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10 0. Modelling Processes in Porous Media with Differential Equations

At the interface of two liquid phases α 1 and α 2 we observe a difference of
the pressures, the so-called capillary pressure, that turns out experimentally
to be a function of the saturations:

(S w ,S o ,S g ) . (0.22)
p c α 1 α 2 := p α 1 − p α 2 = F α 1 α 2
A general model for multiphase flow, formulated for the moment in terms
of the variables p α ,S α , is thus given by the equations

∂ t (φS α α ) −∇ · ( α λ α k(∇p α + α ge z )) = f α (0.23)

with the mobilities λ α := k rα /µ α , and the equations (0.22) and (0.10),
where one of the S α ’s can be eliminated. For two liquid phases w and g,
e.g., water and air, equations (0.22) and (0.10) for α =w, gread p c =
p g − p w = F(S w )and S g =1 − S w . Apparently, this is a time-dependent,
nonlinear model in the variables p w ,p g ,S w , where one of the variables can
be eliminated. Assuming constant densities α , further formulations based
on

∇· q + q g = f w / w + f g / g (0.24)
w
can be given as consequences of (0.10). These equations consist of a sta-
tionary equation for a new quantity, the global pressure, based on (0.24),
and a time-dependent equation for one of the saturations (see Exercise 0.2).
In many situations it is justified to assume a gaseous phase with constant
pressure in the whole domain and to scale this pressure to p g =0. Thus
for ψ := p w = −p c we have

φ∂ t S(ψ) −∇· (λ(ψ)k(∇ψ + ge z )) = f w / w (0.25)
with constant pressure := w ,and S(ψ):= F −1 (−ψ)as a strictly
monotone increasing nonlinearity as well as λ.
With the convention to set the value of the air pressure to 0, the pressure
in the aqueous phase is in the unsaturated state, where the gaseous phase is
also present, and represented by negative values. The water pressure ψ =0
marks the transition from the unsaturated to the saturated zone. Thus
in the unsaturated zone, equation (0.25) represents a nonlinear variant
of the heat conduction equation for ψ< 0, the Richards equation.As
most functional relationships have the property S (0) = 0, the equation

degenerates in the absence of a gaseous phase, namely to a stationary
equation in a way that is different from above.
Equation (0.25) with S(ψ):= 1 and λ(ψ):= λ(0) can be continuedina
consistent way with (0.14) and (0.15) also for ψ ≥ 0, i.e., for the case of a
sole aqueous phase. The resulting equation is also called Richards equation
or a model of saturated-unsaturated flow.