= Solution
\b[Saturation equation.] Introduce intrinsic <permeability of a porous medium> $K$ and <porosity> $\phi$. The relative <porous permeabilities> depend on the wetting <fluid saturation>. Define the <phase mobilities> and <fractional flow>
$$
\lambda_w(s)=\frac{k_w(s)}{\mu_w},\qquad
\lambda_n(s)=\frac{k_{nw}(s)}{\mu_{nw}},\qquad
F(s)=\frac{\lambda_w}{\lambda_w+\lambda_n}.
$$
Use 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 $q_w=-K\lambda_wp_{w,x}$ and $q_n=-K\lambda_n(p_{w,x}+p_c'(s)s_x)$. Eliminate the common pressure gradient using $q_w+q_n=Q$ to obtain
$$
q_w=QF(s)+K\frac{\lambda_w\lambda_n}{\lambda_w+\lambda_n}p_c'(s)s_x
=QF(s)-D_c(s)s_x,
\qquad
D_c(s)=-K\frac{\lambda_w\lambda_n}{\lambda_w+\lambda_n}p_c'(s).
$$
The wetting-phase <mass conservation> law is consequently
$$
\boxed{\phi s_t+Q\partial_xF(s)=\partial_x[D_c(s)s_x].}
$$
For $p_c'\leq0$, $D_c\geq0$ 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.
\b[Shock formation and its speed.] Neglect capillarity first. Smooth saturation values travel along <characteristic curves> at speed $(Q/\phi)F'(s)$. 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 condition>
$$
\boxed{V_s=\frac Q\phi\frac{F(s_s)-F(s_0)}{s_s-s_0}.}
$$
If 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 construction>
$$
\boxed{F'(s_s)=\frac{F(s_s)-F(s_0)}{s_s-s_0}.}
$$
This is the printed equality. It is a tangent-selection condition, not the dimensional speed by itself; the speed includes $Q/\phi$. 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 $s(x-V_st)$, integration gives
$$
D_c(s)s'=Q[F(s)-F(s_0)]-\phi V_s(s-s_0).
$$
The 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.
\b[The specified reciprocal fractional flow.] On its stated branch, $F'=1/s^2$ and $F''=-2/s^3<0$. 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 $a=Q/\phi$ and $\xi=x/t$. The <self-similar solution> is
$$
\boxed{s(x,t)=\begin{cases}
1,&0\leq\xi\leq a,\\
\sqrt{a/\xi},&a<\xi<a/s_0^2,\\
s_0,&\xi\geq a/s_0^2.
\end{cases}}
$$
The 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 $s_0\geq1/2$.
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 gives
$$
s_s=\frac{1+\sqrt{1-2s_0}}2,\qquad
V_s=\frac a{s_s^2}.
$$
The 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.
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