Steady-state response of annular wedge-shaped aquifers to arbitrarily-located multiwells with regional flow

Highlights

• A closed-form Green’s function is constructed for annular wedge-shaped aquifers.

• Stream function is given analytically.

• Extremely simple expressions are derived for stream depletion rates.

• Sensitivity maps are prepared for stream depletion rates.

• Multiple stagnation points are identified on a semi-analytical basis.

Abstract

Wedge-shaped aquifers are sometimes cut by crossing streams so that the areal geometry can be delineated by an annular wedge of infinite radial extent. Despite their worldwide spread, the problem of groundwater flow in such geometrical entities has not received much attention in analytical studies. This paper presents a closed-form analytical solution for the aquifer head distribution in response to a system of multiwells superimposed on a background regional flow. The aquifer is isotropic, homogeneous and in prefect hydraulic connection with the surrounding streams. A steady-state Green’s function is constructed for this laterally bounded aquifer system. The formulation is general in the sense that the aquifer vertex angle as well as number of wells and their positions can be arbitrary chosen. Closed-form expression for the stream function is obtained by integration of the Cauchy-Riemann equations. Three types of hydrogeological boundary conditions are considered, combining streams of constant and linearly varying head and no-flow barrier. The results are applicable to both confined and unconfined flows. Extremely simple expressions are derived to quantify the amount of water exchange across stream-aquifer interfaces. Flow nets are given for a number of hypothetical test cases, demonstrating different aspects of the generated flow field. Sensitivity maps are prepared for the stream depletion rates due to a single pumping well and the role of aquifer vertex angle on the model response is investigated. Multiple stagnation points in the flow field are identified on a semi-analytical basis. This helps to assess how the position of a single stagnation point is affected by changes in the gradient of regional flow. For relatively high gradients, it is observed that the radial position of the stagnation point coincides with that of the well itself. Finally, the formulation is extended to account for rainfall recharge over an unconfined aquifer having the same areal extent. The aquifer response is found to agree well with other analytical and numerical results.

Introduction

Multiple river basins often draw irregularly shaped aquifers between intersecting streams, lakes and alluvial fans. The flow topology is affected not only by the aquifer geometry but also by the way that the components of stream-aquifer system are interconnected. Such geometrical entities are diverse worldwide: the Mekong and Red River deltas (Asadi-Aghbolaghi and Seyyedian, 2010, Mahdavi and Seyyedian, 2013); the trapezoidal region formed by intersection of Karun and Bahmanshir rivers, north-west of Persian Gulf, Iran (Mahdavi and Seyyedian 2014); the U-shaped aquifer bounded by three water-land boundaries, Linyuan District, Taiwan (Huang et al. 2015); the wedge-shaped beach promontory near Bol, Croatia (Kacimov et al. 2016) and the L-shaped aquifer formed by Poonggye stream and bedrock outcrops, Gyeonggi-do, Korea (Kihm et al., 2007, Lin et al., 2018).

The tendency of stream-aquifer system to maintain ecological equilibrium may be disrupted by human activities, for example by urbanization with groundwater being increasingly considered a source of potable water. Besides, extraction/injection systems implemented for groundwater clean-up locally disturb the pattern of background regional flow across the aquifer. Groundwater pumping suffers from cumulative and irreversible impacts whereas excessive groundwater recharge may cause water logging. If pumping continues long enough, the cone of depression may intersect lateral boundaries of the aquifer and pattern of stream depletion rates is drastically altered. When steady-state is reached, depletion of adjacent streams becomes the only source to supply the pumped water. These illustrate the invalidity of assuming an infinite‐extent aquifer as it neglects water exchange across interfaces with adjacent water bodies. Moreover, mutual interaction of multiwells and regional flow should be taken into account in groundwater budget, all depending upon the boundary type and its orientation, among the other mechanisms.

Complexities inherent in hydrogeological processes are often addressed numerically by finite deference and finite element methods. These appear to be computationally expensive in dealing with realistic conditions and require comprehensive databases to attain reliable predictions (Zlotnik and Tartakovsky, 2008). Alternatively, analytical or semi-analytical techniques can be preferred over numerical counterparts due to ease of implementation and less-data requirements. Such solutions have been extensively developed for intended boundary value problems (BVPs) under various types of hydrogeological boundary conditions reflecting zero-drawdown streams, impervious barriers and tidal head fluctuations, among others (Yeh and Chang, 2006, Huang et al., 2015). The method of Green’s function is an example which can efficiently reproduce the aquifer response to extraction/injection wells (Mahdavi, 2019a) and accounting for localized or regional recharge events is straightforward (Mahdavi, 2015).

Semi-analytical solution strategies are widely available for groundwater flow problems and some illustrative examples include: The analytic element method (Strack, 2003) and Adomian's decomposition method (Patel and Serrano, 2011) for irregularly-shaped heterogeneous aquifer with multiple straight or curved boundaries; the variational method of Kantorovich for trapezoidal and triangular–shaped anisotropic aquifers subject to diffusive recharge of rainfall (Mahdavi and Seyyedian, 2014); grid‐free series solution for geometrically complex stratified aquifers interacting with streams or lakes (Ameli and Craig, 2014); the method of image wells for aquifer with a meandering stream (Huang and Yeh, 2015); Laplace-Hankel transforms for unconfined-fractured and leaky wedge-shaped aquifer system (Sedghi and Zhan, 2018); Fourier-Laplace transforms for L-shaped domains composed of two rectangular subregions (Lin et al. 2018), multi-zone wedge-shaped domains (Samani and Sedghi, 2015) and infinite aquifer domains overlying fractured bedrock (Sedghi and Zhan, 2019). In most cases, Laplace-domain head distributions are obtained analytically and inverted by a numerical algorithm, hence the name “semi-analytical”.

Relatively few analytical studies have been devoted to non-rectangular aquifer domains. For BVPs associated with triangular-shaped aquifers, Green’s functions can be obtained by repeating the image wells infinitely to fill entire 2D space, all arranged in square-shaped (Mahdavi and Seyyedian, 2013) or honey-comb lattices (Mahdavi, 2019b). The boundary conditions determine the type of image wells (extraction or injection) operating in each lattice. For wedge-shaped aquifer domains formed by intersection of two linear streams, the method of image wells (Samani and Zarei-Doudeji, 2012) and finite Fourier-Hankel transforms (Yeh and Chang, 2006) are well suited. The former benefits from simple mathematical structure but the applicability is restricted to certain values of wedge angles. Conversely, the latter allows for arbitrary angles but evaluating infinite series of Bessel functions requires substantial effort and convergence accelerating techniques such as Shanks’ method are inevitable.

When a stream crosses the wedge-shaped aquifer, the geometry formed by intersection of borderlines can roughly resemble an annular wedge. This can be observed in various regions of the world and Fig. 1 shows an example. Ideally, the lateral extent of the aquifer is bounded by finite arc of the crossing stream in one side and by straight streams in other two sides (Fig. 2). This study aims to derive closed-form analytical expressions for potential and stream function in such hydrogeological settings. Combined effects of multiwells and background regional flow are addressed under steady-state condition. First, a closed-form Green’s function is constructed for the associated BVP, allowing for annular wedge of arbitrary vertex angle, penetrated by wells of different extraction/injection rates. The resulting expressions are easy to compute and free from infinite series. It is to be noted that the analysis of groundwater flow in annular wedges has not been received much attention in analytical studies. In the wedge with no inner arc (called wedge-shaped aquifer), Chan et al. (1978) obtained a series representation for both the transient and steady state drawdown distributions in finite domains as well as in infinite domains. But in the finite wedge with inner arc (called annular wedge-shaped aquifer), only steady state drawdown distributions were first solved by Chan et al. (1978). Thanks to its simple structure, the present solution is found to be more accurate and computationally efficient than that of Chan et al. (1978) which require high-order partial sums to attain reasonable results. Except for Samani and Zarei-Doudeji (2012), none of the references cited above have reported stream function for the wedge-shaped aquifers and only potential fields were accounted for. In the present framework, the stream function immediately follows from integration of the Cauchy-Riemann equations and streamlines are simply its contour lines. By comparing with Samani and Zarei-Doudeji (2012), the stream lines calculated by both methods appear to be indistinguishable from each other. Next, hypothetical test cases are presented to assess different aspects of groundwater flow in such geometrical entities. Finally, interaction of a single pumping well with uniform rainfall recharge is analyzed and the aquifer response is tested against its numerical counterpart from adaptive finite element method.