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Published online 8 October 2007
Published in Vadose Zone J 6:713-724 (2007)
DOI: 10.2136/vzj2006.0105
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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ORIGINAL RESEARCH

Measurements and Modeling of Variable Gravity Effects on Water Distribution and Flow in Unsaturated Porous Media

Robert Heinsea,*, Scott B. Jonesa, Susan L. Steinbergb, Markus Tullerc and Dani Ord

a Dep. of Plants, Soils and Climate, Utah State Univ., Logan, UT 84322-4820
b Universities Space Research Assoc., Mail Code EC3, NASA/JSC, Houston, TX 77058
c Dep. of Soil, Water & Environmental Science, The Univ. of Arizona, Tucson, AZ 85721
d Lab. of Soil and Environmental Physics, Ecole Polytechnique Fédérale de Lausanne, Switzerland
* Corresponding author (heinse{at}cc.usu.edu).

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.


Received 24 July 2006.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Theoretical Considerations
 Equilibrium Considerations
 Materials and Methods
 Results and Discussion
 Conclusions
 REFERENCES
 
Liquid behavior under reduced gravity conditions is of considerable interest for various components of life-support systems required for manned space missions. High costs and limited opportunities for spaceflight experiments hinder advances in reliable design and operation of elements involving fluids in unsaturated porous media such as plant growth facilities. We used parabolic flight experiments to characterize hydraulic properties under variable gravity conditions deduced from variations in matric potential over a range of water contents. We designed and tested novel measurement cells that allowed dynamic control of water content. Embedded time domain reflectometry probes and fast-responding tensiometers measured changes in water content and matric potential. For near-saturated conditions, we observed rapid establishment of equilibrium matric potentials during the recurring 20-s periods of microgravity. As media water content decreased, the concurrent decrease in hydraulic diffusivity resulted in limited attainment of equilibrium distributions of water content and matric potential in microgravity, and water content heterogeneity within the sample was influenced by the preceding hypergravity phase. For steady fluxes through saturated columns, we observed linear and constant hydraulic gradients during variable gravity, yielding saturated hydraulic conductivities similar to values measured under terrestrial gravity. Our results suggest that water distribution and retention behavior are sensitive to varied gravitational forces, whereas saturated hydraulic conductivity appears to be unaffected. Comparisons between measurements and simulations based on the Richards equation were in reasonable agreement, suggesting that fundamental laws of fluid flow and distribution for macroscopic transport derived on Earth are also applicable in microgravity.

Abbreviations: ESA, European Space Agency • NASA, National Aeronautics and Space Administration • TDR, time domain reflectometry


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Theoretical Considerations
 Equilibrium Considerations
 Materials and Methods
 Results and Discussion
 Conclusions
 REFERENCES
 
The National Aeronautics and Space Administration's (NASA) and European Space Agency's (ESA) vision of life support for space missions considers growing plants as part of long-term, bioregenerative life support systems that would typically operate under microgravity conditions. Efficient and resilient growth systems require monitoring and accurate fluid management in the porous rooting environment to meet plant needs under the intense energy, mass, and volume constraints imposed during space flight (Steinberg et al., 2005). To date, water for plant-growth systems was delivered through porous tubes and membranes (e.g., Bingham et al., 2002; Burtness et al., 2002; Dreschel et al., 1994; Hoehn et al., 2000; Morrow and Crabb, 2000; Morrow et al., 1993; Scovazzo et al., 2001). The design and operation of these root-zone hydration systems implicitly assumed that water content in the porous medium could be uniformly maintained at a prescribed optimal level unaffected by microgravity conditions. Various systems have been designed to use suction in the porous membrane to indirectly manage the water content in the porous medium through matric potential control (Dreschel and Sager, 1989; Hoehn et al., 2003; Morrow et al., 1992). Some researchers have even controlled matric potentials in the media assuming that Earth-determined water retention characteristics are not altered in microgravity (Monje et al., 2005; Morrow et al., 1994; Steinberg and Henninger, 1997). In juxtaposition with these hydration systems, the control of liquid and gaseous fluxes in the root zone for optimal plant growth (Jones and Or, 1998) begs the question whether hydraulic properties of the porous plant-growth medium quantified under terrestrial gravity (Steinberg and Poritz, 2005) can be transferred to microgravity.

Under microgravity conditions, fluid flow and distribution in porous media could be altered as a result of changes in buoyancy, dominance of capillary forces, particle rearrangement, and vehicle vibration. Several studies have presented anecdotal evidence to support this hypothesis. For example, Podolsky and Mashinsky (1994) showed differences between water content distributions measured in microgravity (µg) aboard the MIR space station and measurements on Earth. They demonstrated reduced rates of capillary transport and enhanced preferential wetting in 1.5- to 2.5-mm baked clay aggregates (Perlite). These early observations pointed to enhanced capillary transport and promotion of phase entrapment in reduced gravity; the consequences for liquid behavior at the root module scale, however, remained unknown.

Recently, Levine et al. (2003) reported that reduced gravity accentuates the role of prewetted surfaces for transport and the rearrangement of loosely packed particles. The authors partially confirmed the postulation of Jones and Or (1999) that microgravity enhances liquid- or gas-phase entrapment and affects macroscopic wetting and drainage processes, as well as sample-scale hydraulic properties. The sample-scale distribution of water was studied by Shah et al. (1993), who reported that during parabolic flight, fluid is pushed downward to the bottom of the substrate in hypergravity (1.8 times Earth's gravity) phases and rises and distributes more uniformly under reduced gravity conditions. Mohamed et al. (2002) hypothesized that changes in the microstructure of porous media could significantly alter water retention and hydraulic conductivity. More recently, Reddi et al. (2005) studied the behavior of particles and pore fluid blobs at residual water contents in variable gravity. They concluded that microgravity had little effect on individual particles or the size of individual fluid blobs (i.e., blobs did not coalescence or break apart), but clusters of particles adhering to fluid blobs appeared to rearrange themselves. The preliminary observation of fluid flow and liquid configuration in porous micromodels during parabolic flight experiments by Or et al. (2004) suggested that the impact of reduced gravity is manifested at the mesoscale (cluster of pores) rather than at the single-pore level. Accordingly, sample-scale behavior of the spatial and temporal distribution of water in porous media depends on the physical behavior of water at the mesoscale.

Notwithstanding these recent advances, predictive capabilities for definitive design of plant-growth modules remain ambiguous because of limited opportunities for reduced-gravity experiments. Short-term reduced-gravity tests during parabolic flights providing 20 s of reduced gravity per parabola offer a more accessible alternative to desirable long-duration space-flight experiments (Ivanova and Dandolov, 1992; Podolsky and Mashinsky, 1994). Rare opportunities and high costs that limit the number of trials and repetitions required for obtaining reliable experimental data illustrate why results from past parabolic flight experiments are often inconclusive and qualitative in nature. Additionally, the suitability and representativeness of short-term microgravity tests for long-term behavior under microgravity remains an open question. Parabolic flight experiments are far from ideal for the task because of the short period of induced microgravity (~20 s) with limitations for hydraulic equilibration, the confounding background vibrations (g-jitter), and the ensuing hypergravity phase (1.8 g) required for proper flight trajectory.

Our study comprises results from more than 640 parabolas logged, with significant improvements in our understanding of the working environment and in measurement capabilities. In the following, we report results from water retention and saturated hydraulic conductivity experiments that were conducted during parabolic flight. We focus on spatial and temporal analyses of matric potentials under different water contents and variable gravitational acceleration, commenting on the applicability of hydraulic parameters obtained from short-term microgravity tests. We used estimated parameters and measured data for modeling transient matric potential distributions under microgravity with Hydrus-2D (Simunek et al., 1999). The specific objectives of this study were (i) to measure and analyze matric potentials as a function of water content in porous media during parabolic flight induced microgravity, (ii) to observe and quantify dynamic and nonequilibrium unsaturated flow in variable gravity, (iii) to assess the applicability of existing numerical models for unsaturated water flow and distribution in porous media to simulate hydraulic dynamics in reduced gravity, and (iv) to measure saturated hydraulic conductivity in earth-, hyper-, and microgravity.


   Theoretical Considerations
TOP
 ABSTRACT
 INTRODUCTION
 Theoretical Considerations
Equilibrium Considerations
 Materials and Methods
 Results and Discussion
 Conclusions
 REFERENCES
 
Governing Flow Equations
We briefly introduce here established models and parameterization schemes for describing water transport in unsaturated porous media. In this, we tacitly assume that basic laws of fluid flow for macroscopic transport derived on Earth apply to microgravity. The fundamental functional relationships include the volumetric water content {theta}, matric potential h, pressure potential p, and the saturated hydraulic conductivity Ks. Total hydraulic potential H is the sum of the matric, pressure, and gravitational potential z, where matric (–) and pressure (+) potentials are mutually exclusive (i.e., H = h + p + z).

The Buckingham–Darcy law (Buckingham, 1907) relates the flux density with the hydraulic potential gradient dH/dx in the x direction and the unsaturated hydraulic conductivity K(h). Richards' (1931) equation, expressed in terms of water content, results from combining conservation of mass with the Buckingham–Darcy law:

Formula 1[1]
where t is time, {rho}w is the density of water, g is the acceleration due to gravity, and D({theta}) is the soil-water diffusivity defined as D({theta}) = K({theta})dh/d{theta}, with dh/d{theta} the slope of the water retention function (i.e., reciprocal soil water capacity). To complete Eq. [1], water retention and unsaturated hydraulic conductivity functions describing porous-media fluid behavior are cast in parameterized functional forms. The water retention may be parameterized with the van Genuchten (1980) model:

Formula 2[2]
The subscripts s and r refer to the saturated and residual water contents, respectively, and {alpha}, n, and m are empirical fitting parameters, where we use the simplification m = 1 – 1/n. The unsaturated hydraulic conductivity is then parameterized using the Mualem–van Genuchten expression in terms of these same parameters (Mualem, 1976; van Genuchten, 1980).

With numerical modeling, the challenges, then, are simulating the gravitational forces that induce the hydrodynamic conditions and adequately modeling the porous-medium properties that describe the matric potential–water content relationship that drives transient and equilibrium conditions, especially when gravity forces approach zero. We selected Hydrus-2D (Simunek et al., 1999) because of the capability to simulate 1- or 0-g conditions using the vertical or horizontal flow option (neglecting the gravity dependent term in Eq. [1]), respectively.


   Equilibrium Considerations
TOP
 ABSTRACT
 INTRODUCTION
 Theoretical Considerations
 Equilibrium Considerations
Materials and Methods
 Results and Discussion
 Conclusions
 REFERENCES
 
Characteristic Time and Length Scale
The extraction of static equilibrium matric potential values in microgravity under the dynamic conditions experienced in parabolic flight is constrained by the 20 s period of microgravity. Attainment of equilibrium following the transition from a fundamentally different distribution of matric potentials and water content in the preceding period of hypergravity may therefore be limited. For unsaturated conditions, some of these concerns can be addressed through the observation of matric potential measurements, which provide a rapid tensiometric response (i.e., <1 s) and indicate the level of equilibrium considering the temporal rate of change, where equilibrium conditions in 0 g are defined as:

Formula 3[3]
Because of minor gravitational accelerations during parabolic flight and due to the decreasing diffusivity with decreasing water content, we refer to quasi-steady state conditions indicating that dh/dt approaches zero and becomes relatively small during each period of microgravity.

Considering one-dimensional flow in 0 g and assuming that D is constant (i.e., uniform water content) and a unique function of water content (conveniently taken to be the drainage water retention function), Eq. [1] takes the form of the following differential equation (Childs and George, 1948):

Formula 4[4]
Upon introduction of a dimensionless length {bigcirc}=x/L with the constant cell height L and dimensionless time Formula 4=t/t0, where t0 is a characteristic time scale, we can write:

Formula 5[5]
By inspection of Eq. [5] we may now define the characteristic time scale that undimensionalizes the differential equation:

Formula 6[6]
The characteristic time scale can be thought of as a screening time. A perturbation at the boundary of an initially equilibrated hydrostatic system will decay with time with a characteristic time t0. The question as to the characteristic length of that process may be deduced from the solution. A general solution to Eq. [5] and [6] for D = constant and with a fixed boundary condition at {bigcirc}=0 has the form

Formula 7[7]
from which we may infer that the characteristic diffusion length scale x0 is proportional to the square root of diffusivity and time and state that

Formula 8[8]
The greater the lapse of time, the greater is the distance x0 the perturbation has penetrated for a given diffusivity (i.e., at a given water content). Considering constant and uniformly distributed D implies x/Formula 8 is constant; hence, a perturbation would progress as 1/Formula 8 along x.

Gravity and Capillarity Forces
In parabolic flight, the gravity-induced perturbation is not confined to a boundary but is exerted on the sample as a whole where gravitational and capillary forces strive toward the establishment of a new equilibrium. The static bond number Bos describes this scaling of effects related to the gravity-driven buoyancy force with respect to capillary force, given as (Auradou et al., 1999; Birovljev et al., 1991; Meheust et al., 2002)

Formula 9[9]
where {Delta}{rho} describes the density difference between the liquid and gaseous phases, {sigma} is the surface tension, r is the pore radius, and {xi} is the length scale. The characteristic capillary length for Bos = 1 on Earth is {xi}c = 2.7 mm for the water–air interface; hence, at the pore scale ({xi} = r) for pores with radii larger than {xi}c, gravity is expected to dominate (and conversely, for {xi} < {xi}c, capillary forces dominate). For commonly used particulate porous media with typical pore sizes of 0.26 and 0.14 mm (i.e., Turface and Profile; see section on particulate porous media), the role of reduced gravity on pore-scale processes (e.g., liquid–gas interface configuration) is minor (Bos = 9.1 x 10–3 and 2.7 x 10–3 at 1 g, respectively). However, at the sample scale and in the absence of viscous forces, we consider two length scales for the porous-media samples: the typical pore size r and the system scale L (cell height or observation height). Substituting Eq. [9] into Eq. [1] and defining the characteristic length at the sample scale to be Lr (Nakajima and Stadler, 2006) yields:

Formula 10[10]
When the gravity force approaches zero (i.e., Bos -> 0) the right-hand term vanishes, and Eq. [10] predicts a horizontal flow condition. In reality, the onset of instabilities in this regime (Auradou et al., 1999; Meheust et al., 2002) that are not predicted by Eq. [10] adds complexity in the form of fingering. This additional complexity observed on Earth is likely to affect the flow and distribution of water in parabolic flight and space travel where the gravity force never equals zero but reduces to values between 10–2 and 10–6 g.


   Materials and Methods
TOP
 ABSTRACT
 INTRODUCTION
 Theoretical Considerations
 Equilibrium Considerations
 Materials and Methods
Results and Discussion
 Conclusions
 REFERENCES
 
Parabolic Flight–Induced Variable Gravity
We conducted water retention and saturated hydraulic conductivity measurements during a 4-d flight campaign in February 2004 onboard NASA's parabolic flight aircraft stationed at the Johnson Space Center in Houston, TX. In May 2006, a second campaign provided additional water retention data, which were obtained with cells of smaller vertical extent (1 cm tall). Each flight (1 per day) comprised four sets of 10 parabolas with intermittent 1-g turnaround periods. During each parabola, microgravity conditions existed for 20 to 25 s during free fall in the apex of the parabolic aircraft trajectory (http://jsc-aircraft-ops.jsc.nasa.gov/Reduced_Gravity/index.html). An accelerometer onboard the aircraft measured gravitational accelerations in x (fore to aft), y (wing to wing), and z directions. The resulting normalized effective gravitational acceleration in z direction (g = gobserved/gEarth) had an oscillatory character, sequentially increasing to about 1.8 g (hypergravity) and then decreasing to near 0 g (microgravity). We also experienced small horizontal accelerations and variations in the vertical acceleration (g-jitter). Looking ahead to Fig. 6a, we see the change in gravitational acceleration for two representative parabolas. In addition, Fig. 6a depicts changes in the aircraft cabin pressurization that compensate for variations in pressure with the parabolic flight trajectory. The pressure changes are in phase with the gravitational acceleration, decreasing nearly linearly in µg and increasing at a slightly accelerated rate in the beginning of 1.8 g traversing a pressure change of approximately 40 hPa.


Figure 6
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  		FIG. 6. Measured and modeled matric potentials as a response to variable gravity in the 7-cm cell for porous ceramic aggregate Profile. Figures on the left side (b–e) show measured matric potentials for three observation heights at z = 1, 3, and 5 cm (Fig. 2) as a function of time for different water contents {theta}, where (a) depicts the gravitational acceleration and change in aircraft cabin pressure for plot (b). Figures on the right side compare measured (lines) and simulated (symbols) matric potentials in microgravity following hypergravity (g–j). In (f), measured normalized gravitational accelerations are depicted for measurements shown in (g).

 
Particulate Porous Media
The porous media used in this study were porous ceramic aggregates (Profile Products, Buffalo Grove, IL), sieved to particle-size fractions of 0.25 to 1 mm (Profile), 0.25 to 2 mm (Mix), and 1 to 2 mm (Turface), and glass beads (MO-SCI, Rolla, MO) of 0.35- to 0.5-mm, 0.6- to 1-mm, 1- to 2-mm and 2.5- to 3.5-mm size fractions. The particle density of both the glass and porous ceramic aggregates is 2.5 g cm–3. Porous ceramic aggregates have been widely used in microgravity plant experiments (Levine et al., 2003; Norikane et al., 2004; Steinberg and Henninger, 1997; Stutte et al., 2005). The aggregates are stable and have moderate surface area for nutrient storage. Aggregated ceramics exhibit two distinct pore spaces: interaggregate and intra-aggregate pores. Only the interaggregate pore retention characteristics, particularly in the optimal range for liquid and gas supply of 0 to –25 cm H2O (1 cm H2O = 97.96 Pa) matric potential, are of interest for this study. Measured and fitted porous-media water retention characteristics for Turface, Mix, and Profile are shown in Fig. 1 , illustrating the hysteretic behavior of drainage and wetting processes. To obtain drainage and wetting data, we conducted repetitive measurements with shallow, 1-cm tall by 4-cm diameter cells. A syringe pump controlled water content, and tensiometers measured matric potentials. The van Genuchten (1980) model was fitted to measured data using the RETC software (van Genuchten et al., 1991). The fitted model parameters listed in Table 1 are in good agreement with those determined by Steinberg and Poritz (2005).


Figure 1
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  		FIG. 1. Steady-state water retention curves measured in 1 g for porous ceramic aggregates (a) Profile (0.25–1 mm), (b) Mix (0.25–2 mm), and (c) Turface (1–2 mm). Solid lines represent the van Genuchten water retention model (Eq. [2]) fitted to six replicate measurements of drainage (•) and wetting ({circ}) for processes within interaggregate pores. Dashed and dotted lines indicate the 95% confidence interval for the fitted retention curves.

 

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  		TABLE 1. Porous-medium hydraulic parameters for the van Genuchten (1980) water retention model. The prediction is based on measurements from the 1-cm cell.

 
Experimental Design and Procedures
We designed four novel sample cells with differing vertical dimensions for measuring water retention during parabolic flight (Fig. 2 ). The first cell had a vertical dimension of 1 cm and was connected to a fast-responding transducer tensiometer (PX40-50BHG5V, Omega Engineering, Stamford, CT). The second cell (Fig. 2b) had a vertical dimension of 2 cm and was equipped with a time domain reflectometry (TDR) probe (custom 10-cm three-rod probe connected to a TDR100 [Campbell Scientific, Logan, UT]) in the center and three fast-responding transducer tensiometers. The third cell (Fig. 2c) had a height of 4 cm with TDR probes centered at elevations of 1 and 3 cm and four evenly distributed tensiometer ports. The fourth cell (Fig. 2d) was 7 cm high and equipped with three transducer-style tensiometers. To measure saturated hydraulic conductivity, cell inlets were connected to a syringe pump that provided alternating fluxes, and cell outlets were connected to collapsible reservoirs (Fig. 3 ).


Figure 2
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  		FIG. 2. (a) 1-cm water retention cell with pressure transducer connected to a porous cup embedded in, and in hydraulic contact with, the porous media. The porous cup was connected to a syringe pump that provided metered water addition and removal. (b) 2-cm water retention cell with (1) water inlet connected to sintered porous plate, (2) water outlet, pressure transducer ports, and centered time domain reflectometer (TDR) probe. (c) 4-cm water retention cell showing the vertical location of the two TDR probes. (d) 7-cm cell showing the pressure transducer positions and water inlet/outlet.

 

Figure 3
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  		FIG. 3. Saturated hydraulic conductivity cells showing the pressure transducer locations. The water inlet was connected to a syringe pump for constant water fluxes using a bidirectional feed. The water outlet was connected to a collapsible reservoir.

 
The use of taller cells allowed monitoring of accentuated variable gravity effects due to increased hydrostatic forces in 1.8 g and more pronounced transition in matric potentials between 1.8 g and microgravity. The design of the shallower 1- and 2-cm cells, on the other hand, aimed at minimizing the time for steady-state conditions by reducing hydrostatic induced water content differences within the cells. In the following sections, we briefly describe the various cell designs and experimental procedures. For detailed descriptions, see Heinse et al. (2005).

Retention Cell: 1 cm
The 1-cm cell consists of a cylindrical polyethylene container, 1 cm tall by 3.8 cm i.d., that was open to the atmosphere through a small hole (~1 mm in diameter) in the removable lid (Fig. 2a). A porous cup (5-µm pore size, Mott Metallurgical, Farmington, CT) connected to a hypodermic needle was centered in the cell and connected to a syringe. Water addition and removal was controlled and monitored by means of a high-resolution syringe pump (KDS 230, KD Scientific, Holliston, MA) connected to a datalogger (CR23x, Campbell Scientific, Logan, UT). During the experiment, 0.67 mL of water was step-by-step withdrawn from the sample during 1.8-g periods. Upon desaturation of macropores, the pumping direction was reversed to add water to the sample until satiation. Changes in matric potential as a function of water content and gravitational acceleration were monitored with a tensiometer positioned in the line connecting the porous cup with the syringe (i.e., measuring line pressure). Water content was inferred from pumped volumes of water with the porous media being completely saturated at the beginning of the experiment. Postflight analysis provided measures of bulk density for what were typically nine independent cells on each test day.

Retention Cell: 2 cm
The 2-cm cell consists of a 12- by 5- by 2-cm rigid wall container that is open to the atmosphere (Fig. 2a). The water addition and removal inlet that is connected to the syringe is in close hydraulic contact with the particulate medium through a sintered porous plate with 5-µm pore size (Mott Metallurgical, Farmington, CT) positioned at, and filling the entirety of, the bottom of the cell. Water addition and removal was controlled and monitored by means of a high-resolution syringe pump connected to a datalogger. During the experiment, a fixed volume of water (5 mL) was pumped stepwise into the sample during periods of 1.8 g. The changes in matric potential as a function of water content and gravitational acceleration were monitored with three tensiometers positioned at 0.5, 1, and 1.5 cm height and a TDR probe centered at z = 1 cm. The water content was inferred from TDR measurements using the following water content/dielectric relationship for 2- to 5-mm porous ceramic aggregates:

Formula 11[11]

Retention Cell: 4 cm
The 4-cm cell (Fig. 2b) consists of a 12- by 5- by 4-cm rigid wall container. The changes in matric potential as a function of water content and gravitational acceleration were monitored by a set of four tensiometers located at 1-, 1.5-, 2.5-, and 3-cm heights and two TDR probes centered at z = 1 and 3 cm. Water addition and removal was achieved in the same fashion as outlined for the 2-cm cell.

Retention Cell: 7 cm
The 7-cm cell (Fig. 2c) consists of a vertically oriented clear-polycarbonate cylinder with 2.5 cm i.d. Two stainless steel screens confine the particulate porous medium. The screen at the bottom of the cell is connected to a syringe, and the screen at the top of the cell is open to the atmosphere. Water removal was achieved by manually withdrawing water with a syringe in four steps at volume increments of 5 mL to establish volumetric water contents ranging from 0.7 to 0.48. Matric potentials were measured with tensiometers positioned at 1-, 3-, and 5-cm heights.

Saturated Hydraulic Conductivity Cell
For the hydraulic conductivity experiments, the porous medium was packed into horizontal 48-cm-long acrylic cylinders with 1.9 cm i.d. (Fig. 3). Water was pumped from the syringe end into a flexible-wall bag at the distal end at alternating fixed fluxes. The flexible-wall bag was used to minimize the hydrostatic forces of water exerted by the reservoir and allow the system to be in equilibrium with the cabin pressure. Five pressure transducers along the flow domain (10 cm apart) measured the gradient in pressure potential. Experiments ran continuously to cover both microgravity and 1.8-g conditions.


   Results and Discussion
TOP
 ABSTRACT
 INTRODUCTION
 Theoretical Considerations
 Equilibrium Considerations
 Materials and Methods
 Results and Discussion
Conclusions
 REFERENCES
 
We first discuss the hypothetical distribution of sample matric potential in hyper- and zero-gravity phases of a parabolic flight and illustrate redistribution after transition between the two stages. Next, we use a numerical model, Hydrus-2D, to evaluate the applicability of Richards' equation for modeling liquid behavior in microgravity. We compare model calculations with measured water content and matric potential data. We then focus on water retention characteristics measured in microgravity and compare results to measurements obtained in 1 g. In the final section, we discuss saturated hydraulic conductivity measured in microgravity.

Matric Potential Distribution in Variable Gravity
Vertical distributions of matric potentials under hydrostatic equilibrium for various gravitational conditions experienced during a parabolic flight are depicted in Fig. 4 . In this hypothetical experiment, we consider a 4-cm-tall cell packed with Turface and assume an average matric potential of –4 cm maintained at the center of the sample. The system is then subjected to different gravitational accelerations ranging from 0 to 1.8 g to generate the static equilibrium distributions shown in Fig. 4.


Figure 4
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  		FIG. 4. Hypothetical diagram of the equilibrium distribution of water content and matric potentials in a 4-cm-tall sample of porous ceramic aggregate Turface subjected to (a) Earth's gravity, (b) 1.8 g, and (c) 0 g. The average matric potential (i.e., at the midpoint) is constant at –4 cm. The shaded area shows regions of validity for water content and potentials where the gravitational force scales the hydrostatic equilibrium distribution.

 
At 1 g we find a linear distribution of matric potentials ranging from –2 cm at the bottom of the cell to –6 cm at the top. The sample water content distribution follows the primary drainage or wetting curves described by Eq. [2] and using parameters listed in Table 1.

For 1.8 g the potential distribution changes and corresponds to the increase in the gravitational force. The potential at the bottom of the cell attains a value of –0.4 cm and at the top –7.6 cm. The corresponding water content distribution is obtained by scaling the matric potentials of the primary retention curves [i.e., {theta}(h1.8 g) = {theta}(1.8·h1 g)]. Note that for the same average matric potential, the water in the system is distributed differently. More water accumulates at the bottom of the cell, and less at the top.

If the cell were subjected to 0 g, the matric potential distribution would be constant and uniform throughout the porous medium. Consequently, the water content would have values that lie between the boundaries of the primary drainage and wetting curves. It would be possible to obtain secondary scanning curves between the water content values predicted by the primary curves. Hence, the spatial distribution of water content in the system may vary considerably as deviations between the top and bottom parts of the sample may follow different secondary retention curves. We therefore hypothesize that the water content in 0 g depends on the wetting and drainage history of individual pores and pore clusters prior to the 0 g phase.

For the experimental observations, the rapid transitions in gravitational acceleration between 1.8 g and microgravity may preclude attainment of equilibrium, and dynamic fluid behavior could dominate observations. In the following section, the transition of matric potentials between 1.8 g and microgravity that drive the time-dependent redistribution of water and the consequences for measuring static-equilibrium water retention in variable gravity are discussed in more detail.

Transient and Equilibrium Analysis
To capture the effect of microgravity on hydraulic properties in unsaturated porous media, the transition following a perturbation caused by the ensuing hypergravity phase needs to decay reasonably fast, and a new equilibrium should be established. Clearly, the size of the sample and the hydraulic properties of the media determine this decay, but in addition, the establishment of this new microgravity equilibrium involves a shift in the relative influence of capillary and gravitational forces. Equation [9] states that at the pore scale under 1 g, capillarity dominates the shape of the interfacial configuration over gravity. Figure 5a shows that this remains true even at 1.8 g. On the other hand, if one considers the effect of gravity at the sample scale ({xi} >> r), one could argue that pores at different heights in the sample are interconnected and experience feedback suggesting that the sample height or observation height determines the relative influence of gravitational to capillary force. Figure 5a shows the Bond number using the pore scale and sample scale perspective. At the sample scale, for increasing cell heights, the dominance of gravitational forces occurs in reduced gravitational fields compared with shallower cells. For the observation of equilibrium conditions in microgravity, it follows that the perturbation caused by the preceding hypergravity phase will be more pronounced as cell height increases. Keeping in mind that the time for the decay of this perturbation is limited to 20+ s, minimization of sample vertical extent for experiments in parabolic flight seems advantageous. This aspect is further illustrated in Fig. 5b, depicting the characteristic time scale. Because the movement of water is controlled by the hydraulic diffusivity (Eq. [1]), the transition toward equilibrium is retarded at reduced water contents. The time required for reaching equilibrium conditions increases with cell height and pore size, where equilibrium conditions are reached relatively quickly (within seconds) at high water contents and shallow cell heights, while at low water content and for taller cells, the transition to equilibrium can take considerably longer than 20 s.


Figure 5
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  		FIG. 5. (a) Dependence of static Bond numbers Bos on the normalized gravity force considering the pore scale and system/sample scale influence of gravity vs. capillarity. Solid lines represent porous ceramic aggregate Turface; dashed lines represent Profile. (b) Characteristic time scales for cell heights of 1 and 7 cm after Eq. [6], respectively.

 
Measurements in Parabolic Flight
Measured matric potentials obtained with the 7-cm cell during rapid transitions from 1.8 g to microgravity and vice versa (Fig. 6a ) are shown for average water contents of 0.48, 0.56, 0.63, and 0.7 cm3 cm–3, with potentials measured at 1-, 3-, and 5-cm heights (Fig. 6b–6e). In the 1.8-g phase, the matric potentials diverged between the three monitoring elevations in accordance with the gravitational potential. In microgravity the gradient rapidly decreased due to diminishing body forces and transitioned toward equilibrium. After the microgravity phase, the initial 1.8-g potential distribution was rapidly reestablished. Under near-saturated conditions ({theta} = 0.7), all tensiometers measured a uniform matric potential, indicating that equilibrium in microgravity was quickly attained, whereas the transition toward equilibrium at lower water contents ({theta} = 0.63–0.48) became gradually slower. This progressive delay in attaining equilibrium is attributable to the decrease in unsaturated hydraulic conductivity with decreasing water content scaled by the hydraulic capacity (Fig. 5). In this respect, it is noteworthy that subtle differences in the response rates at different elevations (e.g., compare 1-cm vs. 5-cm observation) indicate water content and unsaturated hydraulic conductivity heterogeneity at lower water contents within the sample (Fig. 6b–6e). Temporal tensiometer responses at different sample elevations further indicate that different volume elements of the porous medium undergo different drainage or wetting processes simultaneously. At the lowest water content (Fig. 6b), the bottom tensiometer shows a reduction in matric potential (drainage) during microgravity, while the top tensiometer indicates an increase in matric potential (wetting). The middle tensiometer, on the other hand, at first shows an increase, followed by a decrease, in potential. This apparent overcompensation likely arises from nonuniform water retention following different secondary curves within the profile coupled with the transition from drainage to wetting conditions.

Similar results as shown for the 7-cm retention cell were obtained for the 1-, 2- and 4-cm cells. The differences in observed matric potentials at different sample elevations and transitions to equilibrium, however, were more pronounced in the 7-cm cell.

Simulation of Fluid Behavior in Variable Gravity
In the following example, Richards' equation (Eq. [1]) and the van Genuchten–Mualem parametric models for water retention (Eq. [2]) and hydraulic conductivity are applied to calculate the transient matric potential distributions shown in Fig. 6g–6j. In this simulation, we attempted to model the transition of matric potentials from an initially equilibrated matric potential distribution in 1.8 g to microgravity using Hydrus-2D, concentrating on the first microgravity period shown in Fig. 6b–6e. Hydrus-2D does not explicitly allow the simulation of 1.8-g conditions or variable gravity. However, the transition from 1.8 g to microgravity can be simulated using measured matric potentials in the sample at the end of the 1.8-g period as the initial condition, where the potentials at the upper and lower boundary of the domain were estimated based on a 1.8-g force (1.8 times distance) assuming static equilibrium. The initial process was specified to be drainage. For the van Genuchten parameterization, we used the Earth-based (1-g) parameters for Profile given in Table 1. For the saturated hydraulic conductivity, we used 0.2 cm s–1 to allow for a better fit. This conductivity is in the range of conductivities given in Table 2, which varied between 0.14 and 0.3 cm s–1 for packing densities between 0.62 and 0.67 g cm–3.


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  		TABLE 2. Saturated hydraulic conductivity (Ks) measured in variable gravity. In the case of multiple measurements at different bulk densities, a range of conductivities is given.

 
A comparison of measured and modeled matric potentials showing the vertical distribution in microgravity as a function of time at three observation heights is shown in Fig. 6g–6j. The results show good agreement between the measured and modeled matric potentials with respect to the transition following 1.8 g at the indicated water contents. Some of the perceptible discrepancies in Fig. 6g–6i for the simulations at 1, 3, and 5 cm are partially attributed to the inability to specify simultaneous wetting and drainage as the initial conditions in Hydrus-2D. As shown in Fig. 6b, in 1.8 g most of the sample is following a drainage curve, with the exception of the lowest location (1 cm). In the transition to microgravity, this scheme reverses and the bottommost location now appears to lose water, which results in drainage conditions at 1 cm and wetting conditions at 3 and 5 cm. Thus, the modeled response at 1 cm more closely resembles the measurement because of a continuation of the initial drainage process (i.e., water moving upward). The modeled responses at 3 and 5 cm, however, transition to a wetting process immediately at the onset of microgravity, thus indicating that the simulation could improve if a variable gravity option were available.

Although these limited simulations show reasonable agreement with the measurements, suggesting the applicability of Earth-based macroscopic transport models, they fail to describe a consistent bias toward a linear increase in measured matric potentials in microgravity. This bias appears to be a water content–dependent migration of the matric potential readings toward less negative potentials present in all three tensiometer readings, and it is particularly perceptible at water contents of 0.63 and 0.56 (Fig. 6h–6i). Changes in cabin pressure could contribute to this bias if some buffering of pressure changes were to occur within the porous medium. Discontinuous pockets of air could then absorb pressure changes, slowing the tensiometers' response compared to the backside of the pressure transducer responding immediately to cabin pressure. Measurements in the 2- and 4-cm cells showed a similar bias, but at a much reduced amplitude (<2 mm), while this bias was not observed in measurements obtained with the 1-cm cell

Water Retention Characteristics in Microgravity
Figures 7 and 8 depict short-term (obtained at the end of the 20-s microgravity period) microgravity water retention characteristics for Profile, Mix, and Turface obtained with the 1-, 2- and 4-cm retention cells. The 7-cm cell was not used because of difficulties in attaining equilibrium at lower water contents (Fig. 5) and the large differences in water content between top to bottom. Drainage and wetting data were obtained under quasi-steady-state conditions at the end of the microgravity phase, where the distinction between drainage and wetting for microgravity conditions is solely based on the external addition or removal of water to the cell. As such, the classification does not consider the actual processes at the specific locations of potential measurement (i.e., drainage vs. wetting at the cell ends) and may be a poor image of the actual process.


Figure 7
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  		FIG. 7. Comparison of quasi-steady-state microgravity (µg) drainage and wetting water retention and steady-state 1-g porous media water retention for Profile, Turface, and a mixture of these obtained in the 1-cm cell. The solid lines are predicted 1-g retention curves (upper = drainage, lower = wetting). Dashed or dotted lines indicate the 95% confidence interval for the 1-g retention curves. Indicated water contents are shown as average sample values determined from pumping volumes. Matric potentials were measured at z = 0.5 cm.

 

Figure 8
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  		FIG. 8. Comparison of quasi-steady-state microgravity drainage and wetting water retention data with steady-state 1-g porous media water retention curves for Turface. In (a) measurements in a 2- cm-tall sample with water contents shown as average sample values determined from pumped volumes are shown. Matric potentials were measured at z = 0.5 and 1 cm. Measurements using a 4-cm-tall sample with water contents shown as TDR measurements are depicted in (b) and (c), where water contents were measured at z = 1 cm (bottom) and z = 3 cm (top), respectively. The solid lines are predicted 1-g retention curves (upper = drainage, lower = wetting). Dashed or dotted lines indicate the 95% confidence interval for the 1-g retention curves.

 
In Fig. 7, we present water retention data obtained with the 1-cm cell. For the convenience of comparison, we also show the primary 1-g wetting and drainage retention curves (Fig. 1). Parameterizations for Profile, Mix, and Turface microgravity retention data of the van Genuchten (1980) model are given in Table 1. Wetting water retention data for Profile and Mix in Fig. 7a and b fell within the 95% confidence intervals of the primary wetting curves obtained at 1 g. For the drainage data, lower microgravity values were observed compared with the 1-g primary drainage curve. For Turface wetting and drainage data in Fig. 7c, significantly reduced potentials were observed compared with the 1-g curves, where all microgravity data fell below the 1-g primary wetting curve. This contrasts with Turface wetting microgravity data measured in the 2-cm cell (Fig. 8a), which generally fell near the 1-g primary wetting curve. We attribute the observed decrease in microgravity potentials in the 1-cm cell partially to larger void spaces along the container wall, where the reduced response was likely enhanced by the larger particles sharing fewer contact points with the porous cup (i.e., Turface). For the measurements in the 2-cm cell, reduced microgravity drainage potentials compared with the 1-g curve were similar to the findings in Profile and Mix in Fig. 7. While microgravity wetting data found a lower bound near the 1-g primary wetting curve, the drainage data were not well described by, and fell below, the 1-g primary drainage curve. No drainage microgravity data mimic the 1-g data, which suggests that either we cannot reproduce the 1-g curves under these limited experimental conditions or the results are significantly different. While the dynamic and partially equilibrium limited measurements contribute to the disparity, Steinberg and Poritz (2005) argued against the solitary contribution based on comparison of dynamic and static water retention in 1 g. They found more variation in dynamic measurements, especially in 1- to 2-mm Turface, but the measurements followed the static primary retention curves.

The observed disparities between the predicted 1-g curve and microgravity measurements are more likely attributed to the simultaneous wetting and drainage processes occurring spatially within different volume elements during microgravity (microgravity following 1.8 g). In a typical parabolic cycle, either we kept the cell water content constant or water was added or removed. Addition or removal of water occurred in the 1.8-g period and induced changes in water retention response as it transitioned between 1.8 g and microgravity. Water accumulated at the bottom of the cells during the 1.8-g phase would then move upward in response to the reduced gravitational force during microgravity. This would result in simultaneous drainage conditions in the lower part of the cell and wetting conditions in the upper part of the cell. For the addition of water in 1.8 g, wetting conditions dominate most volume elements in the cell. On the other hand, following the withdrawal of water in 1.8 g, most volume elements will be in a drainage mode at the end of the 1.8-g period and potentially transition to a wetting mode in microgravity. Consequently, the measurements classified as drainage do not correspond well with the 1-g predicted primary drainage curve, while the wetting data correspond much better in Fig. 7a and 7b and Fig. 8a. Measured temporal changes in the vertical distribution of matric potentials may indicate which process (i.e., drainage or wetting) is occurring at a particular height in the cell. The measurements, however, do not uniquely indicate the water content or the movement of water but rather track a response curve that may or may not involve water content changes.

Using localized rather than cell-averaged water contents may then elucidate the water redistribution and water retention response. For this, TDR-sensed water contents in Fig. 8b and 8c depict water retention curves for a 4-cm-tall sample of Turface as a function of local water contents at elevations of z = 1 and 3 cm. Sensed water contents at these heights primarily reflected water content changes in response to the external addition and removal of water in the cell. The varying gravitational forces, however, did not appear to affect the water content readings. We observed a general lack of redistribution of water between the measurement heights, where the addition and removal of water did not result in equal responses in the top and bottom location. The sensed water content at z = 3 cm (top) varied significantly over the duration of the experiment, whereas the water content at z = 1 cm (bottom) remained elevated. Minute changes in water content at the bottom location were sensed only at the lowest cell-average water content. We believe the reason for the apparent low sensitivity to gravity-induced redistribution of water is partially attributed to the averaging volume of the TDR probe, which is most heavily weighted around the rods (Robinson et al., 2003). More important, the TDR readings indicate an unanticipated lack of water redistribution between gravity force changes. The vertical heterogeneity in the water content distribution (Fig. 8b and 8c) is manifested in the measured water retention data for the bottom and top probe locations within the cell. These measurements show a wide distribution of water contents at comparable matric potentials in µg. This can be attributed to the energy state of water rapidly approaching a uniformly distributed quasi-static equilibrium with the dissipation of hydrostatic potentials, leaving no gradient to drive water movement (Fig. 4). The resulting water content distribution varies widely due to the narrow pore size distribution, hysteresis, and to the hypergravity-induced distribution preceding microgravity. When transitioning to microgravity, the draining pores below and wetting pores above do not require significant changes in localized water content to reach sample-scale equilibrium potentials, in which case, upper and lower region water contents (referring to the vertical position in the cell) vary over the entire interaggregate water content range (compare Fig. 4) due to the relatively narrow pore-size distribution.

To further illustrate this argument, sensed water content and matric potential response data are shown in Fig. 9 . The data represent matric potentials measured during two consecutive parabolas starting at 1.8 g, where consecutive readings of matric potential with time from two heights in the cell (z = 1 and 3 cm) are plotted against TDR-sensed water contents at these heights.


Figure 9
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  		FIG. 9. Dynamic matric potential-water content response measured in Turface in the 4-cm cell during two consecutive parabolas at two vertical locations (z = 1 and 3 cm, bottom and top, respectively). Five mL of water was added in the 1.8-g period in-between the parabolas with a syringe pump. Solid lines indicate the 1-g wetting and drainage water retention curves. Dotted lines indicate scaled water response curves for 1.8-g. Solid symbols (•) indicate 1.8-g, and open-faced symbols ({square}) indicate microgravity-measured data.

 
For the two consecutive parabolas, it is evident that each data set (each at a different water content) experiences significant changes in matric potential that result in little movement of water based on TDR-determined water contents. The disparity in matric potentials between 1.8 g and microgravity is much greater at lower water contents and shows less disparity when water content is increased to near saturation. Note that water was pumped into the bottom of the cell during 1.8 g. Thus, the top and bottom cell regions were not treated the same. Nevertheless, the measured response in the transition from 1.8 g to microgravity and back traverses gravity-induced alterations that are opposite in direction for the observations at the top and bottom locations. Notably, measured potentials in 1.8 g do not scale by a factor of 1.8 alone. For the bottom location, this can be explained by stating that the condition for scaling assumes a seepage face at the bottom boundary of the cell. Yet in the experiment, water was not allowed to drain in 1.8 g. This resulted in more positive potentials than would be predicted under seepage conditions. For the top location, it is apparent that in 1.8 g more negative potentials are experienced than would be suggested by a 1.8-g force alone. We attribute this to local water content changes around the tensiometer cup that are not detected by the TDR sensors because of the disparity in their sampling volumes. The observed response in variable gravity indicates that matric potentials may fulfill quasi-steady-state conditions involving minimal changes in water content that can be partially explained by hydrostatically scaling the measured response. Depending on the wetting and drainage history, sample volumes separated by only a few pore lengths can differ significantly in water content and direction of the wetting/drainage response that are more pronounced because of the repeated transition of 1.8 g to microgravity.

Saturated Hydraulic Conductivity
Hydraulic gradients were measured for prescribed fluxes through horizontally oriented saturated porous-media columns. Several flux rates were used for each medium, with measurements performed continuously under the variable gravity conditions. Results are summarized in Fig. 10 , where, despite a certain degree of measurement error, the experimental data plot within 1 standard deviation of the indicated lines of proportionality (indicating the mean saturated hydraulic conductivity). The results indicate that for the range of prescribed fluxes and porous media tested, the relationship between hydraulic gradient and flux can be described with a linear function (Buckingham–Darcy's law), corroborating the expected gravity-independence of saturated flow. In Heinse et al. (2005), we stressed that this independence was not observed when loose packing allowed changes in the arrangement of particles and pores during cycles of gravity. The coefficient of proportionality, the saturated hydraulic conductivity, for each medium is reported in Table 2, where the measured saturated hydraulic conductivities for Profile and Turface were found to be comparable to measurements obtained by Steinberg and Poritz (2005).


Figure 10
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  		FIG. 10. Saturated-flow pressure gradients as a function of hydraulic flux for glass beads (GB), Profile, and Turface measured in variable gravity. Error bars indicate the standard deviation, while solid lines denote the mean saturated hydraulic conductivity.

 

   Conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Theoretical Considerations
 Equilibrium Considerations
 Materials and Methods
 Results and Discussion
 Conclusions
REFERENCES
 
In this study, we considered dynamic measurements of hydraulic properties in hysteretic porous media during parabolic flight–induced variable gravity, where the flight represents a challenging experimental environment in which acceleration frequently changes during the experiment. Consequently, the parabolic flight data require careful interpretation to account for the influence of dynamic accelerational forces, especially under unsaturated conditions. We developed a variety of cells to facilitate water retention and saturated hydraulic conductivity measurements, primarily for the microgravity portions of the flight. During dynamic measurements of water retention, we observed a transition of matric potentials during microgravity, where the nonlinear response of the transition toward static equilibrium was a function of average water content with the transition slowing at decreased water contents. When transitioning from 1.8 g to microgravity and vice versa, matric potentials tracked secondary drainage or wetting curves simultaneously depending on the vertical position in the experimental cells. Minimal changes in water content fulfilled both the 1.8-g and the microgravity matric potential conditions, which resulted in significant spatial heterogeneity in the distribution of water contents maintained during variable gravity. The porous-media water retention characteristics were approximated for microgravity conditions from quasi-steady-state conditions at the end of the microgravity phase. Despite the apparent influence of the hypergravity phase on dynamic water retention during microgravity, measurements suggested similar water retention characteristics in microgravity compared to 1 g for wetting conditions. Drainage water retention data generally fell below the 1-g measured data. This was attributed to the uncertain redistribution of water following the ensuing hypergravity phase. Long-term microgravity testing is recommended for more reliable and rigorous measurements of porous-media water retention due to the prolonged steady-state period and absence of dynamic constraints of the parabolic flight environment. The experiments and simulations described in this study present evidence of altered dynamics of liquid behavior in porous media but seem to corroborate the applicability of the Buckingham–Darcy law and of Richards' equation for describing macroscopic porous-medium fluid behavior under microgravity conditions with the gravity term approaching zero. Long-term microgravity questions remain, especially regarding the fate of air-entrapment and its potential impact on fluid fluxes in porous media. Air entrapment alone could result in major differences in water retention, satiation, and unsaturated hydraulic conductivity with enhancement of hysteretic behavior in sustained microgravity. Reduced gravity environments considering Lunar and Martian conditions add to the list of future research needs in anticipation of plant-growth facilities and porous-media applications.


   ACKNOWLEDGMENTS
 
The authors gratefully acknowledge funding from NASA-JSC grants NAG 9-1284, NAG 9-1399, and NASA's Advanced Life Support Flight Program. We express appreciation for assistance from Seth Humphries, Bill Mace, Shane Topham, Kelly Lewis, Becky Newman, Tim Doyle, Nihad Daidzic, and Fred Ogden. We thank the associate editor and three anonymous reviewers for their helpful comments in improving the quality of this manuscript. This research was supported by the Utah Agricultural Experiment Station, Utah State University, Logan, Utah, and approved as journal paper no. 7837.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Theoretical Considerations
 Equilibrium Considerations
 Materials and Methods
 Results and Discussion
 Conclusions
 REFERENCES
 




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D. Or, M. Tuller, and S. B. Jones
Liquid Behavior in Partially Saturated Porous Media under Variable Gravity
Soil Sci. Soc. Am. J., February 6, 2009; 73(2): 341 - 350.
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