Underground Flow of Water A company has been disposing of some water containing a small amount of a chemical (1,4–dioxane) by injecting it into a permeable stratum of the ground via a vertical well, A, of radius a. Consequently, the groundwater has become slightly contaminated. As shown in Fig. E7.

Question

Underground Flow of Water

A company has been disposing of some water containing a small amount of a chemical (1,4-dioxane) by injecting it into a permeable stratum of the ground via a vertical well, A, of radius [latex]a[/latex]. Consequently, the groundwater has become slightly contaminated. As shown in Fig. E7.5.1(a), the company proposes to rectify the situation in a cleanup operation by drilling a second well, B, of the same radius, and separated by a distance [latex]L[/latex] from A. Contaminated water will be withdrawn at a volumetric flow rate [latex]Q[/latex] per unit depth from A, detoxified by irradiating it with ultraviolet light, and injecting the purified water back into the ground via well B. By modeling A as a line sink and B as a line source, derive an expression for the pressure difference [latex]p_{\mathrm{B}}-p_{\mathrm{A}}[/latex] between the two wells, in terms of [latex]Q, a, L, \kappa[/latex], and [latex]\mu[/latex]. Sketch several streamlines and isobars.

Accepted Answer

For incompressible liquid flow in a porous medium, the superficial velocity vector (volumetric flow rate per unit area) [latex]\mathbf{v}[/latex] is given by Darcy's law:

[latex]\mathbf{v}=-\frac{\kappa}{\mu} abla p.[/latex] (E7.5.1)

Laplace's equation governs variations of pressure, which may be treated in virtually the same manner as the velocity potential in irrotational flow.

A plan of the permeable stratum is shown in Fig. E7.5.1(b), in which a general point such as [latex]\mathrm{P}[/latex] is seen to lie at radial distances [latex]r_{A}[/latex] and [latex]r_{B}[/latex] from the centers of wells [latex]\mathrm{A}[/latex] and [latex]\mathrm{B}[/latex], respectively.

If [latex]\mathrm{B}[/latex] is considered to be a line source, with a total flow rate of [latex]Q[/latex] per unit vertical depth (normal to the plane of Fig. E7.5.1(b)), the radially outward velocity [latex]v_{r B}[/latex] at any distance [latex]r_{B}[/latex] from it is obtained from continuity and Darcy's law:

[latex]v_{r B}=\frac{Q}{2 \pi r_{B}}=-\frac{\kappa}{\mu} \frac{\partial p_{B}}{\partial r_{B}}.[/latex] (E7.5.2)

The corresponding pressure is obtained by integration:

[latex]p_{B}=-\frac{\mu Q}{2 \pi \kappa} \ln r_{B}+f(z).[/latex] (E7.5.3)

For flow into the line sink at well A, the corresponding pressure is:

[latex]p_{A}=\frac{\mu Q}{2 \pi \kappa} \ln r_{A}+f(z).[/latex] (E7.5.4)

In these last two equations, the function of integration [latex]f(z)[/latex] recognizes that the pressure also varies with the vertical distance [latex]z[/latex]. Considering a fixed depth, the function effectively becomes a constant of integration, [latex]p_{0}[/latex].

The combined effect of both injection and withdrawal wells leads to the pressure distribution:

[latex]p=p_{0}+\frac{\mu Q}{2 \pi \kappa} \ln \frac{r_{A}}{r_{B}},[/latex] (E7.5.5)

in which the added pressure [latex]p_{0}[/latex] depends on the depth of the stratum and the overall level to which it is pressurized.

Equation (E7.5.5) is next applied at the outer radius of each well in turn, namely, for [latex]r_{B}=a, r_{A} \doteq L[/latex] (at well B) and [latex]r_{A}=a, r_{B} \doteq L[/latex] (at well A):

[latex]p_{B}=p_{0}+\frac{\mu Q}{2 \pi \kappa} \ln \frac{L}{a}, \quad p_{A}=p_{0}+\frac{\mu Q}{2 \pi \kappa} \ln \frac{a}{L} .[/latex] (E7.5.6)

The required pressure difference between the wells is therefore:

[latex]p_{B}-p_{A}=\frac{\mu Q}{2 \pi \kappa} \ln \frac{L^{2}}{a^{2}}=\frac{\mu Q}{\pi \kappa} \ln \frac{L}{a}.[/latex] (E7.5.7)

Finally, Fig. E7.5.2 shows several isobars (which are almost circular in the vicinity of the wells), together with the corresponding streamlines.

Question 7.5: Underground Flow of Water A company has been disposing of so...

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