Transport in Porous Media

, Volume 18, Issue 1, pp 15–36

# An integral equation formulation for the unconfined flow of groundwater with variable inlet conditions

• Z. -X. Chen
• P. A. Witherspoon
• Y. C. Yortsos
Article

## Abstract

We combine an integral equation formulation with a hodograph transformation to solve self-similar problems describing the unconfined flow of groundwater with variable inlet conditions. A class of new semi-analytical solutions is obtained for both rectilinear and radial flow geometries. The solutions are in general agreement with those derived by Barenblatt, although there are some discrepancies for the case of radial flow. The formulation presented provides additional analytical insight, and for computational purposes is simpler than Barenblatt's. In addition, the method proposed can be successfully used for the solution of a host of other nonlinear problems that admit self-similarity.

## Key words

Analytical solution unconflned flow groundwater

## Nomenclature

a

parameter defined in (3) [L/T]

f

relative total discharge function for the case of constant head

g

gravitational constant [L/T2]

h

H

k

permeability [L2]

K

hydraulic conductivity [L/T]

Q

total discharge per unit width [L2/T]

R

relative total discharge function

r

t

time [T]

v

dimensionless total discharge

V

cumulative volume [L3]

w

x

distance [L]

α

exponent of time variation of inlet head

λ

dimensionless constant defined in (9)

Μ

viscosity [M/TL]

ξ

similarity variable

ρ

density [M/L3]

Φ

porosity

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• Z. -X. Chen
• 1