Journal of Hydrology 591 (2020) 125706

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Journal of Hydrology
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Research papers

HYPROP measurements of the unsaturated hydraulic properties of acarbonate rock sample
Tatiana Lipovetsky a, Luwen Zhuang b, c, Wenceslau G. Teixeira d, Austin Boyd a,
Elizabeth May Pontedeiro a, b, Luca Moriconi a, e, José L.D. Alves a, Paulo Couto a, Martinus
Th. van Genuchten b, f, *a

LRAP, Department of Civil Engineering, Federal University of Rio de Janeiro, UFRJ, Rio de Janeiro, Brazil
Department of Earth Sciences, Utrecht University, Utrecht, Netherlands
School of Civil Engineering, Sun Yat-sen University, Zhuhai, ChinadEmbrapa Soils, Rio de Janeiro, BrazileDepartment of Physics, Federal University of Rio de Janeiro, UFRJ, Rio de Janeiro, BrazilfCenter for Environmental Studies, CEA, São Paulo State University, UNESP, Rio Claro, Brazilbc

A R T I C L E I N F O

A B S T R A C T

This manuscript was handled by Jiri Simunek,
Editor-in-Chief

The unsaturated hydraulic properties provide important theoretical and practical information about fluid flow insoils and rocks for a range of soil, environmental and engineering applications. In this study we used theevaporation (HYPROP) and chilled-mirror dew point (WP4C) methods to estimate the water retention and un­
saturated hydraulic conductivity curves of an Indiana Limestone carbonate rock sample. The obtained data wereanalyzed in terms of unimodal and bimodal VG (van Genuchten, 1980) and PDI (Peters et al., 2015) typefunctions. Bimodal functions were found to produce excellent descriptions for the unsaturated hydraulic data wemeasured, slightly better than the standard unimodal formulations. For our particular test using an Indiana
Limestone rock sample, we did not find much improvement when accounting for film and corner flow using the
PDI formulation, relative to the VG model. As far as we know, this is the first time the HYPROP methodology wasapplied to a rock sample.

Keywords:
Unsaturated hydraulic Properties
HYPROP evaporation method
Carbonate rock
WP4C
Bimodal hydraulic functions

1. Introduction
The hydraulic properties of unsaturated porous media provide
important information about single- and multi-phase fluid flow in thesubsurface. They are critical to addressing many theoretical and prac­
tical studies in the soil, hydrogeologic, agricultural, civil and petroleum
engineering disciplines. A large number of experimental methodologies
has been developed and tested over the years to estimate the water
retention, θ(h), and unsaturated hydraulic conductivity, K(h) or K(θ),
relationships, where θ is the volumetric water content, h is pressure
head, and K the hydraulic conductivity. Most of the standard techniques,
especially for the hydraulic conductivity, are suitable for intermediate or
relatively wet conditions (Dane and Hopmans, 2002; Durner and Lip­sius, 2005; Looney and Falta, 2000), although some can also be applied
to the dry range, such as hot-air, centrifugation, or dew-point method­
ologies (Arya, 2002; Nimmo et al., 2002; Scanlon et al., 2002; Durnerand Lipsius, 2005).

One popular experimental approach has long been the evaporation
method, starting with early studies by Gardner and Miklich (1962) andWind (1968), and many others later (e.g., Wendroth et al. (1993) and
Iden and Durner (2008), and references therein). Measurements of theevaporation rate and pressure head at multiple depths in the sample
permit then the simultaneous estimation of the water retention andhydraulic conductivity curves, either using direct measurements or in­verse procedures (e.g., Simunek et al., 1998). A popular recent version of
the evaporation method involving semi-automated direct measurements
is the HYPROP system (Peters et al., 2015; Peters and Durner, 2008;
Schindler et al., 2010; among others), commercialized by the METER
group (München, Germany). The HYPROP system involves pressure
head measurements versus time at two depths within a short 5-cm
sample as water evaporates from its surface, with the evaporation rate
determined by continuous weighing of the column setup.
Several studies successfully tested the HYPROP evaporation
approach against multistep outflow and other methods (e.g., Schelle* Corresponding author at: Department of Earth Sciences, Princetonlaan 8a, 3584 CB, Utrecht University, Netherlands.
E-mail addresses: rvangenuchten@hotmail.com, m.t.vanGenuchten@uu.nl (M.Th. van Genuchten).
https://doi.org/10.1016/j.jhydrol.2020.125706
Received 8 July 2020; Received in revised form 22 October 2020; Accepted 23 October 2020
Available online 28 October 20200022-1694/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

T. Lipovetsky et al.

Journal of Hydrology 591 (2020) 125706

Fig. 1. (A) Schematic of the HYPROP setup (after Schindler et al., 2010) for the Indiana Limestone sample. (B) Actual sample showing the two drilled holes for thetensiometers. (C) Plugs used to cap the holes after installation of the tensiometers.

(Indiana Limestone Handbook, 1975). Petrographic and X-ray diffrac­tion (XRD) analyses indicated that Indiana Limestone is made up nearlyexclusively by calcite (99%), with very small amounts (about 1%) ofquartz (Churcher et al., 1991; Drexler et al., 2019).

et al., 2010; Zhuang et al., 2017), including tests against synthetic data
generated with the Richards unsaturated flow equation (e.g., BezerraCoelho et al., 2018; Iden et al., 2019). Most of these and other studies
used van Genuchten-Mualem (VG) type expressions (van Genuchten,
1980; van Genuchten and Nielsen, 1985) for the unsaturated hydraulicfunctions, while Peters et al. (2015) used the equations of Kosugi (1996)
as well as the PDI model of Peters and colleagues (Iden and Durner,
2014; Peters, 2014) to account for the effects of film and corner flow invery dry soils.
Most of the above techniques, including the HYPROP approach,typically work well for unconsolidated media, generally soils. Whilesome of the techniques can be applied also to consolidated rocks,
different approaches are more suitable for such media, such as mercury
intrusion porosimetry, water adsorption or desorption measurements,
centrifuge methods, or core flooding (Purcell, 1949; Dullien, 1979; Liuet al., 2016; Haghi et al., 2020). In this study we extend the HYPROP
approach to a carbonate rock sample. The HYPROP results, augmented
with WP4C (chilled-mirror dew point) water retention data in the dry
range, are analyzed in terms of van Genuchten type hydraulic functions,assuming the presence of single-porosity (unimodal) or double-porosity(bimodal) pore systems. The data are for an Indiana Limestone car­bonate rock sample extracted from a quarry in the USA, similar to thosefound in Brazilian and other oil-bearing pre-salt reservoirs.

2.1. The HYPROP measurement system
To evaluate the relationship between volumetric water retention andthe pressure head (negative capillary pressure, given in cm in this study)of the rock samples, we used the HYPROP evaporation method for thewet range (between saturation and pressure heads of about − 850 cm)
and the WP4C psychrometer method for the dry range. Instrumentation
for both approaches were obtained from the METER Group AG (Mün­chen, Germany).
Fig. 1A provides a schematic of the HYPROP setup involving pressure
head measurements at two depths within a 5-cm long fluid-saturated
sample. Water contents and fluid fluxes versus time are determined byautomated weighing of the sample as water evaporates from the surface.
Observed pressure heads, water contents, and evaporation fluxes aresubsequently used to estimate the water retention and unsaturated hy­
draulic conductivity functions (Pertassek et al., 2015; Schindler et al.,2010). The measurement range of the hydraulic conductivity at the wet
side is restricted by limitations of the tensiometers to register very smallpressure head differences, while limitations at the dry side are due towater outgassing, bubble formation and subsequent bubble expansion in
the tensiometers, usually at about − 850 cm.
Pressure heads versus time within the sample are recorded at two
locations (1.25 cm and 3.75 cm from the evaporating surface), while theevaporation rate is obtained by automated weighing. Water retention
and hydraulic conductivity data points are subsequently estimated fromthe average water content, the pressure heads provided by the twotensiometers, and the mean total head gradient between the two tensi­
ometers. The approach assumes that linear vertical distributions arepresent of the pressure head and the water content within the sample.
These assumptions have been shown to be very much acceptable(Bezerra-Coelho et al., 2018; Peters et al., 2015).
The standard HYPROP setup is suitable for cylindrical samples
measuring 5 cm in height, and either 5 or 8 cm in diameter. For our
experiments we used an Indiana Limestone sample having a diameter of

2. Experimental procedures
Here we briefly review the HYPROP measurement system, as well assummarize WP4C and related experimental protocols used for thisstudy, including how the data were analyzed in terms of functionaldescriptions of the hydraulic properties. Our studies were carried out
using two samples of Indiana Limestone, a calcite-cemented grainstone
carbonate rock. One standard rock sample was used for the HYPROP
measurements, and a much smaller separate sample for the WP4C ex­
periments (taken from the same larger Indiana Limestone core we had).
In addition, another standard sample was used to drill out plugs needed
to cap the holes drilled for the two tensiometers (see Fig. 1). These plugs
also originated from the same larger Indiana Limestone core used for thepermeability and routine core analysis. The rock consisted of fossil
fragments and concentric lamellar calcium carbonate particles (oolites)2

T. Lipovetsky et al.

Journal of Hydrology 591 (2020) 125706

Fig. 2. Observed HYPROP (circles) and WP4C (triangles) water retention (A) and hydraulic conductivity data as a function of volumetric water content (B) andpF (C).

only 3.85 cm. Since the sample’s diameter was smaller than the innerdiameter of 5 cm, we used a 3D printer to construct an impermeable
outside ring. A small rotary saw was used to drill circular holes with adiameter of 0.5 cm vertically through the rock sample to receive thetensiometers and a holder pin at the bottom (Fig. 1B). The tensiometers
shafts were placed within the holes such that the middle of the tensi­
ometers cups were located at 1.25 and 3.75 cm from the bottom. Verysmall, fine silt-sized, crushed rock particles that remained from drilling,
were used to ensure tight lateral contact between the tensiometer cups
and the sample. The remaining open parts of the two holes above thetensiometers were capped with drilled-out plugs (Fig. 1C) from aneighboring sample of the same overall Indiana Limestone core we used.
Wetted ground sample material was first placed around the tensiometercups using a small syringe to again guarantee close contact between theplugs and the tensiometer cups, as well as laterally with the rock sample.
The ground rock particles were compressed as much as possible to avoid
creating artificial macropores. Additional details about the standard
HYPROP system and the evaporation methodology are provided by
Pertassek et al. (2015) and METER (2015). For our experiments we usedthe HYPROP setup for hydraulic measurements down to pressure headsof about − 850 cm. For measurements in the dry range we used the WP4C
chilled-mirror dew point method.

and the samples using a thermal equilibration plate as recommended by
METER (München, Germany). WP4C evaluations started at approxi­
mately pF = 3.5 (about − 3000 cm pressure head), where pF = − log|h|,with h expressed in cm. The WP4C measurements continued until pF =5.5 (about − 27,000 cm) when the weight variations could no longer be
detected accurately. After the last WP4C measurements the samples
were dried in an oven at 105 ◦ C to check the dry weight of the samples.
A DV-4000 Poropermeameter, an automatic steady-state gas
permeameter-porosimeter system from Weatherford Laboratories(Houston, USA), was further used to measure the absolute permeability
and porosity of the sample. The porosity was measured using Helium(He) gas and the absolute permeability, k, using Nitrogen gas (N2), withcorrections for the Klinkenberg effect to account for the slippage of air inthe sample (Klinkenberg, 1941). The corrected air permeability was
automatically obtained internally by the DV-4000 equipment bymeasuring the permeability at several pressures and extrapolating to
infinite pressure (WL, 2017). Measurements had a reliability range be­
tween 0.001 and 40,000 mD (millidarcies) for the absolute permeability.
The saturated hydraulic conductivity for water was subsequently ob­tained using the relation Ks = ρwgk/μ, where k is the permeability, ρw is
the fluid density, g is gravitational acceleration and μ the fluid viscosity.
These basic petrophysical measurements of porosity and the hydraulicconductivity at full saturation were fixed during analysis of the unsat­
urated data in terms of the hydraulic functions.

2.2. WP4C and other measurements
Psychrometer (WP4C) measurements were performed on two 0.5 cm
long Indiana Limestone samples having the same lateral dimensions as
used for the HYPROP measurements. Using drippers, the sample were
wetted to a water content at or slightly below full saturation to avoid
hysteresis effects in the drying curves by starting at relatively wet con­ditions. Drying was done stepwise using a desiccator. All WP4C mea­
surements were preceded by equilibrating the temperatures of the cup

2.3. Unsaturated hydraulic functions
The measured data pairs of water content (θ) versus pressure head(h) as obtained with the HYPROP and WP4C experiments were fitted
jointly using the HYPROP-FIT (Pertassek et al., 2015). Soil hydraulicproperties for most scenarios were initially described using the standard
van Genuchten-Mualem (VG) formulation given by (van Genuchten,
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Journal of Hydrology 591 (2020) 125706

Fig. 3. Observed HYPROP (circles) and WP4C (triangles) volumetric water retention data fitted with the standard (m = 1–1/n) and variable (independent m and n)
van Genuchten and PDI hydraulic functions. The plots show results for (A) the unimodal van Genuchten functions, (B) the unimodal PDI functions, (C) the bimodal
van Genuchten functions and (D) the bimodal PDI functions.

1980)
θ(h) − θr1=mθs − θr[1 + |αh|n ]

(1)

[
(
)m ]2
K(Se ) = Ks SeL 1 − 1 − Se1/m(2)

Se (h) =complete mathematical formulations of the extensions involving thevariable m and n cases and the PDI formulation.
Once the pressure heads and actual evaporation rates (estimatedfrom the monitored sample weights) were obtained, the HYPROP-FIT
analysis was applied to the measured data. HYPROP-FIT uses the root
mean square error (RMSE) to quantify differences between the measured(yi) and calculated (yci ) water retention and hydraulic conductivity data:√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
np∑
1 c
RMSE =[5](yi − yi )2npi=1

where Se is effective saturation, θs and θr are the saturated and residual
water contents, respectively, Ks is the saturated hydraulic conductivity,
α and n are semi-empirical shape parameters, m = 1–1/n, and L is a poreconnectivity parameter. We additionally explored the performance of adual-porosity (bimodal) extension of the hydraulic functions to accountfor the presence of distinct but interacting macropore and micropore
regions. The functions we used are given by Durner (1994) and Priesackand Durner (2006):
Se (h) =θ(h) − θr
w1
w2
=
m +
m
θs − θr[1 + |α1 h|n1 ] 1 [1 + |α2 h|n2 ] 2

L

K(Se )=Ks(w1 S1 +w2 S2 ){where np is the number of data points, and yi refers to either watercontent, θi, or the logarithm of the unsaturated hydraulic conductivity, i.
e., log(Ki). The HYPROP-FIT analysis also provides values of the cor­rected Akaike Information Criterion (AICc), usually a negative value.
The larger the absolute AICc number, the more appropriate the model.

(3)

3. Results and discussion
Fig. 2 shows the observed HYPROP and WP4C water retention data,
and the HYPROP derived hydraulic conductivity data, the latter plottedversus volumetric water content as well as the pF (=− log|h|, with hexpressed in cm). The Indiana Limestone retention data reflect asomewhat bimodal pore size distribution, presumably in part due to finecalcite crystals lining the pores and creating microporosity, and in part
perhaps due to intra-particle microporosity in some of the fossil frag­ments and oolites. A study by Churcher et al. (1991) using thin sections
and scanning electron microscopy showed that the porosity andpermeability of these rocks are controlled mostly by the distribution ofcoarse pore-filling calcite cement. As compared to the water retention
data, the hydraulic conductivity data did not show a similar clear
bimodal behavior, mostly because they are outside of the micropore
range identified with the WP4C data. The dual-porosity nature of ourIndiana Limestone samples is consistent with several previous studies)m1[ ()]m2 }2(1/m
1/m
w1 α1 [1− 1− S1 1
+w2 α2 1− 1− S2 2
(w1 α1 +w2 α2 )2(4)where Si, αi, ni, and mi (i = 1, 2) are the same as in Eqs. (1) and (2) for themacropore and micropore regions, respectively (or for the fracture andmatrix regions if interpreted for unsaturated fractured rock), while w
defines the division of the porous medium in macropore and micropore
regions. We further tested the van Genuchten formulation assuming
variable m and n values in Eq. (1) as described by van Genuchten andNielsen (1985), as well as the PDI formulation by Peters et al. (2015), thelatter accounting for capillary flow, film and corner flow and vapor flow.
Because of their complex nature, we decided not to restate here the4

T. Lipovetsky et al.

Journal of Hydrology 591 (2020) 125706

unimodal van Genuchten and PDI models (including the van Genuchten
model with variable m and n parameters) did not match the data well
due to the bimodal nature of the curves, while the bimodal equivalents
provided excellent fits to the data. Fitted parameter values of the various
models are listed in Table 1, while Table 2 compares the statistical re­sults of the different optimizations. We also included in Table 1 the PDI
parameters. While the unimodal PDI model improved slightly upon theunimodal van Genuchten equation, far better results were obtainedusing the bimodal functions. For these reasons we focus below only on
the standard unimodal and bimodal van Genuchten functions assuming
m = 1–1/n. The VG and PDI scenarios assuming variable m,n parameters
are hence not further shown.
Fig. 4 shows results for the standard unimodal and bimodal van
Genuchten hydraulic conductivity functions, assuming m = 1–1/n,plotted again versus volumetric water content as well as the pressure
head (the latter in terms of pF values). Results for the variable m,nfunctions were only marginally better, which was the case also for the
PDI functions compared to the bimodal VG functions. The negligible
differences between the classical VG functions and the PDI modifications
within the range of available conductivity data is not surprising, since
the expected differences between the VG and PDI formulations should
occur primarily in the dry range of the conductivity function (>pF 3),
which is not covered by our measurements.
We note here that the various optimizations were carried out
assuming fixed values of the saturated water content (θs) and the satu­
rated hydraulic conductivity (Ks) of the Indiana Limestone sample.
These values were obtained by means of poropermeameter measure­ments on samples IH2 and IH3. The values obtained for IH2 were 311
mD (or 25.9 cm/d) for air, and 287 mD (or 23.9 cm/d) for water. IH3
presented values of 400 mD (33.3 cm/d) for air, and 372 mD (30.9 cm/d) for water. IH2 and IH3 further showed helium porosities of 0.185 and0.201 respectively. For our studies we used the IH3 data.
The relative accuracy of the different formulations is best demon­
strated by comparing RMSE values of the fitted water content (RMSEθ)and the hydraulic conductivity (RMSElogK) data, as well as AICc values.
Results for all scenarios are shown in Table 2. The data indicate indeed
very little improvement when adopting variable m,n van Genuchten and
PDI models, at least for our Indiana Limestone sample. This was themain reason for us not to further pursue those two approaches in thispaper, which are far more complicated numerically since the formula­tions lead to incomplete beta functions or hypergeometric functions(Dourado Neto et al., 2011; van Genuchten and Nielsen, 1985).

Table 1Fitted parameter values for the VG and PDI unimodal and bimodal hydraulicformulations assuming the VG constraint that m = 1–1/n. Fixed values areindicated by an asterix (*).
Parameter(unit)

Unimodel VG
model

Bimodal VG
model

Unimodal PDI
model

Bimodal PDI
modelα (1/cm)n (–)θr (cm3/cm3)
θs (cm3/cm3)
Ks (cm/day)L (–)
w1, w2 (–)α1, α2 (1/cm)

0.02701.449
0.008
0.215*
30.9*− 0.611––0.02363.286
0.119
0.215*
30.9*
0.251
–
–

n1, n2 (–)θr (cm3/cm3)
θs (cm3/cm3)
pFdry
ω (–)
A (–)

–
0.008
0.215
–
–
–

–
–
0.0
0.215*
30.9*
0.159
0.445; 0.555
0.00048;0.0221
1.712; 3.146
0.0
0.215
–
–
–

–
–
0.0
0.215*
30.9*
2.43
0.427; 0.573
0.00048;0.02171.645; 2.938
0.0
0.215
6.01
0.00508− 1.681

–
0.119
0.215
5.0
0.00730− 1.721

Table 2
Statistical analysis for the fitted water retention and hydraulic conductivity
models.
Statistical
Criterion

Unimodel VG
model- - - - m = 1–1/n restriction - - - - RMSEθ
0.00990.3803
RMSElogK
AICc− 1156- - - - variable m and n - - - - RMSEθ
0.0068RMSElogK
0.5419AICc− 1189Bimodal VG
model

Unimodal PDI
model

Bimodal PDI
model

0.00240.1678− 14660.00300.0460− 14780.00200.0412− 15630.00180.0571− 15850.0027
0.0944− 14870.00190.0515− 1572

showing the bimodal nature of Indiana Limestone rocks. For example,
Mercury Injection Capillary Pressure (MICP) tests by Churcher et al.
(1991) on Indiana Limestone samples showed very clear bimodal poresize distributions. An NMR study by Dunn et al. (1994) similarly showedbimodality in the relaxation time and associated pore-size distributions,with the peak of the smaller pores being associated with irreducible
saturation.
We next analyzed the observed water retention and conductivity
data in terms of the functional hydraulic descriptions in Section 2.2.
Fig. 3 shows results for the water retention curve. Clearly, the traditional

Fig. 4. Observed HYPROP hydraulic conductivity curves fitted with the unimodal and bimodal van Genuchten equations assuming m = 1–1/n.
5

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T. Lipovetsky et al.

4. Conclusions

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Simunek, J., Wendroth, O., van Genuchten, M.Th., 1998. Parameter estimation analysisof the evaporation method for determining soil hydraulic properties. Soil Sci. Soc.
Am. J. 62, 894–905.
van Genuchten, M.Th., 1980. Closed-Form equation for predicting the hydraulicconductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44, 892–898. https://doi.org/
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To the best of our knowledge, this is the first time HYPROP and
WP4C were used in conjunction to obtain estimates of the water reten­tion and hydraulic conductivity functions of a rock sample. The tested
rock, Indiana Limestone, is a carbonate formation sample showing
bimodal porosity behavior as evidenced from both our measured
retention data, and confirmed by several previous studies. Goodness of
fit values showed better results when the van Genuchten bimodal
models were fitted to the data, as reflected visually from the fitted data
and also seen from RMSE and AICc statistical analyses. Bimodal models
indicate the presence of interacting macropore and micropore parts ofthe medium, in our case each occupying approximately 50% of the porespace. The HYPROP and WP4C data relating volumetric water content
and pressure head could be described well with the bimodal van Gen­uchten hydraulic functions.
CRediT authorship contribution statement
Tatiana Lipovetsky: Investigation, Formal analysis, Writing - orig­
inal draft. Luwen Zhuang: Conceptualization, Methodology, Investi­
gation, Visualization. Wenceslau G. Teixeira: Methodology,
Validation. Austin Boyd: Validation, Writing - review & editing. Eliz­abeth May Pontedeiro: Methodology, Software. Luca Moriconi:
Formal analysis, Supervision. José L.D. Alves: Supervision. Paulo
Couto: Resources, Funding acquisition, Supervision. Martinus Th. van
Genuchten: Conceptualization, Methodology, Writing - review &editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgements
This research was carried out in association with the ongoing R&D
project ANP n◦ 20163-2, “Análise Experimental da Recuperação dePetróleo para os Carbonatos do Pré-sal do Brasil através de Injeção
Alternada de CO2 e Água” (UFRJ/Shell Brasil/ANP), sponsored by Shell
Brasil under the ANP R&D levy as “Compromisso de Investimentos com
Pesquisa e Desenvolvimento”. This study was financed in part also by
CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior,
Brazil) and CNPq (National Council for Scientific and Technological
Development, Brazil. We further would like to thank Maira Lima Santo,
Fernanda Hoerlle, Edmilson Rios, Adriano Dias and Carlos Jr. from
LRAP/UFRJ, Rodrigo Bagueira from UFF, and Raquel Lagarto and Ana
Sara Oliveira from LAMAS/Embrapa for their help.
Appendix A. Supplementary data
Supplementary data to this article can be found online at https://doi.
org/10.1016/j.jhydrol.2020.125706.
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