Deduction of the Dimensionless Groups and Type Curves of Temperature Profiles in Two-Layer Soils with Water Flow at Depth

Abstract

In the common hydrogeologic scenarios of horizontal groundwater flow and a water table below the surface, the steady-state 2D thermal field resulting from the coupling between water flow and heat flow and transport gives rise to a vertical temperature profile that develops progressively over a finite extent of the domain. Beyond this region, the temperature profiles are linear and independent of horizontal position. Such profiles are related to the groundwater velocity so they can be usefully used to estimate this velocity in the form of an inverse problem. By non-dimensionalization of the governing equations and boundary conditions, this manuscript formally derives the precise dimensionless groups governing the main unknowns of the problem, namely, (i) extent of the profile development region, (ii) time required for the steady-state temperature profile solution to be reached and (iii) the temperature–depth profiles themselves at each horizontal position of the development region. After verifying the mathematical dependencies of these unknowns on the deduced dimensionless groups, and by means of a large number of accurate numerical simulations, the type curves related to the horizontal extension of the development of the steady-state profiles, the characteristic time to develop such profiles and the dimensionless vertical temperature profiles inside the characteristic region are derived. These universal graphs can be used for the estimation of groundwater horizontal velocities from temperature–depth measurements.

1. Introduction

The 2D temperature field in aquifers with a geothermal gradient, horizontal water flow and a water table below their surface is a problem of great interest in hydrogeology as knowledge of the vertical temperature profiles would allow estimation of the water velocity. Otherwise, direct determination of this velocity would require the installation of costly and inefficient measurement equipment [1] due to the very low flow rates that generally occur in these scenarios, in the order of 10⁻⁶–10⁻⁷ m/s [2,3,4]. On the other hand, temperature–depth profiles measured in the field could potentially allow the estimation of other physical or geometrical parameters by applying protocols based on the inverse problem [5]. This type of scenario occurs in soils with lower layers of high hydraulic conductivity or in homogeneous soils with a water table below the surface.

Several authors have investigated the dependence of the thermal field on velocity, and many papers have been published in this field, particularly in specialized hydrogeological journals [6,7,8]. In general, these studies refer to real scenarios solved by means of numerical techniques or specific programs whose conclusions are only applicable to such scenarios [9,10] or to more theoretical works referring to single-layer scenarios and vertical flow [11,12,13,14].

In relation to horizontal flow, it is worth mentioning the work of Lu and Ge [15] on shallow semi-confined aquifers assuming the existence of a horizontal thermal gradient throughout the domain, a severe assumption since this gradient is caused by the heat balance between water transport and thermal conditions at the boundaries. Regarding the investigation of universal solutions, Jiménez-Valera and Alhama [16] have recently presented a study based on the dimensionless characterization that derives the dimensionless groups of this problem but referring to single-layer scenarios with horizontal flow and a water table at the surface. In this scenario of a dry soil layer superimposed on a saturated soil layer, the existence of three boundary temperatures—at the top, at the bottom and that of the inlet water—together with the geometric and thermal characteristics of the scenario, plus the groundwater velocity itself, mean that the overall number of parameters influencing the solution of the problem is certainly high.

The dimensionless treatment carried out to derive the smallest number of independent dimensionless groups is based on the non-dimensionalization of the governing equations and includes the groups from the boundary conditions. In order to carry out the non-dimensionalization process, the problem variables must first be defined in dimensionless form, using references such that these variables are normalized. Once the dimensional variables of the governing equation have been replaced by their corresponding dimensionless variables, and the terms of the equation have been simplified (eliminating the derivative factors of these terms), the dimensionless groups as independent quotients between each pair of terms can immediately by obtained. To these dimensionless groups, the form factors derived from the equations defining the boundary conditions must be added [16,17]. According to the pi theorem, such zero-dimensional groups, and not the individual parameters of the problem of non-zero dimension in general, are the only ones that govern or determine the solution of the problem [18].

The verification of the deduced groups is carried out by means of precise numerical simulations in which, while preserving the numerical value of the groups, the particular values of two or more individual parameters are modified. It is thus verified that the solutions are identical for scenarios with the same numerical values of the dimensionless groups. It should be pointed out that the technique used in this and other recent works prevents parameters that are already dimensionless in themselves, such as the porosity of the medium, from being considered independent groups just because they are dimensionless. This is a rule generally assumed in the theory of dimensional analysis [19]. Such parameters appear in combination with others within more complex dimensionless groups.

In this problem, the diffusion and heat transport processes of the lower saturated layer couple at the interlayer boundary with the diffusion process in the upper dry layer, resulting in different steady-state vertical temperature profiles at each horizontal position that converge to a single profile beyond a certain horizontal extent or characteristic length. For this, a period of time or transient must have elapsed which overcomes the effects of the initial temperature conditions on the process and stabilizes the profiles at each position. This horizontal extension of the development of the profiles, as well as the duration time of the transient, is also unknown depending on the deduced dimensionless groups (pi theorem). Thus, at different horizontal positions within this development length of the profile, different profiles whose shape is unique and therefore universal for the same value of the dimensionless groups can be recognized. Beyond the development length, the profiles are always linear and equal in any scenario, as corresponds to the conditions imposed on the problem, and are not useful for velocity estimation.

As a technique for the solution of the scenarios studied and for the verification of the universal dependencies deduced, a numerical model based on the network method [20] has been used. This tool has recently been very successfully used in flow and transport problems of similar or greater complexity than those addressed here [21,22,23]. Following the basic rules of circuit theory and thanks to the existence of a large variety of electronic devices integrated in circuit computational codes (such as Ngspice [24]), a network model or circuit is designed whose equations are formally equivalent to those of the physical process. The computational numerical execution of the network model provides its quasi-exact solution thanks to the powerful and advanced computational algorithms contained in these codes.

After presenting the physical and mathematical model of the problem (Section 3) and discussing the different physical processes that emerge in it (Section 4), the independent dimensionless groups are deduced and the dependencies of the variables of interest—hidden characteristic quantities and temperature–depth profiles—are established as a function of these groups (Section 5). The obtained solutions are verified by numerical simulations to depict the type curves related to the horizontal extent of profile development, characteristic time and temperature profiles corresponding to different horizontal positions within the characteristic region (Section 6). Finally, the conclusions of this work are presented.

2. Physical and Mathematical Model

The physical model studied corresponds to porous media with two overlapping layers (I and II, upper and lower, respectively) subjected to water flow and temperature conditions at their contours, as shown in Figure 1.

The water flow enters at the left boundary of the lower layer (limited by the water table at its upper part), and is maintained at a constant velocity through this layer to exit at its right boundary. The upper and lower boundaries of this layer are assumed impervious so that the mechanical problem is reduced to a constant field of horizontal velocities in layer II. As for the thermal conditions, isothermal boundaries (Dirichlet or first-class condition) are imposed on the lower and upper surfaces of the domain as well as at the water inlet boundary in layer II. An adiabatic or homogeneous Neumann (or second-class condition) is imposed on the left vertical boundary of layer I. The horizontal extent of the domain has to be large enough to ensure that the steady-state temperature profiles are fully developed to become linear. This being so, the boundary condition at the right vertical boundary of layers I and II is irrelevant, and an adiabatic condition has been adopted. Finally, the initial temperature throughout the media is that which would correspond to the linear profile of the solution of the problem with zero inlet velocity.

According to the above description, the mathematical model consists of the following governing equations and boundary conditions [11,25]:

$$\mathrm{L}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r} \mathrm{I}\hspace{1em}\hspace{1em}{\mathrm{k}}_{\mathrm{I}}\frac{{\partial }^2\mathrm{T}}{\partial {\mathrm{x}}^2}+{\mathrm{k}}_{\mathrm{I}}\frac{{\partial }^2\mathrm{T}}{\partial {\mathrm{y}}^2}={\left({\mathsf{\rho }}_{\mathrm{e}}{\mathrm{c}}_{\mathrm{e}}\right)}_{\mathrm{I}}\frac{\partial \mathrm{T}}{\partial \mathrm{t}}\hspace{1em}\hspace{1em}\mathrm{x},0\le \mathrm{y}\le {\mathrm{l}}_{\mathrm{I}},\mathrm{t}$$

(1)

$$\mathrm{L}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r} \mathrm{I}\mathrm{I}\hspace{1em}\hspace{1em}{\mathrm{k}}_{\mathrm{I}\mathrm{I}}\frac{{\partial }^2\mathrm{T}}{\partial {\mathrm{x}}^2}+{\mathrm{k}}_{\mathrm{I}\mathrm{I}}\frac{{\partial }^2\mathrm{T}}{\partial {\mathrm{y}}^2}+{\left({\mathsf{\rho }}_{\mathrm{e}}{\mathrm{c}}_{\mathrm{e}}\right)}_{\mathrm{w}}{\mathrm{v}}_{\mathrm{o}}\frac{\partial \mathrm{T}}{\partial \mathrm{x}}={\left({\mathsf{\rho }}_{\mathrm{e}}{\mathrm{c}}_{\mathrm{e}}\right)}_{\mathrm{I}\mathrm{I}}\frac{\partial \mathrm{T}}{\partial \mathrm{t}}\hspace{1em}\hspace{1em}\mathrm{x},{\mathrm{l}}_{\mathrm{I}}\le \mathrm{y}\le {\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}},\mathrm{t}$$

(2)

$${\left.{\mathrm{k}}_{\mathrm{I}}\frac{\partial \mathrm{T}}{\partial \mathrm{x}}\right|}_{\left(\mathrm{x},\mathrm{y}={\mathrm{l}}_{\mathrm{I}},\mathrm{t}\right),\mathrm{L}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r} \mathrm{I}}={\mathrm{k}}_{\mathrm{I}\mathrm{I}}{\left.\frac{\partial \mathrm{T}}{\partial \mathrm{x}}\right|}_{\left(\mathrm{x},\mathrm{y}={\mathrm{l}}_{\mathrm{I}},\mathrm{t}\right),\mathrm{L}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r} \mathrm{I}\mathrm{I}}=0$$

(3)

$${\mathrm{v}}_{\mathrm{x},\mathrm{t}}={\mathrm{v}}_{\mathrm{o}} \left(\mathrm{horizontal}\right)\hspace{1em}\hspace{1em}\mathrm{x},{\mathrm{l}}_{\mathrm{I}}\le \mathrm{y}\le {\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}},\mathrm{t}$$

(4)

$${\mathrm{v}}_{\mathrm{x},\mathrm{y},\mathrm{t}}=0\hspace{1em}\hspace{1em}\mathrm{x},0\le \mathrm{y}\le {\mathrm{l}}_{\mathrm{I}},\mathrm{t}$$

(5)

Boundary and initial conditions:

$${\mathrm{T}}_{(\mathrm{x},\mathrm{y}=0,\mathrm{t})}={\mathrm{T}}_{\mathrm{s}}$$

(6)

$${\mathrm{T}}_{(\mathrm{x},\mathrm{y}=\mathrm{H},\mathrm{t})}={\mathrm{T}}_{\mathrm{b}}$$

(7)

$${\left.\frac{\partial \mathrm{T}}{\partial \mathrm{x}}\right|}_{(\mathrm{x}=0,0\le \mathrm{y}\le {\mathrm{l}}_{\mathrm{I}},\mathrm{t})}=0$$

(8)

$${\left.\frac{\partial \mathrm{T}}{\partial \mathrm{x}}\right|}_{(\mathrm{x}=\mathrm{L},\mathrm{y},\mathrm{t})}=0$$

(9)

$${\mathrm{T}}_{(\mathrm{x}=0,{\mathrm{l}}_{\mathrm{I}}\le \mathrm{y}\le {\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}},\mathrm{t})}={\mathrm{T}}_{\mathrm{w}}$$

(10)

$${\mathrm{T}}_{(\mathrm{x},\mathrm{y},\mathrm{t}=0)}={\mathrm{T}}_{\mathrm{s}}+\left(\frac{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\right)\mathrm{y}$$

(11)

The processes of diffusion in layer I and diffusion and advection in layer II are given in Equations (1) and (2). Equation (1) results from substituting the constitutive Fourier conduction equation, $\mathbf{j}=-\mathbf{k}·\nabla \mathbf{T}$, into the heat conservation equation, $\mathrm{k}\left(\frac{\partial {\mathrm{j}}_{\mathrm{x}}}{\partial \mathrm{x}}+\frac{\partial {\mathrm{j}}_{\mathrm{y}}}{\partial \mathrm{y}}\right)={\mathsf{\rho }}_{\mathrm{e}}{\mathrm{c}}_{\mathrm{e}}\frac{\partial \mathrm{T}}{\partial \mathrm{t}}$. Likewise, Equation (2) comes from substituting the Fourier equation into the conservation equation, which includes the advection term, $\mathrm{k}\left(\frac{\partial {\mathrm{j}}_{\mathrm{x}}}{\partial \mathrm{x}}+\frac{\partial {\mathrm{j}}_{\mathrm{y}}}{\partial \mathrm{y}}\right)+{\left({\mathsf{\rho }}_{\mathrm{e}}{\mathrm{c}}_{\mathrm{e}}\right)}_{\mathrm{w}}{\mathrm{v}}_{\mathrm{o}}\frac{\partial \mathrm{T}}{\partial \mathrm{x}}={\mathsf{\rho }}_{\mathrm{e}}{\mathrm{c}}_{\mathrm{e}}\frac{\partial \mathrm{T}}{\partial \mathrm{t}}$. The conservation of heat flux at the boundary between layers is implemented by Equation (3). Equations (4) and (5) establish the velocity field. Finally, the thermal boundary conditions are established by Equations (6)–(10), while the initial temperature in the domain is given by Equation (11).

The assumptions are: (i) homogeneous and isotropic porous media, (ii) specific thermal heat of each layer and of the water of the same value, an assumption quite close to reality, and (iii) the medium is long enough for the linear temperature profile to develop completely under steady-state conditions, an assumption that can be verified after numerical simulations of the different scenarios.

3. Preliminary Discussion

The temperature field, and, thus, the profiles ($\mathrm{T}-\mathrm{y}$) at each horizontal position, is strongly influenced by the boundary temperatures, the groundwater velocity, the diffusivity of the porous media and the geometry of the scenario.

Table 1. Parameter values for scenarios 1 to 5. $\mathsf{\alpha }$ = 10⁻⁶ m²/s, ${\mathrm{T}}_{\mathrm{s}}$ = 20 °C, ${\mathrm{T}}_{\mathrm{b}}$ = 10 °C.

Scenario	${\mathbf{l}}_{\mathbf{I}}$ (m)	${\mathbf{l}}_{\mathbf{I}\mathbf{I}}$ (m)	${\mathbf{l}}_{\mathbf{I}}/{\mathbf{l}}_{\mathbf{I}\mathbf{I}}$	$\mathbf{L}$ (m)	${\mathbf{T}}_{\mathbf{w}}$ (°C)	${\mathbf{v}}_{\mathbf{o}}$ (m/s)
1	1.0	1.0	1.0	5.0	15.0	10−5
2	0.5	1.5	1/3	5.0	12.0	10−5
3	0.5	1.5	1/3	5.0	12.0	5 × 10−5
4	1.0	1.0	1.0	5.0	5.0	10−6
5	4.0	4.0	1.0	100.0	15.0	10−5

shows five different scenarios with parameters conveniently chosen to illustrate the shape of the resulting local steady-state temperature field, as shown in Figure 2. In all cases, there is clearly a region where the largest horizontal thermal gradient changes occur, which corresponds to the inflow region. Larger vertical gradients may appear in this region, according to the temperature conditions at the contours, but such gradients remain approximately constant and tend to zero moving away from the inflow region. Also, it is immediately observed in all scenarios that the vertical temperature profile tends to be linear away from the inflow region. Finally, it is possible to appreciate that, in all scenarios, the horizontal temperature profile for each depth has substantially different changes depending also on the boundary temperatures. In the latter profiles, as expected, the temperature ${\mathrm{T}}_{\mathrm{x}=0,\mathrm{y},\mathrm{s}\mathrm{t}}$ is that of the contour itself, while the temperature at a sufficiently distant horizontal position is ${\mathrm{T}}_{\mathrm{x}\approx \mathrm{L},\mathrm{y},\mathrm{s}\mathrm{t}}=\left(\frac{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\right)\mathrm{y}$, a value independent of the water velocity.

If scenario 1 is taken as a reference, the temperature field of scenario 2 differs essentially in the profiles corresponding to the inflow region of layer I due to the change in the value of the $\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}$ ratio. However, scenario 3 shows a much larger temperature profile development region because the velocity in this scenario is five times higher than in scenarios 1 and 2. Scenario 5 is again identical to scenario 1 except for the water inflow temperature, which is reflected in the layer II inflow region. However, the comparison between scenarios 1 and 4 shows that the length of development of the profiles is approximately the same. Finally, scenario 5, for which it has been necessary to extend its length to 100 m, illustrates the sensitive influence of the layer thicknesses on the horizontal length of the temperature profile development.

To reach steady-state solutions of the previous scenarios, two transient processes have to take place: (i) that of pure diffusion, of duration (${{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{*}$), which takes place in layer I, whose lower contour—in the inlet region—does not have a constant temperature but strongly depends on ${\mathrm{v}}_{\mathrm{o}}$ and on the inlet temperature of the water, and (ii) the advection, of duration (${{\mathrm{t}}_{\mathrm{a}\mathrm{d}\mathrm{v}}}^{*}$), required by the water particles to travel the length where the profiles develop, ${{\mathrm{l}}_{\mathrm{x}}}^{*}$. The values of ${{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{*}$ and ${{\mathrm{t}}_{\mathrm{a}\mathrm{d}\mathrm{v}}}^{\mathrm{*}}$ can be estimated from the expressions

$${{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{*}~\frac{{\left(\raisebox{1ex}{${\mathrm{l}}_{\mathrm{I}}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^2}{\mathsf{\alpha }},\hspace{1em}\hspace{1em}{{\mathrm{t}}_{\mathrm{a}\mathrm{d}\mathrm{v}}}^{\mathrm{*}}=\frac{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}{{\mathrm{v}}_{\mathrm{o}}}$$

(12)

These values are not correlated with each other as they depend on different parameters. These times must have elapsed to ensure that the temperature profiles are indeed stationary. These times can be very long when ${\mathrm{l}}_{\mathrm{I}}$ is large, as for the case of ${{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{*}$, or when groundwater velocities are very small, as for the case of ${{\mathrm{t}}_{\mathrm{a}\mathrm{d}\mathrm{v}}}^{\mathrm{*}}$. Note that the range of groundwater velocities, which depends on the hydraulic conductivity of the porous media, is very wide, [10⁻⁵–10⁻⁷] m/s.

Assuming a constant thermal diffusivity of value $\mathsf{\alpha }$ = 10⁻⁶ m²/s, which is very approximate for most soils [26,27,28], and common scenarios with different values of parameters ${\mathrm{l}}_{\mathrm{I}}$ and ${\mathrm{v}}_{\mathrm{o}}$,

Table 2. Values of ${{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{\mathrm{*}}$ in ten scenarios with different values of parameters l_I and v_o.

Scenario	${\mathbf{l}}_{\mathbf{I}}$ (m)	${\mathbf{v}}_{\mathbf{o}}$ (m/s)	${{\mathbf{t}}_{\mathbf{d}\mathbf{i}\mathbf{f}}}^{\mathit{*}}$ (s)	${\mathbf{l}}_{{{\mathbf{t}}_{\mathbf{d}\mathbf{i}\mathbf{f}}}^{\mathbf{*}}}$ (m)
6	4.0	10−5	4.00 × 106	40.000
7	2.0	10−5	1.00 × 106	10.000
8	2.0	10−6	1.00 × 106	1.000
9	2.0	10−7	1.00 × 106	0.100
10	1.0	10−5	2.50 × 105	2.500
11	1.0	10−5	2.50 × 105	0.250
12	1.0	10−7	2.50 × 105	0.025
13	0.5	10−5	6.25 × 104	0.625
14	0.5	10−6	6.25 × 104	0.063
15	0.5	10−7	6.25 × 104	0.006

shows the value of the diffusion time (${{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{*}$) and the length of aquifer (${\mathrm{l}}_{{{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{\mathrm{*}}}$) that the water would have traveled in this time, animated by the advection velocity ${\mathrm{v}}_{\mathrm{o}}$.

On the one hand, the diffusion time ${{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{*}$, relatively small (in the order of a few days) for thicknesses of ${\mathrm{l}}_{\mathrm{I}}$ below 2 m, increases noticeably (≈50 days) for thicknesses of 4 m, and much more for larger thicknesses in less frequent real scenarios.

On the other hand, the length ${\mathrm{l}}_{{{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{\mathrm{*}}}$, negligible or very small for ${\mathrm{l}}_{\mathrm{I}}$ thicknesses up to 2 m, could be much larger for greater thicknesses. As we will see later, it will always be the case that ${{\mathrm{l}}_{\mathrm{x}}}^{*}$ > ${\mathrm{l}}_{{{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{\mathrm{*}}}$ and ${{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{*}$ < ${\mathsf{\tau }}^{\mathrm{*}}$ (the characteristic time of the whole process to which we will refer later). The steady-state profiles in the region x < ${\mathrm{l}}_{{{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{\mathrm{*}}}$ have been stabilized for t = ${{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{*}$, while those farther than ${\mathrm{l}}_{{{\mathrm{t}}_{\mathrm{d}\mathrm{i}\mathrm{f}}}^{\mathrm{*}}}$ have not begun their development. In summary, while T–y profiles develop faster the closer the vertical is to the inflow region, such profiles are more changeable in the vicinity of the inflow region than beyond it, due to the influence of the water inflow temperature.

4. Obtaining Dimensionless Groups and Dependency Relationships

To determine the minimum number of dimensionless groups governing the solution of the problem (pi theorem), the protocol of dimensionless governing equations and boundary conditions will be followed, a technique whose application has proven its effectiveness and reliability in complex benchmark scenarios such as the Henry, Elder and Bénard [29], Graetz [30] and other flow and transport problems [31].

The non-dimensionalization of equations in partial derivatives requires an adequate choice of references for the dependent and independent variables in order to ensure that the range of values of these variables, once made dimensionless, is confined to the interval [0, 1], i.e., they are normalized. This ensures two hypotheses: (i) that by averaging the dimensionless equations over the entire range of values of these variables—as well as by averaging their derivatives—values are obtained that can approximate unity or an order of magnitude unity, and (ii) that the equation, once the previous approximation has been incorporated, is characterized by a group of as many coefficients as there are terms in the equation. The pairs of independent quotients that can be chosen from among these coefficients are the dimensionless groups sought.

In problems such as the one addressed in this paper, or in others of the same nature, such as Bénard’s, there are no explicit references in their setting that allow immediate references to be chosen for the independent variables. This is the case for the variables horizontal position ($\mathrm{x}$) and time ($\mathrm{t}$). In these cases, it is necessary to introduce implicit references derived from the physical process under study, such as, in this case: the horizontal extension (${{\mathrm{l}}_{\mathrm{x}}}^{*}$) of the region of development of the temperature profile, for the position variable, and a characteristic time for the duration of the process (${\mathsf{\tau }}^{\mathrm{*}}$) for the time variable. These are hidden references which are incorporated into the dimensionless equation and which, after the non-dimensionalization process, are incorporated into one or more of the resulting dimensionless groups.

Eventually, a total number of dimensionless groups will emerge which is the sum of the groups from the governing equations—as many as the number of terms in the equations minus one—plus the groups from the boundary conditions. The choice of these groups is not unique since, by means of simple operations between them, new sets of equivalent groups can be formed. Thus, it is possible and desirable to adopt a final choice in which the hidden references introduced in the process appear separately in a single dimensionless group. This results in distinguishing between groups that do not contain these references, consisting only of known parameters of the problem, and others that do. The former will be the only groups on which the solution of any unknown quantity expressed in dimensionless form depends, including the hidden quantities themselves that have been used as references. If we are dealing with local or non-stationary unknowns, such as the temperature field $\mathrm{T}(\mathrm{x},\mathrm{y},\mathrm{t})$, the dimensionless position and time monomials must be added to the aforementioned groups without unknowns.

In the first place, then, to make the variables $\mathrm{x}$, $\mathrm{y}$, $\mathrm{t}$ and $\mathrm{T}$ dimensionless, the magnitudes ${{\mathrm{l}}_{\mathrm{x}}}^{*}$, ${\mathrm{l}}_{\mathrm{I}}$, ${\mathrm{l}}_{\mathrm{I}\mathrm{I}}$, ${\mathsf{\tau }}^{\mathrm{*}}$ and ${\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}$, respectively, have be chosen as references. We will precisely define ${{\mathrm{l}}_{\mathrm{x}}}^{*}$ as the horizontal extension of the aquifer for which the temperature on the line $\mathrm{y}$ = ${\mathrm{l}}_{\mathrm{I}}$ has reached 90% of its final steady-state value, ${\mathrm{T}}_{\mathrm{m}}={\mathrm{T}}_{\mathrm{s}\mathrm{t},\mathrm{y}={\mathrm{l}}_{\mathrm{I}},\mathrm{n}\mathrm{o} \mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}}=\left(\frac{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\right){\mathrm{l}}_{\mathrm{I}}$. The choice of the line $\mathrm{y}$ = ${\mathrm{l}}_{\mathrm{I}}$ is justified because it is along this line that temperatures are in steady–state conditions later—thanks to the combined processes of conduction and convection around the boundary between layers—which ensures that the entire transient process is covered, while the choice of the percentage is a compromise between the accuracy of the results and the long computational times that would require scenarios of greater extent. The characteristic time (${\mathsf{\tau }}^{\mathrm{*}}$) is defined as the time required for the temperature corresponding to 95% of its value (0.95·${\mathrm{T}}_{\mathrm{m}}$) to be reached at point ($\mathrm{x}$ = ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$, ${\mathrm{y}}_{\mathrm{I}}$ = ${\mathrm{l}}_{\mathrm{I}})$, a time that is essentially determined by the diffusive process in the upper layer.

Thus, the variables ${\mathrm{x}}^{\prime }$, ${\mathrm{y}}_{\mathrm{I}}^{\prime }$, ${\mathrm{y}}_{\mathrm{I}\mathrm{I}}^{\prime }$ and ${\mathrm{T}}^{\prime }$ will be defined in the following dimensionless and normalized form:

$${\mathrm{x}}^{\prime }=\frac{\mathrm{x}}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}\hspace{1em}\hspace{1em}{\mathrm{y}}_{\mathrm{I}}^{\prime }=\frac{\mathrm{y}}{{\mathrm{l}}_{\mathrm{I}}}\hspace{1em}\hspace{1em}{\mathrm{y}}_{\mathrm{I}\mathrm{I}}^{\prime }=\frac{\mathrm{y}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\hspace{1em}\hspace{1em}{\mathrm{t}}^{\prime }=\frac{\mathrm{t}}{{\mathsf{\tau }}^{\mathrm{*}}}\hspace{1em}\hspace{1em}{\mathrm{T}}^{\prime }=\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}$$

(13)

Their introduction into governing Equations (1) and (2) leads to the dimensionless equations:

$$\mathrm{L}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r} \mathrm{I}\hspace{1em}\left[\frac{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}{{\left({{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}\right)}^2}\right]\frac{{\partial }^2{\mathrm{T}}^{\prime }}{\partial {\left({\mathrm{x}}^{\prime }\right)}^2}+\left[\frac{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}{{\left({\mathrm{l}}_{\mathrm{I}}\right)}^2}\right]\frac{{\partial }^2{\mathrm{T}}^{\prime }}{\partial {\left({\mathrm{y}}_{\mathrm{I}}^{\prime }\right)}^2}=\left[\frac{\mathsf{\alpha }\left({\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}\right)}{{\mathsf{\tau }}^{\mathrm{*}}}\right]\frac{\partial {\mathrm{T}}^{\prime }}{\partial {\mathrm{t}}^{\prime }}$$

(14)

$$\mathrm{L}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r} \mathrm{I}\mathrm{I}\hspace{1em}\left[\frac{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}{{\left({{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}\right)}^2}\right]\frac{{\partial }^2{\mathrm{T}}^{\prime }}{\partial {\left({\mathrm{x}}^{\prime }\right)}^2}+\left[\frac{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}{{\left({\mathrm{l}}_{\mathrm{I}\mathrm{I}}\right)}^2}\right]\frac{{\partial }^2{\mathrm{T}}^{\prime }}{\partial {\left({\mathrm{y}}_{\mathrm{I}\mathrm{I}}^{\prime }\right)}^2}+\left[\frac{\mathsf{\alpha }{\mathrm{v}}_{\mathrm{o}\left({\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}\right)}}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}\right]\frac{\partial {\mathrm{T}}^{\prime }}{\partial {\mathrm{x}}^{\prime }}=\left[\frac{\mathsf{\alpha }}{{\mathsf{\tau }}^{\mathrm{*}}}\right]\frac{\partial {\mathrm{T}}^{\prime }}{\partial {\mathrm{t}}^{\prime }}$$

(15)

In these expressions, the coefficients of the derivative terms enclosed in brackets are those that characterize the solution of the problem thanks to the approximation to the unit order of magnitude of the derivative terms. The independent ratios between these coefficients are the dimensionless groups sought. After simplifications and combinations between groups so that the unknowns ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$ and ${\mathsf{\tau }}^{\mathrm{*}}$ appear in only one of the groups, the following independent groups are obtained:

$${\mathsf{\pi }}_1=\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\hspace{1em}\hspace{1em}{\mathsf{\pi }}_2=\frac{{\mathrm{v}}_{\mathrm{o}}{\mathrm{l}}_{\mathrm{I}}}{\mathsf{\alpha }}\hspace{1em}{\mathsf{\pi }}_{3\left({{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}\right)}=\frac{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}{{\mathrm{l}}_{\mathrm{I}}}\hspace{1em}\hspace{1em}{\mathsf{\pi }}_{4\left({\mathsf{\tau }}^{\mathrm{*}}\right)}=\frac{{{{\mathrm{l}}_{\mathrm{I}}}^{\mathrm{*}}}^2}{\mathsf{\alpha }{\mathsf{\tau }}^{\mathrm{*}}}$$

(16)

Of this final set of groups, the first two do not contain unknowns and are, therefore, groups that intervene in the functional dependencies of any unknown of the problem. Finally, the existence of three different boundary temperatures, ${\mathrm{T}}_{\mathrm{s}}$, ${\mathrm{T}}_{\mathrm{b}}$ and ${\mathrm{T}}_{\mathrm{w}}$, gives rise to a new dimensionless group independent of the previous ones. Among the possible options, and given that, in general (although not necessarily), ${\mathrm{T}}_{\mathrm{s}}\ge {\mathrm{T}}_{\mathrm{w}}\ge {\mathrm{T}}_{\mathrm{b}}$, we will define it in the form ${\mathsf{\pi }}_5=\frac{{\mathrm{T}}_{\mathrm{w}}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}$. This group does not affect the solutions of ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$ and ${\mathsf{\tau }}^{\mathrm{*}}$, obviously independent of the boundary temperatures, but rather the values of the global temperature field in the whole domain, and hence the shape of its transient and stationary profiles. On the other hand, the group ${\mathsf{\pi }}_2$, which contains the velocity ${\mathrm{v}}_{\mathrm{o}}$ and therefore the effect of the advective term in layer II, does not intervene in the dependence of ${\mathsf{\pi }}_{4\left({\mathsf{\tau }}^{\mathrm{*}}\right)}$ since ${\mathsf{\tau }}^{\mathrm{*}}$ has been defined as the time associated to the diffusion in layer I, time that characterizes the complete development of temperature profiles. In soils where ${\left({\mathsf{\rho }}_{\mathrm{e}}{\mathrm{c}}_{\mathrm{e}}\right)}_{\mathrm{w}}$ ≠ ${\mathsf{\rho }}_{\mathrm{e}}{\mathrm{c}}_{\mathrm{e}}$, no new dimensionless group emerges, and the group ${\mathsf{\pi }}_2$ appears in the form ${\mathsf{\pi }}_2=\frac{{\mathrm{v}}_{\mathrm{o}}{\mathrm{l}}_{\mathrm{I}}}{\mathsf{\alpha }}\frac{{\mathsf{\rho }}_{\mathrm{e},\mathrm{w}}{\mathrm{c}}_{\mathrm{e},\mathrm{w}}}{{\mathsf{\rho }}_{\mathrm{e}}{\mathrm{c}}_{\mathrm{e}}}$, while the rest of the groups are the same.

According to the pi theorem, the dependencies that can be written between monomials with and without unknowns are ${\mathsf{\pi }}_3={\mathsf{\Psi }}_1({\mathsf{\pi }}_1$, ${\mathsf{\pi }}_2)$ and ${\mathsf{\pi }}_4={\mathsf{\Psi }}_2\left({\mathsf{\pi }}_1\right)$, or, in terms of ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$ and ${\mathsf{\tau }}^{\mathrm{*}}$:

$${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}=\left({\mathrm{l}}_{\mathrm{I}}\right){\mathsf{\Psi }}_1\left\{\frac{{\mathrm{v}}_{\mathrm{o}}{\mathrm{l}}_{\mathrm{I}}}{\mathsf{\alpha }},\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\right\},$$

(17a)

$${\mathsf{\tau }}^{\mathrm{*}}=\frac{{ {\mathrm{l}}_{\mathrm{I}}}^2}{\mathsf{\alpha }}{\mathsf{\Psi }}_2\left\{\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\right\}$$

(17b)

The unknown of greatest interest is the steady-state vertical temperature profile for each horizontal position, $\mathrm{P}{(\mathrm{T},\mathrm{y})}_{\mathrm{x}}$, or, in dimensionless form, $\mathrm{P}{\left\{\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}},\frac{\mathrm{y}}{{\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\right\}}_{\frac{\mathrm{x}}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}}$. For this unknown, the horizontal position is limited to $\mathrm{x}$ ≤ ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$ since, beyond this region, the profiles do not change.

$$\mathrm{P}{\left\{\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}},\frac{\mathrm{y}}{{\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}}} \right\}}_{\frac{\mathrm{x}}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}}={\mathsf{\Psi }}_3\left\{\frac{\mathrm{x}}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}},\frac{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{w}}-{\mathrm{T}}_{\mathrm{b}}},\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}} \right\}$$

(18)

In this expression, it is not necessary to include the monomial $\frac{{\mathrm{v}}_{\mathrm{o}}\left({\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}}\right)}{\mathsf{\alpha }}$ since its influence on the solution is captured by the parameter ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$. This is the main advantage of having introduced ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$ in the equations, which, in turn, makes it possible, as we will see later, for the solutions to be represented by means of type curves. However, the group $\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}$ does appear in $\mathrm{P}{\left\{\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}},\frac{\mathrm{y}}{{\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\right\}}_{\frac{\mathrm{x}}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}}$ since this is a group coming from boundary conditions. Finally, and although it is not of major practical interest, we can write the expression for the local dependence of the steady-state temperature field, ${\mathrm{T}}_{\mathrm{s}\mathrm{t},(\mathrm{x},\mathrm{y})}$. For this, it is enough to add to the argument ${\mathsf{\pi }}_5$ the relative positions $\frac{\mathrm{y}}{{\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}$ and $\frac{\mathrm{x}}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}$. Thus,

$$\mathrm{T}(\mathrm{x},\mathrm{y})={\mathsf{\Psi }}_4\left\{\frac{{\mathrm{T}}_{\mathrm{w}}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}},\frac{\mathrm{y}}{{\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}}},\frac{\mathrm{x}}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}},\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\right\}$$

(19)

5. Verification of Results and Representation of the Type Curves

The monomials deduced in the previous section can be easily verified by checking that Expressions (17)–(19) give rise to the same solutions for scenarios with the same numerical values of the monomials and different values for the individual parameters involved in their expressions. The geometric values of ${\mathrm{l}}_{\mathrm{I}}$ and ${\mathrm{l}}_{\mathrm{I}\mathrm{I}}$ are illustrative and sufficiently changeable for the monomials to be clearly verified.

Figure 3 shows the local temperature field of the three scenarios. Scenarios 16 and 17 have the same solution for the length of profile development (${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}})$ and thus the same profiles at each horizontal position. Their temperature fields, shown here in more detail, are identical to that of scenario 1 in Table 1. Scenario 18 is twice as long as the previous scenarios, a result consistent with Expression (17a), because ${\mathrm{l}}_{\mathrm{I}}$ has been doubled. Its temperature field is shown in the same figure, and the new value of ${\mathrm{l}}_{\mathrm{I}}$ is visible. The horizontal profile of temperatures at $\mathrm{y}={\mathrm{l}}_{\mathrm{I}}$, the line on which the characteristic length is measured, for the three scenarios is shown in Figure 4. As expected, this profile is identical for scenarios 16 and 17, which have the same ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$, and clearly different for scenario 18, which has an ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$ of double value. Thus, it is immediately seen that the temperatures at the same relative positions, $\frac{\mathrm{x}}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}$, are identical in the three scenarios, as predicted by Expression (19). The results for ${\mathsf{\tau }}^{\mathrm{*}}$ are also consistent with Expression (17b) for this time. In scenario 17, ${\mathsf{\tau }}^{\mathrm{*}}$ is halved compared to its value in scenario 16 because the factor $\frac{{{\mathrm{l}}_{\mathrm{I}}}^2}{\mathsf{\alpha }}$ is also halved, while, in scenario 18, this time is doubled compared to scenario 16 by doubling that factor.

Figure 3 shows the local temperature field of the three scenarios. Scenarios 16 and 17 have the same solution for the length of profile development ${\mathrm{l}}_{\mathrm{x}}^{\mathrm{*}}$ and thus the same profiles at each horizontal position. Their temperature fields, shown here in more detail, are identical to that of scenario 1 in Table 1. Scenario 18 is twice as long as the previous ones, a result consistent with Expression (17a) due to the doubling of ${\mathrm{l}}_{\mathrm{I}}$. Its temperature field is shown in the same figure, and the new value of ${\mathrm{l}}_{\mathrm{I}}$ can be seen. The horizontal temperature profile at $\mathrm{y}={\mathrm{l}}_{\mathrm{I}}$, the line on which the characteristic length is measured, for the three scenarios is shown in Figure 4. As expected, this profile is identical for scenarios 16 and 17, which have the same ${\mathrm{l}}_{\mathrm{x}}^{\mathrm{*}}$, and clearly different for scenario 18, which presents an ${\mathrm{l}}_{\mathrm{x}}^{\mathrm{*}}$ of double value. Thus, it is immediately seen that the temperatures at the same relative positions, $\mathrm{x}/{\mathrm{l}}_{\mathrm{x}}^{\mathrm{*}}$, are identical in the three scenarios, as predicted by Expression (19). The results for ${\tau }^{*}$ are likewise consistent with Expression (17b) for this time. In scenario 17, ${\tau }^{*}$ is halved with respect to its value in scenario 16 by the factor $\frac{{ {\mathrm{l}}_{\mathrm{I}}}^2}{\mathsf{\alpha }}$ also being halved, while, in scenario 18, this time doubles its value with respect to scenario 16 by doubling $\frac{{ {\mathrm{l}}_{\mathrm{I}}}^2}{\mathsf{\alpha }}$. These times, of approximately 11 days for scenarios 16 and 17 and 5.5 days for scenario 18, as mentioned above, are complementary information for the researcher to provide guidance on whether or not the steady-state temperature field or $\mathrm{T}–\mathrm{y}$ profiles have been reached.

The type curves of the dependencies (17) and (18) are shown below. We do not include the type curves of the temperature field because of their limited usefulness. The curves of the expressions ${\mathrm{l}}_{\mathrm{x}}^{\mathrm{*}}$ and ${\mathsf{\tau }}^{\mathrm{*}}$ are obtained point by point by a large number of numerical simulations, shown in Figure 5 and Figure 6, respectively. The first one represents ${\mathrm{l}}_{\mathrm{x}}^{\mathrm{*}}$ in its dimensionless form, ${\mathsf{\pi }}_{3\left({{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}\right)}=\frac{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}{{\mathrm{l}}_{\mathrm{I}}}$, as a function ${\mathsf{\pi }}_2=\frac{{\mathrm{v}}_{\mathrm{o}}{\mathrm{l}}_{\mathrm{I}}}{\mathsf{\alpha }}$ using ${\mathsf{\pi }}_1=\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}$ as the abacus parameter. The second is a single universal curve representing ${\mathsf{\tau }}^{\mathrm{*}}$ in its dimensionless form versus ${\mathsf{\pi }}_1=\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}$.

Regarding the hidden magnitude ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$, shown in Figure 5, its value increases appreciably as the ratio $\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}$ decreases. This can be explained by the fact that, for identical values of ${\mathrm{l}}_{\mathrm{I}}$ and ${\mathsf{\pi }}_2=\frac{{\mathrm{v}}_{\mathrm{o}}{\mathrm{l}}_{\mathrm{I}}}{\mathsf{\alpha }}$, an increase in ${\mathrm{l}}_{\mathrm{I}}$ implies that the profiles need a larger extent for their development.

Thanks to the introduction of ${{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}$ in the dimensionless process to obtain the dimensionless groups, the temperature profiles $\mathrm{P}{\left\{\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}},\frac{\mathrm{y}}{{\mathrm{l}}_{\mathrm{I}}+{\mathrm{l}}_{\mathrm{I}\mathrm{I}}},\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}\right\}}_{\frac{x}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}}$, Expression (18), depend on three monomials: the horizontal position, expressed in its dimensionless form $\frac{\mathrm{x}}{{{\mathrm{l}}_{\mathrm{x}}}^{\mathrm{*}}}$, the temperature monomial $\frac{{\mathrm{T}}_{\mathrm{w}}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}$ and the ratio $\frac{{\mathrm{l}}_{\mathrm{I}}}{{\mathrm{l}}_{\mathrm{I}\mathrm{I}}}$. Given the huge range of values that the dimensionless set of temperatures could take, it is not possible to cover the whole range, and only two values have been chosen for the group, $\frac{{\mathrm{T}}_{\mathrm{w}}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}$ = 0 and 0.75. The profiles are shown in Figure 7.

As can be seen from this figure, the profiles separate from each other more sharply in the region where the water flows and are significantly farther away from the steady-state linear profile the closer we are to the water inflow region. For the same vertical position, temperatures are always below their stationary value when $\frac{{\mathrm{T}}_{\mathrm{w}}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}$ = 0, i.e., when ${\mathrm{T}}_{\mathrm{w}}={\mathrm{T}}_{\mathrm{b}}$, which is a common scenario. However, they remain above their stationary value for the value $\frac{{\mathrm{T}}_{\mathrm{w}}-{\mathrm{T}}_{\mathrm{b}}}{{\mathrm{T}}_{\mathrm{s}}-{\mathrm{T}}_{\mathrm{b}}}$ = 0.75. No values are included for the case ${\mathrm{T}}_{\mathrm{w}}<{\mathrm{T}}_{\mathrm{b}}$, but, according to the temperature fields in Figure 2 (scenario 4), the profiles would have more complex shapes with maxima or minima temperatures within each profile. In any case, these are always profiles that allow a clear discrimination and therefore could be very useful for water flow estimations.

6. Conclusions

The non-dimensionalization of the governing equations of groundwater flow and heat transport in porous soils in which there is a dry layer above the layer with water flow has made it possible to formally deduce the dimensionless groups that govern the solutions to the problem. The introduction of normalized dimensionless variables in the governing equations makes it possible to simplify the dimensionless mathematical model, to establish the coefficients on which it depends and, from there, to obtain the dimensionless groups sought.

Among the solutions of greatest interest is that of universal temperature–depth profiles at each horizontal position of the porous medium. Such profiles, which always are clearly discriminated in the temperature profile development region, would allow direct estimation of the water flow velocity from experimental measurements, a difficult and costly measurement by other methods. These applications of the curves may be the subject of further work, as well as the construction of universal type curves for field case problems. The dimensionless groups obtained by this precise protocol are verified by numerical simulations of different scenarios, checking that they produce the same solutions for identical numerical values of the groups, even if the values of some of the individual parameters change in each scenario.

The introduction of two characteristic quantities of the problem as references for making the horizontal position and time variables dimensionless, as well as the choice of suitable references for the vertical position in each layer, leads to precise dimensionless groups being obtained which, suitably expressed, allow the universal dependencies of any quantity of interest with the deduced groups to be established. In particular, the dependence on the value of the horizontal length where the steady-state temperature profile progressively develops until it reaches a constant linear form can be deduced. This dependence is shown in an abacus of type curves. In relation to the characteristic times, it is very useful information in hydrogeology to know the duration of these processes and therefore to know whether the steady-state profiles have been reached or whether they are in a phase of local development of these profiles.

Thanks to the introduction of the development length of the steady-state temperature profiles, such profiles depend on the horizontal position relative to this length and can therefore also be represented by simple type curves with two dependencies: the ratio between the thickness of the layer and the dimensionless group formed by the boundary temperatures of the problem. Without the introduction of this length, the universal profiles would have depended on three dimensionless groups, making it impossible to represent them by means of practically useful type curves.

Among the type curves derived from this problem, there are: (i) the abacus of the extension of the horizontal length of development of the steady-state profiles, (ii) the type curve of the characteristic time to develop such profiles, (iii) the type curves of vertical temperature profiles corresponding to each horizontal position for different values of the ratio between the thickness of the layer and two values of the dimensionless group formed by the three boundary temperatures.