An Application of Inverse Problem and Universal Solutions for Pumping Wells in Unconfined Aquifers

Abstract

As far as we know, universal solutions (or type-curves) for scenarios of flow through anisotropic unconfined aquifers due to pumping wells cannot be found in the literature. On the contrary, those theoretical solutions in hydrogeological manuals are commonly based on Dupuit solutions for isotropic soils or simplifying other characteristics of the chosen medium. In this study, the application of the discriminated nondimensionalization technique allowed for the inclusion of vertical and radial hydraulic conductivities in the data set, with which the monomials ruling unknown variables of the problem, pumping flow and seepage surface in their dimensionless form are obtained. One of the main findings of this research is depicting these relationships as type-curves from a large number of precise numerical simulations based on the Network Simulation Method. The other main finding is an easy-to-apply methodology to estimate vertical and radial hydraulic conductivities employing these type-curves. This methodology can be considered as an inverse problem. In addition, an example of the problem is presented, in which the influence that measure deviations may have on the estimated values of the hydraulic conductivities in anisotropic soils is also studied and discussed.

1. Introduction

Water flow through porous media due to the presence of a pumping well has been a widely reviewed topic [1,2] in hydrogeology and geologic engineering. Applications such as water supply [3] and water management [4] are issues that are strongly related to this research area. When studied from a macroscopic point of view, the flow depends on the gradient of the hydraulic potential and on the hydraulic conductivity of the soil, which, in turn, is a function of the characteristics of both medium (grain size, porosity, tortuosity) and fluid (density, viscosity) [5]. Nevertheless, although Darcy’s law rules the phenomenon, specific formulations on pumping have been developed to achieve results in a rather simple manner.

This paper studies unconfined aquifers, whose traditional study is based on several restrictions. Firstly, the aquifer is supposed to be infinite, homogeneous and isotropic, with horizontal bottom and constant thickness. Moreover, the diameter of the pumping well is negligible, and the well is fully penetrating. Another important hypothesis is only considering radial flow without vertical velocity (‘Dupuit-Forcheimer hypothesis’). Finally, there is only one pumping well in the area that is working with a constant discharge rate [6]. Once the steady state has been reached, water flow due to pumping wells in unconfined aquifers can be calculated according to Dupuit formulation [7], which states that it depends on the difference of the squared values of potential in two points of the soil.

Some of the hypotheses that are taken for solving problems of flow due to pumping wells in unconfined aquifers lead to non-realistic results: considering isotropic media when soils are generally anisotropic, and exclusively radial flow when the flow compulsory presents a vertical component in free aquifers in the surroundings of the well (although the importance of this vertical component depends on value of the potential variation due to the pumping). Focusing on analytic and theoretical solutions, several authors have proposed to study anisotropic media calculating the equivalent isotropic hydraulic conductivity [8]. However, one of the objectives of this research is to obtain easy-to-use type-curves that allow engineers and researchers to include different characteristics of the problems of flow through anisotropic unconfined aquifers due to pumping wells, especially radial and vertical hydraulic conductivities.

The second simplification that is exposed, not considering the vertical flow in the well surroundings, has led to obviate the analytical study of the seepage surface, which is the wet area in the well wall [9]. This area, as the free surface, is also under atmospheric pressure.

In this paper, the scenario of flow in unconfined aquifers due to a pumping well has been approached and characterized employing the discriminated nondimensionalization technique, which conjugates the scale analysis [10] with the spatial discrimination [11]. According to this method, both the governing equations and boundary conditions are transformed into dimensionless expressions to obtain the lowest number of independent dimensionless monomials that rule the problem [12]. Examples of problems that have been approached employing the discriminated nondimensionalization technique are flow under dams [13], solute and temperature flow [14] and soil consolidation [15]. In addition, this work is also an approach to an inverse engineering problem [16], as it aims to infer the value of some properties, in this case, the hydraulic horizontal and vertical conductivity values from measured variables that are unknown in directs problems (in these scenarios, pumping flow and seepage surface). The accuracy of the proposed methodology is presented, analyzing the influence of measure deviations of pumping flow and seepage surface on the precision of the sought parameters and affecting the variables by statistic errors of 0.5–2%. In this way, the deviation between the real and the estimated values of K_r and K_z can be quantified.

2. Materials and Methods

2.1. Mathematical Model

Unconfined aquifers present the characteristic that its upper border (phreatic level) is at atmospheric pressure. This means that, if the reference for the vertical coordinates, z = 0, is set at the bottom of aquifer, the hydraulic potential values at the phreatic level coincide with the vertical coordinate at those points. As for any other problems of flow through porous media, a Laplace’s type expression result of substituting Darcy´s law in the continuity equation describes this phenomenon. In cylindrical coordinates, the water velocity and continuity equations take, respectively, the forms

$${\mathrm{v}}_{\mathrm{r}}=-{\mathrm{K}}_{\mathrm{r}}\frac{\partial \mathrm{h}}{\partial \mathrm{r}} \mathrm{and} {\mathrm{v}}_{\mathrm{z}}=-{\mathrm{K}}_{\mathrm{z}}\frac{\partial \mathrm{h}}{\partial \mathrm{z}}$$

(1)

$$\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}{\mathrm{v}}_{\mathrm{r}}\right)+\frac{\partial {\mathrm{v}}_{\mathrm{z}}}{\partial \mathrm{z}}=0$$

(2)

In Equations (1) and (2), r is the radial coordinate, z the vertical coordinate, v_r the radial velocity, v_z the vertical velocity, K_r the radial hydraulic conductivity and K_z the vertical hydraulic conductivity.

From these, governing equation in terms of the hydraulic potential for anisotropic aquifers writes as

$${\mathrm{K}}_{\mathrm{r}}\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}\frac{\partial \mathrm{h}}{\partial \mathrm{r}}\right)+{\mathrm{K}}_{\mathrm{z}}\frac{{\partial }^2\mathrm{h}}{{\partial \mathrm{z}}^2}=0$$

(3)

which, for isotropic domain (K_r = K_z = K, where K is the isotropic hydraulic conductivity), reduces to

$$\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}\frac{\partial \mathrm{h}}{\partial \mathrm{r}}\right)+\frac{{\partial }^2\mathrm{h}}{{\partial \mathrm{z}}^2}=0$$

(4)

Traditional formulation does not consider vertical flow either far from the well center or in its surrounding. If that term is removed from Equation (4), the traditional expression for modeling these problems reduce to the simple equation $\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}\frac{\partial \mathrm{h}}{\partial \mathrm{r}}\right)=0$ which give rises to the Dupuit´s solution. Figure 1 shows a sketch of the scenario with the nomenclature and the variables and parameters involved. Dupuit´s expression is

$${\mathrm{H}}^2-{\mathrm{h}}^2=\frac{\mathrm{Q}}{\mathsf{\pi }\mathrm{K}}\mathrm{l}\mathrm{n}\frac{\mathrm{R}}{\mathrm{r}}$$

(5)

In Figure 1, R is the aquifer radius, r_w the well radius, H the aquifer thickness, h_w water height in the well and h_s the seepage surface. The well wall is the vertical contact between the aquifer and the well, and its length is the aquifer thickness. This length includes the water height in the well and the seepage surface, and it has to be permeable.

The second term in Equation (3) involves the vertical variation of the potential (and, therefore, vertical flow), which is the reason why the mathematical resolution of the problem is very complex, as the seepage surface appears, and the possibility of reaching an analytical solution is almost impossible without assuming other simplifications. For Equation (3), two different types of boundary conditions must be set. First-type conditions, also known as Dirichlet conditions, mean that a constant value of hydraulic water potential is set, and second-type conditions, called Neumann conditions (or homogeneous Neumann in the case of impervious borders), mean that the normal derivative of the hydraulic potential is constant. As the borders these conditions are applied to are considered as impervious, the value of the constant is zero, so the flow across them cannot happen. These conditions are also presented in Figure 1.

2.2. Discriminated Nondimensionalization Technique for Flow towards Pumping Wells

2.2.1. Discriminated Nondimensionalization Procedure

Employing the discriminated nondimensionalization technique leads to study seepage scenarios in a summarized manner. This procedure is an extension of the well-known Pi theorem [12], generally applied to isotropic domains, which states that the dimensionless unknowns of a problem can be expressed as an arbitrary function of the dimensionless monomials in which the geometrical and hydrogeological parameters are involved. Dimensionless numbers have been traditionally used in different study fields, although approaching this technique from the discriminated point of view is relatively new. According to this methodology, groups can only be derived from the governing equations and boundary conditions (or by experimentation) and represent balances between pairs of quantities referred to the whole domain or to a part of it. Discrimination is a concept that here is applied in two different ways: from the ‘spatial’ point of view, which forces choosing geometrical parameters in such a way that their orientation assumed different dimensional equations for each of the spatial directions, and from a ‘general’ point of view, which allows assuming that the hydraulic potential has its own dimension in terms of energy, avoiding that the gradient of the potential is dimensionless. The number of these monomials is the lowest possible.

In addition, the application of the discriminated nondimensionalization technique has allowed studying problems of flow through anisotropic unconfined aquifer in a rather accurate way because:

• New groups that remove classical ones appear. This occurs as a consequence of including spatial and general discrimination.
• Correct references for the problem variables are chosen, since ratios of lengths must include parameters in the same spatial direction, and aspect ratios can only appear modified by the ratio of other parameters.
• Universal solutions can be presented as universal type-curves in which the relationship between unknowns and data is depicted.
• Inverse problem methodologies can be developed using the universal solutions.

2.2.2. Discriminated Dimensionless Groups Achievement

According to the nomenclature and parameters of Figure 1, the references for variables r, z and h are R, H and Δh, respectively. Thus, the dimensionless variables are defined in the form

$${\mathrm{r}}^{\prime }=\frac{\mathrm{r}}{\mathrm{R}}; {\mathrm{z}}^{\prime }=\frac{\mathrm{z}}{\mathrm{H}}; {\mathrm{h}}^{\prime }=\frac{\mathrm{h}}{∆\mathrm{h}}$$

(6)

where r′ is the dimensionless radial coordinate, z′ the dimensionless vertical coordinate and h′ the dimensionless hydraulic potential.

Introducing these in Equation (3), results

$${\mathrm{K}}_{\mathrm{r}}\frac{1}{{\mathrm{r}}^{\prime }\mathrm{R}}\frac{\partial }{\mathrm{R}\partial {\mathrm{r}}^{\prime }}\left({\mathrm{r}}^{\prime }\mathrm{R}\frac{∆\mathrm{h}\partial {\mathrm{h}}^{\prime }}{\mathrm{R}\partial {\mathrm{r}}^{\prime }}\right)+{\mathrm{K}}_{\mathrm{z}}\frac{∆\mathrm{h}{\partial }^2{\mathrm{h}}^{\prime }}{{\mathrm{H}}^2{\partial \mathrm{z}}^2}=0$$

(7)

It is visible from this equation that the solution does not depend on the variation of water potential. The derivatives of r′, z′ and h′ are assumed to be of order of magnitude unity because of the range chosen to find these dimensionless variables. Since the governing equation (Equation (3) in its dimensional form and Equation (7) in its dimensionless form) equals to 0, and the derivatives are of order of magnitude unity, the coefficients that multiply them must be of the same order of magnitude. Therefore, their ratio must be of order of magnitude unity, and the first dimensionless group can be obtained.

$${\mathsf{\pi }}_1=\sqrt{\frac{{\mathrm{K}}_{\mathrm{r}}}{{\mathrm{K}}_{\mathrm{z}}}}\frac{\mathrm{H}}{\mathrm{R}}$$

(8)

According to discriminated nondimensionalization technique, monomials based on length ratios can only relate lengths in the same direction, so the final monomial is dimensionless. In the scenarios of flow through unconfined aquifers, there are boundary conditions applied to the two horizontal contours, so their two lengths, r_w and R, can be related in a discriminated monomial

$${\mathsf{\pi }}_2=\frac{{\mathrm{r}}_{\mathrm{w}}}{\mathrm{R}}$$

(9)

In addition, this scenario also presents boundary conditions applied to the vertical contours, which means that a monomial relating their lengths, H and h_w, can also be obtained:

$${\mathsf{\pi }}_3=\frac{{\mathrm{h}}_{\mathrm{w}}}{\mathrm{H}}$$

(10)

Applying discriminated nondimensionalization technique, more monomials cannot be obtained, since it does not allow to relate lengths in different directions, so aspect ratios such as $\frac{\mathrm{R}}{\mathrm{H}}$ or $\frac{{\mathrm{r}}_{\mathrm{w}}}{{\mathrm{h}}_{\mathrm{w}}}$ cannot rule the scenario. The only way they can be part of the set of ruling monomials is by being part of dimensionless groups that include the ratio of other parameters of the problem, in this case a ratio of hydraulic conductivities, so the final monomial is dimensionless. If this occurs, monomial π₁ (or an equivalent one) would be obtained again.

Each one of the three monomials that rule the scenario of flow through anisotropic unconfined aquifers influences the flow (either radial or vertical) in a different way. A high value of π₁ means that radial flow is more important than vertical one, and the opposite happens if the value is low. Referring to π₂, the lower the value of this monomial is, the more relevant the horizontal flow. Finally, π₃ shows the importance of vertical flow, as if considering that the well only works pumping water out of the aquifer, low values of this group are equivalent to a higher importance of vertical flow.

The first unknown to be studied is the water flow pumped out of the system. Its dimensionless form can be obtained from different ways with a reference whose unit is that of water flow. For example, we can take ${\mathrm{Q}}_{\mathrm{r}\mathrm{e}\mathrm{f}}={\mathrm{v}}_{\mathrm{r}\mathrm{e}\mathrm{f}}·{\mathrm{S}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ where ${\mathrm{Q}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is the reference pumping flow, ${\mathrm{v}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ the reference velocity and ${\mathrm{S}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ the reference surface. In this expression,

$${\mathrm{v}}_{\mathrm{r}\mathrm{e}\mathrm{f}}={\mathrm{K}}_{\mathrm{r}}\frac{\mathrm{H}-{\mathrm{h}}_{\mathrm{w}}}{\mathrm{R}-{\mathrm{r}}_{\mathrm{w}}}$$

(11)

$${\mathrm{S}}_{\mathrm{r}\mathrm{e}\mathrm{f}}=2\mathsf{\pi }\mathrm{R}\mathrm{H}$$

(12)

where the discriminated dimensions of the parameters are:

$$\left[{\mathrm{K}}_{\mathrm{r}}\right]=\frac{{{\mathrm{L}}_{\mathrm{r}}}^2}{\mathrm{T}{\mathrm{L}}_{\mathrm{w}\mathrm{c}}}\hspace{1em}\hspace{1em}\left[\mathrm{H}\right]=\left[{\mathrm{h}}_{\mathrm{w}}\right]={\mathrm{L}}_{\mathrm{w}\mathrm{c}},\hspace{1em}\hspace{1em}\left[\mathrm{R}\right]=\left[{\mathrm{r}}_{\mathrm{w}}\right]={\mathrm{L}}_{\mathrm{r}}$$

(13)

With this,

$$\left[{\mathrm{v}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right]={\mathrm{L}}_{\mathrm{r}}{\mathrm{T}}^{-1}\hspace{1em}\hspace{1em}\left[{\mathrm{S}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right]={\mathrm{L}}_{\mathsf{\alpha }}{\mathrm{L}}_{\mathrm{z}}\hspace{1em}\hspace{1em}{[\mathrm{Q}}_{\mathrm{r}\mathrm{e}\mathrm{f}}]={\mathrm{L}}_{\mathrm{r}}{\mathrm{L}}_{\mathrm{z}}{\mathrm{L}}_{\mathsf{\alpha }}{\mathrm{T}}^{-1}$$

(14)

In these dimensional equations, L_α is a length along the circumferential perimeter while L_r and L_z are lengths in the radial and vertical directions, respectively. In short, we can define the dimensional flow as

$${\mathsf{\pi }}_{\mathrm{Q}}=\frac{\mathrm{Q}}{{\mathrm{Q}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}=\frac{\mathrm{Q}}{{\mathrm{K}}_{\mathrm{r}}\frac{\mathrm{H}-{\mathrm{h}}_{\mathrm{w}}}{\mathrm{R}-{\mathrm{r}}_{\mathrm{w}}}2\mathsf{\pi }\mathrm{R}\mathrm{H}}$$

(15)

where Q is the pumping flow.

The second unknown to be studied is the seepage surface (h_s − h_w in Figure 1), which is the wet length over the water height in the well through which water also flows. This is the length of the well wall in which the hydraulic potential is just higher or equal than its vertical position. In this study, well entry loss is not included, and the seepage surface appears because both vertical and radial flow are considered. The variable h_s is the whole contact surface between the well and the aquifer, and in this surface the potential is higher than the position, as authors such as [17] have considered in their work. The deduction of the dimensionless group which includes the seepage surface is simpler than that of the water flow, as it must include somehow a ratio of this variable and one of the boundary hydraulic potentials. After trying different options to set the dimensionless form of h_s, the one which leads to more understandable results is

$${\mathsf{\pi }}_{{\mathrm{h}}_{\mathrm{s}}}=\frac{{\mathrm{h}}_{\mathrm{s}}-{\mathrm{h}}_{\mathrm{w}}}{\mathrm{H}}$$

(16)

This monomial includes both the initial hydraulic potential in the aquifer and the potential imposed in the well.

2.2.3. Verification of the Dimensionless Groups

In this section, the previous monomials (π₁, π₂ and π₃) are verified, i.e., it is demonstrated that they work as independent groups to rule the solution of the problem. In addition, it is also confirmed that they are the least number of independent groups that influence the solution. To do this, we choose a significate number of typical different scenarios for which the groups take the same value, for different values of the physical and geometrical parameters involved. After simulation, for all these scenarios, the dimensionless value of the unknown must remain the same.

The numerical simulation is carried out in the free software Ngspice [18] following the guides of the Network Simulation Method. This procedure designs an electric circuit for a volume element or cell whose differential equation is formally equivalent to that of the physical model [19]. The reliability of the Network Method has been successfully applied in other engineering fields, such as coupled solute and heat transport in porous media [20]. In this research, the solutions calculated with the numerical tool have been contrasted with theoretical results found in the literature.

Table 1. Physical and hydrogeological parameters and variables for the dimensional scenarios.

Case	Kr (m/s)	Kz (m/s)	R (m)	rw (m)	H (m)	hw (m)	Q (m3/s)	hs (m)
1	0.0001	0.0001	10	1	10	5	0.0103	7.736
2	0.000225	0.0001	15	1.5	10	5	0.0231	7.736
3	0.0001	0.0001	50	5	10	5	0.0104	5.343
4	0.0001	0.000025	50	5	5	2.5	0.0260	2.672
5	0.0001	0.0001	10	2.5	10	5	0.0171	7.349
6	0.0005	0.0001	22.36	5.59	10	5	0.0854	7.349
7	0.0001	0.0001	10	1	10	7	0.0070	8.240
8	0.0003	0.0001	10	1	5.77	4.07	0.0070	4.753

shows the dimensional geometric and hydraulic parameters of the different scenarios and dimensional results from the simulations. The synthetized solutions in terms of the values of the dimensionless data and unknown groups are shown in

Table 2. Data and unknown monomials for the dimensionless scenarios.

Case	π1	π2	π3	πQ	πhs
1	1	0.1	0.5	0.295	0.274
2	1	0.1	0.5	0.295	0.274
3	0.2	0.1	0.5	0.297	0.034
4	0.2	0.1	0.5	0.297	0.034
5	1	0.25	0.5	0.408	0.235
6	1	0.25	0.5	0.408	0.235
7	1	0.1	0.7	0.334	0.124
8	1	0.1	0.7	0.334	0.123

. So, Case 1 and Case 2 are related to different dimensional scenarios, but they are the same dimensionless one for which π₁ = 1, π₂ = 0.1 and π₃ = 0.5. For Cases 3 and 4, π₁ is changed from 1 to 0.2, keeping π₂ and π₃ the same values as in Cases 1 and 2. In this way, the interaction between the global vertical and radial hydraulic conductivities is modified. Cases 5 and 6 have the same values of π₁ and π₃ as cases 1 and 2, while π₂ takes a value of 0.25. With this change, the well radius has been decreased. Finally, cases 7 and 8 have the same values of π₁ and π₂ as in cases 1 and 2 while π₃ is 0.7, so scenarios with a higher value of water height in the well are represented.

As seen in Table 1 and Table 2, although the dimensional scenarios in each pair are different, when turned into dimensionless, the values of the dimensionless unknown are the same, which verifies the technique employed throughout this paper. The only values that differ are those of π_hs in Cases 7 and 8, but this small difference (below 1%) is attributable to the numerical simulation employing the Network Method, since dividing the scenarios in cell or volume elements may lead to small deviations in the results. Furthermore, it is also interesting to remark the little difference between the values of π_Q for π₁ = 1, π₂ = 0.1 and π₃ = 0.5 and for π₁ = 0.2, π₂ = 0.1 and π₃ = 0.5, which are 0.295 and 0.297, respectively. The effect of monomial π₂ is the following: as the well radius is increased, then higher values of dimensional and dimensionless pumping flow are obtained. In the case of π₃, its effect is the opposite. As h_w increases (which means that π₃ also does), the pumping rate decreases because the water potential difference value, H − h_w, is lower. This fact is easily observed studying Dupuit’s expression (Equation (5), giving h the value of h_w). In the study of the dimensionless pumping rate, Equation (15) is considered. If the dimensional flow value is kept constant, increasing h_w leads to a decrease in the reference flow (lower part of the ratio in Equation (15)) and, therefore, an increase in the dimensionless flow.

3. Results and Discussion

3.1. Universal Curves

Two type-curves are presented for the dimensionless characterization of the problem of flow in unconfined aquifers due to a pumping well: one for the groundwater flow (Figure 2) and another one for the seepage surface (Figure 3). The last value of π₃, 0.001, is employed to model the problem of a well where the hydraulic potential is zero.

After the simulations, it was evident that the differences of π_Q in relation to π₁, for the same values of π₂ and π₃, were low enough to consider that the group involving hydraulic conductivities does not affect the dimensionless water flow (the highest difference is around 2.3%). This fact leads to a simplification of the type-curve (Figure 2) as it only includes one curve for each π₂.

Apart from the ‘lack of effect’ of π₁ monomial, it is also relevant to know how the geometry (reflected in π₂ and π₃) influences the solution. Referring to the effect on water flow, it can differ whether the real or the dimensionless water flow is being considered. Studying the impact of π₂, it is observed that, as its value increases (which means that the well radius is wider), the pumping capacity is also increased. This influences both real and dimensionless water flow.

π₃ affects differently the behavior of real and dimensionless water flow. If studying the real variable, the lower the value of π₃, the higher the hydraulic potential difference is, which generates more water flow. Nevertheless, when comparing this with the reference flow, Q_ref, as it also grows with the decrease of π₃, the difference between the two water flows also increases. This is translated into a reduction of π_Q.

Figure 3 shows the type-curve for the dimensionless seepage surface, π_hs, and, unlike the dimensionless groundwater flow, this dimensionless variable is affected by the permeability monomial, π₁.

For this dimensionless group, the effect of π₁ is the one expected: the higher the horizontal flow is, the higher the value of the seepage surface, because it is easier for the flow to go on in the horizontal direction. Monomial π₂ has the opposite effect: higher values of π_hs are obtained as π₂ decreases. This means that the seepage surface increases as the well radius presents a lower value because the flow has a longer horizontal length to keep flowing in that direction. Finally, the effect of π₃ is the same as that of π₂:π_hs increases its value as π₃ decreases, which means that the seepage surface is higher as the water height in the well has a lower value. It occurs because the potential difference increases and there is a longer length to develop the seepage surface.

3.2. Illustrative Example—Influence of Measure Deviations in the Calculation of Conductivity Values

This section shows how to obtain the values of vertical and radial hydraulic conductivities employing the type-curves presented in Section 3.1. Knowing the geometrical parameters of the aquifer and well, as well as pumping rate and seepage surface values, conductivities can be calculated. Therefore, this problem falls into the category of ‘inverse problem’, since those parameters that were known in the previous section are now the sought values (hydraulic conductivities) and pumping rate and seepage surface are part of the data set. The resolution of this kind of inverse problem is explained through the following example, in which the data are:

• H = 10 m
• h_w = 5 m
• R = 10 m
• r_w = 1 m
• Q = 0.002308 m³/s
• h_s = 8.471 m

In order to compare the estimated values with the real ones, we also know the hydraulic conductivities:

• K_r = 0.0000225 m/s (named K_r,real, as it is the real value of the parameter in the example).
• K_z = 0.00001 m/s (named K_z,real, as it is the real value of the parameter in the example).

From all these data, we can calculate the values of the data groups, which now are π₂, π₃ and π_hs.

$${\mathsf{\pi }}_2=\frac{{\mathrm{r}}_{\mathrm{w}}}{\mathrm{R}}=\frac{1}{10}=0.1$$

(17)

$${\mathsf{\pi }}_3=\frac{{\mathrm{h}}_{\mathrm{w}}}{\mathrm{H}}=\frac{5}{10}=0.5$$

(18)

$${\mathsf{\pi }}_{\mathrm{h}\mathrm{s}}=\frac{{\mathrm{h}}_{\mathrm{s}}-{\mathrm{h}}_{\mathrm{w}}}{\mathrm{H}}=\frac{8.471-5}{10}=0.347$$

(19)

Using π₂ and π₃ in Figure 2, we obtain the value of π_Q, which is 0.295. As the pumping rate, Q, is part of the problem data, the horizontal conductivity, K_r, is calculated from Equation (15).

$${\mathrm{K}}_{\mathrm{r}}=\frac{\mathrm{Q}}{{\mathsf{\pi }}_{\mathrm{Q}}\frac{\mathrm{H}-{\mathrm{h}}_{\mathrm{w}}}{\mathrm{R}-{\mathrm{r}}_{\mathrm{w}}}2\mathsf{\pi }\mathrm{H}\mathrm{R}}=\frac{0.002308}{0.295·\frac{10-5}{10-1}·2\mathsf{\pi }·10·10}=0.0000224 \mathrm{m}/\mathrm{s}$$

(20)

This value is the calculated horizontal or radial conductivity, K_r,cal. Once it is known, the deviation must be calculated.

$$\frac{{\mathrm{K}}_{\mathrm{r},\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}-{\mathrm{K}}_{\mathrm{r},\mathrm{c}\mathrm{a}\mathrm{l}}}{{\mathrm{K}}_{\mathrm{r},\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}100\mathrm{\%}=\frac{0.0000225-0.0000224}{0.0000225}·100\%=0.44\%$$

(21)

Knowing the value of K_r allows calculating K_z from the type-curve in Figure 3, from which, for the values of π_hs, π₂ and π₃, π₁ is 1.5. Therefore, according to Equation (8), K_z is calculated as

$${\mathrm{K}}_{\mathrm{z}}=\frac{{\mathrm{K}}_{\mathrm{r}}}{{{\mathsf{\pi }}_1}^2}\frac{{\mathrm{H}}^2}{{\mathrm{R}}^2}=\frac{0.0000224}{{1.5}^2}\frac{{10}^2}{{10}^2}=0.00000996 \mathrm{m}/\mathrm{s}$$

(22)

This value is the calculated vertical conductivity, K_z,cal, and its deviation is calculated according to Equation (23).

$$\frac{{\mathrm{K}}_{\mathrm{z},\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}-{\mathrm{K}}_{\mathrm{z},\mathrm{c}\mathrm{a}\mathrm{l}}}{{\mathrm{K}}_{\mathrm{z},\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}100\%=\frac{0.00001-0.00000996}{0.00001}·100\%=0.4\%$$

(23)

It is observed that, in both cases, the deviation is much lower than 1%, which shows the reliability of the methodology.

It is interesting now to study how possible measure errors may affect the sought conductivities, for which statistic errors, ξ, up to 2% are applied to the field variables, pumping rate and seepage surface. The values of the horizontal and vertical conductivities, as well as the maximum deviation for each of the errors are shown in

Table 3. Maximum deviations of K_r and K_z as a function of ξ.

Deviations ξ (%)	Kr × 10−5 (m/s)	Max Deviations Kr (%)	Kz × 10−5 (m/s)	Max Deviations Kz (%)
0	2.24	0.44	0.996	0.40
0.5	2.23/2.25	0.89	1.07/0.99	7.02
1	2.22/2.26	1.33	1.12/0.89	11.92
1.5	2.21/2.28	1.78	1.19/0.84	18.69
2	2.20/2.29	2.22	1.24/0.78	24.36

, and Figure 4 displays the deviations of both conductivities with respect the statistic error.

From these results, we can see that the deviations due to measure errors up to 2% present low values for horizontal conductivity (2.2%) while it is more than 10 times higher for vertical conductivity (24.4%).

4. Conclusions

The dimensional discrimination technique applied to hydrogeology has allowed approaching the problem of steady flow through unconfined aquifers overcoming one of the most limiting restrictions in order to obtain more realistic results: the anisotropy of soil, in terms of hydraulic conductivity values, in vertical and horizontal direction. This, as well as being able to model vertical flow in the simulation tool, has led to more accurate values of seepage surface.

The unknowns of interest in these anisotropic problems, which are pumping flow and seepage surface, have been expressed in their dimensionless form as functions of the derived groups and their solutions are displayed in universal type-curves. A large number of numerical simulations, carried out with models based on the Network Method, have been necessary to depict these type-curves. It was observed that the dimensionless pumping flow is hardly influenced by monomial π₁ (conductivity ratio affected by an aspect ratio).

The use of the type-curves is verified by numerical simulation of scenarios in which physical and geometric parameters were varied, maintaining the values of the dimensionless groups. As expected, the unknown monomials are identical for scenarios with the same values of the dimensionless groups.

The proposed methodology in the form of inverse problem, with the use of two measures (pumping flow and extension of the seepage surface) read at a single well, allows obtaining precise estimates of the hydraulic conductivities in anisotropic unconfined aquifers. The reliability of the type-curves is verified by the calculation of an illustrative example whose values of hydraulic conductivities are already known in order to be compared to those estimated. The deviation between real and estimated horizontal hydraulic conductivities is very low (lower than 1%), while that for the vertical hydraulic conductivity is higher, although among acceptable values (lower than 5%). The effect of possible deviations on the measure of the variables pumping flow and seepage surface has also been studied (values up to 2% have been applied). The maximum deviation that would appear in the horizontal conductivity would be around 2%, while for the vertical conductivity the deviation is around 24%. These highlight the importance of measuring the seepage surface in the most precise way.