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0.3. Fluid Flow in Porous Media 9

whichis termedthe Laplace equation for f =0. This model and its more
general formulations occur in various contexts. If, contrary to the above as-
sumption, the solid matrix is compressible under the pressure of the water,
and if we suppose (0.13) to be valid, then we can establish a relationship
φ = φ(x, t)= φ 0 (x)φ f (p)

with φ 0 (x) > 0 and a monotone increasing φ f such that with S(p):= φ (p)
f
we get the equation
φ 0 S(p) ∂ t p + ∇· q = f
and the instationary equations corresponding to (0.17)–(0.19), respectively.
For constant S(p) > 0 this yields the following linear equation:

φ 0 S∂ t h −∇· (K∇h)= f, (0.20)
which also represents a common model in many contexts and is known from
corresponding fields of application as the heat conduction equation.
We consider single phase flow further, but now we will consider gas as
fluid phase. Because of the compressibility, the density is a function of the
pressure, which is invertible due to its strict monotonicity to

p = P( ) .
Together with (0.13) and (0.15) we get a nonlinear variant of the heat
conduction equation in the unknown :
2
∂ t (φ ) −∇ · K( ∇P( )+ ge z ) = f, (0.21)
which also contains derivatives of first order in space. If P( )= ln(α )holds
for a constant α> 0, then ∇P( ) simplifies to α∇ . Thus for horizontal
flow we again encounter the heat conduction equation. For the relationship
2
P( )= α suggested by the universal gas law, α ∇ = 1 α∇ remains
2
2
nonlinear. The choice of the variable u := would result in u 1/2 in the
time derivative as the only nonlinearity. Thus in the formulation in the
coefficient of ∇ disappears in the divergence of = 0. Correspondingly,
the coefficient S(u)= 1 φu −1/2 of ∂ t u in the formulation in u becomes
2
unbounded for u = 0. In both versions the equations are degenerate,whose
treatment is beyond the scope of this book. A variant of this equation has
gainedmuchattentionasthe porous medium equation (with convection) in
the field of analysis (see, for example, [42]).
Returning to the general framework, the following generalization of
Darcy’s law can be justified experimentally for several liquid phases:
k rα
q = − k (∇p α + α ge z ) .
α
µ α
Here the relative permeability k rα of the phase α depends upon the
saturations of the present phases and takes values in [0, 1].