Ground Water Flow Equation

When the fluid mass flows through various porous medium it will follow the physical property and nature of media. With the combination of Darcy’s law and equation of continuity, we can describe the conservation of fluid mass during flow through porous medium

CONFINED AQUIFER

Steady State Saturated Flow

Here the flow is taking place in all the three directions. Hence the rate of inflow any elemental control volume is equal to the rate of out flow elementary control volume. Hence for the steady state flow through isotropic and homogeneous medium is given by following equation

This is also called as Laplace’s equation

Transient Saturated Flow

The main principle for the flow equation is the rate of flow into any elemental control volume is equal to the time rate of change of fluid mass storage within the element



In special case of horizontal confined aquifer of thickness b, S = Ss b and T = Kb, then the above equation simplified as



UNCONFINED AQUIFER

Transient flow in unconfined aquifer

Here the flow distributions govern by the water table shape. To find a solution Dupit’s given two assumptions

Flow line are horizontal, equi potential lines are vertical
The horizontal k is equal to the slope of the free surface and is invariant with depth



Solution for Ground Water Flow Equation

To solve the flow equation either analytical or numerical methods are used. In analytical methods the actual filed conditions are so complex, it becomes to obtain solution, whereas the numerical solutions are much more versatile and with widespread availability of computer, they are much easier to use than complex analytical methods

In general numerical methods, such as the finite element method (FEM) and Integrate finite difference method (IFDM) are most commonly used.

While solving the groundwater flow equation is more real time problem, boundary condition have to be considered. Basically three-boundary condition, such as variable head boundary, constant head and no flow boundary exists.


Application

Ground water is used to understand the quantities of flow and its direction. The main application is flow net analysis, water balance study, and Ground water flow modeling planning of ground water management strategies.

ELASTICITY AND COMPRESSIBILITY OF FORMATION

Rocks possessing void space are susceptible to volume change in response to external force (load) acting on it. The compressibility of the rock is expressed in term of bulk modulus of elasticity.


E - Bulk modules kg/m2
r - Mass in metric slugs/cm2
p - Pressure in kg/m2
V - Volume in m3

The reciprocal of bulk modulus of elasticity is called compressibility


It always expressed in m2/kg. In general b is individual grain in negligible and granular rocks with high porosity are more compressible than dense rocks

Relation between b and storage coefficient

In confined aquifer, change in the pressure head reflects changes in the pressure exerted on the aquiclude and the resulting elastic change in the aquifer system. The force at contact of confining layer artesian aquifer may be expressed as

Pa = Pw + ps

Pa - Total load exerted on a unit area of aquifer
Pw - Part of total load born by the confined water
ps - Part of total load born by the structural skeleton of the aquifer

During pumping in a confined aquifer water is discharged due toe the expansion of water and compression of rock formation. Conversely when pumping stopped, the pressure head builds up gradually due to the transfer of land water itself undergoes slight contraction.

Effect of Elasticity of Confined Aquifer on Water Level in Wells

In confined aquifer, changes in external load on them is reflected as variation in water levels in the wells, which are tapping them


Atmospheric Pressure

In confined aquifer, when the increase in the atmospheric pressure produces decreases in the water level and conversely. This is expressed as Barometric efficiency, which is



The barometric efficiency is interpreted as a measure of the competence of overlying confining bed to resist pressure changes.

Tides

The responses to the tides are recorded in sinusoidal fluctuations. Contrary to the barometric effect, in the case of ocean tides, as the sea level rises, ground water also increases. Tidal efficiency C is related it the barometric efficiency B by

C = 1 – B

Earthquakes

Earthquakes shocks produce small fluctuations, hydroseisms, in well penetrating confined aquifer. The passage of seismic wave through the confined layer of aquifer resulting compression and expansion of layer. Hence fluctuations appear after little more than one hour even from the most distant earthquake centers.

External loads

Changes in loading results in Change in hydrostatic pressure in confined aquifer because of its elastic property. For example in wells located near railway lines, passing train produce measurable fluctuations of the piezometric surface.