# Transmissivity Estimates by Specific Capacity Data of Some Fractured Italian Carbonate Aquifers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{c}, Equation (1)).

- Q—discharge of pumping well (L
^{3}/T); - s
_{w}—steady-state drawdown (L).

_{c}data are typically much more abundant and readily available than time-drawdown data [3]. Several studies in the literature presented empirical relationships (T = f(S

_{c})) for different aquifer types (fractured and karst carbonates, sandstones, metamorphic, volcanic, alluvial, etc.). Most of the relationships proposed are log–log equations [2,4,5,6,7,8,9,10,11,12,13] even if some linear functions were also proposed [14,15,16]. As reported by Verbovšek [16], studies of T-S

_{c}relationships for fractured or karst rocks are scarcer than those of alluvial aquifers. Mace [2]—based on data from southwest Texas (USA)—developed a log–log empirical equation, which was validated on data from Florida and Ohio aquifers, suggesting its potential application to other similar fractured karst aquifers. Central Italy is characterized by a wide outcropping of fractured and karst carbonates hosting large aquifers pumped by wells; information useful to enrich the discussion about this topic is therefore available. Coupling new data from thirty-two pumping tests (PTs) carried out in the last thirty years with those available from the literature; this work presents a new equation describing the T-S

_{c}relationship for some Italian carbonate aquifers. Results are compared and discussed with other relationships on similar aquifers, improving the knowledge on transmissivity of fractured-karst aquifers.

## 2. Materials and Methods

#### 2.1. Hydrogeological Setup and Well Characteristics

#### 2.2. Analytical Methods for Pumping Test Analysis

_{1}and r

_{2}from the pumping well. Theis [36] derived an analytical equation (Equation (3)) for the non-steady flow considering fully penetrating well in a confined homogeneous and isotropic aquifer having an infinite areal extent. This equation can also be used for unconfined aquifers if the vertical component of the flow can be neglected (Dupuit-Forchheimer assumption). The Theis’ equation has been simplified by Cooper-Jacob [37] Equation (4), truncating the infinite Taylor series, which is used for estimating the well function W(u) of Equation (3). This method is valid for smaller values of u, generally less than 0.01–0.05 [38,39,40]. A straight line through the data in s-log t plot identifies the range of validity of Cooper-Jacob equation. In the case of steady state, radial flow to a pumping well, for both confined and unconfined aquifers.

- s—drawdown (L);
- Q—pumping discharge (L
^{3}/T); - T—transmissivity (L
^{2}/T); - W(u)—Theis well function, $\mathrm{u}=\frac{{\mathrm{r}}^{2}\xb7\mathrm{S}}{4\xb7\mathrm{T}\xb7\mathrm{t}}$ (dimensionless);
- t—time from the beginning of pumping (T);
- S—storage coefficient (dimensionless);
- r—radial distance from the pumping well (L).

#### 2.3. Empirical Relationships between Specific Capacity and Transmissivity

_{c}[4,6]. Referring to a pumping test in a confined aquifer at steady-state conditions, Equation (2) can be rewritten as Equation (6), considering r

_{2}= R (radius of influence) with s

_{2}= 0 and r

_{1}= r

_{w}(radius of the well) with s

_{1}= s

_{w}(drawdown in the well). Therefore, T can be calculated by knowing S

_{c}data and the coefficient c. Equation (6) can be used for unconfined aquifers by correcting drawdown data using the Jacob equation [51]. To test the efficiency of the well, step-drawdown tests are carried out. Generally, the drawdown at low flow rates is not affected by well losses, i.e., the validity of the Equation (6) is verified. As Mace [2] reported, the number of transmissivity data available for certain aquifer types is not so many as to allow a spatial description. In many cases, well reports only indicate the specific capacity value and not the aquifer parameters. Therefore, the use of empirical relationships between S

_{c}and T values can be useful to increase the number of data for aquifer characterization, a very important task in fractured carbonate aquifers. A set of at least 25 T-S

_{c}data covering a wide range of transmissivity values is needed to build the relationships [3].

## 3. Results

_{c}values: some of T-S

_{c}pairs are taken from the literature (Table 1). These data come from pumping tests carried out during the last thirty years, previously not systematically analyzed together. They were obtained on carbonate aquifers with different karst and fracturing degree related to tectonic activity and presence of fault zones nearby the well areas of the well. A synthesis of hydrogeological parameters of some PTs carried out in the main Umbria Region aquifers is also reported in [54]. Table 3 shows the results of the pumping tests indicating the methods used for the interpretation.

^{2}/day, falling within the literature range for carbonate-fractured aquifers worldwide (Table 2). In order to create a more representative dataset involving different Italian carbonate aquifers, our data have been integrated with those of the Euganean aquifer, northern Italy [7]. A total of 77 T-S

_{c}pairs have been collected: according to the Kolmogorov-Smirnov method for the goodness of fit (K-S test), both variables are log-normally distributed.

^{2}) than that of the linear regression, 0.94 and 0.91, respectively. This is because the two parameters S

_{c}and T are log-normally distributed [6]. To check the performance of the two relationships, the Relative Mean Absolute Error (RMAE) has been used. The RMAE value for the log-log relationship is 26.0% while that of the linear relationship is 80.0%, indicating that the log–log Equation (7) is much more accurate than the linear one.

## 4. Discussions

_{c}data, can improve the knowledge of estimate of T values in fractured and karstified carbonate aquifers. This approach has to be considered for first rough estimates of transmissivity values, especially in hydrogeological systems with wells not provided by pumping tests but with known S

_{c}data. In these conditions, S

_{c}values may improve the hydrogeological characterization of data-scarce aquifers. The equation here proposed for some Italian carbonate aquifers Equation (7) was derived on the basis of transmissivity values determined mostly from the analysis of data collected during pumping tests in thick confined aquifers. As shown in Figure 1, Maiolica complex and Basal Limestones complex of Central Apennines are several hundred meters thick. In these aquifers, due cost–benefit constraints, pumping wells often penetrate the aquifer thickness just partially. This approach is typically used in mountain regions where wells are drilled in the productive part of aquifers, characterized by high transmissivity, without drilling deeper [16]. As shown in Table 3, most of the step-drawdown pumping tests can be interpreted by the Cooper-Jacob method which, according to [56,57], is affected only minimally by partial penetration in confined aquifers. Moreover, as Verbovšek [16] reported, pumping wells, which penetrate more than 70% of the entire aquifer thickness, can be treated as fully penetrating, as they activate the whole aquifer thickness. Among the wells analyzed in the Umbria region, those sited in unconfined aquifers have a screen length, which penetrates more than 70% of the aquifer thickness: in addition, transmissivity values have been estimated on drawdown curves at low flow rates, so that the vertical component of the flow can be considered negligible.

^{3}/s, while that in the area where TR wells are located is 0.60 m

^{3}/s. The one order of magnitude difference in transmissivity estimated in VN compared to TR wells (about 2500 m

^{2}/day vs. about 125 m

^{2}/day) can be attributed to the higher degree of aquifer fracturing in the VN area, which contributes to the high rivers discharge increases.

_{c}pairs coming from similar hydrogeological systems can help check the typical transmissivity ranges for carbonate aquifers, including their distribution. Figure 5 shows the distribution of a large T-S

_{c}dataset (about 180 pairs) obtained on fractured karstified carbonate aquifers worldwide, which confirm the lognormal distribution. Figure 6 plots the T-S

_{c}pairs available from the literature, highlighting that the data from carbonate aquifers in Central Italy agrees with those obtained on similar aquifers, falling on the Mace [2] relationships. As Mace [3] reported and observed in Figure 6, for S

_{c}values lower than 10 m

^{2}/day, the relationship tends to overestimate the transmissivity values. Moreover, S

_{c}values lower than 10 m

^{2}/day and higher than 3000 m

^{2}/day represent only 7% of the dataset, indicating that the aquifers having these values are few and the T estimates are less reliable. As expected, the Italian carbonates’ $\mathsf{\alpha}$ and β parameters Equation (7) differ from those found by [2]. The Absolute Mean Error (MAE) computed for the Mace equation is 154 m

^{2}/day while that for the specific equation for the Italian carbonates is 142 m

^{2}/day. However, it should be noted that the latter equation gives Relative Mean Absolute Error (RMAE) just slightly lower than the Mace one, i.e., 26.0% instead of 26.5%. This indicates that the Mace’s model [2] remains valid and applicable in similar geological environments. Results here obtained differ from that proposed by Verbovšek [16] for confined and unconfined dolomite aquifers, where the prediction accuracy was higher for the linear than for the log-transformed relationship.

## 5. Conclusions

_{c}pairs of carbonate aquifers of central Italy are log-normally distributed, and transmissivity values range from about 10 to 2700 m

^{2}/day (pumping tests carried out during the last thirty years). A new empirical correlation for estimating T based on Sc values has been presented. The equation compared to that presented by Mace [2] shows the same log–log form but with different coefficients α and β. The development of specific equations for a certain geological environment, such that here presented, is recommended. Despite this, the Mace equation’s predictions—even if slightly less accurate—help to a regional characterization of fractured and karstified carbonates of different regions, especially when few data from pumping tests are available, or it is difficult to obtain them as tests are very dated. In conclusion, it is necessary to test and improve the proposed equations in other hydrogeologically similar areas. It should be noted that the results of the present work and those from previous studies confirm that a general equation can be used for a rough estimation of transmissivity values in carbonate aquifers of different kind. Pumping test analysis remains the most accurate tool for the estimation of the hydrogeological parameters, even if the development of empirical equations is necessary for data scarce areas.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Lithological map of the Umbria region (central Italy) with wells in fractured and karstified carbonate aquifers.

**Figure 2.**Cooper-Jacob straight-line method on two pumping tests carried out in Maiolica and Scaglia-Maiolica aquifers. (

**a**) Drawdown data recorded in an observation well during the first step-drawdown pumping test in MCUC1 well. (

**b**) Drawdown data recorded in SUB well (first step-drawdown curve at low discharge Q = 0.005 m

^{3}/s).

**Figure 3.**Diagnostic plot of drawdown monitored during unsteady state constant-rate pumping tests. (

**a**) Data from piezometer of well TR5: double porosity model; (

**b**) data of well MCAL2: triple porosity model.

**Figure 5.**Frequency distribution (

**a**,

**b**) and Q-Q plots (

**c**,

**d**) of specific capacity and transmissivity of fractured and carbonate aquifers worldwide. Data are picked from Mace [3] by using Engauge Digitizer free software and are expressed as logarithm of the values.

**Table 1.**Main characteristics of pumping wells in the fractured carbonate aquifer in the Umbria region. The location of wells is in Figure 1.

Name | Well Depth (m) | Hydrogeological Complex | Source |
---|---|---|---|

BOT1 | 200 | BL | Umbra Acque Company (ATI2 Umbria) |

BOT2 | 200 | BL | Umbra Acque Company (ATI2 Umbria) |

MCUC1 | 250 | MA | Umbria Region |

MCUC2 | 200 | MA | Umbria Region |

MIG1 | 85 | MA | Umbra Acque Company (ATI2 Umbria) |

MIG2 | 102 | MA | Umbra Acque Company (ATI2 Umbria) |

MIG3 | 109 | MA | Umbra Acque Company (ATI2 Umbria) |

MIG4 | 105 | BL | Umbra Acque Company (ATI2 Umbria) |

SUB | 445 | SC-MA | Umbra Acque Company (ATI2 Umbria) |

MMAR1 | 429 | SC | Umbria Region |

MMAR2 | 436 | MA | Umbria Region |

MAM | 240 | MA | Umbra Acque Company (ATI2 Umbria) |

SPL | 300 | BL | Present work |

ACQ1 | 45 | TRA | Present work |

ACQ2 | 60 | TRA | Present work |

ACQ3 | 48 | TRA | Present work |

VN2 | 140 | MA | Umbria Region |

VN3 | 150 | MA | Umbria Region |

TR1 | 300 | BL | SII (ATI4 Umbria) |

TR2 | 300 | BL | SII (ATI4 Umbria) |

TR3 | 300 | MA-CD | SII (ATI4 Umbria) |

TR4 | 300 | MA | SII (ATI4 Umbria) |

TR5 | 300 | CD-BL | SII (ATI4 Umbria) |

TR6 | 300 | CD | SII (ATI4 Umbria) |

MCAL1 | 280 | BL | Umbria Region |

MCAL2 | 270 | BL | Umbria Region |

**Table 2.**Summary of empirical relationships between well specific capacity (S

_{c}) and transmissivity (T) of fractured and karst aquifers. The unit of all equations is m

^{2}/day.

Author | Setting | Empirical Relationships | n. Data | Range of Application (m ^{2}/day) |
---|---|---|---|---|

Reference [52] * | Fractured carbonate (Northwestern Ohio aquifer, USA) | T = 3.24 S_{c}^{0.81} | - | 10–2000 |

Reference [53] | Fractured/karstic carbonate (Floridan aquifer, USA) | T = 1.23 S_{c}^{1.05} | 14 | 100–100,000 |

Reference [6] | Fractured carbonate (Amman-Wadi Sir aquifer, Jordan) | T = 1.81 S_{c}^{0.917} | 237 | 3–20,000 |

Reference [2] | Fractured/karstic carbonate (Edwards aquifer, USA) | T = 0.76 S_{c}^{1.08} | 71 | 1–100,000 |

Reference [7] | Fractured carbonate (Euganean basin, North-East Italy) | T = 0.85 S_{c}^{1.07} | 45 | 6–2500 |

Name | PTs Type | Method of PTs Analysis | S_{c}(m ^{2}/day) | T (m ^{2}/day) |
---|---|---|---|---|

BOT1 | step-drawdown | 2 | 1800 | 1361 |

BOT2 | step-drawdown | 2 | 1469 | 1400 |

MCUC1 | step-drawdown | 2 | 86 | 100 |

MCUC2 | step-drawdown | 2 | 9 | 5 |

MIG1 | step-drawdown | 2 | 62 | 71 |

MIG2 | step-drawdown | 2 | 115 | 132 |

MIG3 | step-drawdown | 2 | 13 | 15 |

MIG4 | step-drawdown | 2 | 59 | 67 |

SUB | step-drawdown | 2 | 27 | 49 |

MMAR1 | step-drawdown | 2 | 26 | 10 |

MMAR2 | step-drawdown | 2 | 778 | 1050 |

MAM | step-drawdown | 2 | 186 | 86 |

SPL | constant-rate | 2 | 50 | 53 |

ACQ1 | step-drawdown | 2 | 140 | 130 |

ACQ2 | step-drawdown | 2 | 112 | 94 |

ACQ3 | step-drawdown | 2 | 47 | 84 |

VN2 | step-drawdown | 1 | 1350 | 2318 |

VN2 | step-drawdown | 1 | 1412 | 2317 |

VN3 | step-drawdown | 1 | 2667 | 2578 |

VN3 | step-drawdown | 1 | 2373 | 2635 |

VN3 | step-drawdown | 1 | 2304 | 2635 |

VN3 | constant-rate | 1 | 2036 | 2693 |

TR1 | step-drawdown | 2 | 94 | 84 |

TR1 | constant-rate | 2 | 95 | 300 |

TR2 | step-drawdown | 2 | 23 | 20 |

TR2 | step-drawdown | 2 | 20 | 11 |

TR3 | step-drawdown | 2 | 84 | 154 |

TR4 | step-drawdown | 2 | 105 | 106 |

TR5 | constant-rate | 3 | 97 | 140 |

TR6 | constant-rate | 2 | 141 | 190 |

MCAL1 | step-drawdown | 2 | 86 | 100 |

MCAL2 | constant-rate | 4 | 48 | 43 |

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## Share and Cite

**MDPI and ACS Style**

Valigi, D.; Cambi, C.; Checcucci, R.; Di Matteo, L. Transmissivity Estimates by Specific Capacity Data of Some Fractured Italian Carbonate Aquifers. *Water* **2021**, *13*, 1374.
https://doi.org/10.3390/w13101374

**AMA Style**

Valigi D, Cambi C, Checcucci R, Di Matteo L. Transmissivity Estimates by Specific Capacity Data of Some Fractured Italian Carbonate Aquifers. *Water*. 2021; 13(10):1374.
https://doi.org/10.3390/w13101374

**Chicago/Turabian Style**

Valigi, Daniela, Costanza Cambi, Roberto Checcucci, and Lucio Di Matteo. 2021. "Transmissivity Estimates by Specific Capacity Data of Some Fractured Italian Carbonate Aquifers" *Water* 13, no. 10: 1374.
https://doi.org/10.3390/w13101374