# Thiem Equation for Steady-State Flow in a Confined Aquifer (2 monitoring wells)

Application: Geotechnics

## Summary

### Notes on Thiem Equation for Steady-State Flow in a Confined Aquifer (2 monitoring wells)

- Select the required output, then set the other parameters to the required values.
- Transmissivity (T) = K * D

For equations, assumptions and references see equations below.

For determination of steady state conditions see procedures below.

### Tool Parameters

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### Thiem Equation for Steady-State Flow in a Confined Aquifer^{1}

^{1}Thiem, G., Hydrologische Methoden; Gebhardt, Leipzig, 1906.

### Two piezometers:

where:

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

*KD* = transmissivity of aquifer (m^{2}/d)

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

*s*= respective steady-state drawdowns in the piezometers (m)

_{1}, s_{2}

### Assumptions^{2}:

- The aquifer is confined.
- 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 is screened over the entire thickness of the aquifer ensuring entirely horizontal flow.
- The hydraulic gradient between the pumping well and monitoring wells is at steady-state.
- The water removed from storage is discharged instantaneously with decline of head.
- Storage in the well can be neglected (the diameter of the well is small).

### References:

^{1}Thiem, G., Hydrologische Methoden; Gebhardt, Leipzig, 1906.

^{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*s*_{1,2}into the tool*or*

Determine the difference of drawdown (Ds) per log cycle of r (note log (*r*_{1}/*r*_{2}) = 1) and substitute into a simplified version of the Thiem equation:

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

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