# Gringarten and Witherspoon Solution for Fractured Aquifers with a Single Vertical Plane Fracture

- Assumptions
- Equations
- Data requirements
- Solution options
- Estimated parameters
- Curve matching tips
- Benchmark
- References

Related Solution Methods

Additional Topics

A mathematical solution by Gringarten and Witherspoon (1972) is useful for determining the hydraulic properties (hydraulic conductivity, specific storage, hydraulic conductivity anisotropy and fracture length) of **fractured** aquifers with a single vertical plane fracture intersecting the pumped well. Analysis involves matching the solution to drawdown data collected during a pumping test. The solution estimates hydraulic conductivity anisotropy in a horizontal (x-y) plane.

AQTESOLV provides two configurations for simulating a vertical fracture using the Gringarten and Witherspoon (1972) solution: **uniform flux** and **infinite conductivity**. Select the appropriate condition when you choose a solution.

You are not restricted to constant-rate tests with the Gringarten and Witherspoon solution. AQTESOLV incorporates the principle of superposition in time to simulate variable-rate and recovery tests with this method.

The early-time response of a pumped well intersecting a vertical fracture has a distinct signature that you can diagnose with a linear flow plot.

Due to anisotropy, use of a distance-drawdown plot with the Gringarten and Witherspoon solution is limited to wells located on the x-coordinate axis.

## Assumptions

- aquifer has infinite areal extent
- aquifer has uniform thickness
- aquifer potentiometric surface is initially horizontal
- pumping and observation wells are fully penetrating
- fractured aquifer represented by anisotropic nonleaky confined system with a single plane vertical fracture that fully penetrates aquifer
- flow is unsteady
- water is released instantaneously from storage with decline of hydraulic head
- diameter of control well is very small so that storage in the well can be neglected

## Equations

The following equation by Gringarten and Witherspoon (1972) predicts drawdown in a nonleaky confined aquifer with a uniform-flux vertical fracture:

$$s=\frac{Q}{4\pi \sqrt{{T}_{x}{T}_{y}}}\frac{\sqrt{\pi}}{2}{\int}_{0}^{{t}_{D}}\left(\mathrm{erf}\frac{1-{x}_{D}}{2\sqrt{\tau}}+\mathrm{erf}\frac{1+{x}_{D}}{2\sqrt{\tau}}\right)\xb7{e}^{-{y}_{D}^{2}{T}_{x}/\left(4\tau {T}_{y}\right)}\frac{d\tau}{\sqrt{\tau}}\text{(1)}$$ $${x}_{D}=x/{x}_{f}\text{(2)}$$ $${y}_{D}=y/{x}_{f}\text{(3)}$$ $${t}_{D}=\frac{{T}_{x}t}{S{x}_{f}^{2}}\text{(4)}$$ $${s}_{D}=\frac{4\pi \sqrt{{T}_{x}{T}_{y}}}{Q}s\text{(5)}$$where

- $\mathrm{erf}\left(\right)$ is the error function
- $Q$ is pumping rate [L³/T]
- $s$ is drawdown [L]
- $S$ is storativity ($={S}_{s}b$) [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- ${T}_{x}$ is transmissivity in x direction ($={K}_{x}b$) [L²/T]
- ${T}_{y}$ is transmissivity in y direction ($={K}_{y}b$) [L²/T]
- $\tau $ is a variable of integration
- $x$ is distance in x direction [L]
- ${x}_{f}$ is half-length of fracture in x direction [L]
- $y$ is distance in y direction [L]

### Notes

- Equations (1) through (5) assume a uniform-flux condition along the fracture.
- Gringarten, Ramey and Raghavan (1974) found that equations (1) through (5) could be used to predict drawdown in an
**infinite-conductivity fracture**by simply using ${x}_{D}$ = 0.732 to compute drawdown in the pumped well.

## Data Requirements

- pumping and observation well locations
- pumping rate(s)
- observation well measurements (time and displacement)
- saturated thickness
- length of fracture

## Solution Options

- variable pumping rates
- multiple pumping wells
- multiple observation wells
- boundaries

## Estimated Parameters

- ${K}_{x}$ (hydraulic conductivity in x direction)
- ${S}_{s}$ (specific storage)
- ${K}_{y}/{K}_{x}$ (hydraulic conductivity anisotropy ratio)
- ${L}_{f}$ (length of fracture)

## Curve Matching Tips

- Use linear flow plots to help diagnose linear flow.
- Match the Cooper and Jacob (1946) solution to late-time data to obtain preliminary estimates of aquifer properties.
- Choose Match>Visual to perform visual curve matching using the procedure for type curve solutions.
- Use active type curves for more effective visual matching with variable-rate pumping tests.
- Select values of ${K}_{y}/{K}_{x}$ from the
**Family**and**Curve**drop-down lists on the toolbar. - Use parameter tweaking to perform visual curve matching and sensitivity analysis.
- Perform visual curve matching prior to automatic estimation to obtain reasonable starting values for the aquifer properties.

## Benchmark

## References

Gringarten, A.C. and P.A. Witherspoon, 1972. A method of analyzing pump test data from fractured aquifers, Int. Soc. Rock Mechanics and Int. Assoc. Eng. Geol., Proc. Symp. Rock Mechanics, Stuttgart, vol. 3-B, pp. 1-9.