Groundwater energy balance

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The

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.

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

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

The mathematics is simplified using the Dupuit assumption

.

The hydraulic friction losses can be described in analogy to

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 (iteration

s), 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.

The trial and error procedure is cumbersome and therefore computer program

s may be developed to aid in the calculations.

, 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.

**groundwater energy balance**is the energy balance of a groundwaterGroundwater

Groundwater is water located beneath the ground surface in soil pore spaces and in the fractures of rock formations. A unit of rock or an unconsolidated deposit is called an aquifer when it can yield a usable quantity of water. The depth at which soil pore spaces or fractures and voids in rock...

body in terms of incoming hydraulic energy associated with groundwater inflow

Groundwater flow

Groundwater flow is defined as the "...part of streamflow that has infiltrated the ground, has entered the phreatic zone, and has been discharged into a stream channel, via springs or seepage water". In hydrogeology it is measured by the Groundwater flow equation.- See also :*Subsurface...

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 assumption

Dupuit assumption

The Dupuit assumption holds that groundwater moves horizontally in an unconfined aquifer, and that the groundwater discharge is proportional to the saturated aquifer thickness...

.

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 resistanceElectrical resistance

The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

of the material through which the current occurs. In groundwater hydraulics (fluid dynamics

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, hydrodynamics) one often works with hydraulic conductivity

Hydraulic conductivity

Hydraulic conductivity, symbolically represented as K, is a property of vascular plants, soil or rock, that describes the ease with which water can move through pore spaces or fractures. It depends on the intrinsic permeability of the material and on the degree of saturation...

(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

Water table

The water table is the level at which the submarine pressure is far from atmospheric pressure. It may be conveniently visualized as the 'surface' of the subsurface materials that are saturated with groundwater in a given vicinity. However, saturated conditions may extend above the water table as...

between drains under specific aquifer

Aquifer

An aquifer is a wet underground layer of water-bearing permeable rock or unconsolidated materials from which groundwater can be usefully extracted using a water well. The study of water flow in aquifers and the characterization of aquifers is called hydrogeology...

conditions. For this a numerical solution can be used, taking small steps along the impermeable base. The drainage equation

Drainage equation

A drainage equation is an equation describing the relation between depth and spacing of parallel subsurface drains, depth of the watertable, depth and hydraulic conductivity of the soils. It is used in drainage design....

is to be solved by trial and error (iteration

Iteration

Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.-Mathematics:Iteration in...

s), 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.

The trial and error procedure is cumbersome and therefore computer program

Computer program

A computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...

s 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 programComputer program

A computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...

*EnDrain*compares the outcome of the traditional drain spacing equationDrainage equation

A drainage equation is an equation describing the relation between depth and spacing of parallel subsurface drains, depth of the watertable, depth and hydraulic conductivity of the soils. It is used in drainage design....

, based on Darcy's law

Darcy's law

Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...

cum continuity equation

Continuity equation

A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...

(i.e. conservation of mass

Conservation of mass

The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

), 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.

## See also

- HydrogeologyHydrogeologyHydrogeology is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust, . The term geohydrology is often used interchangeably...
- Groundwater discharge
- Groundwater flow equationGroundwater flow equationUsed in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar to that used in heat transfer to describe the flow...
- Drainage equationDrainage equationA drainage equation is an equation describing the relation between depth and spacing of parallel subsurface drains, depth of the watertable, depth and hydraulic conductivity of the soils. It is used in drainage design....

## External links

- Articles on the energy balance of groundwater flow can be downloaded from : http://www.waterlog.info/articles.htm under nr. 3 and 4.