= Solution
Use the common-density, equal-volume-heat-capacity model implicit in the stated speed law. Let the <liquid fraction> be $\phi$ ahead of the melting front and $\phi+\Delta\phi$ behind it. Melting adds exactly the liquid needed to fill the newly created pores, so total <mass conservation> makes <Darcy flux> continuous. Consequently,
$$
\boxed{u_{\rm ahead}=U,\qquad v_{\rm ahead}=U/\phi,\qquad v_{\rm behind}=U/(\phi+\Delta\phi).}
$$
Here $u$ denotes <Darcy velocity> and $v$ the pore-liquid velocity in the stationary rock frame. Treating the pore-space increase as storage without its simultaneous melting source would incorrectly change the Darcy flux.
Relative to the cold unmelted material, the bulk <enthalpy> increase behind the front is $\rho c_p\Delta T+\rho L\Delta\phi$: all phases gain sensible heat and the melted ice consumes <latent heat>. The advective heat-flux difference is $\rho c_pU\Delta T$. Applying the <Rankine-Hugoniot condition> to this energy balance therefore gives
$$
V(\rho c_p\Delta T+\rho L\Delta\phi)=\rho c_pU\Delta T,
$$
so the <advection-driven melting front in a porous matrix> moves at
$$
\boxed{V=\frac{U}{1+S\Delta\phi},\qquad S=\frac{L}{c_p\Delta T}.}
$$
The advected latent contribution of the liquid is the same on both sides and cancels; the denominator measures sensible heating plus phase-change energy per unit bulk volume.
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