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

Let $X(t)$ denote the polymer–oil material interface, measured from the injection well, and let $U(t)$ be the Darcy velocity. Incompressibility and uniform cross-sectional area make $U$ independent of $x$ in this planar model. Integrating Darcy's law through the injected layer and the oil layer givesThe material interface moves at the pore velocity, so $X^{\prime }=U/\varphi$. Put $a=\mu -{\mu }_h$ and $A={\mu }_{h}L$. Then $(A+aX)X^{\prime }=k\Delta P/\varphi$, which integrates from $X(0)=0$ toThus the fixed-pressure planar Darcy displacement hasFor $a\ne 0$, $X=[\sqrt{A^2+2ak\Delta Pt/\varphi }-A]/a$; its continuous $a=0$ limit is $X=k\Delta Pt/(\varphi {\mu }_{h}L)$. Positive layer dynamic viscosities ensure the resistance stays positive until breakthrough, which occurs at $t_b=\varphi L^2(\mu +{\mu }_h)/(2k\Delta P)$. The formulas stop there, since the two-layer geometry then changes.

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.”

At the porous thermal front, the cold polymer solution behind has dynamic viscosity $\mu$, while the hot solution ahead has dynamic viscosity $b\mu >\mu$. Thus a more mobile region pushes a less mobile one. A forward protrusion locally reduces the length of high-resistance hot fluid and concentrates Darcy flux into itself; it advances faster, amplifying the protrusion. This is the mechanism of the Saffman–Taylor instability.

The leading polymer–oil interface can nevertheless be stable when the hot polymer dynamic viscosity is at least the oil dynamic viscosity. Stability of that leading interface does not guarantee stability of the trailing porous thermal front. Slow thermal conduction makes the latter sharp and offers weak short-wave smoothing; in practice it is the adverse dynamic viscosity contrast across this front that can invalidate the assumed planar stable-flow calculation.

Take constant Darcy velocity $U$ and work locally near a planar porous thermal front moving at $V_T=\Gamma U$. Let $\xi =x-V_{T}t$ and perturb its position by $\eta ={\eta }_0{e}^{\sigma t+iay}$, with wavelength $\lambda =2\pi /|a|$. Neglect thermal conduction and regard the adjoining layers as semi-infinite on this disturbance scale. In each region, Darcy's law and incompressibility imply that the pressure perturbation is harmonic, so the bounded normal modes areHere the minus side is cold with dynamic viscosity $\mu$, and the plus side is hot with dynamic viscosity $b\mu$. The base pressure gradients are $p_{-,x}^0=-\mu U/k$ and $p_{+,x}^0=-b\mu U/k$. Linearizing pressure continuity at the displaced front yieldsContinuity of the perturbed normal Darcy velocity givesso $A_{+}=-b{A}_{-}$. Therefore $\delta U=U|a|(b-1){\eta }_0/(b+1)$. The thermal-front kinematic law is $\sigma {\eta }_0=\Gamma \delta U$, proving the thermal-front viscous-fingering dispersion relationIt is positive for $b>1$, zero for $b=1$, and negative for $b<1$. Without thermal conduction the model has no finite fastest wavelength: the positive growth rate increases without bound as $\lambda \to 0$. A resolved thermal transition and diffusion regularize that short-wave limit.

For a material interface the kinematic coefficient would be $1/\varphi$ rather than $\Gamma$. Thus, if the interface of interest were instead the hot polymer–oil interface, the same calculation would give $\sigma =2\pi (U/\varphi )({\mu }_h-b\mu )/[\lambda ({\mu }_h+b\mu )]$, the usual planar viscous-fingering dispersion relation. This distinguishes the two interfaces and their speeds. The boxed thermal result uses the local planar approximation; distant wells or the other front modify the pressure eigenfunctions for wavelengths comparable to those separations.