Mathematical Model of Water Alternated Polymer Injection

Abstract

Chemical enhanced oil recovery (EOR) methods include the injection of aqueous polymer solutions slugs driven by water. Polymer solutions increase water viscosity, decreasing the water phase mobility and improving oil recovery through better sweep efficiency. In this paper, we present the water alternated polymer EOR technique, which is based on the injection of successive polymer slugs alternated by water slugs. The mathematical problem is composed by two conservation equations: one of them is related to the water volume and the other one to the polymer mass. We assume that the polymer may be adsorbed by the rock, and the relation between the concentration in the aqueous solution and the solid is governed by a Langmuir type adsorption isotherm. The water viscosity is a function of the polymer concentration in water. The 2 × 2 system of hyperbolic equations was decoupled by introducing a potential function instead of time as an independent variable. The water alternated polymer injection is represented by a varying boundary condition. The analytical solution presents interactions between waves of different families. It is shown that the polymer slugs always catch up each other along the porous media generating a single slug. As a consequence, the water slugs will disappear. This solution is new and was compared to numerical results with close agreement. It also can be used for the selection of the most suitable enhanced oil recovery technique for a particular oil field.

Appendix

The shock path resulting from the interaction between two simple waves is described in Rhee et al. (1989a) when the rarefaction waves arise from the coordinate axis. Here, the same approach is used to build the trajectory of shocks when the rarefaction waves arise from any curve. Considering that along the path shock $x$ and $\varphi$ are functions of $U^{ - }$ and $U^{ + }$, we obtain the following expression to the shock path:

$$\frac{{{\text{d}}\varphi }}{{{\text{d}}x}} = \frac{{F\left( {U^{ + } ,0} \right) - F\left( {U^{ - } ,0} \right)}}{{U^{ + } - U^{ - } }} = \frac{{\left( {{{\partial \varphi } \mathord{\left/ {\vphantom {{\partial \varphi } {\partial U^{ - } }}} \right. \kern-0pt} {\partial U^{ - } }}} \right)\left( {{{{\text{d}}U^{ - } } \mathord{\left/ {\vphantom {{{\text{d}}U^{ - } } {{\text{d}}U^{ + } }}} \right. \kern-0pt} {{\text{d}}U^{ + } }}} \right) + \left( {{{\partial \varphi } \mathord{\left/ {\vphantom {{\partial \varphi } {\partial U^{ + } }}} \right. \kern-0pt} {\partial U^{ + } }}} \right)}}{{\left( {{{\partial x} \mathord{\left/ {\vphantom {{\partial x} {\partial U^{ - } }}} \right. \kern-0pt} {\partial U^{ - } }}} \right)\left( {{{{\text{d}}U^{ - } } \mathord{\left/ {\vphantom {{{\text{d}}U^{ - } } {{\text{d}}U^{ + } }}} \right. \kern-0pt} {{\text{d}}U^{ + } }}} \right) + \left( {{{\partial x} \mathord{\left/ {\vphantom {{\partial x} {\partial U^{ + } }}} \right. \kern-0pt} {\partial U^{ + } }}} \right)}} = \tilde{\omega }\left( {U^{ - } ,U^{ + } } \right)$$

(63)

Equation (63) can be rewritten as

$$\frac{{{\text{d}}U^{ - } }}{{{\text{d}}U^{ + } }} = - \frac{{\tilde{\omega }\left( {{{\partial x} \mathord{\left/ {\vphantom {{\partial x} {\partial U^{ + } }}} \right. \kern-0pt} {\partial U^{ + } }}} \right) - \left( {{{\partial \varphi } \mathord{\left/ {\vphantom {{\partial \varphi } {\partial U^{ + } }}} \right. \kern-0pt} {\partial U^{ + } }}} \right)}}{{\tilde{\omega }\left( {{{\partial x} \mathord{\left/ {\vphantom {{\partial x} {\partial U^{ - } }}} \right. \kern-0pt} {\partial U^{ - } }}} \right) - \left( {{{\partial \varphi } \mathord{\left/ {\vphantom {{\partial \varphi } {\partial U^{ - } }}} \right. \kern-0pt} {\partial U^{ - } }}} \right)}}$$

(64)

At an arbitrary point $\left( {x,\varphi } \right)$ on the shock path we have two characteristics intersecting, one from the curve $\varphi_{A} \left( x \right)$ and another from the curve $\varphi_{B} \left( x \right)$ (Fig. 19). These characteristic curves are straight lines:

$$\varphi = \omega^{ + } \left( {x - \xi^{ + } } \right) + \eta^{ + } \quad {\text{and}}\quad \varphi = \omega^{ - } \left( {x - \xi^{ - } } \right) + \eta^{ - }$$

(65)

where $\omega^{ \pm } = F^{\prime}_{U} \left( {U^{ \pm } ,0} \right)$, $\left( {\xi^{ + } = \xi^{ + } \left( {U^{ + } } \right),\eta^{ + } = \eta^{ + } \left( {U^{ + } } \right)} \right)$ and $\left( {\xi^{ - } = \xi^{ - } \left( {U^{ - } } \right),\eta^{ - } = \eta^{ - } \left( {U^{ - } } \right)} \right)$ are points on the curves $\varphi_{A} \left( x \right)$ and $\varphi_{B} \left( x \right)$, respectively.

Fig. 19

Shock path arising from the interaction of two rarefaction waves of the same family

From Eq. (65) we write $\varphi$ and $x$ as functions of $U^{ - }$ and $U^{ + }$:

$$x = \frac{{\eta^{ - } - \eta^{ + } + \omega^{ + } \xi^{ + } - \omega^{ - } \xi^{ - } }}{{\omega^{ + } - \omega^{ - } }}\quad {\text{and}}\quad \varphi = \frac{{\omega^{ + } \eta^{ - } - \omega^{ - } \eta^{ + } + \omega^{ - } \omega^{ + } \left( {\xi^{ + } - \xi^{ - } } \right)}}{{\omega^{ + } - \omega^{ - } }}$$

(66)

Deriving $x$ and $\varphi$ with respect to $U^{ + }$ and $U^{ - }$ and substituting into Eq. (64), we obtain an ordinary differential equation relating $U^{ - }$ and $U^{ + }$:

$$\frac{{{\text{d}}U^{ - } }}{{{\text{d}}U^{ + } }} = \frac{{\tilde{\omega } - \omega^{ - } }}{{\tilde{\omega } - \omega^{ + } }}\frac{{\left( {\omega^{ + } - \omega^{ - } } \right)\left( {\eta^{\prime + } - \omega^{ + } \xi^{\prime + } } \right) + \left[ {\eta^{ - } - \eta^{ + } + \left( {\xi^{ + } - \xi^{ - } } \right)\omega^{ - } } \right]\omega^{\prime + } }}{{\left( {\omega^{ + } - \omega^{ - } } \right)\left( {\eta^{\prime - } - \omega^{ - } \xi^{\prime - } } \right) + \left[ {\eta^{ - } - \eta^{ + } + \left( {\xi^{ + } - \xi^{ - } } \right)\omega^{ + } } \right]\omega^{\prime - } }}$$

(67)

where $\omega^{\prime \pm } = F^{\prime\prime}_{U} \left( {U^{ \pm } ,0} \right)$, $\eta^{\prime \pm } = {{{\text{d}}\eta^{ \pm } } \mathord{\left/ {\vphantom {{{\text{d}}\eta^{ \pm } } {{\text{d}}U^{ \pm } }}} \right. \kern-0pt} {{\text{d}}U^{ \pm } }}$ and $\xi^{\prime \pm } = {{d\xi^{ \pm } } \mathord{\left/ {\vphantom {{d\xi^{ \pm } } {{\text{d}}U^{ \pm } }}} \right. \kern-0pt} {{\text{d}}U^{ \pm } }}$.

The solution of Eq. (67) allows the determination of $U^{ - }$ for an specified $U^{ + }$. Replacing $U^{ - }$ and $U^{ + }$ in Eq. (66), we obtain the shock path.