Delayed polymer diversion in parallel porous layers ($q_i=k_i\Delta P/[\mu (L+(\lambda -1){\ell }_i)]$)

For a fixed pressure difference across two noncommunicating layers, an activated viscous polymer segment of length ${\ell }_i$ adds hydraulic resistance in series with the remaining fluid. The Darcy flux is the displayed expression when polymer raises dynamic viscosity by a factor $\lambda$. Tracking particle entry time determines activation and exit, while ${\dot{X}}_i=q_i/\varphi$ gives each displacement front. Preferentially occupying the more permeable layer shifts the relative inflow toward the other layer, but cannot increase that layer's absolute flux at fixed pressure. It can delay its breakthrough and leave original reservoir fluid flowing after the unmodified reservoir would be fully swept.