The groundwater energy balance
is the energy balance of a groundwater body in terms of incoming hydraulic energy
associated with groundwater inflow into the body, energy
associated with the outflow, energy conversion into heat due to friction of
flow, and the resulting change of energy status and groundwater level.
Theory
When multiplying the horizontal
velocity of groundwater (dimension, for example, m3/day per m2 cross-sectional
area) with the groundwater potential (dimension energy per m3 water, or E/m3)
one obtains an energy flow (flux) in E/day per m2 cross-sectional area.
Summation or integration of the
energy flux in a vertical cross-section of unit width (say 1 m) from the lower
flow boundary (the impermeable layer or base) up to the water table in an unconfined aquifer gives the energy flow Fe
through the cross-section in E/day per m width of the aquifer.
While flowing, the groundwater
loses energy due to friction of flow, i.e. hydraulic energy is converted into
heat. At the same time, energy may be added with the recharge of water coming
into the aquifer through the water table. Thus one can make an hydraulic energy
balance of a block of soil between two nearby cross-sections. In steady state,
i.e. without change in energy status and without accumulation or depletion of
water stored in the soil body, the energy flow in the first section plus the
energy added by groundwater recharge between the sections
minus the energy flow in the second section must equal the energy loss due to
friction of flow.
In mathematical terms this balance
can be obtained by differentiating the cross-sectional integral of Fe in
the direction of flow using the Leibniz rule, taking into account that the
level of the water table may change in the direction of flow. The mathematics
is simplified using the Dupuit–Forchheimer assumption.
The hydraulic friction losses can
be described in analogy to Joule's law in electricity (see Joule's law#Hydraulic equivalent), where the
friction losses are proportional to the square value of the current (flow) and
the electrical resistance of the material through
which the current occurs. In groundwater hydraulics (fluid
dynamics, hydrodynamics) one often works with hydraulic conductivity (i.e. permeability of the soil for water),
which is inversely proportional to the hydraulic resistance.
The resulting equation of the
energy balance of groundwater flow can be used, for example, to calculate the
shape of the water table between drains under specific aquifer
conditions. For this a numerical solution can be used, taking small
steps along the impermeable base. The drainage
equation is to be solved by trial and error (iterations),
because the hydraulic potential is taken with respect to a reference level
taken as the level of the water table at the water divide midway between the
drains. When calculating the shape of the water table, its level at the water
divide is initially not known. Therefore this level is to be assumed before the
calculations on the shape of the water table can be started. According to the
findings of the calculation procedure, the initial assumption is to be adjusted
and the calculations are to be restarted until the level of the water table at
the divide does not differ significantly from the assumed level.
Shape of water table between
drains
The trial and error procedure is
cumbersome and therefore computer programs may be developed to aid in the
calculations.
Application
The energy balance of groundwater
flow can be applied to flow of groundwater to subsurface drains.The computer
program EnDrain compares the
outcome of the traditional drain
spacing equation, based on Darcy's
law cum continuity equation (i.e. conservation of mass), with the solution
obtained by the energy balance and it can be seen that drain spacings are wider
in the latter case. This is owing to the introduction of the energy supplied by
the incoming recharge.
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