Geoscience Reference
In-Depth Information
becomes in the notation of porous media flow
=−
∂θ
∂
∇·
q
(8.54)
t
Conservation of mass and momentum: Richards's equation
Substitution of Darcy's law (8.19) in the equation of continuity (8.54) produces imme-
diately
=
∂θ
∂
∇·
(
k
∇
h
)
(8.55)
t
or, written out in full,
∂
∂
k
∂
k
∂
k
∂
h
∂
∂
h
∂
∂
h
=
∂θ
∂
+
+
(8.56)
x
∂
x
y
∂
y
z
∂
z
t
which is now usually referred to as the Richards (1931) equation. As such, Equation (8.55)
is valid only for isotropic materials; it would be a straightforward exercise to extend the
formulation to anisotropic materials. Under conditions of steady flow or under conditions
of fully saturated flow, the right-hand side of (8.55) becomes zero. Under conditions of
fully saturated flow in a uniform material, so that
θ
=
θ
0
and
k
=
k
0
=
constant, (8.55)
2
h
reduces to the equation of Laplace, that is
∇
=
0, or written out in full
2
h
2
h
2
h
∂
x
2
+
∂
+
∂
=
0
(8.57)
∂
∂
y
2
∂
z
2
8.4.2
General case of two immiscible fluids in an elastic porous material
Biot (1941, 1955, 1956a, b) was probably the first to present a general theory of elasticity
of a porous material saturated with an elastic fluid for the three-dimensional case with an
arbitrary and variable load. This theory was subsequently (Brutsaert, 1964; Brutsaert and
Luthin, 1964) extended to describe the elasticity of an unconsolidated granular material,
containing two fluids in its interstices. Later Verruijt (1969) showed that Biot's theory for
a saturated material can be simplified to describe groundwater movement in most cases of
practical interest; he thus demonstrated that Biot's theory can often be reduced to Jacob's
(1940) simple equation but also that in some cases the general theory is the only one that
succeeds in explaining experimental results. In what follows, Verruijt's (1969) develop-
ment is combined with Brutsaert's (1964) two-fluid extension of the theory to obtain a
general formulation for unconfined and confined groundwater flow. Although the matter is
straightforward, a careful exposition is desirable to bring out the significance of the under-
lying assumptions of the various more special groundwater equations used in the technical
literature.
Strains
It is convenient to consider a fixed (Eulerian), infinitesimally small cubic element of a porous
material, containing both water and air (or, more generally, a wetting fluid and a non-wetting
fluid) in a Cartesian coordinate system. In this section the displacement vector of the solid
part relative to its initial position is denoted by
u
(
=
u
x
i
+
u
y
j
+
u
z
k
). The corresponding
displacement of the water
w
(
=
w
x
i
+
w
y
j
+
w
z
k
) and that of the air
v
(
=
v
x
i
+
v
y
j
+
v
z
k
)
