Published in Vadose Zone Journal 2:602-610 (2003)
© 2003 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SPECIAL SECTION - ADVANCES IN MEASUREMENT AND MONITORING METHODS
Gas Diffusion Measurement and Modeling in Coarse-Textured Porous Media
Scott B. Jones*,a,
Dani Orb and
Gail E. Binghamc
a Dep. Plants, Soils and Biometeorology, Utah State University, Logan, UT 84322-4820
b Civil and Environmental Engineering Department, University of Connecticut, Storrs, CT 06269-2037
c Space Dynamics Laboratory, 1695 North Research Park Way, North Logan, UT 84341
* Corresponding author (scott.jones{at}usu.edu).
Received 19 March 2003.
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ABSTRACT
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Conventional gas diffusion measurements in coarse-textured and aggregated porous media are severely limited due to hydrostatically induced variations in water content and air-filled porosity. Motivated by the need to measure gas diffusion in coarse-textured plant growth media designed for use in microgravity (e.g., aboard the International Space Station), our objectives were (i) to develop and test an automated diffusion measurement system on earth with water content adjustment capability and that minimizes hydrostatic effects, and (ii) to model characteristics of gas diffusion in partially saturated aggregated porous media. The horizontally oriented O2 diffusion cell design for reducing the gravitational effect was based on a thin profile rectangular cell. Continuous measurement of O2 in sealed dual-chamber diffusion cells provided concentration data for fitting diffusion coefficients where water content was controlled volumetrically using a porous membrane with an imposed pressure for incremental addition or removal of water. Gas diffusion was modeled as a function of air-filled porosity in millimeter-sized aggregated particles exhibiting a substantial difference between internal and external aggregate pore sizes. For this case, the internal aggregate porosity contribution to diffusion compared with external aggregate pore space was minor as described by a dual-porosity diffusion model. The measurement approach described can be used in other coarse-textured and structured porous media.
Abbreviations: WLR, water-induced linear reduction
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INTRODUCTION
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COARSE-TEXTURED plant growth media are commonly used in containerized systems where plant roots are restricted to a volume much smaller than commonly available in a native soil. The porous medium serves both as a structural support and provides the supply network for resources (water, O2, and nutrients). The small and relatively shallow volume presents two problems, (i) a reduced water storage capacity (and reduced surface area for root absorption) and (ii) potential for a perched water table at the container bottom leading to potential aeration problems (Bunt, 1988). For well-graded media and fine-textured soils, assumptions of uniform water content and one-dimensional flux may be sufficient for modeling gas diffusion under quasi steady-state conditions. For coarse-textured materials with a narrow pore-size distribution, however, the hydrostatic component of water potential induces substantial changes in water content within a shallow sample thickness (T) such as in the soil core illustrated in Fig. 1a. Under equilibrium conditions the vertical water content distribution follows that predicted by the retention curve for the coarse-textured material with higher water content (
) at the bottom and decreasing with sample thickness resulting in the water content difference shown in Fig. 1a. The air-filled porosity (
) maps directly to through total porosity (i.e., 
= + ) and the gas diffusion coefficient, Ds, can be described in terms of h, , or (Fig. 1b). The gaseous diffusion, like , exhibits a vertical distribution proportional to sample thickness (Fig. 1c). In coarse-textured media this distribution may result in significant differences in the measured gas diffusion coefficients compared to measurements with similar but spatially uniform water content (e.g., in microgravity). Later we compare the impact of sample thickness of modeled diffusivities for these two conditions. These differences become important for design and testing of conditions in plant growth media for reduced gravity conditions as components of NASA's advanced life support systems for long duration space missions.
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Fig. 1. Illustration of (a) a conventional soil core attached to a diffusion cell with a hypothetical coarse-textured water retention curve depicting the distribution of water content, , and difference,  , between the core profile top and bottom. The relationship (b) between water retention, h(), and soil gas diffusion, Ds, is linked through air-filled porosity, . Similar to water content distribution in the sample profile, (c) Ds is also distributed within the sample thickness, where a thinner sample exhibits less variation in Ds.
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The inference of air-filled porosity under non-uniform hydrostatic conditions could be based on the measured water retention curve, which in coarse textured materials using conventional measurement techniques requires corrections to obtain true ‘point’ estimates (Jalbert and Dane, 2001; Liu and Dane, 1995). Tokunaga et al. (2002) found water retention corrections necessary in 3-cm tall samples of 5- to 9-mm gravels but less necessary in 2-mm particles. Customized water retention measurement techniques have addressed some of these challenges in coarse-textured materials using slow drying coupled with tensiometric and dielectric measurements (Rassam and Williams, 2000) as well as shallow sample techniques for reducing hydrostatic effects (Jones and Or, 1998).
Gas diffusion measurements are needed to design and model improved plant-rooting environments for the microgravity environment of space. Past work focused on improving the understanding and control of plant–fluid–porous medium interactions in a weightless environment (Bugbee and Salisbury, 1989; Ivanova and Dandolov, 1992; Mashinsky et al., 1994; Morrow et al., 1994; Podolsky and Mashinsky, 1994). The challenge of designing and implementing advanced plant growth facilities is to maximize the inversely related processes of gas and liquid (nutrient) transport within the root zone (Jones and Or, 1998). Present state-of-the-art plant growth facilities for microgravity utilize coarse texture growth media such as calcined and aggregated clays and zeolites, which have the advantage of providing a nutrient storage capability. The coarse-textured (millimeter-sized) particles yield a pore-size distribution where water is readily removed using low-pressure systems inducing matric suctions of 10 or 20 cm (-1 or -2 kPa). For control or adjustment of water content in these media, it is convenient to employ a porous membrane in hydraulic contact with the sample to impose a matric suction facilitating partial drainage without sample disturbance. This method has been successfully employed as a water control system in microgravity for testing and evaluating plant growth systems for space (Jones and Or, 1999; Morrow et al., 1993; Scovazzo et al., 2001). Developments in microgravity research discussed here were incorporated in the objectives and plan of work presented here. The first objective was to design and test an O2 diffusion measurement system on earth capable of automated water content adjustment with minimal hydrostatic effects on the sample. The second objective was to model characteristics of gas diffusion in partially saturated coarse-textured and aggregated porous media on earth and eventually in a reduced gravity environment.
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THEORY
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Substrate-Water Characteristic
The fundamental physical relationship for soils and plant growth media describing water content and matric potential is also useful for describing other physical processes such as gas diffusion. Gas diffusion being air-filled porosity () dependent, may be described in terms of the total porosity () less the volumetric water content, . An effective and commonly used parametric model for relating to the matric suction (h) was proposed by van Genuchten (1980), given as
 | [1] |
This relationship describes the saturation, 
, in terms of the residual (r) and saturated (s) water contents or in terms of the empirical parameters 
, n, and m. These parameters are directly dependent on the shape of the (h) curve. For aggregated porous media, it may be sufficient to characterize water retention only within the macro pore domain since water held within the internal aggregate pores may not be available to plant roots. As mentioned earlier, water held between millimeter-sized aggregates may be removed using modest suction (e.g., 5–25 cm). Figure 2 illustrates water retention curves for (i) four different size classes of sand and (ii) two classes of aggregated-calcined clays, revealing the low range of matric suction at which water is drained. Note the large water content change that occurs over a relatively small matric suction range, which for hydrostatic conditions in potting media relates directly to height above the drainage point in containerized media. The parameters for these media are listed in Table 1 showing the draining curve parameters for the sand and both wetting and draining curve parameters for the calcined clay.
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Fig. 2. Modeled substrate water characteristic curves for different particle size ranges of (a) sand (Schroth et al., 1996) and (b) calcined clay are plotted using Eq. [1]. In (b) both measured draining (filled symbols) and wetting (empty symbols) retention curves for the aggregate external (macropore) water are shown.
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Table 1. Parameters of the van Genuchten (1980) water retention model for different particle sizes and soil classes where the assumption m = 1 - (1/n) is used.
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Dual-Chamber Gas Diffusion Model
Analytical solutions to specific diffusion problems relate Fick's laws of diffusion to a physical system such as a single or double chamber device (Rolston and Moldrup, 2002). For conditions where atmospheric pressure fluctuations may induce unwanted convective fluxes, a sealed dual chamber device is preferable. An analytical solution (Glauz and Rolston, 1989) was derived for determining the gas diffusion coefficient, Ds, using a dual-chamber diffusion apparatus (Fig. 3). Assuming a constant cross-sectional area, the model was derived in terms of chamber dimensions H, K, and L. Parameters describing the relative lengths were omitted in the paper of Glauz and Rolston (1989) and are given as (Rolston and Moldrup, 2002)
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Fig. 3. Gas diffusion chamber design showing source (O2) and sink (N2) chambers on either side of the substrate chamber, each of specified length assuming each with the same cross-sectional area as the substrate chamber. Each air chamber is equipped with a galvanic O2 sensor and dual solenoid valves for air and N2 priming. The substrate chamber is underlain by a porous stainless steel sheet providing water to the substrate and with a heat pulse moisture probe and tensiometer for matric suction determination.
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Using measured concentration as a function of time, t, Ds is optimized to c(t) data where co is the initial gas (O2) concentration.
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The additional parameters A, B, and 1, neglecting second-order effects and concentration in the source chamber, are given as
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 | [6] |
and for monitoring concentration in the sink chamber, Eq. [5] is replaced with
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Gas Diffusion in a Porous Medium with Unimodal Pore-Size Distribution
The major mechanism for gas exchange and transport within a porous medium lacking convective forces is by diffusion through the gaseous and the liquid phases. This spontaneous process results from the thermal motion of gas molecules in air or solution. The driving force for diffusion is a gradient of concentrations or partial pressures of gas within the soil air. For the simple case of a sand exhibiting a unimodal pore-size distribution illustrated in Fig. 2, the relative gas diffusion coefficient can be described in terms of bulk medium properties such as air-filled porosity, , and total porosity, . A recent modification of the Marshall (1959) gas diffusion model was termed the water-induced linear reduction (WLR) model (Moldrup et al., 2000):
 | [8] |
where Da is the gas diffusion coefficient in air (1.9 x 10-5 m2 s-1). This model accounts for increased tortuosity exhibited in two different media, one made up of water and solids and the other of only solids, both having the same total air-filled porosity. The WLR model uses the ratio of air-filled porosity to total porosity to account for this difference and was tested using six differently textured and repacked soils. The Marshall (1959) version of the WLR model outperformed other diffusion models to which the WLR concept was applied.
Gas Diffusion in Bimodal (Aggregated) Porous Media
The influence of pore-size distribution on fluid permeability was conceptualized by Childs and Collis-George (1950) using a statistical ‘cut and rejoin’ theoretical approach. The cut and rejoin approach bases gaseous diffusion on the probability of pore continuity along a common plane. The idealized porous medium composed of capillary tubes is sliced and apposed with varying degrees of freedom of reattachment. The probability of continuity after relocation was described in terms of the porosity, , and was said to vary between x and 2x where 0.5 < x < 1 (Millington and Quirk, 1961). An implicit relation where is replaced by describes the air-filled diffusing (j) and solid- or water-filled non-diffusing (1 - j) fractions of the jth phase given as
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For convenience in modeling, a polynomial expression for xj fit (r2 = 0.997) to iterated solutions of j in increments of 0.05 from 0 to 1 is given as
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This theoretical framework, applicable to variably saturated porous media, was extended to describe gas diffusion through aggregated media where the internal and external aggregate porosities were accounted for separately (Millington and Shearer, 1971). The system is illustrated for idealized spherical aggregates in Fig. 4 where the internal aggregate porosity (i) contains pores that are much smaller than pores making up the external porosity (e), leading in general to saturation of the internal pores before external pore wetting. The Millington and Shearer (1971) expression is rewritten here in terms of air-filled porosity of each pore domain
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where the internal and external air-filled porosities (i, e) characterize gas diffusion within the two pore regimes. Exponents, x0, x1, and x2 describe the effective pore cross-sectional areas for diffusion relating pore connectivity and tortuosity of the inner-aggregate and intra-aggregate pore systems. The exponents are determined from Eq. [9] and [10] using j values defined in a bimodal pore system as (Collin and Rasmuson, 1988)
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where s is the solid fraction and both 0 and 2 are water content dependent. Note that these values of air-filled porosity differ from i and e in Eq. [11]. For saturated conditions, the diffusion rate goes to zero in Eq. [11] and does not account for gas diffusion through the liquid phase. The diffusion coefficient through pure water is roughly four orders of magnitude smaller than through air and as such provides a lower limit for gas diffusion. Readers are referred to the work of Collin and Rasmusen (1988) who derived an expression similar to the Millington and Shearer (1971) relationship, including diffusion through the water phase.
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MATERIALS AND METHODS
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Calcined clay (Profile Products, Buffalo Grove, IL) was sieved to size fractions of 1 to 2 and 0.25 to 0.85 mm. The particle density is 2.5 g cm-3 according to the manufacturer and the bulk density varies from about 0.6 to 0.65 g cm-3. Wetting and draining water retention curves were generated using a hanging water column (Dane and Hopmans, 2002) packed to a 1.5-cm depth and at a bulk density of 0.65 g cm-3. For this bulk density, the internal and external aggregate porosities are approximately half (0.37) of the total porosity (0.74).
Diffusion Cell Design
For higher water contents (macropore water) where gravitational forces compete with capillary forces in coarse-textured media, a thin horizontal diffusion pathway serves to minimize gravitational effects on the vertical water content distribution in the sample. The problem for a vertical (series) diffusion path is that the wettest layer at the bottom of the sample controls the resulting diffusion coefficient. For the horizontal (parallel) diffusion scenario the driest region would dictate the diffusion coefficient assuming a horizontally stratified water content distribution making sample height an issue in either case. The minimum sample height should also provide a representative volume, which is a function of particle size and packing. For our granular media of up to 2-mm diameter a sample depth of 1.9 cm assures a minimum ratio of the chamber height/particle diameter of 10 (Cumberland and Crawford, 1987).
Based on a thorough analysis of the dual chamber diffusion cell design (Glauz and Rolston, 1989), the optimal chamber dimensions are intimately tied to the air-filled porosity of the sample. As sample air-filled porosity is reduced, so does the optimal (i.e., for minimum measurement error) sample length. Using an average value of air-filled porosity 0.2 where the total macro porosity was 0.4, the soil chamber length was 12 cm (2L in Fig. 3). This design criteria was based on their analysis for minimizing the measurement error. Two different dual chamber designs were developed with a rectangular cell providing the minimum sample height for studies on earth and the cylindrical diffusion cell being developed for microgravity (Jones et al., 2002). Here our focus is on the rectangular diffusion cell in which the substrate sample is contained on both ends by a stainless steel screen (74 mesh) whose open area for diffusion is 52.7% with a mean opening size of 0.249 mm. A porous stainless steel plate (5 µm, Mott Corp., Farmington, CT) at the bottom is used as a water supply/removal membrane. Control, monitoring and automation of diffusion measurements were performed using a CR10X data logger (Campbell Sci., Logan, UT). A microliter metered peristaltic pump (Instech Laboratories, model P625, Plymouth Meeting, PA) controls water additions or removal at 144 µL per revolution. A heat-pulse moisture sensor and tensiometer provide confirmation of the substrate water content and energy status, respectively (Bingham et al., 2002). Galvanic oxygen sensors (Model R22D, Teledyne Analytical Inst., Los Angeles, CA) on the top of each gas chamber continually monitor O2 concentrations during the diffusion process. The inlet and outlet ports for N2 and O2 priming are normally closed solenoid valves that are actuated during fluid exchange (i.e., either water or air).
Design Limitations
The mathematical solution to the dual chamber diffusion problem satisfies assumptions of Fick's law where the initial boundary conditions specify a diffusing gas concentration of zero in the soil and sink (N2) chambers and the tracer gas (O2) concentration is initially C0 in the source chamber. To accomplish this, a mechanical gate is required to seal the source chamber during priming (Rolston and Moldrup, 2002). For an automated measurement system made up of multiple diffusion cells, a gate adds considerable complexity to the system control and maintenance. Because of mechanical and power restrictions on the International Space Station where automated measurements will be made, we found it necessary to develop a modified design. In a previous study using a single chamber diffusion cell, gate and no-gate measurements were compared using two different diffusion coefficient analysis techniques. In that study no significant differences were found in the resulting diffusion coefficients (Jones et al., 2002). For the dual-chamber diffusion cells used here, a similar comparison of diffusion measurements were made where the gate could either be closed using the conventional approach (Rolston and Moldrup, 2002) or left open during gas priming. Results of measurements in 1- to 2-mm calcined clay made using Cell 1 equipped with a gate and Cell 2, which had no gate, are shown in Table 2. For purposes of reproducibility, each diffusion cell was brought to the water content transition between the two pores regimes yielding an air-filled porosity of 0.37 (i.e., macropores are completely air-filled). Mean diffusion coefficients measured with and without the gate closed during priming were not significantly different (error of 1 standard deviation) from Cell 2, which had no gate installed, and were comparable with Ds (0.37) predicted by the Moldrup et al. (2000)WLR model. The lack of a gate alters the initial substrate gas concentration assumed in the theory and poses a short-term perturbation in the gas concentration profile adjacent to the source chamber. This discrepancy between model and measurement appears to be minor based on the agreement between measurements made without the gate and the probabilistic model of Millington and Shearer (1971). The lack of a gate does require care in the gas priming procedure. We equilibrate the air priming flow rates in both chambers to minimize convective gas transfer between air chambers and maximize the initial concentration gradient. No air mixing fan was used because the largest air chamber volume was similar (120 cm3) to the 100-cm3 threshold suggested by Currie (1960).
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Table 2. Comparison of gas diffusion coefficient mean and standard deviations measured in gated (Cell 1) and non-gated (Cell 2) dual chamber cells using 1- to 2-mm calcined clay.
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Galvanic oxygen sensors (Model R22D, Teledyne Analytical Inst., Los Angeles, CA) are available for less than $100 and provide a continuous monitoring capability. They act like a DC battery whose output voltage is linear and proportional to O2 concentration, but whose potential decreases under atmospheric conditions requiring periodic recalibration. The sensor life under normal atmospheric conditions is about 2 yr. While these sensors provide a convenient measurement of the gas of interest, the use of O2 as the tracer gas is a disadvantage because of its reactivity where, for example, significant microbial activity could lead to an over prediction of the oxygen diffusion rate. Our analysis showed that using moist calcined clay, there was a slow but steady decrease in O2 with time of up to 1% per day in a sealed 70-mL volume, depending on moisture level. Given a supplemental source of C and nutrients, the O2 concentration would generally drop to 0% within 1 to 2 d. A correction in the diffusing rate could be derived from monitoring the rate of O2 reduction after priming the entire chamber with atmospheric air. This correction technique will be evaluated as part of our ongoing research.
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RESULTS
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A demonstration of the gas diffusion measurement system illustrates the water content adjustment capability and the gas priming and diffusion coefficient determination capability. Finally, measured air-filled porosity-dependent diffusion coefficients are plotted in comparison with the WLR model of Moldrup et al. (2000) and to the Millington and Shearer (1971) diffusion model for aggregated porous media.
Water Content Control
Water content in the porous medium was controlled using a precision peristaltic pump in combination with tensiometric measurements. The heat-pulse moisture sensor was also useful for verification of relative moisture level but the pump provides a more accurate measurement system. This is based on several conditions; the first being that pump suction does not exceed the bubbling pressure of the porous water control membrane and this was achieved using microliter incremental pumping coupled with continuous pressure monitoring in the water line. The second requirement is to have negligible evaporation (occurring only during gas priming) and establish a reference water content that could be zero if initially dry or some repeatable value. Considering the water retention curves shown in Fig. 2, which are common for coarse-textured media, the reference could occur at the asymptotic portion of the curves, preferably at the dry end since there is considerable variation in the saturation value due to air entrapment. In the calcined clay material shown in Fig. 2b, four samples of each size were presaturated and drained to a matric suction of 20 cm yielding volumetric water content average and standard deviations of 0.373 ± 0.004 for 1 to 2 mm and 0.374 ± 0.006 for 0.25- to 0.85-mm particles. A reference water content of 0.37 cm3 cm-3 for the calcined clay was used. Steady-state water content was established after pumping a prescribed volume and by monitoring output from the external tensiometer in the outer cell wall. The water retention curve for the 0.25–0.85 mm calcined clay material is regenerated in Fig. 5 by plotting tensiometer readings against pumping volume (converted to ) for consecutive wetting and draining processes. After initial wetting and draining to prime the initially dry media, the first diffusion measurement began with Point 1 and proceeded from there along the wetting curve to near saturation returning along the draining curve to Point 10 shown in Fig. 5. The disparity between the modeled curves from Fig. 1 and the measurements is in part attributable to the different sample packing and slower wetting process in the case of the measured data, which occurred over several weeks compared with several days for data in Fig. 1.
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Fig. 5. Modeled water retention for the calcined clay (0.25–0.85 mm plotted in Fig. 2) compared with the wetting and draining process measured by pumping and tensiometric readings showing the expected hysteresis in water retention. Order of measurement is indicated by numbers and pressure jumps arise from ambient air pressure changes during diffusion.
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Determination of Gaseous Diffusion Coefficient
Before the initiation of a diffusion measurement, the two chambers are flushed with N2 and air until the O2 concentrations are relatively steady state near 0 and 20% shown in an example in Fig. 6, at which point, the solenoid valves are closed to seal the diffusion cell. Details on the algorithm for determining priming time and diffusion time are given elsewhere (Jones et al., 2003). The diffusion coefficient is defined by the changing O2 concentration in each chamber recorded as a function of time. The diffusion coefficient model was fit to the measured data in Fig. 6 by adjusting Ds in Eq. [3]. In this case the medium was 1- to 2-mm calcined clay with = 0.22 and the resulting value of Ds = 0.88 cm2 min-1. The equilibrium concentration in the system at infinite time may be estimated from the following expression (Glauz and Rolston, 1989).
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Fig. 6. Concentration change of O2 (%) measured in the sink and source chambers of the diffusion cell showing both the gas priming stage and diffusing stage (note different time scales). The modeled diffusion coefficient was fit using Eq. [3] to the measured sink chamber concentration for 1 to 2 mm calcined clay with = 0.22 and the resulting value of Ds = 0.88 cm2 min-1.
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Diffusion Dependency on Air-filled Porosity
The gas diffusion coefficient determination procedure described in the previous section was performed for each measurement cycle consisting of up to ten points along the wetting/draining water retention curve illustrated in Fig. 5. Oxygen diffusion measurements made in the two sizes of aggregated media are plotted in Fig. 7 as a function of air-filled porosity. In Fig. 7a measurements were made in 1- to 2-mm particles earlier in the development of the diffusion cell when water content was manually controlled, which may account for some of the dispersion in the data compared with Fig. 7b in which complete automation was available. Internal aggregate pore wetting had little impact on diffusion as evidenced by the ‘repacked’ data measured with a conventional single chamber diffusion device. Modeled results of the Millington and Shearer (1971) equation are in agreement with this trend. In the transition from internal pore wetting to external pores measured diffusion coefficients are greatly reduced as described by the Millington and Shearer model. Making the assumption that the total porosity is equal to the external aggregate porosity (e = 0.37), the WLR model of Moldrup et al. (2000) shows excellent correlation to the diffusion data in the external pore water region. This result is not surprising given the nature of the WLR correction, which accounts for a diffusion path reduction due to water phase using the ratio of air-filled porosity to total porosity. The success of the Millington and Shearer model in capturing the trend of diffusion across these two pore domains in this medium is surprising considering the only required inputs are the internal and external air-filled and total porosities. The model describes diffusion in the internal pore domain with the assumption that flux between aggregates must pass along contact points, greatly reducing the internal pore diffusion (Collin and Rasmuson, 1988). The large disparity between the internal and external aggregate pore sizes, satisfy the requirement for the extreme case of this model where complete saturation of the internal pores occurs before water is retained in the external pores.
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Fig. 7. Measured oxygen diffusion coefficients in (a) 1- to 2-mm aggregates using manual adjustment of sample water content and (b) 0.25- to 0.85-mm aggregates using automated control. The WLR model (Moldrup et al., 2000) assumes a total porosity of 0.37 (Eq. [8]) and models the external pore diffusion only. The M&S model (Millington and Shearer, 1971) describes the entire pore system including measurements made in the internal pore water domain using a conventional diffusion chamber where samples are pre-wet and repackaged for each measurement. Two identical diffusion cells were used for measurements in the external pore domain where wetting and draining indicate the process of addition or removal of water between measurements.
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Influence of Sample Thickness and Gravitational Force on Diffusion Coefficient
We illustrate the influence of sample thickness, T, and gravitational force, g, on diffusion coefficient determination using the assumption of one-dimensional gas flux. The ratio of gravity affected (1 g) and gravity-free (0 g) diffusivities computed for a relative water content at the sample base (z = 0) of 0.99 is presented in Fig. 8. This calculation was made by solving Eq. [1] in terms of and substituting ‘ - ’ for in Eq. [8]. The 1 g gas diffusion is the integral of diffusivities computed using Eq. [8] along the vertical sample height where is defined in terms of h and z, and z is the variable of integration taken from 0 to T (see Fig. 1c). The 0 g gas diffusion is computed similarly for the same sample thickness, but assuming a uniform water content, which is computed as the integral of water contents from 0 to T. The reference in either case is the matric potential, h0, located at the bottom of the sample (i.e., corresponding to = 0.99). This ensures the sample is just beginning to desaturate at the bottom and represents the maximum ratio or difference for our interests. The diffusion ratio is given as
 | [14] |
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Fig. 8. Sample thickness- and gravity-dependent gas diffusivities computed as the ratio of the integrated diffusion coefficient [Ds(1 g)] and the diffusion coefficient computed at the mean water content of the sample [Ds(0 g)]. Both the integration of Ds(1 g) and the integration of water content at which Ds(0 g) is computed are taken over the sample thickness, T, (e.g., 1.9 cm in the diffusion cell) using the draining retention curves shown in Fig. 2b. The relative water content, , at the sample base is 0.99 for each of the five media shown. Representative sample volume indicators described by ratios of ten times the mean particle diameter, d, to sample thickness (i.e., 10 x d x T-1) are plotted for d = 1.5 and 0.55 mm, representing the 1- to 2- and 0.25- to 0.85-mm particles, respectively.
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This ratio indicates the magnitude of the expected diffusion on earth relative to the diffusion in a sample of uniform water content.
The effect of the gravity-induced hydrostatic water distribution on the gas diffusion coefficient is compared with the expected diffusion coefficient in a uniform water content distribution in 0 gravity given in terms of sample thickness shown in Fig. 8. Five different porous media are analyzed, with four of the five media showing the peak difference occurring for sample thickness between 3 and 6 cm. For the 1.9-cm thick sample in the diffusion cells used here, the expected ratio of Ds(1 g)/Ds(0 g) is 1.5 to 2. This difference may be difficult to measure given the several order-of-magnitude change in diffusion coefficient as a function of or . The magnitude of the error is proportional to the parameter n of Eq. [1] listed in Table 1. The n parameter is related to the media particle-size distribution (also pore-size distribution), suggesting narrow distributions lead to greater differences in gas diffusion for this case where the base water content, = 0.99. For reduced water contents the magnitude of the difference is reduced and the sample thickness where the peak difference occurs is greater.
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SUMMARY
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The automated gas diffusion measurement system presented is based on a closed dual-chamber diffusion cell design for minimizing hydrostatic effects on the sample profile air-filled porosity. The system developed for coarse-textured porous media relies on a porous membrane to control the sample water content where matric suction is monitored using an in situ tensiometer and a pressure transducer in the water supply line. Water retention data derived from a tensiometer and a microliter peristaltic pump used to add or remove water were reasonably correlated to measured retention data using a hanging water column. Continuous monitoring of oxygen concentration in both gas chambers by galvanic cells provide data both during gas purging (N2 and air) and throughout the diffusion measurement. Oxygen diffusion was modeled across the dual pore system of the aggregated calcined clay using the Millington and Shearer (1971) model, which required only internal and external aggregate air-filled and total porosities as inputs. The WLR model (Moldrup et al., 2000) was better correlated to diffusion data than the model of Millington and Shearer (1971). Measurements made using the dual chamber system described will be used for comparison with data collected in microgravity on the International Space Station to advance model development and the understanding of porous media physics in space. The diffusion measurement system has application to coarse-textured porous media where water content can be controlled within the bubbling pressure range of the porous membrane.
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ACKNOWLEDGMENTS
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The authors gratefully acknowledge funding from NASA-JSC NRA award 99HEDS-02, contract NAG9-1284. We thank the Associate Editor, Gerard Kluitenberg, and two anonymous reviewers for their helpful comments.
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REFERENCES
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ABSTRACT
INTRODUCTION
