Transient-state Analytical Solution for Arbitrarily-Located Multiwells in Triangular-Shaped Unconfined Aquifer

Abstract

This work presents an analytical solution for the linearized Boussinesq equation describing the nature of well hydraulics in equilateral triangular-shaped unconfined aquifer. This homogeneous, isotropic, fully-saturated porous media is hydraulically connected to three surrounding streams of constant-head. The solution enables modeling aquifer response to a system of arbitrarily-located, fully-penetrating multiwells (injection, extraction or combination of both), each characterized by stepwise time-varying rate. First, a fundamental solution is provided for multiwell-induced head distribution in an infinite aquifer domain. Image well theory is then efficiently implemented to create an equivalent flow field for the intended domain. Spatiotemporal head distribution is obtained in the form of fivefold series involving exponential integrals. Expressions are also derived to quantify stream depletion rates caused by a single pumping well, under both transient and steady-state conditions. As a hypothetical example, an aquifer remediation scheme is planned by combining two extraction wells with an injection one. The computed head profiles reveal strictly close agreement with numerical results obtained by finite element method. Sensitivity map for stream depletion rate is also discussed. The present results are found to exactly reproduce those available for the wedge-shaped domain, under certain geometric constraint. Finally, the solution is extended to the case of hemi-equilateral triangular-shaped aquifer with or without an impervious boundary line.

Appendix 1

A triple integral transform is adopted here to derive an analytical solution for Eq. (1) subject to conditions (3) and (4). The method includes Laplace transform with respect to time followed by double Fourier transform with respect to two spatial variables. This closely follows the method developed by Watanabe (2014) for 2-D wave equation.

Hereafter, the overbar “−” and tilde “~” will denote the Fourier transform of indicated function with respect to x and y coordinates, respectively. With this notation, the symbol “≃” will stand for double Fourier transform. Likewise, the asterisk “∗” refers to the Laplace transform.

Eq. (1) simplifies to a linear algebraic equation taking the triple integral transforms successively. It is given by:

$$ p{\tilde{\overline{H}}}^{\ast }+a\left({\xi}^2+{\eta}^2\right){\tilde{\overline{H}}}^{\ast }=\frac{2a}{K}\sum \limits_{j=1}^N{Q}_j^{\ast }(p)\exp \left( i\xi {x}_j\right)\exp \left( i\eta {y}_j\right) $$

(14)

which is simply rearranged to obtain

$$ {\tilde{\overline{H}}}^{\ast}\left(\xi, \eta, p\right)=\frac{2a}{K\left(p+a\left({\xi}^2+{\eta}^2\right)\right)}\sum \limits_{j=1}^N{Q}_j^{\ast }(p)\exp \left( i\xi {x}_j\right)\exp \left( i\eta {y}_j\right) $$

(15)

where $i=\sqrt{-1}$; ξ [L⁻¹] and η [L⁻¹] refer to the Fourier transform variables corresponding to x and y, respectively while p[T⁻¹] is the Laplace transform variable. Eq. (15) expresses the groundwater head in the transformed domain (ξ, η, p) and inverse integral transforms are needed to covert this back into the original variables (x, y, t). The first Fourier inversion is performed with respect to the variable η, leading to:

$$ {\overline{H}}^{\ast}\left(\xi, y,p\right)=\frac{1}{\pi K}\sum \limits_{j=1}^N{Q}_j^{\ast }(p)\exp \left( i\xi {x}_j\right){\int}_{-\infty}^{\infty}\frac{a\exp \left(- i\eta y\right)\exp \left( i\eta {y}_j\right)}{a\left({\xi}^2+{\eta}^2\right)+p}\mathrm{d}\eta $$

(16)

Eq. (16) is manipulated to obtain a semi-infinite integral as follows:

(17)

The integral I₁ can be evaluated from a standard table of integrals (Erdélyi 1954, pp. 8, Eq. (11)):

$$ {I}_1=\frac{\pi }{2\sqrt{\xi^2+\frac{p}{a}}}\exp \left(-\sqrt{\xi^2+\frac{p}{a}}|y-{y}_j|\right) $$

(18)

Fourier inversion is then performed with respect to the variable ξ:

$$ {H}^{\ast}\left(x,y,p\right)=\frac{1}{2\pi K}\sum \limits_{j=1}^N{Q}_j^{\ast }(p){\int}_{-\infty}^{\infty}\frac{\exp \left(-\sqrt{\xi^2+\frac{p}{a}}|y-{y}_j|\right)\exp \left(- i\xi x\right)\exp \left( i\xi {x}_j\right)}{\sqrt{\xi^2+\frac{p}{a}}}\mathrm{d}\xi $$

(19)

or

(20)

Therefore, the integral I₂ can be evaluated as (Erdélyi 1954, pp. 17, Eq. (27)):

$$ {I}_2={\mathrm{K}}_0\left[{R}_j\sqrt{\frac{p}{a}}\right] $$

(21)

where K₀ denotes the second kind, zero-order modified Bessel function. Noting that ${\left[\frac{1}{2}{t}^{-1}\exp \left(-{R}_j^2/4 at\right)\right]}^{\ast }={K}_0\left[{R}_j\sqrt{p/a}\right]$ when (x, y) ≠ (x_j, y_j) (Erdélyi 1954, pp. 283, Eq. (35)) and applying the convolution theorem for the Laplace inversion, one obtains:

$$ H\left(x,y,t\right)=\frac{1}{2\pi K}\sum \limits_{j=1}^N{\int}_0^t\frac{Q_j\left(\tau \right)}{t-\tau}\exp \left[\frac{-{R}_j^2}{4a\left(t-\tau \right)}\right]\mathrm{d}\tau $$

(22)

which leads to Eq. (5) after performing the integration with the flow rates given by Eq. (2).