# Dupuit Equation for Steady-State Flow in an Unconfined Aquifer (1 monitoring well)

Application: Geotechnics

## Summary

### Notes on Dupuit Equation for Steady-State, Radial Flow in an Unconfined Aquifer (1 monitoring well)

- 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.

### Tool Parameters

Values displayed with a may be changed ... click on a value to display an entry form ... or use the arrows right of a value to select from a list.

### Tool Behaviour

This tool exhibits spreadsheet-like behaviour ... change a value and related values are re-calculated automatically.

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

*sup>1Dupuit, 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.*

### One piezometer:

where:

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

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

*r _{w}* = radius of the pumping well (m)

*r*= distance from piezometer to the pumping well (m)

_{1}h

_{w}= steady-state head in the pumping well (m)

h

_{1}= steady-state head in the piezometer (m)

### Assumptions^{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.

- Responsible Tailings Management 101
- Empirical Design Methods For Underground Mines
- Rock Mass Characterization for Mine Design
- Fundamentals of Mine Water Management
- Open Pit Slope Design and Implementation
- Advanced Tailings and Mine Waste Facility Design, Construction, Operation, and Closure
- Risk Assessment and Decision Making for Mine Geowaste Facility Management
- Advanced Mine Water Management and Treatment
- Groundwater Modelling for Mining
- Geosynthetics for Geowaste Facilities
- Geosynthetics in Mining - Practice and Developments
- Open Pit Slope Design
- Tailings Management 101
- Bench Face Design in Rock
- Covers for Mine Geowaste Facilities - 1: Principles, Practice, and Selection
- Covers for Mine Geowaste Facilities - 2: Design and Performance Analysis
- Design for Underground Metal Mines 1 - Design Parameters
- Design for Underground Metal Mines 2 - Design Guidelines
- Risk Assessment, Decision-Making, and Engineering Management for Mine Geowaste Facilities
- Geosynthetics in Mining
- Geotechnical Engineering for Mine Geowaste Facilities
- Groundwater in Mining
- Groundwater Modelling for Mining 1 - Model Conceptualization
- Groundwater Modelling for Mining 2 - Numerical Modelling
- Geotechnical Data 1 - Rock Fabric and Structures
- Geotechnical Data 2 - Rock Material Properties
- Rock Mass Classification for Mine Design
- Introduction to Groundwater Modeling for Mines and Mining
- Guidelines for Open Pit Slope Design 1 - Fundamentals and Data Collection
- Guidelines for Open Pit Slope Design 2 - Modelling
- Guidelines for Open Pit Slope Design 3 - Design
- Guidelines for Open Pit Slope Design 4 - Operation
- A Rock Engineering Primer for Non-Engineers in Mining
- Design and Operation of Large Waste Dumps
- Practical Rock Engineering 1 - Introduction
- Practical Rock Engineering 2 - Structural Analysis
- Practical Rock Engineering 3 - Slope Stability and Rockfalls
- Practical Rock Engineering 4 - Stress Analysis
- Practical Rock Engineering 5 - Excavation and Support
- Reinforcement Design for Excavation in Rock
- Stereographic Projections
- Surface Water Management at Mines
- Tailings Facility Design, Operation, and Closure
- Tailings Management
- Mine Water and Chemical Balance Analysis
- Empirical Design Methods For Underground Mines
- Groundwater Modelling for Mining
- Open Pit Slope Design
- Rock Mass Characterization for Mine Design
- Rock Mass Classification for Mine Design
- Advanced Tailings and Mine Waste Facility Design, Construction, Operation, and Closure
- Advanced Topics in Mine Water Management
- Mine Water Management: Surface and Groundwater
- Stereographic Projections
- Stereographic Projections
- Tailings Management 101

- Unit Conversions for the Geosciences
- Darcy Equation for Flow in a Confined Aquifer
- Dictionary of Mining, Mineral and Related Terms
- Dupuit Equation for Steady-State Flow to a Trench in an Unconfined Aquifer (No Recharge)
- Dupuit Equation for Steady-State Flow to a Trench in an Unconfined Aquifer (With Recharge)
- Jacob Method for Transient Flow in a Confined Aquifer
- Periodic Table with Minerals, Isotopes and Water Quality
- Spatial Analysis - Area of an Irregular Polygon
- Principle Stress Calculations
- Rock Mass Rating (RMR) Calculations
- Rock Mass Rating (RMR) Design Tables
- Rock Tunneling Quality Index Design Tables
- Rock Property Tables for Specific Gravity, Density and Porosity
- Geometric Properties of Circle Components
- Geometric Properties of Parabolic Components
- Tool for Slurry Conversions
- Phase Relationships for Soils
- Thiem Equation for Steady-State Flow in a Confined Aquifer (2 monitoring wells)
- Thiem Equation for Steady-State Flow in a Confined Aquifer (1 monitoring well)
- Dupuit Equation for Steady-State Flow in an Unconfined Aquifer (2 monitoring wells)
- Dupuit Equation for Steady-State Flow in an Unconfined Aquifer (1 monitoring well)