Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 73 / 2 / ii

Let $Y(t)$ be the porous thermal front and $X(t)$ the polymer front. With the literal stated convention for Darcy velocity,Therefore $Y=\beta X$ with $\beta =\Gamma \varphi$. This is thermal-front retardation in Darcy flow. The region $0<x<Y$ is cold injected fluid of dynamic viscosity $\mu$; $Y<x<X$ is heated injected fluid of dynamic viscosity $b\mu$; $X<x<L$ is oil of dynamic viscosity ${\mu }_h$. The pressure drop is consequentlywhere ${\mu }_{\mathrm{eff}}=\mu [\beta +b(1-\beta )]$. Apply the preceding fixed-pressure planar Darcy displacement calculation with $\mu$ replaced by this effective dynamic viscosity. The polymer-front speed isIts position solves ${\mu }_{h}LX+({\mu }_{\mathrm{eff}}-{\mu }_h)X^2/2=k\Delta Pt/\varphi$, and $Y=\Gamma \varphi X$. These expressions hold while both interfaces remain in the field, the porous thermal front is sharp and stable, and $0\le \Gamma \varphi \le 1$. For $b=1$ the original constant-viscosity answer is recovered.

Sometimes a thermal retardation factor is defined relative to the pore velocity instead. Under that alternative definition one substitutes $\beta =\Gamma$ throughout. The factor of $\varphi$ in the boxed answer follows specifically from the PDF's phrase “fraction of the Darcy speed.”