pollutant parameters. They can be applied to aquifer contamination problems, and they are also important for the verification of numerical solutions.

Appendix A: Mass of Penetrating Oil Pollutant into Semi-infinite Aquifer

The mass of oil pollutant entering a semi-infinite aquifer which is initially free of oil may be defined as

$$\begin{aligned} M\left( {t_\mathrm{D} } \right) =\mathop \int \nolimits _0^{x_{\mathrm{D}_{\mathrm{f}}} \left( {t_\mathrm{D} } \right) } \phi S\hbox {d}x_\mathrm{D} =-\mathop \int \nolimits _0^{S_0 \left( {t_\mathrm{D} } \right) } \phi S\frac{\hbox {d}x_\mathrm{D} }{\hbox {d}S}\hbox {d}S \end{aligned}$$

(A.1)

where $x_{\mathrm{D}_{\mathrm{f}}} \left( {t_\mathrm{D} } \right) =\vartheta t_\mathrm{D} =S_*^{n-1} t_\mathrm{D}$ is the oil pollutant front position at time $t_\mathrm{D}$ and $S_0 \left( {t_\mathrm{D} } \right)$ is the oil saturation at the left boundary of the aquifer.

The value $\hbox {d}x_\mathrm{D} /\hbox {d}S$ appearing in the right-hand side of (A.1) can be calculated from (36) by deriving with respect to S (i.e., note that ${\varPhi }=S$, $\xi =x_\mathrm{D} -\vartheta t_\mathrm{D}$ and $\xi _f =0)$. We obtain after some mathematical manipulations

$$\begin{aligned} \frac{\hbox {d}x_\mathrm{D} }{\hbox {d}S}=-\frac{S^{n-m-1}}{N_\mathrm{c} S_*^{n-2} }\frac{1}{1-\left( {S/S_*} \right) ^{n-1}}. \end{aligned}$$

(A.2)

Substituting (A.2) into (A.1) we get after making the change of variable $u=\left( {S/S_*} \right) ^{n-1}/\left( {1-\left( {S/S_*} \right) ^{n-1}} \right)$

$$\begin{aligned} M\left( {t_\mathrm{D} } \right) =\frac{\phi }{\left( {n-1} \right) N_\mathrm{c} S_*^{m-3} }\mathop \int \nolimits _0^\zeta \frac{u^{\left( {n-m+1} \right) /\left( {n-1} \right) -1}}{\left( {1+u} \right) ^{\left( {n-m+1} \right) /\left( {n-1} \right) }}\hbox {d}u \end{aligned}$$

(A.3)

where $\zeta =\left( {S_0 /S_*} \right) ^{n-1}/\left( {1-\left( {S_0 /S_*} \right) ^{n-1}} \right)$.

The integral in (A.3) can be evaluated analytically by using equation (1) of 3.194, p. 315 of Gradshteyn and Ryzhik (2007) and the Pfaff transformation. We finally obtain

$$\begin{aligned} M\left( {t_\mathrm{D} } \right) =\frac{\phi S_0^{n-m+1} }{\left( {n-m+1} \right) N_\mathrm{c} S_*^{n-2} }{ }_2F_1 \left( {\frac{n-m+1}{n-1},1;1+\frac{n-m+1}{n-1};\left( {\frac{S_0 }{S_*}} \right) ^{n-1}} \right) .\nonumber \\ \end{aligned}$$

(A.4)