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HYDC02 12/5/05 5:38 PM Page 56

56 Chapter Two

and upon substitution in equation 2.37: Changes in fluid density and formation porosity are
both produced by a change in hydraulic head and the
∂ ⎛ ∂h⎞ ∂ ⎛ ∂h⎞ ∂ ⎛ ∂h⎞ volume of water produced by the two mechanisms
s
∂x ⎝ ⎜ K x ∂x⎠ ⎟ + ∂y ⎝ ⎜ K y ∂y⎠ ⎟ + ∂z ⎝ ⎜ K z ∂z⎠ ⎟ = 0 for a unit decline in head is the specific storage, S .
Hence, the time rate of change of fluid mass storage
eq. 2.39 within the control volume is:

For an isotropic and homogeneous porous material, ∂ h
ρS s
K = K = K and K(x, y, z) = constant, respectively. By t ∂
x y z
substituting these two conditions in equation 2.39 it
can be shown that: and equation 2.41 becomes:
2
2
2
∂ h ∂ h ∂ h ∂(ρq ) ∂(ρq y ) ∂(ρq ) ∂h
+
+
=
+
+
2 2 = 0 eq. 2.40 x z ρS s eq. 2.42
2
∂x ∂y ∂z ∂x ∂y ∂z ∂t
Thus, the steady-state groundwater flow equation By expanding the terms on the left-hand side of equa-
is the Laplace equation and the solution h(x, y, z) tion 2.42 using the chain rule (eliminating the smaller
describes the value of the hydraulic head at any point density gradient terms compared with the larger spe-
in a three-dimensional flow field. By solving equation cific discharge gradient terms) and, at the same time,
2.40, either in one, two or three dimensions depend- inserting Darcy’s law to define the specific discharge
ing on the geometry of the groundwater flow prob- terms, then:
lem under consideration, a contoured equipotential
map can be produced and, with the addition of flow ∂ ⎛ ∂h⎞ ∂ ⎛ ∂h⎞ ∂ ⎛ ∂h⎞ ∂h
lines, a flow net drawn (Box 2.3). ⎜ K ⎟ + ⎜ K ⎟ + ⎜ K ⎟ = S
∂x ⎝ x ∂x⎠ ∂y ⎝ y ∂y⎠ ∂z ⎝ z ∂z⎠ s ∂t

Transient saturated flow eq. 2.43
The law of conservation of mass for transient flow in If the porous material is isotropic and homogeneous,
a saturated porous material requires that the net rate equation 2.43 reduces to:
of fluid mass flow into the control volume (Fig. 2.24)
is equal to the time rate of change of fluid mass stor- ∂ h ∂ h ∂ h S s ∂h
2
2
2
=
+
+
age within the control volume. The equation of con- ∂x 2 ∂y 2 ∂z 2 K ∂t eq. 2.44
tinuity is now:
or, expanding the specific storage term, S (eq. 2.33):
s
ρ
n
∂(ρq ) ∂(ρq ) ∂(ρq ) ∂() ∂ρ ∂n
=
+
+
+
=
x y z n ρ
∂x ∂y ∂z ∂t ∂t ∂t 2 2 2 ρα + β
∂ h ∂ h ∂ h g( n ) ∂h
+
=
+
eq. 2.45
eq. 2.41 ∂x 2 ∂y 2 ∂z 2 K ∂t

The first term on the right-hand side of equation 2.41 Equations 2.43, 2.44 and 2.45 are all transient ground-
describes the mass rate of water produced by expan- water flow equations for saturated anisotropic (eq.
sion of the water under a change in its density, ρ, 2.43) and homogeneous and isotropic (eqs 2.44 and
and is controlled by the compressibility of the fluid, 2.45) porous material. The solution h(x, y, z, t)
β. The second term is the mass rate of water pro- describes the value of hydraulic head at any point in a
duced by the compaction of the porous material as three-dimensional flow field at any time. A solution
influenced by the change in its porosity, n, and is requires knowledge of the three hydrogeological
determined by the compressibility of the aquifer, α. parameters, K, α and n, and the fluid parameters,