# Gringarten and Ramey Solution for Fractured Aquifers with a Single Horizontal 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 Ramey (1974) is useful for determining the hydraulic properties (hydraulic conductivity, specific storage, hydraulic conductivity anisotropy and fracture radius) of **fractured** aquifers with a single horizontal plane fracture intersecting the control well. Analysis involves matching the solution to drawdown data collected during a pumping test. The solution estimates hydraulic conductivity anisotropy in a vertical (x-z) plane.

You are not restricted to constant-rate tests with the Gringarten and Ramey 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 horizontal fracture has a distinct signature that you can diagnose with a linear flow plot.

## Assumptions

- aquifer has infinite areal extent
- aquifer has uniform thickness
- aquifer potentiometric surface is initially horizontal
- fractured aquifer represented by anisotropic nonleaky confined system with a single plane horizontal fracture
- 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 Ramey (1974) predicts drawdown at a fixed point (piezometer) in an anisotropic confined aquifer with a uniform-flux horizontal fracture:

$$s=\frac{Q}{4\pi \sqrt{{K}_{r}{K}_{z}}}\frac{2}{{H}_{D}}{\int}_{0}^{{t}_{D}}P\xb7Z\frac{d\tau}{\tau}\phantom{\rule{1em}{0ex}}\text{(1)}$$ $$P={e}^{-\frac{{r}_{D}^{2}}{4\tau}}{\int}_{0}^{1}{I}_{0}\left(\frac{{r}_{D}\upsilon}{2\tau}\right){e}^{-\frac{{\upsilon}^{2}}{4\tau}}\upsilon d\upsilon \phantom{\rule{1em}{0ex}}\text{(2)}$$ $$Z=1+2\sum _{n=1}^{\infty}{e}^{-\frac{{n}^{2}{\Pi}^{2}\tau}{{H}_{D}^{2}}}\mathrm{cos}\left(n\pi \frac{{z}_{f}}{b}\right)\mathrm{cos}\left(n\pi \frac{z}{b}\right)\phantom{\rule{1em}{0ex}}\text{(3)}$$ $${H}_{D}=\frac{b}{{r}_{f}}\sqrt{\frac{{K}_{z}}{{K}_{r}}}\phantom{\rule{1em}{0ex}}\text{(4)}$$ $${r}_{D}=r/{r}_{f}\phantom{\rule{1em}{0ex}}\text{(5)}$$ $${t}_{D}=\frac{{K}_{r}t}{{S}_{s}{r}_{f}^{2}}\phantom{\rule{1em}{0ex}}\text{(6)}$$where

- $b$ is aquifer saturated thickness [L]
- ${I}_{0}$ is modified Bessel function of first kind, zero order
- ${K}_{r}$ is radial hydraulic conductivity [L/T]
- ${K}_{z}$ is vertical hydraulic conductivity [L/T]
- $Q$ is pumping rate [L³/T]
- $r$ is radial distance from pumping well to observation well [L]
- ${r}_{f}$ is radius of horizontal fracture [L]
- $s$ is drawdown [L]
- ${S}_{s}$ is specific storage [L
^{-1}] - $t$ is elapsed time since start of pumping [T]
- $\tau $ and $\upsilon $ are variables of integration
- $z$ is distance in z direction [L]

## Data Requirements

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

## Solution Options

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

## Estimated Parameters

- ${K}_{r}$ (radial hydraulic conductivity)
- ${S}_{s}$ (specific storage)
- ${K}_{z}/{K}_{r}$ (hydraulic conductivity anisotropy ratio)
- ${R}_{f}$ (radius 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 ${K}_{z}/{K}_{r}$ 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 H.J. Ramey, 1974. Unsteady state pressure distributions created by a well with a single horizontal fracture, partial penetration or restricted entry, SPE Journal, pp. 413-426.