Page 74 - Hydrogeology Principles and Practice

HYDC02 12/5/05 5:38 PM Page 57

Physical hydrogeology 57

ρ and β. A simplification is to take the special case of a The first term on the right-hand side of equation 2.47
horizontal confined aquifer of thickness, b, storativ- is insignificantly small and by inserting the unsatur-
ity, S (= S b), and transmissivity, T (= Kb), and substi- ated form of Darcy’s law, in which the hydraulic
s
tute in equation 2.44, thus: conductivity is a function of the pressure head, K(ψ),
then equation 2.47 becomes, upon cancelling the ρ
2
2
∂ h ∂ h S ∂h terms:
+
=
2 eq. 2.46
2
∂x ∂y T ∂t
∂ ⎛ ∂h⎞ ∂ ⎛ ∂h⎞
+
ψ
ψ
The solution of this equation, h(x, y, t), describes the ∂x ⎝ ⎜ K() ∂x⎠ ⎟ ∂y ⎝ ⎜ K() ∂y⎠ ⎟
hydraulic head at any point on a horizontal plane
through the horizontal aquifer at any time. A solution ∂ ⎛ ∂h ⎞ θ ∂
ψ
requires knowledge of the aquifer parameters T and + ⎜ K () ⎟ = eq. 2.48
∂z ⎝ ∂z ⎠ ∂t
S, both of which are measurable from field pumping
tests (see Section 5.8.2).
It is usual to quote equation 2.48 in a form where the
independent variable is either θ or ψ. Hence, noting
Transient unsaturated flow
that h = z + ψ (eq. 2.22) and defining the specific
A treatment of groundwater flow in unsaturated moisture capacity, C, as dθ/dψ, then:
porous material must incorporate the presence of an
air phase. The air phase will affect the degree of con- ∂ ⎛ ∂ψ⎞ ∂ ⎛ ∂ψ⎞
+
ψ
ψ
nectivity between water-filled pores, and will there- ⎜ K() ⎟ ⎜ K() ⎟
∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠
fore influence the hydraulic conductivity. Unlike in
saturated material where the pore space is com- ⎛ ⎞⎞
pletely water filled, in unsaturated material the par- + ∂ ⎜ K ψ ⎛ ∂ψ + 1 ⎟⎟ = C ψ ∂ψ eq. 2.49
⎜
()
()
tial saturation of pore space, or moisture content (θ), ∂z ⎝ ⎝ ∂z ⎠⎠ ∂t
means that the hydraulic conductivity is a function of
the degree of saturation, K(θ). Alternatively, since the
This equation (eq. 2.49) is the ψ-based transient
degree of moisture content will influence the pres-
unsaturated flow equation for porous material and is
sure head (ψ), the hydraulic conductivity is also a
known as the Richards equation. The solution ψ(x, y,
function of the pressure head, K(ψ). In soil physics,
z, t) describes the pressure head at any point in a flow
the degree of change in moisture content for a change
field at any time. It can be easily converted into a
in pressure head (dθ/dψ) is referred to as the specific
hydraulic head solution h(x, y, z, t) through the rela-
moisture capacity, C (the unsaturated storage prop-
tion h = z + ψ (eq. 2.22). To be able to provide a solu-
erty of a soil), and can be empirically derived from the
tion to the Richards equation it is necessary to know
slope of a soil characteristic curve (see Section 5.4.1).
the characteristic curves K(ψ) and C(ψ) or θ(ψ) (see
Returning to Fig. 2.24, for flow in an elemental
Section 5.4.1).
control volume that is partially saturated, the equa-
tion of continuity must now express the time rate of
change of moisture content as well as the time rate of
2.12 Analytical solution of one-dimensional
change of storage due to water expansion and aquifer
groundwater flow problems
compaction. The fluid mass storage term (ρn) in
equation 2.41 now becomes ρθ and:
The three basic steps involved in the mathematical
analysis of groundwater flow problems are the same
θ
ρ
∂(ρq ) ∂(ρq ) ∂(ρq ) ∂() ∂ρ ∂θ
=
+
x + y + z = θ ρ whatever the level of mathematical difficulty and are:
∂x ∂y ∂z ∂t ∂t ∂t (i) conceptualizing the problem; (ii) finding a solu-
eq. 2.47 tion; and (iii) evaluating the solution (Rushton 2003).