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58 Chapter Two

Simple one-dimensional problems can be solved using 2.13 Groundwater flow patterns
ordinary differential and integral calculus. Two-
dimensional problems or transient (time-variant) flow Preceding sections in this chapter have introduced
problems require the use of partial derivatives and the fundamental principles governing the existence
more advanced calculus. and movement of groundwater, culminating in the
In conceptualizing the groundwater flow prob- derivation of the governing groundwater flow equa-
lem, the basic geometry should be sketched and the tions for steady-state, transient and unsaturated flow
aquifers, aquitards and aquicludes defined. Simplify- conditions. Within the water or hydrological cycle,
ing assumptions, for example concerning isotropic groundwater flow patterns are influenced by geolo-
and homogeneous hydraulic conductivity, should gical factors such as differences in aquifer lithologies
be stated and, if possible, the number of dimensions and structure of confining strata. A further influence
reduced (for example, consider only the horizontal on groundwater flow, other than aquifer heterogene-
component of flow or look for radial symmetry or ity, is the topography of the ground surface. Topo-
approximately parallel flow). If the groundwater flow graphy is a major influence on groundwater flow at
is confined, then a mathematically linear solution local, intermediate and regional scales. The elevation
results which can be combined to represent more of recharge areas in regions of aquifer outcrop, the
complex situations. Unconfined situations produce degree to which river systems incise the landscape
higher order equations. and the location and extent of lowland areas experi-
As a first step, an equation of continuity is written encing groundwater discharge determine the overall
to express conservation of fluid mass. For incom- configuration of groundwater flow. As shown by
pressible fluids this is equivalent to conservation Freeze and Witherspoon (1967), the relative positions
of fluid volume. Water is only very slightly com- and difference in elevation of recharge and discharge
pressible, so conservation of volume is a reasonable areas determine the hydraulic gradients and the
approximation. By combining the equation of con- length of groundwater flowpaths.
tinuity with a flow law, normally Darcy’s law, and To understand the influence of topography, con-
writing down equations that specify the known con- sider the groundwater flow net shown in Fig. 2.25 for
ditions at the boundaries of the aquifer, or at specified a two-dimensional vertical cross-section through a
points (for example, a well), provides a general differ- homogeneous, isotropic aquifer. The section shows a
ential equation for the specified system. Solving the single valley bounded by groundwater divides and an
problem consists of finding an equation (or equa- impermeable aquifer base. The water table is a sub-
tions) which describes the system and satisfies both dued replica of the topography of the valley sides.
the differential equation and the boundary condi- The steady-state equipotential and groundwater flow
tions. By integrating the differential equation, the lines are drawn using the rules for flow net analysis
resulting equation is the general solution. If the con- introduced in Box 2.3. It is obvious from the flow net
stants of integration are found by applying the bound- that groundwater flow occurs from the recharge
ary conditions, then a specific solution to the problem areas on the valley sides to the discharge area in the
is obtained (for examples, see Box 2.9). valley bottom. The hinge line separating the recharge
The final step in the mathematical analysis is the from the discharge areas is also marked in Fig. 2.25.
evaluation of the solution. The specific solution is For most common topographic profiles, hinge lines
normally an equation that relates groundwater head are positioned closer to valley bottoms than to catch-
to position and to parameters contained in the prob- ment divides with discharge areas commonly com-
lem such as hydraulic conductivity, or such factors prising only 5–30% of the catchment area (Freeze &
as the discharge rate of wells. By inserting numerical Cherry 1979). Tòth (1963) determined a solution
values for the parameters, the solution can be used to to the boundary-value problem represented by the
evaluate groundwater head in terms of position in the flow net shown in Fig. 2.25. The solution provides an
co-ordinate system. The results might be expressed as analytical expression for the hydraulic head in the
a graph or contour diagram, or as a predicted value of flow field for simple situations of an inclined water
head for a specified point. table of constant slope and cases in which a sine curve