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8 0. Modelling Processes in Porous Media with Differential Equations

If the density of water is assumed to be constant, due to neglecting
the mass of solutes and compressibility of water, equation (0.13) simplifies
further to the stationary equation

∇· q = f, (0.14)
where f has been replaced by the volume source density f/ , keeping the
same notation. This equation will be completed by a relationship that
can be interpreted as the macroscopic analogue of the conservation of mo-
mentum, but should be accounted here only as an experimentally derived
constitutive relationship. This relationship is called Darcy’s law,which
reads as
q = −K (∇p + ge z ) (0.15)

2
and can be applied in the range of laminar flow. Here p [N/m ] is the intrinsic
2
average of the water pressure, g [m/s ] the gravitational acceleration, e z the
unit vector in the z-direction oriented against the gravitation,
K = k/µ , (0.16)

a quantity, which is given by the permeability k determined by the solid
phase, and the viscosity µ determined by the fluid phase. For an anisotropic
solid, the matrix k = k(x) is a symmetric positive definite matrix.
Inserting (0.15) in (0.14) and replacing K by K g,known as hydraulic
conductivity in the literature, and keeping the same notation gives the
following linear equation for
1
h(x, t):= p(x, t)+ z,
g
the piezometric head h [m]:

−∇ · (K∇h)= f. (0.17)
The resulting equation is stationary and linear. We call a differential equa-
tion model stationary if it depends only on the location x and not on the
time t,and instationary otherwise. A differential equation and correspond-
ing boundary conditions (cf. Section 0.5) are called linear if the sum or a
scalar multiple of a solution again forms a solution for the sum, respectively
the scalar multiple, of the sources.
If we deal with an isotropic solid matrix, we have K = KI with the d×d
unit matrix I and a scalar function K. Equation (0.17) in this case reads

−∇ · (K∇h)= f. (0.18)
Finally if the solid matrix is homogeneous, i.e., K is constant, we get from
division by K and maintaining the notation f the Poisson equation
−∆h = f, (0.19)