# HYPROP-FIT to Model Rock Water Retention Curves Estimated by Different Methods

^{*}

## Abstract

**:**

## 1. Introduction

^{3}·m

^{−3}), yields information for evaluating, analyzing, and predicting unsaturated flow [1].

## 2. Materials and Methods

_{3}≥ 97%). Minor constituents form the insoluble residue, which is commonly represented by clay minerals with negligible quartz, feldspar, and hydrous iron and aluminum oxides. Their main physical properties are reported in Table 1.

#### 2.1. The Suction Table Method

#### 2.2. The Evaporation Method

#### 2.3. The Quasi-Steady Centrifuge (QSC)

#### 2.4. Dewpoint Potentiometer

^{5}kPa with an accuracy of ±100 kPa from 0 to −1 × 10

^{4}kPa, and 1% from −1 × 10

^{4}to −3.0 × 10

^{5}kPa. Cylindrical samples, 3.5 cm in diameter and 0.6 cm in height, were used. The value of θ corresponding to each ψ value identifies the experimental data points of the WRC. For each lithotype, the pair values (θ, ψ) were obtained as an average computed for each θ value on 10 tested rock samples.

#### 2.5. Model

_{r}is the maximum water content for the water adsorption, and θ

_{s}is the saturated water content.

^{cap}(ψ) is a capillary retention term, θ

^{ad}(ψ) is an adsorptive retention term, and S

^{cap}and S

^{ad}are the capillary and the water adsorption saturation functions.

^{cap}by $\Gamma \left(\psi \right)$, that is, the basic saturation function for the vG model, Equation (2) becomes:

_{0}is the basic function at ψ = ψ

_{0}where the water content reaches zero, and

_{a}and x

_{0}are pF log values at ψ

_{a}and ψ

_{0}, respectively, where ψ

_{a}is the matric potential at air entry value for the adsorptive retention, and b is the shape parameter given by:

^{ad}= 1):

_{i}is the weighting factor for the subfunction i and is 0 < w

_{i}< 1 and ∑w

_{i}= 1.

^{ad}:

_{0}parameter is set in HYPROP-FIT equal to 10

^{6.8}pF (pF = −log|ψ| with ψ expressed in cm), which is the matric potential at oven dryness for 105 °C [23].

_{θ}) to evaluate the difference between the measured (θ

_{im}) and estimated (θ

_{ie}) water contents:

_{p}indicates the number of data points.

_{θ}, and the more accurate the model.

## 3. Results and Discussion

#### 3.1. Rock Water Retention Data

^{3}m

^{−3}) versus ψ (kPa), represented on a logarithmic scale, measured by suction table, evaporation, QSC, and WP4-T dewpoint potentiameter methods.

^{3}·m

^{−3}for C, and between 0.338 and 0.309 m

^{3}·m

^{−3}for M. This method, in fact, allows one to measure retention data near the saturation even if not exactly the θ

_{s}value.

^{3}·m

^{−3}and from 0.318 to 0.060 m

^{3}·m

^{−3}for C and M, respectively.

^{3}·m

^{−3}and θ = 0.193 m

^{3}·m

^{−3}, respectively. For M, the measured ψ values range from −1.07 to −46.91 kPa corresponding to θ = 0.408 m

^{3}·m

^{−3}and θ = 0.135 m

^{3}·m

^{−3}, respectively. It is important to highlight that the QSC method can expand the measurement range by simply increasing the combinations of cakes, the run speed, and the duration of the run in the centrifuge.

^{5}kPa and from −110 to 1.33 × 10

^{5}kPa for C and M, respectively; that is, in the very dry range of the WRCs. The corresponding θ values range between 0.047 and 0.001 m

^{3}m

^{−3}and between 0.041 to 0.003 m

^{3}·m

^{−3}for C and M, respectively.

_{s}, hence close to the porosity value, with these rocks having all pores interconnected, as mentioned above [11]. Specifically, the maximum θ values measured by QSC, which are 0.420 m

^{3}m

^{−3}for C and 0.408 m

^{3}m

^{−3}for M, are almost equal to the porosity values, Φ = 0.432 and Φ = 0.410 for C and M, respectively. This is because, according to the QSC method, water is supplied to the sample continuously from the upper reservoir [18], thus keeping the θ value high in the sample before starting the runs in the centrifuge. Differently, the maximum θ values measured by the other methods are far from the θ

_{s}value because of the water loss by gravity that occurs when handling the samples, in almost saturated conditions, during the preliminary sample preparation methods.

^{3}m

^{−3}, especially for M. This discrepancy is due to the different way of measuring θ and ψ values in the mentioned methods. In fact, the evaporation method measures θ and ψ values quasi-continuously, whereas the QSC method measures (θ, ψ) pair values when the sample reaches equilibrium.

_{s}to determine the WRCs in the whole θ range and capture the bimodality of the pore-size distribution of the studied rocks. This result implies future studies in order to test the QSC method with fractured rocks in order to verify its capability to capture multiple porosity scales of actual rocks and support numerical investigations with laboratory tests [24,25].

#### 3.2. Fitting Functions and Statistical Parameters

_{θ}for bimodal functions is always lower than the corresponding unimodal ones. Specifically, the goodness of fit is higher or equal for the bimodal PDI model compared to the bimodal vG one.

_{θ}is affected due to the non-equidistance of experimental points measured by the QSC method. In addition, since the greater the number of methods combined to determine WRCs, the greater the RMSE

_{θ}value, it is advisable to choose only one method among several that measure points in the same range of WRCs. This consideration suggests that either the evaporation or QSC data points should be used for a better fit.

_{θ}is also affected by the data points measured by the suction table such that, by depicting the saturated range of WRCs with θ values lower than θ

_{s}, the fitting is worse. For this reason, the suction table method should be avoided, thus reducing the experimental test time.

## 4. Conclusions

_{s}. In addition, the QSC method is the only one able to capture the bimodal behavior of the tested media with respect to the others.

_{θ}and AICc values.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Macroscopic appearance of studied calcarenites: (

**a**) lithotype C, medium-grained grainstone, and (

**b**) lithotype M, fine-grained packstone. Microscopic appearance in plane-polarized light of thin sections: (

**c**) Lithotype C and (

**d**) lithotype M.

**Figure 2.**Pore-size distributions for the rock samples, C and M, measured by the mercury intrusion porosimetry (MIP) method expressed as (

**a**) percentage of pore volume with respect to porosity; (

**b**) specific pore volume (mL·g

^{−1}), that is the incremental pore volume occupied from the mercury during the MIP test (Turturro et al., 2021).

Lithotype | C * | M * |
---|---|---|

Bulk density, ρ_{b} (g/cm^{3}) | 1.50 | 1.56 |

Particle density, ρ_{p} (g/cm^{3}) | 2.65 | 2.65 |

Porosity, Φ | 0.43 | 0.41 |

**Table 2.**Fitted parameters values and statistical analysis for the vG and PDI unimodal and bimodal water retention functions. Fixed values are indicated by an asterisk (*).

Parameter (Unit) | C | M | ||||||
---|---|---|---|---|---|---|---|---|

Unimodal vG | Bimodal vG | Unimodal PDI | Bimodal PDI | Unimodal vG | Bimodal vG | Unimodal PDI | Bimodal PDI | |

α (1/cm) | 0.028 | - | 0.0298 | - | 0.0329 | - | 0.0329 | - |

n | 1.41 | - | 1.389 | - | 1.626 | - | 1.635 | - |

θ_{r} (cm^{3}/cm^{3}) | 0.001 | 0.0001 | 0.001 | 0.0001 | 0.003 | 0.0001 | 0.008 | 0.0001 |

θ_{s} (cm^{3}/cm^{3}) | 0.432 * | 0.432 * | 0.432 * | 0.432 * | 0.410 * | 0.410 * | 0.410 * | 0.410 * |

w_{2} | - | 0.225 | - | 0.791 | - | 0.655 | - | 0.385 |

α_{1} (1/cm)α _{2} (1/cm) | - | 0.0389 0.00117 | - | 0.00113 0.0391 | - | 0.5 0.0169 | - | 0.0165 0.5 |

n_{1}n _{2} | - | 1.492 2.263 | - | 2.379 1.473 | - | 1.267 1.944 | - | 2.017 1.233 |

RMSE_{θ} | 0.016 | 0.0118 | 0.016 | 0.0118 | 0.0223 | 0.0207 | 0.0224 | 0.0207 |

AICc | −1657 | −1773 | −1657 | −1774 | −1553 | −1577 | −1552 | −1578 |

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**MDPI and ACS Style**

Caputo, M.C.; De Carlo, L.; Turturro, A.C. HYPROP-FIT to Model Rock Water Retention Curves Estimated by Different Methods. *Water* **2022**, *14*, 3443.
https://doi.org/10.3390/w14213443

**AMA Style**

Caputo MC, De Carlo L, Turturro AC. HYPROP-FIT to Model Rock Water Retention Curves Estimated by Different Methods. *Water*. 2022; 14(21):3443.
https://doi.org/10.3390/w14213443

**Chicago/Turabian Style**

Caputo, Maria Clementina, Lorenzo De Carlo, and Antonietta Celeste Turturro. 2022. "HYPROP-FIT to Model Rock Water Retention Curves Estimated by Different Methods" *Water* 14, no. 21: 3443.
https://doi.org/10.3390/w14213443