Page 39 - Hydrogeology Principles and Practice

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22 Chapter Two

moves. For unconsolidated sand, Krumbein and Monk
(1943) derived the following empirical relationship
where GM is the geometric mean of the grain diam-
d
eter (mm) and ρ is the standard deviation of the grain
size in phi units (−log (grain diameter in mm) ):
2
2 −1.3σ
k = 760(GM ) e = Cd 2 eq. 2.8
i d
As shown, equation 2.8 is more generally expressed
2
as k = Cd where d is equal to the mean pore diam-
i
eter and C represents a dimensionless ‘shape factor’
assessing the contribution made by the shape of the
pore openings, as influenced by the relationship
between the pore and grain sizes and their effect
on the tortuosity of fluid flow. Intrinsic permeability
2
has the dimensions of [L ] and, using nomenclature
common in the petroleum industry, the unit of k i
is the darcy where 1 darcy is equivalent to 9.87 ×
2
10 −13 m .
Now, combining equations 2.5, 2.7 and 2.8 gives a
full expression of the flow through a porous material as:

ρ
2
Q h d Cd g h d
q −
=
== K − eq. 2.9
A l d µ l d

The quotient Q/A, or q, indicates the discharge per
unit cross-sectional area of saturated porous material.
The term q, referred to as the specific discharge, has
−1
the dimensions of velocity [LT ] and is also known
as the darcy velocity or darcy flux. It is important to
Fig. 2.4 Laboratory-determined values of hydraulic conductivity
as a function of grain size for alluvial aquifers in the Rivers remember that the darcy velocity is not the true,
Missouri and Arkansas. Note the log–log scales. After Sharp microscopic velocity of the water moving along
(1988). winding flowpaths within the soil or rock. Instead, by
dividing the specific discharge by the fraction of open
space (in other words, effective porosity, n ) through
e
−3
−2
× 10 Nsm ), although a groundwater flow sys- which groundwater flows across a given sectional
area, this provides an average measure of groundwa-
tem exhibiting such a temperature change would be
ter velocity such that:
considered unusual. An example is groundwater that
penetrates deep in the Earth’s crust, becomes heated
and returns rapidly to the surface as highly mineral- Q = q =
V eq. 2.10
ized hot springs. Equally, in coastal areas, saline intru- An n
e e
sion into fresh groundwater will cause variations in
fluid density such that information about both k and where V is the average linear velocity (Fig. 2.5).
i
K is required in any investigation. As illustrated in Box 2.1, the application of
The intrinsic permeability is representative of the equations 2.5 and 2.10 to simple hydrogeological
properties of the porous material alone and is related situations enables first estimates to be obtained for
to the size of the openings through which the fluid groundwater flow and velocity. More accurate calcula-