Introduction

Gravitational groundwater flow has been studied extensively for many years by numerous researchers. In 1856, Darcy was the first to derive the relationship between flow, head gradient, and hydraulic conductivity, which became known as Darcy’s Law1. In 1935, Theis published his pioneering paper on the unsteady variations in groundwater levels resulting from the pumping of a well in a confined aquifer2. Hubbert3 was the first to introduce the term ‘potential’ in groundwater flow as a physical value governing groundwater movement. He also developed a 2D conceptual groundwater flow system using flow nets4,5.

In his pioneering 1962 paper, Tóth represented the 2D groundwater flow described by Hubbert3 in mathematical form6. Tóth’s mathematical model focused on a specific groundwater system in Alberta, where the groundwater basin is deep, with a sand layer on the top, which provides a continuous source of recharge. He assumed that the water table was parallel and close to the land surface and solved for groundwater potential using Hubbert’s3 potential Equation. This assumption of a known water table location eliminated the need for additional boundaries, such as recharge or discharge into or out of the basin. Consequently, Tóth’s solution does not explicitly account for the influences of groundwater recharge or hydraulic conductivity, nor the shape of the water table.

Vandenberg7 extended Tóth’s work to account for transient conditions of the same problem. Thompson and Moore8 explored how topography controls water-table depth in a forested soil. They concluded the topography has a significant control over water-table depth in forest soils. Some other studies related recharge and water table to topography9,10,11. Haitjema and Mitchell-Bruker10 applied a simple criterion based on recharge, transmissivity, and basin length to assess whether the water table is correlated with the land surface. The study shows that in aquifers with high permeability and low recharge (e.g., productive aquifers), the water table is often controlled by recharge, not topography. However, in low-permeability aquifers with high recharge rates, the water table tends to follow surface topography more closely.

Condon and Maxwell12 explored the effect of topography on groundwater and water table depth in the United States, and classified the water table type based on10 classification. They used a physical hydrology model to simulate groundwater flow and depth to water table. The investigation demonstrates that different water-table types exist in different parts of the United States. Areas with recharge-controlled water tables, like the Southwest and Rocky Mountains, have deeper, more variable water tables and considerable regional groundwater flow, while topography-controlled areas, such as the Northeast, have shallower, more stable water tables with minimal regional flow.

Zhao et al.13 developed a 3D solution for the transient form of Tóth’s problem. As with Vandenberg’s7 study, Zhao et al.13 assumed the shape and the location of the water table are known. A more recent study by Sun et al.14 offered a solution to Tóth’s problem using deep learning. Similar to the original setup, they employed Neumann boundary conditions at the no-flow boundaries, along with Dirichlet and Robin boundary conditions at the upper and lower boundaries. Like Tóth’s model, Sun et al.14 assumed the water table’s shape and location are known.

Zhang et al.15 explored how topography controls topography-driven groundwater flow using a two-dimensional groundwater-surface water coupled model. They found that when rainfall or hydraulic conductivity is reduced, climate or geology may play a more significant role in controlling groundwater flow systems than topography.

Solving the flow problem in an unconfined aquifer presents challenges. On one hand, the water table serves as the upper physical boundary of the flow domain, which is needed for solving the flow problem. On the other hand, the exact location of the water table can only be determined after the flow problem has been solved.

The Dupuit-Forchheimer assumptions were introduced in the mid-19th century, primarily by Jules Dupuit in 1863 and later expanded by Philipp Forchheimer in the early 20th century to simplify groundwater flow problems16,17. They assume that the groundwater flowlines are horizontal and equipotential lines are vertical, and the hydraulic gradient is constant across the depth of the aquifer. While the Dupuit-Forchheimer assumptions are commonly used to simplify the 2D groundwater flow problem to one dimension18,19,20, however, these assumptions are not always valid, especially near wells or when the vertical flow component is significant21.

In this paper, a new formulation of Tóth’s flow problem is presented. In this approach, identifying the water table location is unnecessary, as it is replaced by Neumann boundary conditions. Unlike Tóth’s original model, the mathematical formulation in this study explicitly incorporates both hydraulic conductivity and groundwater recharge/discharge, without defining the water table location in advance. Several case studies are provided to demonstrate groundwater flow regimes, with a comparison between numerical and analytical solutions. The impact of various parameters such as the recharge/discharge rate, hydraulic conductivity and basin dimension on the hydraulic head is analyzed.

Groundwater flow and Tóth unit basin model

The water table location is normally unknown in groundwater flow conceptual models. These models use boundary conditions such as recharge and hydraulic properties to solve for the groundwater head in space and time. The Laplace Equation that describes the flow in isotropic porous media expressed in terms of potential is3:

$$\frac{{\partial ^{2} \phi }}{{\partial x^{2} }} + \frac{{\partial ^{2} \phi }}{{\partial y^{2} }}$$
(1)

where φ is the potential and it is equal the total head (h) multiplied by the acceleration of gravity (g).

Tóth22 defined the unit basin as “the elementary building concept in the theory of gravity-driven regional groundwater flow”. This definition implies the unit basin is homogeneous, isotropic and has definitive boundaries such as no-flow or water divide6. Unlike usual way of flow conceptualisation, in his unit basin model, Tóth assumed the location and the shape of water table are known, as shown in Fig. 1. No flow was assumed at all other boundaries. A linear water table was assumed in the unit basin, and the flow was assumed to be steady. Thus, the solution of flow Eq. (1) for the unit basin was obtained using the separation of variables6,22:

$$\emptyset = gh = g\left( {b - \frac{{ca^{2} }}{2}} \right) + g\frac{{4ca}}{{\pi ^{2} }}\mathop \sum \limits_{{m = 0}}^{\infty } \frac{{\cos \left[ {\left( {2m + 1} \right)\pi x/a} \right]\cosh \left[ {\left( {2m + 1} \right)\pi y/a} \right]}}{{(2m + 1)^{2} \cosh \left[ {\left( {2m + 1} \right)\pi b/a} \right]}}$$
(2)

where c is the slope of the water table, a is length of the unit basin and b is the basin depth at the left end of the basin. The head can be obtained by dividing Eq. (2) by the acceleration of gravity.

Although Tóth’s solution was a breakthrough in hydrogeological science, it has some issues. Having the shape of the water table predetermined is problematic as it is part of the solution. Further, it is not possible to examine how hydraulic conductivity and groundwater recharge affect the lines of produced equipotential, or the shape of the resulting water table with these boundary conditions. The biggest problem, however, is assumption of a linear water table. As Tóth’s model is steady, the groundwater heads and the water table location and shape would not change over time, despite there is a continuous flow from one half of the basin to the other half. The middle point of the unit basin must be a saddle point, else the groundwater head will be the same in the entire basin given the steady state conditions. Mathematically, the middle point (i.e. the saddle point) is a point of singularity23,24.

Fig. 1
figure 1

Tóth problem setup and boundary conditions (adopted after Tóth 1962).

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Analytical versus numerical solution

A comparison was made between the analytical and the numerical solution obtained using MODFLOW25 to represent Tóth’s model with a linear water table, as implemented by Tóth. The numerical model measures 100 by 100 m and is homogeneous and isotropic. All boundaries are no-flow boundaries, except for the upper boundary, which represents a linear water table that gently decreases in level from west to east, in accordance with analytical model (Fig. 1).

Since MODFLOW typically updates the water table with each iteration, a constant head boundary was assigned at the top of the model, with head values equal to the elevation (pressure is by definition equal to 0 at the water table). The water table boundary (i.e. constant head) gradually changes from 100.0 m on the left side to 98.8 m on the right (slope c = 0.012). The hydraulic properties and the slope of the water table were the same for both the analytical problem and the numerical model. A program was written using Python to solve Eq. 2, and the obtained results are presented and compared to those from the numerical model, as shown in Fig. 2.

Fig. 2
figure 2

Left: 2D contour map of groundwater head [m] resulting from the analytical solution using Eq. (2), and right: the numerical model results.

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It is evident that both maps are identical, clearly demonstrating the correctness of Tóth’s model for the given boundary conditions. Both the numerical model and the analytical solutions produce exactly the same contour maps, regardless of the values of hydraulic conductivity or recharge. In Tóth’s conceptual model, neither recharge nor hydraulic conductivity explicitly appears in Eq. (2); instead, they are embedded in the shape of the water table, represented by the variable c in the Equation.

In this problem setup, the use of a predefined water table renders both hydraulic conductivity and recharge irrelevant as they are implicitly included in the model. Furthermore, when examining the flow budget using the numerical model of Tóth’s problem, significant variation in flow is observed just below the water table (i.e., constant head inflow/outflow). The cell-by-cell flow budget in MODFLOW indicates that the highest recharge occurs at the top left of the model and decreases gradually to the right, reaching zero in the middle of the basin. Beyond this point, it becomes negative, indicating discharge, and continues to decrease as we move toward the right side, satisfying the conservation of mass law (Fig. 3).

The distribution of recharge and discharge is unusual, as they typically do not change dramatically over a small area, nor they follow this specific pattern. The smooth transition in recharge and discharge depicted in Fig. 3 rarely occurs in reality in such a manner. In many studies recharge is shown to follow either uniformly distributed over land, especially in wet regions26 or point recharge in arid regions27.

Fig. 3
figure 3

Recharge/discharge at the water table as calculated by cell-by-cell in MODFLOW.

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Reformulation of Tóth’s model

The main issue with water table aquifer is that the location of the water table, which is the top boundary of the aquifer, is unknown. If we predetermine the water table’s location, we assume part of the solution we are trying to find. However, the problem cannot be solved without proper boundary conditions. To address this issue, a new formulation of the flow must be developed. Instead of using Dirichlet boundary conditions (i.e., a specified head) at the water table, it is replaced by Neumann boundary conditions (i.e., a specified flow).

This new formulation explicitly highlights the effects of recharge and hydraulic conductivity, without the need for pre-assigned water table. Figure 4 illustrates the flow problem for Tóth’s unit basin, with the new boundary conditions. The lateral and lower sides are no-flow boundaries, consistent with Tóth’s model. The upper part, representing the water table, is not prescribed; instead, recharge and discharge are applied. The numerical representation of the original Tóth’s problem, which was discussed above, enables identify the rate of recharge/discharge that replaces the water table in the new formulation.

One-unit basin setting

The new setup of the Tóth’s problem has a different boundary condition on the top boundary as shown in Fig. 4. This formulation satisfies the conservation of mass, which is clearly satisfied by applying uniform recharge at the first half of the basin, and equal discharge at the other half. With a uniform recharge rate, N, the upper boundary condition can be set as shown in Fig. 4. As the infiltration and evaporation rates equal N, and using Darcy’s Law, the flow rate is given by K⋅dℎ/dy, where K is the hydraulic conductivity and dℎ/dy is the hydraulic gradient.

At the water table, the flow is in the y direction only, hence qx=0. The value of qy at the water table equals the recharge rate N. Using Darcy’s law, the rate of flow at the water table will be N/K in the left side of the basin (flow into the basin) and -N/K on the right side of the basin (flowing out of the basin).

Fig. 4
figure 4

Re-conceptualization of Tóth’s model of the flow problem in 2D for one-unit basin.

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Analytical solution

The solution of Eq. (1), given the boundary conditions on Fig. 4, is obtained using the separation of variables:

$$\:h={A}_{0}+{\sum\:}_{m=1}^{\infty\:}\frac{4aN{sin}\left(\frac{\pi\:m}{2}\right)}{{m}^{2}{\pi\:}^{2}K{sin}h\left(\frac{\pi\:mb}{a}\right)}{cos}\left(\frac{m\pi\:}{a}x\right){cos}h\left(\frac{m\pi\:}{a}y\right)$$
(3)

where Ao is an arbitrary constant, and all other variables are as defined before. The solution is clearly non-unique, which is expected since the problem has no fixed head boundary (constant head). However, the constant in Eq. (3) does not affect the shape of the resulting groundwater head contours because it is just an addition constant. Knowing the head at any point within the flow domain is sufficient to determine the value of A0. The true value of A0 should produce a head value equal to y at the point (0, b) because by definition the head at the water table is equal to elevation y. By setting h = y at (0, b) in Eq. (3), the value of A0 is found.

A python code was written to solve Eq. (3). The resulting groundwater head contour map is shown in Fig. 5. The shape and values of the contours are similar to those produced by the original Equation of Tóth’s solution (Eq. 2). Unlike Tóth’s original problem, hydraulic conductivity in this case is important and affects the magnitude of the head. For this case and all following cases, the hydraulic conductivity is assumed to be 0.5 m/d. Groundwater recharge rate N was calculated based on flow budget of the numerical model for the original case of Tóth (Fig. 2), and it was found to be 0.0012 m/d. The recharge/discharge rate N = 0.0012 m/d was obtained from the flow budget analysis of the cell-by-cell flow of the numerical model representing Tóth’s original problem with a linear water table. The cell-by-cell flow budget of MODFLOW contains inflow and outflow of each cell in the model domain. As such, recharge and discharge values were obtained from the file of MODFLOW right below the water table, which is assumed in the original model of Tóth, given the hydraulic conductivity of 0.5 m/d. This approach ensures that the recharge/discharge rate is consistent with the boundary conditions of Tóth’s model.

Fig. 5
figure 5

Groundwater head [m] in one-unit basin resulting from analytical solution (Eq. (3)).

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Numerical solution

A numerical model was developed to simulate this new formulation. The model domain is 100 by 100 m, with a hydraulic conductivity of 0.5 m/d. The finite difference model (MODFLOW) supports many packages that can be used to represent various boundary conditions27. Recharge was applied to the left half of the upper boundary using the Recharge Package in MODFLOW, while an equal amount of evaporation was applied to the right half using the Evaporation Package25. Both recharge and discharge rates are 0.012 m/d. A constant head boundary condition was assigned to the top-left cell of the model to avoid non-uniqueness solution issue (by definition, the head at the water table equals the elevation y). The resulting contour lines, shown in Fig. 6, closely match both the values and the shape produced by the analytical solution (Fig. 5), demonstrating consistency between the numerical and analytical models.

Fig. 6
figure 6

Contour map of the produced groundwater head [m] using a numerical model with Neumann boundary conditions for one-unit basin.

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Two-unit basins problem

Similar to the previous case, the formulation of the two-unit basins problem implies two adjacent unit basins6,22. This can be done by changing only the upper boundary conditions as depicted in Fig. 7. In this case, all boundaries are the same as with the previous case except the upper one, which has a boundary as follows:

$$\:\partial\:h\frac{\partial\:h}{\partial\:y}=\frac{N}{K}\:\:\:for\:0\ll\:x<\frac{a}{4},\:and\:for\:\frac{a}{2}\le\:x<\frac{3a}{4}$$
(4a)
$$\:\partial\:h\frac{\partial\:h}{\partial\:y}=-\frac{N}{K}\:\:\:for\:\frac{a}{4}\ll\:x<\frac{a}{2},\:and\:for\:\frac{3a}{4}\le\:x<a$$
(4b)
Fig. 7
figure 7

Re-conceptualization of Tóth’s model of the flow problem in 2D for two-unit basins.

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This formulation satisfies the conservation of mass as the recharge equals discharge.

Analytical solution

Using separation of variables technique, the solution of the flow problem is:

$$\:h={A}_{0}+{\sum\:}_{m=1}^{\infty\:}\frac{4aN\left[2\:sin\left(\frac{\pi\:m}{4}\right)-2\:sin\left(\frac{\pi\:m}{2}\right)+2\:sin\left(\frac{3\pi\:m}{4}\right)-\text{s}\text{i}\text{n}\left(\pi\:m\right)\right]}{{m}^{2}{\pi\:}^{2}K{sin}h\left(\frac{\pi\:mb}{a}\right)}{cos}\left(\frac{m\pi\:}{a}x\right){cos}h\left(\frac{m\pi\:}{a}y\right)$$
(5)

where A0 is arbitrary constant. A python code is used to produce the groundwater head contours using Eq. (5) over a domain of a = 100 m and b = 100 m. The hydraulic conductivity is 0.5 m/d, and recharge/discharge rate is 0.012 m/d. As with the case of one-unit basin, the value of A0 is found by setting h = y at (0, b). The contour map resulting from this solution is shown in Fig. 8.

Numerical solution

A two-dimensional model was developed in MODFLOW with dimensions similar to the one-basin case, measuring 100 by 100 m. All boundaries were set to no-flow, and Neumann boundary conditions were applied on the upper side of the model to represent those depicted in Fig. 7. Recharge was applied to the first and third quarters of the model’s length (25 m each), while the Evaporation Package in MODFLOW was used to simulate discharge in the second and fourth quarters. The first cell in the upper row was assigned a constant head boundary of 100 m, representing A0 in Eq. 6. The hydraulic conductivity was set to 0.5 m/d, and recharge/discharge at 0.012 m/d, consistent with the previous one-basin model.

Fig. 8
figure 8

Groundwater head contours [m] resulted from analytical solution of two-unit basins (Eq. 5).

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The model results, shown in Fig. 9, depict groundwater contours. Similar to the analytical solution, flow occurs from the first top quarter (recharge) to the second quarter (discharge) and repeats symmetrically in the third and fourth quarters. The values obtained are identical to those from the analytical model.

Fig. 9
figure 9

Groundwater head contour map [m] resulted from the numerical model of two-unit basins.

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Solution using stream function

The same problem can be solved using a stream function instead of head. The stream function contours define the steam-lines of groundwater flow, and they are orthogonal to equipotential lines. The stream function is related to specific discharge based on the following Equation29:

$$q_{x} = - \frac{{\partial \psi }}{{\partial y}}\;and\;q_{y} = \frac{{\partial \psi }}{{\partial x}}$$
(6)

The Laplace Equation of the stream function for isotropic and homogenous aquifer is given by:

$$\:\frac{{\partial\:}^{2}\psi\:}{\partial\:{x}^{2}}+\frac{{\partial\:}^{2}\psi\:}{\partial\:{y}^{2}}=0$$
(7)

Equation (7) can be solved easily without the need to solve Eq. (1), as the boundaries are Neumann30.

Stream function for one-unit basin case

The stream function problem for the one-unit basin case can be developed using Eqs. (6) and (7). From Eq. (6) the upper boundary condition for the stream function problem is:

$$\:\psi\:(x,b)={\int\:}_{0}^{a}{q}_{y}(x,b)dx=\left\{\begin{array}{c}Nx/K,\:\:x<a/2\\\:N(a-x)/K,\:\:x>a/2\end{array}\right.$$
(8)

The stream function at all other boundaries has a constant value of ψ0 (i.e. no flow) because stream lines are parallel to the no-flow boundaries. With these boundary conditions, the solution of Eq. (5) is:

$$\:\psi\:\left(x,y\right)={\psi\:}_{0}+{\sum\:}_{n=1}^{\infty\:}{B}_{n}{sin}\left(\frac{\pi\:n}{a}x\right){sin}h\left(\frac{\pi\:n}{a}y\right)$$
(9)

where

$$\:{B}_{n}=\frac{2}{a{sin}h\left(\frac{n\pi\:b}{a}\right)}\left[\frac{{a}^{2}N\left(2{sin}\left(\frac{\pi\:n}{2}\right)-sin\left(\pi\:n\right)\right)}{K{\pi\:}^{2}{n}^{2}}\right]$$

As with groundwater head (Eqs. (3) and (5)), Eq. (9) explicitly demonstrates the linear dependence of groundwater head on both recharge and hydraulic conductivity. A Python program was developed to solve Eq. (9). An example output is shown in Fig. 10, where the basin depth and width are assumed to be 100 m each.

Fig. 10
figure 10

Contours of stream function [m2/d] for one-unit basin obtained by Eq. (9) and draped over the groundwater head contours [m] from Eq. (3)-.

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Stream function for two-unit basins

The stream function can be derived in this case using Eq. (8) but changing the upper boundary conditions to be:

$$\:\psi\:(x,b)={\int\:}_{0}^{a}{q}_{y}(x,b)dx=\left\{\begin{array}{c}\frac{Nx}{K},0\le\:\:x<\frac{a}{4}\\\:\frac{N\left(\frac{a}{2}-x\right)}{K},\frac{a}{4}\le\:\:x<\frac{a}{2}\\\:\frac{N\left(x-\frac{a}{2}\right)}{K},\frac{a}{2}\le\:\:x<\frac{3a}{2}\\\:\frac{N\left(a-x\right)}{K},\frac{3a}{4}\le\:\:x<a\end{array}\right.$$
(10)

The solution is similar to Eq. (9) but with a different coefficient:

$$\:\psi\:\left(x,y\right)={\psi\:}_{0}+{\sum\:}_{n=1}^{\infty\:}{C}_{n}{sin}\left(\frac{\pi\:n}{a}x\right){sin}h\left(\frac{\pi\:n}{a}y\right)$$
(11)

where \(\:{\psi\:}_{0}\) is a constant and:

$$\:{C}_{n}=\frac{2}{a{sin}h\left(\frac{n\pi\:b}{a}\right)}\left[\frac{{a}^{2}N\left(2{sin}\left(\frac{\pi\:n}{4}\right)-2{sin}\left(\frac{\pi\:n}{2}\right)+2{sin}\left(\frac{3\pi\:n}{4}\right)-\text{s}\text{i}\text{n}\left(\pi\:n\right)\right)}{K{\pi\:}^{2}{n}^{2}}\right]$$

Analytical solution of Eq. (11) was obtained using a Python code, and the results are shown in Fig. 11, draped over the contours of groundwater heads from Eq. (5).

Fig. 11
figure 11

Contours of stream function [m2/d] for two-unit basin obtained by Eq. (12) and draped over the groundwater head contours [m] from Eq. (7)-.

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Factors affecting contour shape and head magnitude

The effect of recharge and hydraulic conductivity

Results of all models discussed above depends on the dimensions of the basin (e.g. a and b), recharge/discharge N and hydraulic conductivity K, as appears in all flow equations. To examine the effect of recharge and hydraulic conductivity on the shape of resulting groundwater head, the following formulation was done:

Let \(\:\alpha\:=\frac{\:N}{K}\), then Eq. (3) of the groundwater head in a one-unit basin can be rewritten as:

$$\:h={\sum\:}_{m=1}^{\infty\:}\frac{4\:a\:\alpha\:{sin}\left(\frac{\pi\:m}{2}\right)}{{m}^{2}{\pi\:}^{2}{sin}h\left(\frac{\pi\:mb}{a}\right)}{cos}\left(\frac{m\pi\:}{a}x\right){cos}h\left(\frac{m\pi\:}{a}y\right)$$
(12)

Clearly, the value of ℎ changes linearly with α. To examine this relationship, groundwater head values were calculated within a square domain with sides a=b=100 m, α varies between 0.0001 and 0.02, as shown in Fig. 12. The difference between the maximum and minimum head within a 100 by 100 basin is plotted against α. It is clear this difference would amplify for larger basin dimensions.

Fig. 12
figure 12

Relationship between difference of max–min head and alpha within a unit basin of 100*100 meters basin.

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By examining the resulting contour maps, it is observed that the shape of the resulting contour map remains unchanged, regardless of the value of α, as long as the basin is square (i.e., a=b). These findings suggest that the shape of the contours is not affected by the hydraulic conductivity or recharge/discharge values. For small values of α, which imply a low recharge rate and/or high hydraulic conductivity, the difference between the maximum and minimum head values is minimal.

The basin aspect ratio impact on head

To explore the impact of basin dimensions on the calculated head, Eq. (3) was solved multiple times using varying values of basin depth (b) while keeping the basin length (a) and all other variables constant. The Equation was solved with a=100 m, and b was varied from 10 to 100 m in increments of 2 m. Other parameters were kept the same as in previous cases, specifically K=0.5 m/d and N=0.012 m/d. Figure 13 depicts the results of this analysis. It shows that the difference between the maximum and minimum head values is non-linearly related to the ratio of basin depth to length (b/a). As the basin changes from a rectangular shape to a square, the difference between the maximum and minimum head decreases. At lower aspect ratios (i.e. b/a < < 1), the head gradient increases as the flow becomes more aligned with the horizontal direction and predominantly follows in a single direction. It should be noted the relationship between head gradient and the aspect ratio is exponential, as appears in Fig. 13. As such, the aspect ratio has higher impact on groundwater head than α.

Fig. 13
figure 13

Relationship between maximum difference of groundwater head and aspect ration of the basin (b/a).

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The basin aspect ratio impact on the shape of the contours

To explore the impact of the basin’s aspect ratio on the shape of the contours, Eq. (3) for the one-unit basin was solved for various aspect ratios. Figure 14 shows the resulting contour maps of the one-unit basin head for four cases. In each case, all variables remained the same except for b (the depth of the basin), which took values of 10, 25, 50, and 100 m. The basin length was kept constant at 100 m, with a hydraulic conductivity of 0.5 m/d and a recharge/discharge rate of 0.012 m/d. It is observed that as the aspect ratio (b/a) decreases, the flow becomes more horizontal, and the equipotential lines become nearly vertical.

Fig. 14
figure 14

The resulting groundwater head [m] for one-unit basin with aspect ratios from top to bottom: 0.1, 0.25, 0.5, 1.0.

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Discussion

This study builds on the pioneering groundwater flow concepts, which was developed by Hubbert3 and mathematically presented by6. Equation (3), which represents the solution for with hydraulic head with Neumann boundary conditions enables direct consideration of the hydraulic conductivity and groundwater recharge in the solution, which is missing in Tóth’s formulation of the problem.

The issue with the Tóth’s solution is the assumption of the shape of the water table (i.e. linear in case of unit basin and sinusoidal in another study), which implicitly includes recharge and discharge. This hinders any possibility of exploring their rules in the flow system. By reformulating Tóth’s classic model, the study addresses a long-standing challenge in hydrogeology: the inherent circularity of defining the water table as both a boundary condition and a solution variable. This reformulation, which replaces Dirichlet boundary conditions with Neumann conditions, is more realistic for modeling groundwater flow. This would eliminate the need to predetermine the water table’s location. This advancement enables the explicit incorporation of hydraulic conductivity and recharge/discharge rates, which are fundamental to understanding groundwater behavior in real-world scenarios.

To satisfy the conservation of mass in any model, recharge and discharge must be equal. When examining Tóth’s solution using a numerical model and analyzing the flow from the water table (represented by a constant head boundary), the resulting water budget appears unusual. It shows a gradual decrease in recharge on one side of the basin and a corresponding gradual increase in discharge (i.e., evaporation) on the other side. This is because the water table was assumed to be linear and gradually decreasing.

The new analytical solution of the problem for both head and stream functions show they are linearly and directly related to N/K values (i.e. α = N/K). While any increase in α resulted in an increase in groundwater head magnitude, the shape of the contours remains the same regardless to α value. This is because when considering Eq. (3) for head, we can see the trigonometric and hyperbolic cosine components determine the form of the resulting groundwater head contours. These terms are not directly dependent on N (recharge/discharge) or K (hydraulic conductivity), but rather on the geometry of the basin (i.e., a and b) and the spatial coordinates x and y. As a result, although variations in N and K will impact the contour values (i.e., the head magnitude), they would not modify the contour lines’ overall form.

Recharge affects the head’s magnitude and is present in Eq. (3) numerator. The head values will increase with a higher N, but the general form of the Equation—which is determined by the spatial variables x and y—stays the same. The contours’ values are only scaled by the recharge; their spatial structure remains unchanged. Hydraulic Conductivity (K) appears in the denominator of Eq. (3), altering the velocity of groundwater. The head values will scale in a similar way when K is changed, with a higher K resulting in lower head values. But the contours’ shape, or the pattern of flow, is still determined by the trigonometric and hyperbolic functions, which are independent of K.

The aspect ratio of the basin affects the flow in a different way than K or N, where the contour lines’ shapes would change if the value of aspect ratio changed. The variation of the head in the vertical (or y) direction can be attributed to the length and depth of the basin. Because b modifies the sinh function in Eq. (3), it alters the rate at which the hydraulic head values change with depth. Longer or stretched contour lines are produced by a larger b because it causes head changes with depth to occur more gradually. Conversely, a smaller b compared to a, results in faster changes in head values, which in turn produce more compact and tighter contour lines.

When basin length equals depth (i.e. a = b), the flow contributions from the x and y directions are equal since the basin is square. Because of this, the flow is affected by the boundary conditions equally in both directions, and the equipotential lines typically have curved patterns that represent the impact of the flow components that are horizontal and vertical (Fig. 13- bottom).

As basin depth b becomes smaller compared to length a, the flow becomes more constrained in the vertical direction (y-axis). This happens because as b becomes smaller, the vertical hydraulic gradients weaken, and flow tends to be dominated by horizontal components. Therefore, the equipotential lines become more vertical since the flow is predominantly controlled by the horizontal hydraulic gradient along the x-direction (Fig. 13-top).

Conclusion

This study introduces a new formulation for solving Tóth’s one and two-unit basin groundwater flow problem by considering Neumann boundary conditions at the water table. By removing the assumption of a predefined water table, the model explicitly accounts for the effects of recharge and hydraulic conductivity. This enables exploring the impact of recharge/discharge and hydraulic conductivity on the groundwater flow. This advancement is especially valuable for regions where data on water table fluctuations is limited or unreliable such as arid and semi-arid areas where effective groundwater management is essential.

The comparison between the numerical and analytical solutions show a good match, which confirms the validity of the new developed solutions for the various cases considered in this study. The derived stream function solutions provide a further proof of the head solutions for both one- and two-unit basins. The solutions can be expanded for as many as needed unit basins, and can accommodate various boundary conditions.

The Tóth’s boundary conditions for a unit basin problem may not suite many real groundwater basins, as recharge and discharge do not normally change in the manner resulted from a predefined linear water table. Based on literature data, uniform distribution of recharge/discharge is more likely than the gradual increase or decrease, which stemmed from the linear shape of the water table.

The results reveal that recharge and hydraulic conductivity, along with basin dimensions, affect groundwater flow in different ways. The variations in hydraulic conductivity and recharge/discharge have an impact on the hydraulic head’s magnitude, but not the contour lines’ shape. The smaller the N/K the smaller the head gradient and vice versa. However, the geometry of the equipotential lines is strongly dependent on the basin aspect ratio, in particular the depth-to-length ratio (b/a). The flow becomes more unidirectional (i.e. horizontal) as the aspect ratio (b/a) decreases, and vice versa. The recharge/discharge have nothing to do with the shape of the contours. In addition, the aspect ration affects the head magnitude exponentially.

As the hydraulic head gradient increases with a decrease in the aspect ratio (b/a), the water table in a square basin shows minimal gradient. The shape of the water table is affected more by the basin’s aspect ratio and N/K value than by the topography, as some previous studies suggest. The topography and land slope may affect the rate of recharge/discharge, which in turn affects the boundary conditions. While the increase of head gradient in linear with any increase in N/K, it increases exponentially with the decrease in aspect ratio. As such, it is concluded that the aspect ratio has higher impact on the water table gradient.

The results of this study may help improve our knowledge of how groundwater flow is affected by hydrological parameters and basin geometry. The developed solution in this study can be adapted to a wide range of hydrogeological settings, including basins with variable aquifer geometries, hydraulic conductivities, and recharge rates. This flexibility makes the presented solution applicable to both regional and local groundwater flow studies, from large-scale aquifers to smaller, localized systems. The ability to account for different basin aspect ratios and recharge/discharge rates extends the model’s capabilities to be used in various hydrogeological contexts. Future work may consider more complicated cases, with heterogeneity and anisotropy. In addition, more complex patterns of recharge and discharge can be investigated. This may include non-uniform distribution over the model domain.