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Physical hydrogeology 55

and is that volume of water that an unconfined
aquifer releases from storage per unit surface area of
aquifer per unit decline in the water table (Fig. 2.22b),
and is approximately equivalent to the total porosity
of a soil or rock (see Section 2.2). Specific yield is a
dimensionless term and the normal range is from
0.01 to 0.30. Relative to confined aquifers, the higher
values reflect the actual dewatering of pore space
as the water table is lowered. Consequently, the
same yield can be obtained from an unconfined
aquifer with smaller head changes over less extensive
areas than can be produced from a confined aquifer.
Although not commonly used, by combining the Fig. 2.24 Unit volume (elemental control volume) for flow
aquifer properties of transmissivity (T or K) and storat- through porous material.
ivity (S or S ) it is possible to define a single formation
s
parameter, the hydraulic diffusivity, D, defined as The law of conservation of mass for steady-state flow
either: requires that the rate of fluid mass flow into the control
volume, ρq (fluid density multiplied by specific dis-
T
=
D K eq. 2.34 charge across a unit cross-sectional area), will be equal
or
S S to the rate of fluid mass flow out of the control volume,
s
such that the incremental differences in fluid mass flow,
Aquifer formations with a large hydraulic diffusivity in each of the directions x, y, z, sum to zero, thus:
respond quickly in transmitting changed hydraulic
conditions at one location to other regions in an aquifer, ⎛ ( ∂ ρq ) ⎞ ⎛ ( ∂ ρq ) ⎞
x⎟
for example in response to groundwater abstraction. ⎜ ρq + x – ρq + ⎜ ρq + y – ρq y⎟
⎝ x ∂ ⎠ ⎝ y ∂ ⎠
y
x
⎛ ∂ ⎞
2.11.4 Equations of groundwater flow (ρq z )
⎜
+ ρq z + z ∂ – ρq z ⎟ ⎠ = 0 eq. 2.35
⎝

Equations of groundwater flow are derived from
a consideration of the basic flow law, Darcy’s law
From equation 2.35, the resulting equation of con-
(eq. 2.5), and an equation of continuity that describes
tinuity is:
the conservation of fluid mass during flow through a
porous material. In the following treatment, which
∂(ρq ) ∂(ρq y ) ∂(ρq )
derives from the classic paper by Jacob (1950), steady- x + + z = 0 eq. 2.36
∂x ∂y ∂z
state and transient saturated flow conditions are con-
sidered in turn. Under steady-state conditions, the
If the fluid is incompressible, then density, ρ(x, y, z), is
magnitude and direction of the flow velocity at any
constant and equation 2.36 becomes:
point are constant with time. For transient condi-
tions, either the magnitude or direction of the flow
∂q ∂q ∂q
+
+
=
velocity at any point may change with time, or the x y z 0 eq. 2.37
potentiometric conditions may change as groundwa- ∂x ∂y ∂z
ter either enters into or is released from storage.
From Darcy’s law, each of the specific discharge
terms can be expressed as:
Steady-state saturated flow
First, consider the unit volume of a porous material q =− K ∂ h , q =− K h ∂ , q =− K h ∂ eq. 2.38
(the elemental control volume) depicted in Fig. 2.24. x x x ∂ y y y ∂ z z z ∂