Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 84 / 2 / a / Solution

Let $k_1>k_2$ and let both original and injected fluids initially have dynamic viscosity $\mu$. Assume sharp displacement fronts, constant porosity and negligible exchange between layers. The imposed pressure difference gives Darcy velocities $q_i^0=k_i\Delta P/(\mu L)$ and interstitial velocities $u_i^0=q_i^0/\varphi$. The two arrival times are thereforeIf $A_i$ is the flow cross-section of layer $i$, its volume discharge is $Q_i^0=A_i{q}_i^0$. Equal viscosities make the hydraulic resistance independent of front position, so the total fluid discharge $Q_1^0+Q_2^0$ stays constant. The injected-fluid component at the outflow well isIt jumps from zero to the high-permeability discharge at the first breakthrough, then to the total discharge at the second. Before a layer breaks through, its outflow is original reservoir fluid. Cross-sections are needed to state an absolute volume discharge or its partition; the arrival times do not require them.