= 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> $u_i^0=q_i^0/\phi$. The two arrival times are therefore
$$
\boxed{t_a=\frac{\phi\mu L^2}{k_1\Delta P},
\qquad t_s=\frac{\phi\mu L^2}{k_2\Delta P}
=\frac{k_1}{k_2}t_a.}
$$
If $A_i$ is the flow cross-section of layer $i$, its volume discharge is $Q_i^0=A_iq_i^0$. 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 is
$$
Q_{\rm inj}(t)=Q_1^0\mathbf1_{t\ge t_a}
+Q_2^0\mathbf1_{t\ge t_s}.
$$
It 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.
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