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§2.3 Basic Equation of Groundwater Flow
2.3.1 Basic Equation for Steady Incompressible Flow
The groundwater flow will be in steady state when there is no change of mass storage.In this case,the mass balance states that:
Total mass in=total mass out
Let's consider time derivative=0(Steady State System),the equation of continuity become:
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Furthermore,if the fluid is incompressible,the density ρ is a constant and equation(2.22)reduces to:
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Equation(2.23)is the equation of continuity for steady incompressible groundwater flow.
Substitution of Darcy's law,equation(2.12)and equation(2.23)gives:
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This is the basic differential equation of steady incompressible flow in homogeneous isotropic porous medium.It is noted that equation(2.24)is the standard Laplace equation and is often written in abbreviation form:
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Substitution of Darcy's law,equation(2.16)into equation(2.23)will give the differential equation of steady incompressible flow in anisotropic porous medium,which is:
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When the porous medium is anisotropic but homogeneous,equation(2.26)reduces to:
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2.3.2 Basic Equation for Non-steady Compressible Flow
In the case of non-steady flow,the storage of mass is changed since the total mass in is not equal to the total mass out.
From the principles of Soil Mechanics,it is known that the total pressure(σz)from overlying geological strata is balanced by the effective pressure(σ′z)of grains and pore pressure(p)of water(Figure 2.7).i.e..
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Since the total pressure can be assumed to be independent of time,any increase of pore pressure will result in a decrease of effective pressure,the consequences of such a change are:
(1)Water is compressed due to the increase of pore pressure.Therefore,the density of water is a function of time.
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Figure 2.7 Principle of effective stress
(2)Grain skeleton is expanded due to the decrease of effective pressure.Hence,the porosity and size of grain skeleton are functions of time.
Taking the above effects into consideration,the change of mass storage in equation(2.20)can be expressed as:
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1.Compressibility of water
From the elastic theory,the relative change of density of water is proportional to the change of pore pressure,i.e..
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Where:βw is the coefficient of compressibility of water.Its value is about 0.5×10-9 m2/N.
From equation(2.30)it follows that:
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2.Compressibility of soil
From the theory of elasticity,soil will be compressed under the pressure.When sample is confined horizontally,the vertical relative compression is proportional to the increase of pressure,i.e..
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Where:is vertical pressure on sol with area(ΔxΔy)and height Δz;α is coefficient of compressibility of soil,common values are 10-8~10-7m2/N for sand,and 10-7~10-6 m2/N for clay.
In many practical cases usually the horizontal deformations of the soil skeleton is much smaller than the vertical deformation.Therefore,it can be assumed that Δx,Δy are constant and only Δz changes.Under this assumption,
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Sinceis assumed to be independent of time,i.e.
,so that:
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Substitution of equation(2.34)into equation(2.33)yields:
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3.Change of porosity
Although the size ΔxΔyΔz changes with compression,it is assumed that only pore space is changed,but the total value of grains(Vs)is kept constant since grains can be considered completely incompressible.The assumption of Vs=(1-n)ΔxΔyΔz constant gives:
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From equation(2.36)with equation(2.35),the relation between changes of porosity with change pore pressure is found as:
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4.Equation of continuity of non-steady compressible flow
Substitution of equations(2.31),(2.35)and(2.37)into equation(2.29)gives:
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Substitution of equation(2.38)into equation(2.21)gives the general equation of continuity for non-steady compressible flow as:
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5.Basic equation of non-steady compressible flow
In case of compressible flow in homogeneous isotropic porous medium,Darcy's law of equation(2.12)will be replaced by:
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Substitution of equation(2.40)into equation(2.39)results in:
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If the vertical gradient of fluid density can be neglected(for example in one fluid flow),equation(2.41)reduces to:
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This is a basic equation of non-steady compressible flow in homogeneous isotropic medium.
Using the equation ofan equation in terms of head can be found as:
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Where:Ss=ρg+(α+nβw)is usually called the storativity or specific storage coefficient.It is defined as the volume of water stored in a unit volume of soil by a unit increase of the head.It has a dimension of[L-1].The physical explanation of the storativity is that with increase of head more water is stored in pore space since water is compressed;and pore space is enlarged since the grain skeleton is expanded.Since the expansion and compression of water and soil skeleton is elastic,Ss is called elastic storage.