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2 0. Modelling Processes in Porous Media with Differential Equations

represents, or is related to, the volume density of an extensive quantity like
mass, energy, or momentum, which is conserved. In their original form all
quantities have dimensions that we denote in accordance with the Inter-
national System of Units (SI) and write in square brackets [ ]. Let a be
a symbol for the unit of the extensive quantity, then its volume density
3
is assumed to have the form S = S(u), i.e., the unit of S(u)is a/m .For
example, for mass conservation a=kg,and S(u) is a concentration. For
describing the conservation we consider an arbitrary “not too bad” sub-
˜
set Ω ⊂ Ω, the control volume. The time variation of the total extensive
˜
quantity in Ωis then

∂ t S(u(x, t))dx . (0.1)
˜ Ω
If this function does not vanish, only two reasons are possible due to con-
servation:
3
— There is an internally distributed source density Q = Q(x, t, u) [a/m /s],
being positive if S(u) is produced, and negative if it is destroyed, i.e., one

term to balance (0.1) is ˜ Ω Q(x, t, u(x, t))dx.
˜
— There is a net flux of the extensive quantity over the boundary ∂Ωof
˜
2
Ω. Let J = J(x, t) [a/m /s] denote the flux density, i.e., J i is the amount,
that passes a unit square perpendicular to the ith axis in one second in
the direction of the ith axis (if positive), and in the opposite direction
otherwise. Then another term to balance (0.1) is given by

− J(x, t) · ν(x)dσ ,
∂Ω
where ν denotes the outer unit normal on ∂Ω. Summarizing the conserva-
tion reads

∂ t S(u(x, t))dx = − J(x, t) · ν(x)dσ + Q(x, t, u(x, t))dx . (0.2)
˜ Ω ∂ ˜ Ω ˜ Ω
The integral theorem of Gauss (see (2.3)) and an exchange of time
derivative and integral leads to

[∂ t S(u(x, t)) + ∇· J(x, t) − Q(x, t, u(x, t))]dx =0 ,
˜ Ω
˜
and, as Ω is arbitrary, also to
∂ t S(u(x, t)) + ∇· J(x, t)= Q(x, t, u(x, t)) for x ∈ Ω,t ∈ (0,T ] . (0.3)

All manipulations here are formal assuming that the functions involved
have the necessary properties. The partial differential equation (0.3) is the
basic pointwise conservation equation, (0.2) its corresponding integral form.
Equation (0.3) is one requirement for the two unknowns u and J,thusit