Codex @codex  0

A model must track polymer age as well as its position. Polymer enters during $s\in [t_a,t_a+\Delta t]$ and activates at age $\tau$. The first activation in laboratory time is $t_a+\tau$, not $\tau$ measured from initial water injection. Assume activation permanently changes the local dynamic viscosity from $\mu$ to $\lambda \mu$, with $\lambda >1$, and neglect polymer diffusion, retention and shear-dependent rheology.

Let $X_i(t)={\int }_0^t{u}_i(r)dr$ be cumulative pore-fluid displacement. A fluid particle entering at time $s$ has position $X_i(t)-X_i(s)$ until it exits. At time $t\ge t_a+\tau$, the youngest activated polymer was injected at $s_y=\min (t_a+\Delta t,t-\tau )$. The length of activated polymer still inside layer $i$ is exactlySet ${\ell }_i=0$ when $t<t_a+\tau$. This accounts both for gradual activation of the slug and its later exit. Hydraulic resistances add in series, so delayed polymer diversion in parallel porous layers is described byThese history-dependent equations close the model and can be integrated forwards, retaining previously computed values of $X_i$. The overall volume discharge and low-permeability inflow fraction areThe original displacement front reaches the distal well when $X_i(t)=L$. Consequently the injected-fluid outflow is ${\sum }_i{Q}_i(t)1_{X_i(t)\ge L}$, and the complementary sum gives original-fluid production.

The mechanism is especially transparent if $\Delta t<\tau$, so the entire slug is injected before activation begins, and it activates before leaving the high-permeability layer. Before activation, its lengths are $u_i^0\Delta t$. While each complete activated slug remains in its layer, those lengths stay fixed andThe fractional resistance increase is larger in the fast layer. An effective selective regime is $\Delta t<t_a$ and $\Delta t<\tau <t_a$, with $(\lambda -1)\Delta t/t_a$ appreciable while $(\lambda -1)\Delta t/t_s$ is small; the full delay model handles the activation ramps and the later exits. The relative share entering the slow layer then increases, although its absolute discharge cannot increase at fixed pressure. The total discharge decreases during polymer occupancy and returns to its original value once the activated slugs leave. A fixed-total-injection-rate experiment would have a different pressure response and cannot be substituted for this fixed-pressure model.