A Nonhomogeneous Porous-Medium Equation for Field Scale CO${}_2$ Plume Spreading

We derive a nonlinear diffusion model for field scale CO${}_2$ plume spreading from a Global Buckley–Leverett component balance. The reduced variable $u$ is the vertically averaged mobile gas phase CO${}_2$ content normalized by its maximum column value; under vertical segregation, $u=h\mathrm{/}H$, where $h$ is plume thickness and $H$ is aquifer thickness. The resulting equation is a nonhomogeneous porous medium type equation in which nonlinear lateral spreading is coupled to source/sink terms for injection, dissolution, mineral fixation, and retention. Using the nonlinear diffusivity $D_u(u)\simeq D_0{u}^{1-q}$, we analyze Barenblatt-type profiles with prescribed mobile mass and a capped plume constrained by $0\le u\le 1$. The capped solution contains a ful-thickness core of radius $a(t)$ and a compact plume edge $R(t)$. Constant net mobile injection can sustain the core and gives square-root growth of $R(t)$, whereas shut-in or weak mobile addition causes the core to shrink and disappear. We compare these regimes with equivalent radii from time lapse seismic plume maps at Sleipner, Aquistore, and Weyburn–Midale. The data distinguish injection controlled growth, delayed layer filling, and tail dominated redistribution, but do not determine a unique nonlinear exponent. The model provides an analytical reference for interpreting plume footprint evolution while separating cumulative injected CO${}_2$ from mobile gas phase CO${}_2$.