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Ground water HydraulicsGround water Hydraulics

(Basic equations (Basic equations ii))

Derivation of groundwater flowDerivation of groundwater flow--

RevisionRevision

Ground water occursGround water occurs1.1. Confined Confined 2.2. unconfined unconfined 3.3. leaky conditionsleaky conditions4.4. Isotropic and Isotropic and

homogeneous homogeneous 5.5. Non isotropic Non isotropic

homogeneoushomogeneous6.6. Isotropic and Isotropic and

heterogeneousheterogeneous7.7. Non Isotropic and Non Isotropic and

heterogeneousheterogeneous8.8. Steady state Steady state 9.9. Non steady stateNon steady state10.10.Fully penetrated aquifers Fully penetrated aquifers

(wells)(wells)11.11.Partially penetrated Partially penetrated

aquifers (wells)aquifers (wells)

Conditions for steady Conditions for steady statestate--Theim equationTheim equation

�� Homogeneous and Homogeneous and isotropic aquiferisotropic aquifer

�� Confined Confined

�� Non leaky conditionsNon leaky conditions

�� Fully penetratedFully penetrated

�� Steady state Steady state conditionsconditions

Theim equation for Confined Theim equation for Confined --steady state conditionssteady state conditions Theim equation for Confined Theim equation for Confined --steady state conditionssteady state conditions

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K= q ln (r1/rw)2 Л b (sw-s1)

K= q ln (re/rw)2 Л b sw

If only one observation well is

If the data is only available for pumped well

Rw is the radius of the wellRe=Outer radius of influenceSw= drawdown of the well

Theim equation for Confined Theim equation for Confined --steady state conditionssteady state conditionsUnconfined aquiferUnconfined aquifer--Steady State Theim equationSteady State Theim equation

Unconfined aquiferUnconfined aquifer--Steady StateSteady State--Theim Theim

equationequationq=2 Л r kh dh/dr

dr/r= (2 Л k/q) h dh

Integrating the above equation between limits r1 and r2 and h1 and h2

2 Л k/q) hdh

= ln (r2/r1)= (2Лk/2q)(h2^2-h1^2)

= ln (r2/r1) = (Лk/q)(h2^2-h1^2)

q= (Лk)(h2^2-h1^2)ln (r2/r1)

Unconfined aquiferUnconfined aquifer--Steady StateSteady State-- Theim equationTheim equation

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Three dimensional flow Three dimensional flow Continuity and the Laplace equation for Steady State flow Continuity and the Laplace equation for Steady State flow

If a unit cube is drawn with sides parallel to the x, y, and z axes, conservation of mass requires that any flux into the cube must equal the flux out of the cube. This results in the following:

Note that the divergence operator (the second partial derivatives with respects to the independent variables x, y and z) operates on the directional permeability terms as well as the gradient, both of which are functions of the independent variables. This mathematical difficulty can be relaxed with the following condition:

Laplace equationLaplace equation

�� If The medium is homogeneous and Isotropic If The medium is homogeneous and Isotropic then there is no directional variation in K and then there is no directional variation in K and so it is constant . This implies that Kxx = Kyy so it is constant . This implies that Kxx = Kyy = Kzz = K.= Kzz = K.

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ConfinedConfined--unsteady stateunsteady state-- Three dimensional equationThree dimensional equation

Continuity requires that a flux into the volume be balanced by either a flux out of the volume or a time dependent change in hydraulic head dependent upon the storage coefficient. This results in the diffusion equation:

For Homogeneous and Isotropic aquifers in two dimensions in confined

Radial form for two dimensional equationCartesian form of two dim

For unconfined

Aquifer Parameters

Unconfined Confined

Steady Unsteady Non leaky Leaky

Steady Unsteady Steady UnsteadyTheim Boulton

TheimTheis

match pointCooper &Jacob

Hantushmatch point

Hantush st.line

Hantushmatch point

Hantush Inflection

Aquifer parameters TestsAquifer parameters Tests-- Pumping Pumping

teststests