# Porous Medium Typology Influence on the Scaling Laws of Confined Aquifer Characteristic Parameters

^{*}

## Abstract

**:**

## 1. Introduction

_{e}) so as to take into account also the interconnection conditions of the voids, which have a decisive influence on water flow and on the mass transport phenomena that are located in them [10,15]. Moreover, it seems possible, albeit in an uncertain manner, to affirm the existence of a scalar behavior even for the porosity; however, this does not always consist in a tendency to increase the porosity with the scale but sometimes with a decreasing trend with increasing of this, as verified by some researchers [16,17]. In any case, considerable caution should be used in asserting the existence of a scalar behavior of porosity, due to the great complexity of the phenomena that determine the influence of the heterogeneity of the porous medium [18]. In some studies, the porosity—or, more appropriately, the effective one—is assumed as a scale parameter [7,10]. In this way, it is possible to identify the modalities of the hydraulic conductivity variation when the porosity changes and the corresponding scaling law constitutes a valid alternative to the numerous empirical and semi-empirical formulas based on the grain size distribution [19,20,21,22,23]. The importance of determining a scaling law for a given parameter is also due to the fact that this allows identification of the distribution of the considered parameter in the spatial context taken into consideration, avoiding the use of traditional geostatistical methods [24,25]. However, in the experimental verification of the scaling behavior of the hydraulic conductivity, it is necessary to verify that the significant values of this parameter were obtained in contexts and in conditions to guarantee a correct comparison, namely under identical flow conditions. This is particularly important if a series of K values measured at different scales, in the field and in the laboratory on samples of porous media, are considered. In fact, on the latter, the measure generally supplies the vertical hydraulic conductivity value, while field tests are used to obtain the value of the horizontal one. Therefore, in such cases, the K values must be standardized, trying to obtain the values of the horizontal hydraulic conductivity using suitable methods for the measurements carried out on soil samples in laboratory, often based on the knowledge of the anisotropy of the porous medium under examination [10,23,26,27].

_{e}, this last parameter was assumed as a scale parameter, identifying a single K variation law when n

_{e}, namely the porous medium, changes. The scaling law thus obtained represents a new relation able to give the value of K, known the value of n

_{e}, within the porous media examined and those of similar characteristics, therefore describable also in terms of grain size distribution. The usefulness of a relationship of this type is evident, as it allows users to make a fast and reliable estimate of the hydraulic conductivity, even if limited to the ambit, although vast, of the investigated soils and taking into account some conditions and restrictions specified below.

_{e}scaling laws and a new variation law of K with n

_{e}performing the appropriate comparisons with the results obtained using the grain size distribution methodologies.

## 2. Materials

#### 2.1. Experimental Set-Up, Equipment and Tests Execution

_{s}, as shown for each configuration in Table 1. Table 1 also shows the undisturbed hydraulic head related to the injection well. In order to ensure the confined formation, a thin impermeable plastic panel was laid down over each configuration and then covered with additional sandy material. The impermeability of the confined formation was verified by means of some preliminary tests [25]. In order to guarantee a constant hydraulic head condition during the tests, a perimetric chamber was built along the metal box lateral boundaries, and the perimetric chamber was connected with two external loading reservoirs. More details are given in [28]. Several slug tests were performed for each soil configuration using an injection volume V into the central well of 0.03 L, 0.04 L, 0.06 L, 0.07 L, 0.08 L and 0.09 L, respectively. The initial undisturbed hydraulic head varied between 0.32 m and 0.38 m to ensure that the confined formation remained under pressure, and the complete restoration of the initial loading conditions between tests was always verified [28,29]. In order to measure head changes during the slug tests, 10 submersible transducers, model Druck PDCR1830 (for more details, see [30,31,32]), were positioned at the bottom of each well. The pressure data were recorded using a measurement frequency of 100 Hz and filtered using the Mexican hat wavelet transform to eliminate the experimental high-frequency noise, as suggested by the authors of [28].

#### 2.2. Soil Configurations

_{10}was assumed as an effective diameter, the uniformity coefficient (U = d

_{60}/d

_{10}) and the values of total and effective porosity [24,33]. The graph in Figure 3 shows the granulometric curves characterizing the four types of porous media used for the construction of the corresponding confined aquifers. The highest gravel content is present in the porous medium of type II with 27.70%, while the lower content is relative to that of the type I with 12.01%. Regarding the sand, the highest content was found in the first porous medium type, which contains about 88%, while the lower content of this component is found in the porous medium of type IV with 56.10%. The maximum silt content was found in the configuration of the type IV with 16.40%, while the minimum was found in the configuration type I with 0.60%.

_{10}tends to decrease, varying from 0.19 mm to 0.0055 mm, while the uniformity coefficient tends to increase, going from 5.21 to 163.63. The total porosity for type I configuration assumes the highest value, equal to 37.6%, while, for the other three configurations, it assumes lower and comparable values, ranging between 27.3% and 29.3%. The effective porosity tends to increase, passing from configuration I to IV, assuming values ranging between 5.60% and 19%.

## 3. Methods

#### 3.1. Data Analysis

_{0}the initial variation in the well of the hydraulic head (L), h the variation of the hydraulic head from the undisturbed value at a generic radial distance (L), r

_{w}the effective radius of well screen (L), r

_{c}the effective radius of well casing (L), r the radial distance (L), t the time (T), K

_{r}the component of the hydraulic conductivity in radial direction (LT

^{−1}), S

_{s}the specific storage (L

^{−1}) and B the thickness of the aquifer (L). The relationships (1) and (2) state that at time t = 0, the hydraulic head variation is zero everywhere outside the well and equal to H

_{0}inside the well. Moreover, the relationship (3) establishes that, for r approaching to infinity, the variation of the hydraulic head approaches zero, while, for the relation (4), the hydraulic head in the immediate vicinity of the well is equal to that inside it for t > 0. Finally, the boundary condition (5) requires compliance with the continuity principle for incoming and outgoing flows from the aquifer—well system. In compliance with these boundary conditions, the mathematical model suggested by the authors of [34] is expressed by the following relation:

#### 3.2. Scaling Analysis

^{−1})), x is the scale parameter (as the representative scale dimension (L)), a is a parameter that takes into account the structure and the heterogeneity of the porous medium (having the dimensions to ensure congruence) and b (-) is the scaling index (also called crowding index), which is related to the type of flow in the porous medium [5]. The use of a power-type law to represent the scaling behavior of a parameter requires the fulfillment of the so-called lacunarity condition. This condition consists in identifying minimum and maximum cut-off limits within the range of values of the parameter under examination, within which the hydrological process remains correctly defined. This implies that the scaling parameter considered is representative of a scale-invariant phenomenon [36]. These cut-off limits can be determined identifying, within the investigation context, the range showing the maximum value of the determination coefficient (R

^{2}), as treated in numerous studies in the literature [9,25,36]. In the following, the scale invariance of the phenomenon and the condition of lacunarity were assumed, and on this basis, the scaling behaviors of K and n

_{e}were verified using Equation (7). Initially, this equation was used assuming as scaling parameter the radius of influence (R), the values of which were measured experimentally for each value of K. However, it must be clearly understood that, with the term radius of influence, one means the dimension that characterizes the confined aquifer volume involved in the measurement of the parameter under examination—for example, K—taking into account that the measured value of this parameter is representative only of this volume and that it changes if this varies. This volume can be assumed to be of cylindrical shape, and an estimate of its amplitude can be provided by R. Subsequently, the same effective porosity was assumed as a scaling parameter. Specifically, the total porosity was measured in the laboratory using the densiometric method by the following relationship [37,38]:

_{bulk}is the bulk mass density (ML

^{−3}) and ρ

_{grain}the particle mass density (ML

^{−3}), while the effective one, considering the saturated medium, was obtained based on the following relationship:

^{3}) is the total volume and V

_{w}(L

^{3}) the portion of the water volume which cannot be drained by gravity [39].

#### 3.3. Grain Size Analysis

_{e}as a scaling parameter. Furthermore, other empirical and semiempirical formulas are based on the grain size distribution theory. These formulas commonly follow the general model of [40], represented by the following equation:

^{−1}), n

_{e}is the effective porosity (-), C a general coefficient (-), ν the kinematic viscosity (L

^{2}T

^{−1}), g the acceleration of gravity (LT

^{−2}), f(n

_{e}) the porosity function defining the relationship between the real and modeled porous media and d

_{e}the effective grain diameter (L).

## 4. Results and Discussion

^{−4}m/s (configuration of the type II) and a maximum value 1.34 × 10

^{−3}m/s (configuration of the type IV). The mean value varies between a minimum value of 2.45 × 10

^{−4}m/s in the configuration of the type II and a maximum equal to 1.24 × 10

^{−3}m/s in that of the type IV. The variance (VAR) assumes always very low values, with an order of magnitude between 10

^{−10}and 10

^{−8}. The standard deviation (SD) assumes values with order of magnitude between 10

^{−5}and 10

^{−4}. The standard error (SE) 10

^{−5}shows values with an order of magnitude of 10

^{−5}. The variation coefficient (VC) shows values with an order of magnitude between 10

^{−2}and 2.56 × 10

^{−1}. Similarly, the mean value of R is variable from 0.799 m to 0.90 m, with the variance (VAR) variable between a minimum equal to 0.001 and a maximum equal to 0.015, the standard deviation (SD) variable between a minimum equal to 0.039 and a maximum equal to 0.123, the standard error (SE) variable between a minimum equal to 0.016 and a maximum equal to 0.050 and the coefficient of variation (VC) variable between a minimum value of 0.043 and a maximum value of 0.154. Afterwards, assuming the radius of influence as the scale parameter, it was possible to identify the variation modalities of K with R for each of the four porous medium configurations taken into consideration, determining the corresponding scaling laws by means of power-type relations, according to Equation (7), which, for the specific case, is given by the following relation:

^{2}is relative to configuration II.

_{e}values relative to each of the configurations taken into consideration, shown in Table 2, were correlated with the radii of influence (R) identified experimentally for each injection volume considered for the slug tests, as shown in Table 3. The corresponding scaling laws, determined according to relation (7), are represented in the specific case by the following relation:

^{2}are all high, varying between a minimum value of 0.795, related to the injection volume of 0.08 L and a maximum value of 0.958, relative to the injection volume of 0.09 L. Figure 5 shows the trends of the scaling laws related to the injection volumes considered for the slug tests. The R

^{2}values shown in Table 6 and the trends of the scaling laws represented by Equation (12) and obtained for the injection volumes considered, confirm the existence of a positive scaling behavior of n

_{e}, namely that there is an increase of n

_{e}as R increases. Furthermore, the graph in Figure 5 shows that the R values, measured for the four configurations considered, fall within ranges which have amplitudes tending to decrease with increasing the injection volume considered for slug tests.

_{e}, in order to investigate the variation of K with n

_{e}, it was decided to assume a fixed value of the radius of influence R and to verify the corresponding variation modalities of the parameters under examination, assuming as a scale parameter n

_{e}, to determine the following scaling law:

_{1}= 0.8400 m, R

_{2}= 0.8573 m, R

_{3}= 0.8746 m, R

_{4}= 0.8931 m, R

_{5}= 0.9112 m and R

_{6}= 0.9305 m. For each of these values, the scaling law represented by the relation (13) was determined, and the related parameters a, b and R

^{2}are shown in Table 7.

^{−5}and 2.54 × 10

^{−5}. Additionally, the parameter b assumes very similar values, varying between 1.2513 and 1.3361, while R

^{2}assumes always high values, between a minimum of 0.859 and a maximum of 0.910. Furthermore, it can be seen that, as the fixed value of R increases, the parameter a tends to decrease and b tends to increase, as well as R

^{2}. Figure 6 shows the trends of the scaling laws represented by Equation (13) for the six fixed values of R, according to what is reported in Table 7.

_{e}already found in previous studies [7,10]. As already noted, analyzing the data shown in Table 7, Figure 6 also shows that the six scaling laws representing the variation of K with n

_{e}for the different fixed R values are very similar, even if they show a not marked tendency to accentuate the relative differences. In fact, Figure 6 highlights the tendency to diverge the straight lines representing the scaling laws for the fixed R values on the double logarithmic graph of this figure. On the basis of these results, an attempt was made to determine a single scaling law, also of the type of Equation (13), which allows to determine the variation of K with n

_{e}for all the injection volumes and for the four soils, with the configurations taken into consideration and all those similar to these. For this purpose, determining the mean values of the parameters a and b of the individual scaling laws, the only law representative of the four configurations considered was identified, defined by the following parameter values:

^{−5}and b = 1.29

_{3}and R

_{4}radii of influence. Furthermore, the other straight lines, representing the scaling laws relating to the remaining R values, are thickened to that representative of the global scaling law, and they are all inside the 95% confidence intervals. The global scaling law, represented by Equation (13) and specified by the parameters provided by (14), expresses the variability of K vs. n

_{e}. Since n

_{e}is a characteristic parameter of the porous medium, it was considered appropriate also to consider the empirical and semi-empirical equations deriving from the grain size distribution. Therefore, the general model of [40] was considered. With reference to this model, described by Equation (10), the global scaling law defined above by relations (13) and (14) can also be explained by the following expression:

_{10}(L) is the particle size for which 10% of the sample are finer than and the meaning of other symbols was already specified. This law, represented by Equation (15), presents a field of validity defined by 5.5 × 10

^{−3}mm ≤ d

_{10}≤ 0.19 mm. Although it is evident that relation (14) is simpler and more immediate than relation (15), it is undeniable that, in the presence of fluids other than water, or for values of d

_{10}not falling within the validity range of the (15); both the previous relations must be calibrated again, determining new values of the coefficients a and b of the relation (14) and also of the coefficient C and n

_{e}index of relation (10). Equation (15) was compared to one of the best-known relationships following the model of [40], which is the [19,20] equation which, valid for d

_{10}< 3 mm and not appropriate for clayey soils [44], falls within the validity range of this investigation. The graph in Figure 8 shows the trends of both these variation laws.

## 5. Conclusions

_{e}, assuming as scale parameter R, was made for each injection volume used in the execution of the slug tests. As shown by the data of Table 6 and by Figure 5, the scaling behavior of n

_{e}is also verified for all the cases examined, with an increase of this parameter with the increase of R. Furthermore, by decreasing the injection volume and the n

_{e}value in the investigation interval, the variation range of R tends to decrease. Due to the particular scale of investigation, even the scaling laws determined for n

_{e}are affected by the influence of heterogeneity with the modalities mentioned above. Afterwards, n

_{e}was assumed as a scaling parameter, and the scaling law (13) was determined for each injection volume considered. Since the laws obtained were very similar, as evidenced by the values of Table 7 and Figure 6, a single scaling law was identified, with a level of significance of 5%, representative of all the individual laws relating to each injection volume. The values of parameters a and b of this global scaling law, represented in Figure 7, are given by relation (14). Obviously, even in this case, the scaling laws reported in Table 7 and, consequently, also the one defined by relation (14) are influenced by the heterogeneity with the modalities induced by the particular scale of investigation. Relation (14) represents, therefore, a new law, K = K(n

_{e}), valid within the porous media investigated and in those with similar characteristics, namely that can be defined as coarse-grained. To highlight further the importance of the parameters characterizing the structure of porous media, the new law, K = K(n

_{e}), represented by Equation (14), was also written in terms of grain size analysis, obtaining, on the model of [40], Equation (15), which is very similar to that obtainable with the model of [19,20] and valid always for coarse-grained aquifers.

_{e}but also increase the injection volume, namely approaching the flow conditions proper to those of the field scale, as shown by the laws of Table 7 and highlighted in Figure 6 and in Figure 7. Understanding and verification of these mechanisms is of fundamental importance for a correct description of the flow and transport phenomena in porous media, for which it is desirable to increase the experimental research on these topics, with the acquisition of a greater quantity of significant data. Furthermore, it should be pointed out that the use of grain size analysis is able to provide alternative relationships, also easy to use but only usable on the basis of the structural characteristics of the considered porous medium and certainly not always able to easily define the spatial distribution of the parameter under examination, which is possible, however, using the scaling laws, thus avoiding the need to use of the traditional geostatistical methods. Therefore, even if the scaling laws depend on fewer variables than the relationships provided by the grain size analysis, they also have a limited validity range. In fact, for scaling laws, this validity range is represented by the particular scale for which they were determined. With reference to this study, the scaling laws determined on the basis of Equations (11), (12) and (13) are valid exclusively within the mesoscale of interest defined above.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Planimetric scheme of the metal box in which the confined aquifer was built, with the location of the ten wells (numbered from 1 to 10). (

**b**) Section of the metal box with the associated stratigraphic schemes.

**Figure 2.**Experimental device, with pressure transducers inserted in the wells, located in the confined aquifer, as shown in Figure 1a.

Configurations | Formation Thickness t_{s} (m) | Undisturbed Hydraulic Heads (m) |
---|---|---|

I | 0.25 | 0.38 |

II | 0.25 | 0.32 |

III | 0.25 | 0.32 |

IV | 0.22 | 0.35 |

**Table 2.**Contents in percent of gravel, sand, silt and clay and values of effective diameter, uniformity coefficient, total porosity and effective porosity for the porous media of the four configurations considered.

Textural Parameters and Porosity | Porous Media | |||
---|---|---|---|---|

Type I (%) | Type II (%) | Type III (%) | Type IV (%) | |

Gravel | 12.01 | 27.70 | 23.90 | 22.50 |

Sand | 87.39 | 71.00 | 61.00 | 56.10 |

Silt | 0.60 | 1.30 | 15.10 | 16.40 |

Clay | --- | --- | --- | 5.00 |

Effective diameter (d_{10} mm) | 0.19 | 0.16 | 0.02 | 0.0055 |

Uniformity coefficient (U = d _{60}/d_{10}) | 5.21 | 8.125 | 51.5 | 163.63 |

Total porosity (n) | 37.60 | 27.30 | 29.30 | 27.50 |

Effective porosity (n_{e}) | 5.60 | 8.60 | 13.00 | 19.00 |

**Table 3.**Hydraulic conductivity values and the corresponding radii of influence relative to each injection volume of the slug test and for each configuration.

V (L) | Type I | Type II | Type III | Type IV | ||||
---|---|---|---|---|---|---|---|---|

k (m/s) | R (m) | k (m/s) | R (m) | k (m/s) | R (m) | k (m/s) | R (m) | |

0.03 | 2.15 × 10^{−4} | 0.590 | 1.36 × 10^{−4} | 0.600 | 7.13 × 10^{−4} | 0.820 | 1.07 × 10^{−3} | 0.840 |

0.04 | 2.38 × 10^{−4} | 0.720 | 2.20 × 10^{−4} | 0.750 | 7.20 × 10^{−4} | 0.840 | 1.09 × 10^{−3} | 0.870 |

0.06 | 2.79 × 10^{−4} | 0.835 | 2.60 × 10^{−4} | 0.840 | 7.38 × 10^{−4} | 0.899 | 1.30 × 10^{−3} | 0.910 |

0.07 | 2.82 × 10^{−4} | 0.850 | 2.47 × 10^{−4} | 0.860 | 7.53 × 10^{−4} | 0.910 | 1.31 × 10^{−3} | 0.914 |

0.08 | 2.67 × 10^{−4} | 0.870 | 2.98 × 10^{−4} | 0.906 | 7.40 × 10^{−4} | 0.909 | 1.31 × 10^{−3} | 0.916 |

0.09 | 2.88 × 10^{−4} | 0.930 | 3.10 × 10^{−4} | 0.936 | 7.86 × 10^{−4} | 0.940 | 1.34 × 10^{−3} | 0.950 |

Parameters | Type I | Type II | Type III | Type IV | ||||
---|---|---|---|---|---|---|---|---|

K (m/s) | R (m) | K (m/s) | R (m) | K (m/s) | R (m) | K (m/s) | R (m) | |

min | 2.15 × 10^{−4} | 0.590 | 1.36 × 10^{−4} | 0.600 | 7.13 × 10^{−4} | 0.820 | 1.07 × 10^{−3} | 0.840 |

max | 2.88 × 10^{−4} | 0.930 | 3.10 × 10^{−4} | 0.936 | 7.86 × 10^{−4} | 0.940 | 1.34 × 10^{−3} | 0.950 |

mean | 2.61 × 10^{−4} | 0.799 | 2.45 × 10^{−4} | 0.815 | 7.42 × 10^{−4} | 0.886 | 1.24 × 10^{−3} | 0.900 |

VAR | 8.33 × 10^{−10} | 0.015 | 3.95 × 10^{−9} | 0.015 | 6.80 × 10^{−10} | 0.002 | 1.49 × 10^{−8} | 0.001 |

SD | 2.89 × 10^{−5} | 0.123 | 6.29 × 10^{−5} | 0.123 | 2.61 × 10^{−5} | 0.046 | 1.22 × 10^{−4} | 0.039 |

SE | 1.18 × 10^{−5} | 0.050 | 2.57 × 10^{−5} | 0.050 | 1.06 × 10^{−5} | 0.019 | 4.99 × 10^{−5} | 0.016 |

VC | 1.10 × 10^{−1} | 0.154 | 2.56 × 10^{−1} | 0.151 | 3.52 × 10^{−2} | 0.052 | 9.89 × 10^{−2} | 0.043 |

Kurtosis | −4.98 × 10^{−1} | 0.654 | 1.270 | 1.175 | 9.73 × 10^{−1} | −1.189 | −1.793 | −0.072 |

Skewness | −9.88 × 10^{−1} | −1.100 | −1.078 | −1.225 | 9.27 × 10^{−1} | −0.639 | −9.26 × 10^{−1} | −0.563 |

**Table 5.**Parameters a and b of the scaling laws (11) for each porous medium configurations and relative values of R

^{2}.

Soil | a | b | R^{2} |
---|---|---|---|

I | 3 × 10^{–4} | 0.669 | 0.931 |

II | 3 × 10^{–4} | 1.803 | 0.977 |

III | 8 × 10^{–4} | 0.598 | 0.825 |

IV | 1.6 × 10^{–3} | 2.215 | 0.890 |

**Table 6.**Parameters a and b of the scaling laws n

_{e}= n

_{e}(R) for each injection volume considered and the relative values of R

^{2}.

V (L) | a | b | R^{2} |
---|---|---|---|

0.03 | 25.41 | 2.520 | 0.849 |

0.04 | 39.42 | 5.715 | 0.955 |

0.06 | 48.50 | 11.058 | 0.883 |

0.07 | 54.05 | 13.217 | 0.891 |

0.08 | 88.64 | 20.314 | 0.795 |

0.09 | 394.37 | 57.668 | 0.958 |

**Table 7.**Parameters a and b of the scaling laws K = K(n

_{e}) for each fixed value of R and relative values of R

^{2}.

Section | a | b | R^{2} |
---|---|---|---|

1 | 2.54 × 10^{−5} | 1.251 | 0.859 |

2 | 2.51 × 10^{−5} | 1.268 | 0.871 |

3 | 2.48 × 10^{−5} | 1.285 | 0.881 |

4 | 2.45 × 10^{−5} | 1.302 | 0.891 |

5 | 2.41 × 10^{−5} | 1.319 | 0.900 |

6 | 2.38 × 10^{−5} | 1.336 | 0.910 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fallico, C.; Lauria, A.; Aristodemo, F.
Porous Medium Typology Influence on the Scaling Laws of Confined Aquifer Characteristic Parameters. *Water* **2020**, *12*, 1166.
https://doi.org/10.3390/w12041166

**AMA Style**

Fallico C, Lauria A, Aristodemo F.
Porous Medium Typology Influence on the Scaling Laws of Confined Aquifer Characteristic Parameters. *Water*. 2020; 12(4):1166.
https://doi.org/10.3390/w12041166

**Chicago/Turabian Style**

Fallico, Carmine, Agostino Lauria, and Francesco Aristodemo.
2020. "Porous Medium Typology Influence on the Scaling Laws of Confined Aquifer Characteristic Parameters" *Water* 12, no. 4: 1166.
https://doi.org/10.3390/w12041166