1.1 Background
In structural geology, an anticline, as a result of lateral compression in crustal deformation, is a convex-upward fold in layers of rock. A well-structured anticline formation may be considered as a potential site for waste disposal or carbon sequestration. Ashjari and Raeisi (2006) investigated the groundwater flow in Zagros anticlines in Iran and indicated that the anticline structure of aquifers and the geometry of bedrocks primarily dominate the direction of regional groundwater flow.
Moreover, it can be expected that the flow patterns will be changed on the condition that wastes or water being injected into or pumped from the aquifer. Because of the movement of groundwater carries the contaminants, explicit information such as geological structure and hydrogeological data are necessary to judge the applicability of the potential storage sites or to predict the migration of the contaminant plume in the site.
The drawdown or head data set obtained from a field aquifer testing, e.g., slug test or pumping test, is generally analyzed based on a relevant solution to determine the aquifer parameters. For a constant-flux pumping test, the test well pumps at a constant flow rate during the test time and the drawdown responses are measured in one or more observation wells in the vicinity. Commonly, a drawdown solution is
either incorporated with an optimization technique or applied to generate the type curves for the graphical method to find the best-fit aquifer parameters. An anticline aquifer has curved surfaces on their top and bottom boundaries; moreover, its profile may be asymmetric to its ridge. The complexity of the geometric situation of an anticline makes it challenging to solve the model analytically. In this study, we devoted to derive the analytical solution for the drawdown distribution in the approximated anticline aquifer since it can serve as an invaluable tool for gaining physical insight into the flow behavior affected by geologic and geometric settings.
1.2 Literature Review
The classic Thiem (1906) or Theis (1935) equation may be the most popular way used to estimate the drawdown distribution or to determine the aquifer parameters in an inverse problem for a constant-flux pumping in a confined aquifer. The Thiem equation (1906) described the spatial drawdown distribution within the radius of influence under steady-state condition. The Theis solution (1935) delineated the transient drawdown response in a confined aquifer. However, the assumptions made for developing these equations on well of full penetration and aquifer of infinitely lateral extent may not be capable of describing the flow in an anticline aquifer.
Numerous studies have been made to cope with the groundwater flow problem edged
with peculiar boundaries. Among these studies, the integral transform method is commonly used to obtain the hydraulic head or drawdown solutions for specific boundary conditions in the mathematical model. For example, Chan et al. (1976) used the finite Fourier transform to obtain the transient and steady-state drawdown solutions for pumping in a rectangular aquifer. Chan et al. (1978) and Yeh and Chang (2006) applied the finite sine transform and Hankel transform to obtain the transient and steady-state analytical solutions for head distribution in a wedge-shaped aquifer. On the other hand, some drawdown solutions accounting for various topography boundaries in flow systems are based on the image-well method. The method removes aquifer boundaries and place pumping or recharging image wells at judicious locations. The drawdown in an observation well is calculated by summing up the drawdown or buildup due to the real well and image wells (Ferris et al. 1962;
Streltsova 1988; Kuo et al. 1994; Chen et al. 2009).
The domain decomposition method can be applied to handle the problem with complex geometry or mix-typed boundary. In this method, the problem domain is split into several subdomains. Thereafter, the solutions for each subdomain are derived to satisfy the corresponding boundary conditions as well as the continuities of head and flux at the interface between the connected elements. The concept of domain decomposition method was first presented in Kirkham (1957) to calculate the
electrostatic potential between two concentric coaxial capped cylinders. The procedure was further extended in Kirkham (1959) to obtain the hydraulic head solution for the flow toward a partially penetrating well in a confined aquifer. Later, Javandel and Zaghi (1975) used a similar procedure to obtain the potential distribution in a confined aquifer due to the pumping at a well of vertically full penetrating and radially finite extension on the bottom of the aquifer. A similar decomposition concept was also deployed by Connell et al. (1998) for solving the problem of topographically driven flow in hillslope aquifers by dividing the problem domain into several rectangular elements.
Recently, some studies using numerical or analytical approaches were presented to investigate the head responses in anticline reservoirs due to the well injection or pumping. Al-Mohannadi et al. (2007) used the finite-difference method to simulate the transient pressure responses to horizontal wells in anticline reservoirs and curved wells in slab reservoirs. Yeh and Kuo (2010) proposed a steady-state analytical solution for a constant-head injection via a fully penetrating well into a heterogeneous, anisotropic, and dome-like anticline reservoir. Yet, it seems to lack the consideration of well partial penetration and asymmetric profile of the anticline.
1.3 Objective
The objective of this study is to develop a mathematical model for describing the steady-state drawdown distribution to a constant-flux pumping in an anticline aquifer.
The pumping well is of infinitesimal diameter and can partially or fully penetrate the aquifer. The anticline aquifer is homogeneous, anisotropic and confined by a curved layer on the top and a horizontal impermeable layer at the bottom. Three successive blocks of different heights are used to represent the shape of the top curved boundary.
The solution of the model is then obtained by applying the integral transform techniques including Fourier transform (FT) and finite Fourier cosine transform (FFCT) within each block and the hydraulic continuity requirements between the blocks. The solution is used to predict the spatial drawdown distribution in a wide variety of anticline aquifer system and to investigate the influences of well location, screen length, aquifer geometry and anisotropy on the flow system. Moreover, the present solution is applied to simulate the flow in hillslope and slab-shaped aquifers by assuming some of the adjacent blocks with the same heights. In addition to the analytical approach, the numerical model, MODFLOW, is used to perform simulations and the results are compared with those predicted by the present solution.
The solution can also be employed to estimate the aquifer parameters in an inverse problem if integrated with an optimization algorithm.