Introduction: 
Circular aquifers, such as those found in karst systems or fault-bounded basins, are common in nature and require precise modeling to understand their response to pumping. While extensive analytical solutions exist for rectangular, triangular, and semi-infinite aquifers, a comprehensive closed-form analytical solution for a finite circular aquifer with an arbitrarily located well has been notably absent. Previous studies have primarily focused on simpler geometries or relied on numerical approximations, leaving a gap in the direct analytical treatment of circular domains under pumping conditions. This study addresses this gap by developing a rigorous mathematical model capable of accurately predicting hydraulic head distribution and flow potential in a confined, homogeneous, and isotropic circular aquifer. The proposed solution eliminates the need for numerical discretization, offering a direct and computationally efficient tool for groundwater management in circular or near-circular aquifer systems.

Materials and Methods: 
This study presents a rigorous closed-form analytical solution for modeling groundwater flow to a pumping well in a finite, homogeneous, isotropic, and confined circular aquifer. The aquifer has a fixed radius bb, and the well, with an infinitesimal radius, is positioned arbitrarily at coordinates (r0, θ0) with a constant pumping rate Q. The mathematical formulation begins with the governing partial differential equation in polar coordinates, which is the Poisson equation incorporating a point source term:

where ϕ (r, θ) is the hydraulic head, and the right-hand side uses the Dirac delta function δ to localize the source/sink. The physical constraints are enforced by two boundary conditions: (1) the head remains finite at the origin , and (2) the head is constant (set to zero) at the outer circular boundary ϕ(b,θ)=0.

The solution methodology employs the Green's function approach combined with a Fourier series expansion. First, the unknown potential is expanded into a full trigonometric series:
A similar expansion is applied to the source term. Substituting these series into the governing PDE decomposes the two-dimensional problem into an infinite set of one-dimensional ordinary differential equations for the radial coefficients ϕn(r).
The core of the solution is the construction of the Green's function G(r,θ;ρ,ψ) for the corresponding Dirichlet boundary-value problem on a circle. Due to different fundamental solutions, the cases for n=0 and n≥1 are treated separately. For n=0, the radial component is logarithmic:

For n≥1, the solution involves power laws:
The complete Green's function is then:

The final expression for the hydraulic head is obtained via convolution and simplifies, after applying the properties of the delta function, to a closed-form series solution:

This infinite series can be summed analytically using known identities, resulting in compact algebraic formulas valid in the regions r≤r0 and r≥r0. The analytical results were successfully validated against a finite-element numerical model.


Results and Discussion:
The application of the developed analytical solution to a confined circular aquifer with parameters b=7m, r0=3m, θ0=50∘, Q=−100 m3/d, and T=5 m2/d revealed several key hydraulic behaviors. The primary outcome is the clear asymmetric drawdown distribution resulting from the well’s off-center position. The drawdown cone is deepest immediately around the well location and becomes progressively shallower with increasing radial distance, as visualized in the dimensionless drawdown contour plot (Fig. 2). The non-concentric nature of the equipotential lines is distinctly captured in the hydraulic head contour map (Fig. 4), where contours are elliptical and skewed away from the geometric center of the aquifer, demonstrating that symmetry is broken when the well is not at the center. A detailed analysis of drawdown profiles along various angular directions relative to the well’s azimuth further quantified this asymmetry (Fig. 3). The maximum drawdown occurs precisely along the radial line corresponding to the well’s own azimuth (θ=50∘). As the angular deviation from this direction increases, the drawdown magnitude at a given radial distance systematically decreases. For instance, the drawdown along θ=67.5∘ is less than that at θ=50∘, and the profile along θ=90∘ shows even lower values. This pattern conclusively illustrates how the aquifer’s circular geometry and the well’s specific location jointly control the spatial pattern of the cone of depression. Validation against a numerical model confirmed the high accuracy and reliability of the analytical solution. A finite-element model was constructed for the identical aquifer setup. A comparison of the drawdown versus radial distance profiles along two distinct directions (θ=67.5∘ and θ=90∘) showed nearly perfect overlap between the analytical and numerical results (Fig. 8). The minor discrepancies were within negligible numerical tolerance, verifying that the closed-form solution correctly solves the governing equations and boundary conditions without the approximations required in numerical discretization. This successful validation establishes the solution as a robust and efficient tool for predicting head distributions in finite circular aquifers.

Conclusion: 
This study successfully developed a rigorous closed-form analytical solution for predicting hydraulic head distribution and drawdown in a finite, confined, circular aquifer subjected to pumping from a well at an arbitrary location. By employing Green's function and Fourier series techniques, the governing equation was solved analytically, resulting in a direct mathematical formula. The solution efficiently captures the asymmetric hydraulic response caused by an off-center well, eliminating the need for numerical approximations. Validation against a finite element model demonstrated excellent agreement, confirming the solution's accuracy. This analytical tool provides a practical and efficient means for well placement optimization, capture zone delineation, and management of circular or near-circular aquifers, such as karst systems. The methodological framework established here also holds potential for extension to more complex hydrological conditions, including heterogeneous or anisotropic properties.