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At a fixed location behind the advancing nose, the depth decreases after passage. A fall $-dh>0$ leaves the immobile fluid volume $s\varphi (-dh)$ per unit length, so total local fluid storage changes by $\varphi dh+s\varphi (-dh)=\varphi (1-s)dh$. The thinning-region equation is thereforeThe inlet flux remains $uh(0,t)=Q{e}^{-t/\tau }$. Applying the method of characteristics with thinning-wave speed $u/[\varphi (1-s)]$ gives the residual-trapping attenuation of a porous current:The nose advances into new pore space, which has no residual fluid yet, so its storage factor is still $\varphi$ and its speed is still $u/\varphi$. Consequently $X(t)=ut/\varphi$ and the interior nose trace becomes $h_0{e}^{-st/\tau }$. Using the faster thinning-wave speed for the leading nose would incorrectly erase the attenuation caused by capillary residual trapping. The broken printed reference “(??)” points to the preceding outer profile.

For $0<s<1$, differentiation givesIts unique zero solves $e^{-(1-s)t/\tau }=s$. Therefore the mobile-volume peak of an exponentially forced porous current occurs atThe derivative is initially positive and is negative after this time. More explicitly, at the stationary point $V^{\prime \prime }=-Qs{e}^{-st/\tau }/\tau <0$. Thus it is the global positive maximum, withThe claim of a finite maximum requires nonzero capillary residual trapping. At $s=0$ the mobile volume rises monotonically towards $Q\tau$ and the peak time tends to infinity. As $s\to 1$, the limiting peak time is $\tau$ and the limiting maximum is $Q\tau /e$.

The mobile fluid volume per unit well length includes the porosity:Integrating the residual-trapping attenuation of a porous current gives, for $0\le s<1$,This is mobile volume in the current; retained fluid outside its mobile thickness is not counted. Since $h_t=-h/\tau$ in the thinning region, the rate of accumulation of immobile retained volume is ${\dot{V}}_r=sV/\tau$. Direct differentiation verifies $\dot{V}+{\dot{V}}_r=Q{e}^{-t/\tau }$, so the mobile-volume expression obeys the overall injection balance.

Without capillary residual trapping, $s=0$ gives $V=Q\tau (1-e^{-t/\tau })$. The limiting expression as $s\to 1$ is $V=Qt{e}^{-t/\tau }$, though the local thinning equation degenerates at exactly $s=1$. Both limits provide useful consistency checks.

Take $x$ increasing upslope and let $u$ be the superficial Darcy velocity, so the mobile discharge is $uh$. Its pore transport speed is $u/\varphi$. This interpretation is required by the factors of porosity in the printed equations; a literal pore-speed convention would instead replace $u$ there by $\varphi u$.

For capillary residual trapping, let $M(x,t)={\max }_{0\le \tau \le t}h(x,\tau )$ be the maximum invaded thickness, including the initial state. The stored carbon-dioxide volume per plan area isDuring advance, $M=h$ and new pores fill with mobile carbon dioxide; during recession, $M$ is fixed and the newly vacated pores retain saturation $s$. Conservation of mobile plus trapped fluid isConsequentlyAssume $a,L,u,\varphi >0$ and $0<s<1$. The two speeds differ because the advancing front fills a whole pore volume while the receding tail removes only its mobile fraction.

The method of characteristics keeps height constant on $x=x_0+v_{A}t$ on the leading face and $x=x_0+v_{R}t$ on the trailing face. Matching the two linear profiles gives the triangular current with capillary retention:Before extinction, the rear, front, crest position and crest height areEvery positive height has one trailing and one leading characteristic; their intersections trace the crest. The mobile current disappears at

The final trapped region is the maximum-invasion envelope of a retained current. For $0<x<L$, the initial height is already the maximum, $M_{\infty }=ax$. For $L<x<x_{\ast }$, the maximum occurs when the crest passes $x$, at $t_m=2(x-L)/(v_R+v_A)$. Substituting into the trailing profile givesIf $z$ is distance into the aquifer measured normally from its upper boundary, trapped carbon dioxide occupies $0<z<M_{\infty }(x)$ at residual pore saturation $s$. The occupied pore volume per unit transverse width isexactly the initial mobile volume. This is an independent mass-conservation check of both the envelope and extinction distance. In the limit $s=0$, the current translates without shrinking and nothing is trapped; the finite-extinction formula is not used in that limit.

For capillary residual trapping of an initial triangular current, the method of characteristics translates its leading and trailing faces at $v_A=u/\varphi$ and $v_R=u/[\varphi (1-s)]$. Their intersection has height $a[L-(v_R-v_A)t/2]$. Extinction occurs at $t_{\ast }=2L/(v_R-v_A)$ and position $x_{\ast }=2L/s$. These formulas require $0<s<1$; at zero residual saturation the triangle translates indefinitely.