# Dupuit Equation for Steady-State Flow in an Unconfined Aquifer (2 monitoring wells)

Application: Geotechnics

## Summary

### Notes on Dupuit Equation for Steady-State, Radial Flow in an Unconfined Aquifer (2 monitoring wells)

- Select the required output, then set the other parameters to the required values.
- Caution should be used when interpreting these results as the Dupuit formula fails to give an accurate description of the drawdown curve near the pumping well, where the strong curvature of the watertable contradicts the Dupuit assumption of horizontal flow.

For equations, assumptions and references see equations below.

For determination of steady state conditions see procedures below.

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### Dupuit Equation for Steady-State Flow to a Well in an Unconfined Aquifer^{1}

^{1}Dupuit, J., Etudes theoriques et pratiques sur le mouvement des eaux dans les canaux decouverts et a travers les terrains permeables, 2eme edition; Dunot, Paris, 1863.

### Two piezometers:

where:

*Q* = well discharge rate (m^{3}/d)

*K* = hydraulic conductivity of aquifer (m/d)

*r _{1}, r_{2}* = respective distances of the piezometers from the pumping well (m)

*h*= respective steady-state heads in the piezometers (m)

_{1}, h_{2}- The aquifer is unconfined.
- The aquifer has an infinite areal extent.
- The aquifer is homogeneous, isotropic and of uniform thickness over the area influenced by the test.
- Prior to pumping, the piezometric surface is horizontal over the area that will be influenced by the test.
- The aquifer is pumped at a constant discharge rate.
- The well penetrates the entire saturated thickness of the aquifer.
- The gradient between the pumping well and monitoring wells is at steady-state.
- The velocity of flow is proportional to the tangent of the hydraulic gradient instead of the sine as it is in reality.
^{1} - The flow is horizontal and uniform everywhere in a vertical section through the axis of the well.
^{1}

### References:

^{1}Dupuit, J., Etudes theoriques et pratiques sur le mouvement des eaux dans les canaux decouverts et a travers les terrains permeables, 2eme edition; Dunot, Paris, 1863.

^{2}Kruseman, G.P. and N.A. de Ridder, Analysis and Evaluation of Pumping Test Data (Second Edition), Publication 47; International Institute for Land Reclamation and Improvement, Wageningen, 1994.

### Procedure to Determine if Flow to a Pumping Well is Steady-State^{1}

^{1}Kruseman, G.P. and N.A. de Ridder, Analysis and Evaluation of Pumping Test Data (Second Edition), Publication 47; International Institute for Land Reclamation and Improvement, Wageningen, 1994.

In practice, flow to a pumping well is considered steady-state when the difference in drawdown between piezometers becomes negligible, i.e., when the *gradient* between the piezometers becomes constant. The following procedures can be used to determine whether flow to a well is steady-state.

### Procedure 1:

- Create a semi-log plot of drawdown (s) vs. time (t) for all piezometers (Fig. 1). Drawdown is plotted on the vertical, linear axis, while time is plotted on the horizontal, logarithmic axis.
- Draw a best-fit curve through the data for each piezometer. The curves for the piezometers may become parallel for the data collected late in the pump test. If this is the case, then the gradients are constant and the system has reached steady-state.
- Where two curves are parallel, read the steady-state drawdown (s) for each piezometer at the end of the test and substitute these values with the corresponding radii into the Thiem equation or tool.
- Repeat 3 for any combination of piezometers with parallel time-drawdown curves. If the aquifer is truly homogeneous, then the different combinations of piezometers should yield a similar transmissivity (K x D). Different values of transmissivity suggest that the aquifer is heterogeneous, in which case, the mean of the values is used as a final result.

Figure 1: Sample time-drawdown graph produced in Procedure 1. Note that the gradient between piezometer H215 and the other piezometers is not constant (curves are not parallel) and therefore this piezometer would not be used to calculated transmissivity by the Thiem method.

### Procedure 2:

- Create a semi-log plot of observed drawdown (s) for each piezometer vs. the distance of the piezometer from the well (Fig. 2).
- Draw a best-fit line through the plotted points.
- Pick two colinear points (not necessarily observed data points) and substitute values of
*r*_{1,2}and*h*_{1,2}into the tool (h = saturated thickness - drawdown).

Figure 2: Sample distance-drawdown graph used in Procedure 2.

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