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Eliminating phase pressures with $p_c=p_n-p_w$ gives $\varphi s_t+QF(s{)}_x=(D_c{s}_x{)}_x$. For decreasing capillary pressure, $D_c\ge 0$. This nonlinear diffusion broadens a saturation shock; it can degenerate where a phase mobility vanishes. Endpoint conservation retains the limiting Rankine-Hugoniot condition.

The fraction of total advective Darcy flux carried by one phase when capillary and gravity corrections are omitted. For two phases it is ${\lambda }_w/({\lambda }_w+{\lambda }_n)$. This quantity lies between zero and one for nonnegative phase mobilities. Its nonlinear dependence on fluid saturation determines characteristic speeds and shocks in the Buckley-Leverett equation.

Effective permeability of a porous medium. Work to leading order in the slender-layer ratio $H/L$. Take the growing lower wedge to have porous permeability $k_1$, with interface $y=Hx/L$, and the upper wedge to have porous permeability $k_2$. The leading pressure is independent of $y$; vertical flow is smaller than horizontal flow by $H/L$. DefineDarcy's law and the fixed two-dimensional volume flux giveHere $q_i$ is Darcy velocity; the parcel speed is $q_i/\varphi$. Integrating the pressure gradient over the length and defining $Q=H{k}_{\mathrm{eff}}\Delta p/(\mu L)$ yieldsThe equal-porous permeability limit is $k_{\mathrm{eff}}=k_1=k_2$. This logarithmic mean comes from parallel layers at each cross-section followed by series addition of their local hydraulic resistances. It is a slender-layer result; a full two-dimensional transmission problem has small end and interface corrections.

Parcel paths and travel times. Pressure equalization does not mean that parcels stay at fixed $y$: mass conservation requires a small vertical flow. Define a streamfunction by $\psi (x,0)=0$ and $\psi (x,H)=Q$. Its leading expression isLet $r=y_0/H$ label a parcel released at the inlet. There $\psi =Qr$. At the inclined interface $\psi /Q=k_1\xi /K(\xi )$, so the parcel crosses from the upper wedge into the lower one atBefore crossing, $y/H=1-(1-r)K/k_2$; afterwards, $y/H=rK/k_1$. These expressions show why integrating the speed along a horizontal line would give the wrong parcel time.

Put $d=k_1-k_2$ and ${\tau }_V=\varphi HL/Q$, the pore-volume throughput time. Integrating $dt=\varphi dx/q_i$ on the two portions of the path givesIts derivative with respect to ${\xi }_c$ is ${\tau }_{V}dK/(k_1{k}_2)$, so it is monotone. The limiting streamline times at the lower and upper boundaries are ${\tau }_V(k_1+k_2)/(2k_1)$ and ${\tau }_V(k_1+k_2)/(2k_2)$ respectively. HenceThe printed expression assumes $k_1\ge k_2$. The absolute value is needed for a nonnegative maximum difference without that ordering. For equal porous permeabilities every parcel has time ${\tau }_V$.

Oil recovery. In the ideal passive-displacement model with $k_1/k_2=10$, the earliest and latest travel times are $0.55{\tau }_V$ and $5.5{\tau }_V$, a spread of $4.95{\tau }_V$. Preferential paths through the high-porous permeability wedge therefore give early water breakthrough while oil on slower paths remains unswept. Continued injection sends much water through paths already swept; complete displacement requires several pore volumes. The flux-weighted residence time in a porous layer remains ${\tau }_V$. Real waterflooding also depends on phase mobilities, relative permeabilities, capillary pressure and mixing, so these numbers illustrate heterogeneity rather than a quantitative two-phase recovery prediction.

Saturation equation. Introduce intrinsic permeability of a porous medium $K$ and porosity $\varphi$. The relative porous permeabilities depend on the wetting fluid saturation. Define the phase mobilities and fractional flowUse the conventional capillary pressure $p_c=p_{nw}-p_w$, generally decreasing as wetting saturation increases. Neglect gravity in this horizontal model. The two Darcy fluxes are $q_w=-K{\lambda }_w{p}_{w,x}$ and $q_n=-K{\lambda }_n(p_{w,x}+p_c^{\prime }(s)s_x)$. Eliminate the common pressure gradient using $q_w+q_n=Q$ to obtainThe wetting-phase mass conservation law is consequentlyFor $p_c^{\prime }\le 0$, $D_c\ge 0$ and the capillary pressure provides nonlinear diffusion. The initially uniform saturation is $s_0$. Pure wetting-fluid injection prescribes $q_w(0,t)=Q$ and $q_n(0,t)=0$; in the zero-capillarity, zero-residual-nonwetting idealization this corresponds to inlet saturation one. With residual nonwetting fluid, replace one by the appropriate maximum accessible saturation.

Shock formation and its speed. Neglect capillarity first. Smooth saturation values travel along characteristic curves at speed $(Q/\varphi )F^{\prime }(s)$. When the trailing values have larger characteristic speeds than the values ahead, the curves intersect, and the Buckley-Leverett equation requires a shock selected as an entropy solution. This is typical for the increasing-convex part of a physical fractional-flow curve; shocks are not inevitable for every possible $F$.

Integrating mass conservation across a shock from upstream $s_s$ to downstream $s_0$ gives the Rankine-Hugoniot conditionIf an upstream rarefaction wave joins the shock smoothly in similarity coordinates, its terminal characteristic speed equals $V_s$. Therefore the saturation at that join obeys the fractional-flow tangent constructionThis is the printed equality. It is a tangent-selection condition, not the dimensional speed by itself; the speed includes $Q/\varphi$. An arbitrary shock between prescribed constant states satisfies the jump condition but need not satisfy this additional tangency relation.

Positive capillary diffusion spreads a jump into a transition layer. For a travelling profile $s(x-V_{s}t)$, integration givesThe endpoint states still satisfy the same Rankine-Hugoniot condition. Thus small capillarity primarily gives the front a finite thickness rather than changing the selected limiting speed; vanishing phase mobility at an endpoint can make the regularization degenerate.

The specified reciprocal fractional flow. On its stated branch, $F^{\prime }=1/s^2$ and $F^{\prime \prime }=-2/s^3<0$. If $F$ extends continuously to the initial saturation and the inlet is at one, the decreasing saturation from behind to ahead makes characteristic speeds increase forwards: there is a rarefaction wave, not a compressive shock. Write $a=Q/\varphi$ and $\xi =x/t$. The self-similar solution isThe fan connects continuously to the initial state and broadens linearly in time. This interpretation requires $F(s_0)=2-1/s_0$. A nonnegative physical fractional flow on the entire interval $[s_0,1]$ also requires $s_0\ge 1/2$.

The PDF specifies the formula only for $s>s_0$ and earlier calls $s_0$ residual. If that means an immobile wetting phase with $F(s_0)=0$, the endpoint is an extra constraint and the continuous-branch answer applies without conflict only at $s_0=1/2$. For $0<s_0<1/2$, the printed branch would give negative fractional flow for $s_0<s<1/2$, so it cannot be the complete physical constitutive law. With an additional admissible extension, for example $F=0$ up to $1/2$ and the given positive branch above it, a compound fan–shock is possible. The tangent condition then givesThe fan $s=\sqrt{a/\xi }$ ends at $\xi =V_s$, followed by a jump to $s_0$. This is a conditional physical extension, not data supplied by the question. For $s_0>1/2$ an immobile residual endpoint is likewise incompatible with a continuous version of the stated branch. These distinctions identify what is, and is not, determined by the printed constitutive assumption.

A dimensionless reduction of a porous formation's intrinsic permeability of a porous medium for one fluid phase in multiphase flow. Its effective phase permeability is $K{k}_{ri}$, and its phase mobility is $k_{ri}/{\mu }_i$. It depends on saturation and pore-scale phase configuration; it is not the magnetic use of relative permeability.