Groundwater Flow in a Soil with a Weakly–Random Hydraulic Conductivity Field
Assistant Professor, Department of Applied Mechanics,
Bengal Engineering and Science University, Shibpur, Howrah, India
TECHNICAL NOTE
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Groundwater Flow in a Soil with a Weakly–Random Hydraulic Conductivity Field
Assistant Professor, Department of Applied Mechanics,
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TECHNICAL NOTE |
SYNOPSIS
The present paper develops a formulation for seepage flow in a soil stratum with a weakly–random hydraulic conductivity field, i.e. with a hydraulic conductivity field that is almost deterministic.
INTRODUCTION
A two-dimensional vertical slice through a soil stratum is taken. The superficial seepage velocity components in this vertical plane are vx and vy, where vx and vy are the velocity components in the horizontal and vertical directions respectively. If the velocity potential is denoted by f, then
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(1) |
satisfies Laplace’s equation.
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(2) |
In a homogeneous aquifer, the hydraulic conductivity, K, is the same at all points in the flow domain. In actual practice, due to non-homogeneity of soil properties like void ratio, mean grain diameter etc., K varies from point to point in a manner that is not deterministic. If for an aquifer, the mean value of K is 
and the standard deviation of K is 
, the hydraulic conductivity field may be considered to be weakly-random if << .
Because it is difficult to solve for the velocity potential analytically in a vast range of situations, numerical methods like the Finite Element Method (Mondal, 2001; Maji, et al., 2002) have been employed to compute the velocity potential. In the above two works, three-noded triangular finite elements with Lagrangian Interpolation have been used. This has been extended to six-noded triangular finite elements with Lagrangian Interpolation by Choudhury (2002). Whatever be the exact nature of the element being utilised, ultimately the matrix equation developed comes out to be of the form
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(3) |
where [G(K)] is the global constitutive matrix, {f(K)} is the matrix of the nodal velocity potentials, and {P(K)} is the equivalent of the load matrix in solid mechanics.
If the hydraulic conductivity is deterministic, a definite pattern of equipotential lines is obtained for a definite flow geometry. If the hydraulic conductivity at different points is not deterministic, the equipotential lines become uncertain. The present work addresses this situation.
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(3) |
or,
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(4) |
Now, differentiating equation (4) with respect to K, one gets
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(5) |
or,
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(6) |
where 
is the sensitivity of f with respect to K.
Equation (6) can be written as
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(7) |
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(8)) |
and
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(9) |
Now, undertaking a Neumann expansion,
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(10) |
where 
is the deterministic component of [G (K)]
and 
is the residual component.
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(11) |
, so,
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(12) |
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(13) |
Therefore, the velocity potential matrix can be written as
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(14) |
where 
is the mean of {f(K)}.
This can be written as
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(15) |
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(16) |
Let now a weakly-random hydraulic conductivity field be considered.
Then
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(17) |
and [H] is small.
It is permissible, then, to write
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(18) |
and
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(19) |
Thus,
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(20) |
and
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(21) |
The stochastic nature of the {vx} and {vy} matrices is introduced through the presence of the matrix [H]. However, the two velocity matrices are almost deterministic as [H] is small.
CONCLUSION
It can be concluded that the presence of a very low degree of randomness in the hydraulic conductivity field of a soil stratum does not appreciably alter the velocity distribution in the aquifer. Localised changes in point velocity do occur but these are unlikely to have a major impact on, for example, the discharge into a pumping well. The situation can change totally if the hydraulic conductivity field becomes strongly random.
REFERENCES
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