# Barker Solution for Nonleaky Confined Aquifers

A mathematical solution by Barker (1988) known as the **generalized radial flow (GRF) model** is useful for determining the hydraulic properties of single- and double-porosity aquifers. Analysis involves matching the Barker solution to drawdown data collected during a pumping test.

The **GRF model** may be used to simulate unsteady, n-dimensional flow to a fully penetrating source in an isotropic nonleaky confined aquifer. The source is an n-dimensional sphere (projected through three-dimensional space) of finite radius (${r}_{w}$), storage capacity ($\beta $) and skin factor (${S}_{w}$).

You are not restricted to constant-rate tests with the Barker solution. Using the principle of superposition in time, AQTESOLV can simulate variable-rate and recovery tests with this method.

## Assumptions

- aquifer has infinite extent
- aquifer is homogeneous, isotropic and of uniform extent of flow region
- potentiometric surface is initially horizontal
- aquifer is confined
- flow is unsteady
- wells are fully penetrating
- water is released instantaneously from storage with decline of hydraulic head

## Equations

Barker (1988) derived a generalized radial flow model for unsteady, $n$-dimensional flow to a fully penetrating source in an isotropic nonleaky confined aquifer. The spatial dimension, $n$, determines the change in conduit area with distance from the source (Doe 1990). In a two-dimensional system ($n$=2), the source is a finite cylinder, the typical configuration for analyzing cylindrical flow to a well.

The Laplace transform solution for drawdown in the pumped well (source) is as follows:

$${\overline{h}}_{w}=\frac{Q\left[1+{S}_{w}{\Phi}_{\nu}\left(x\right)\right]}{p\left[p\beta \left[1+{S}_{w}{\Phi}_{\nu}\left(x\right)\right]+K{b}^{3-n}{\alpha}_{n}{r}_{w}^{n-2}{\Phi}_{\nu}\left(x\right)\right]}\phantom{\rule{1em}{0ex}}\text{(1)}$$The following equation is the Laplace transform solution for drawdown in an observation well:

$$\overline{h}=\frac{Q{r}_{D}^{\nu}{K}_{\nu}\left(x{r}_{D}\right)/{K}_{\nu}\left(x\right)}{p\left[p\beta \left[1+{S}_{w}{\Phi}_{\nu}\left(x\right)\right]+K{b}^{3-n}{\alpha}_{n}{r}_{w}^{n-2}{\Phi}_{\nu}\left(x\right)\right]}\phantom{\rule{1em}{0ex}}\text{(2)}$$ $$\nu =1-n/2\phantom{\rule{1em}{0ex}}\text{(3)}$$ $${\lambda}^{2}=p{S}_{s}/K\phantom{\rule{1em}{0ex}}\text{(4)}$$ $$\mu =\lambda {r}_{w}\phantom{\rule{1em}{0ex}}\text{(5)}$$ $${r}_{D}=r/{r}_{w}\phantom{\rule{1em}{0ex}}\text{(6)}$$ $$\Phi \left(z\right)=z{K}_{\nu -1}\left(z\right)/{K}_{\nu}\left(z\right)\phantom{\rule{1em}{0ex}}\text{(7)}$$ $${\alpha}_{n}=2{\pi}^{n/2}/\Gamma \left(n/2\right)\phantom{\rule{1em}{0ex}}\text{(8)}$$ $$\beta =\pi {r}_{c}^{2}\phantom{\rule{1em}{0ex}}\text{(9)}$$ $$x=\mu \phantom{\rule{1em}{0ex}}\text{(10)}$$where

- $b$ is extent of flow region [L]
- $h$ is hydraulic head at time $t$ [L]
- $K$ is hydraulic conductivity [L/T]
- ${K}_{\nu}$ is modified Bessel function of second kind, order $\nu $
- $n$ is flow dimension [dimensionless]
- $p$ is the Laplace transform variable
- $Q$ is pumping rate [L³/T]
- $r$ is radial distance from pumping well to observation well [L]
- ${r}_{c}$ is casing radius [L]
- ${r}_{w}$ is well radius [L]
- ${S}_{s}$ is specific storage [dimensionless]
- ${S}_{w}$ is wellbore skin factor [dimensionless]
- $t$ is elapsed time since start of pumping [T]

The parameter $b$, the flow region extent, has a simple interpretation for integral flow dimensions.

- For $n$=1 (one-dimensional flow), $b$ is the square root of the conduit flow area (normal to the flow direction).
- For $n$=2 (two-dimensional radial flow), $b$ is the thickness of the aquifer.
- For $n$=3 (spherical flow), the parameter $b$, which is raised to the power of 3-$n$, has no significance.

For nonintegral flow dimensions, $b$ has no simple interpretation (Barker 1988).

In the Laplace transform solution, ${S}_{w}$ is limited to positive values; however, using the effective well radius concept, we also may simulate a negative skin (Hurst, Clark and Brauer 1969).

If you enter a radius for downhole equipment, AQTESOLV uses the effective casing radius instead of the nominal casing radius in the equations for this solution.

## Data Requirements

- pumping and observation well locations
- pumping rate(s)
- observation well measurements (time and displacement)
- casing radius and wellbore radius for pumping well(s)
- downhole equipment radius (optional)
- extent of flow region

## Solution Options

- constant or variable pumping rate with recovery
- multiple pumping wells
- multiple observation wells
- boundaries

## Estimated Parameters

AQTESOLV provides visual and automatic methods for matching the Barker method to data from pumping tests and recovery tests. The estimated aquifer properties are as follows:

- $K$ (hydraulic conductivity)
- ${S}_{s}$ (specific storage)
- $n$ (flow dimension)
- $b$ (flow region extent)
- ${S}_{w}$ (dimensionless wellbore skin factor)
- $\mathrm{r(w)}$ (well radius)
- $\mathrm{r(c)}$ (nominal casing radius)

## Curve Matching Tips

- Use radial flow plots to help diagnose wellbore storage.
- 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.
- Match early-time data affected by wellbore storage by adjusting $\mathrm{r(c)}$ with parameter tweaking.
- If you estimate $\mathrm{r(c)}$ for the test well, the estimated value replaces the nominal casing radius and AQTESOLV still performs the correction for downhole equipment.
- 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 $n$ and ${S}_{w}$ 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.

## References

Barker, J.A., 1988. A generalized radial flow model for hydraulic tests in fractured rock, Water Resources Research, vol. 24, no. 10, pp. 1796-1804.

Moench, A.F., 1984. Double-porosity models for a fissured groundwater reservoir with fracture skin, Water Resources Research, vol. 20, no. 7, pp. 831-846.