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14 0. Modelling Processes in Porous Media with Differential Equations

0.3 A frequently employed model for mechanical dispersion is
D mech = λ L |v| 2 Pv + λ T |v| 2 (I − Pv )
2
T
with parameters λ L >λ T ,where v = q/Θand Pv = vv /|v| .Here
2
λ L and λ T are the longitudinal and transversal dispersion lengths.Give a
geometrical interpretation.

0.5 Boundary and Initial Value Problems

The differential equations that we derived in Sections 0.3 and 0.4 have the
common form

∂ t S(u)+ ∇· (C(u) − K(∇u)) = Q(u) (0.33)
with a source term S, a convective part C, a diffusive part K, i.e., a total
flux C − K and a source term Q, which depend linearly or nonlinearly
on the unknown u. For simplification, we assume u to be a scalar. The
nonlinearities S, C, K,and Q may also depend on x and t,which shall be
suppressed in the notation in the following. Such an equation is said to be
in divergence form or in conservative form; a more general formulation is
obtained by differentiating ∇· C(u)= ∂ C(u) ·∇u +(∇· C)(u)orby
∂u
introducing a generalized “source term” Q = Q(u, ∇u). Up to now we have
considered differential equations pointwise in x ∈ Ω(and t ∈ (0,T )) under
the assumption that all occurring functions are well-defined. Due to the
˜
applicability of the integral theorem of Gauss on Ω ⊂ Ω (cf. (3.10)), the
integral form of the conservation equation follows straightforwardly from
the above:

∂ t S(u) dx + (C(u) − K(∇u)) · νdσ = Q(u, ∇u) dx (0.34)
˜ Ω ∂ ˜ Ω ˜ Ω
with the outer unit normal ν (see Theorem 3.8) for a fixed time t or also
in t integrated over (0,T ). Indeed, this equation (on the microscopic scale)
is the primary description of the conservation of an extensive quantity:
˜
Changes in time through storage and sources in Ω are compensated by the
˜
normal flux over ∂Ω. Moreover, for ∂ t S, ∇· (C − K), and Q continuous
˜
on the closure of Ω, (0.33) follows from (0.34). If, on the other hand, F is
˜
ahyperplanein Ω where the material properties may rapidly change, the
jump condition
[(C(u) − K(∇u)) · ν] = 0 (0.35)
for a fixed unit normal ν on F follows from (0.34), where [ · ] denotes the
difference of the one-sided limits (see Exercise 0.4).
Since the differential equation describes conservation only in general,
it has to be supplemented by initial and boundary conditions in order to