Published in Vadose Zone Journal 2:409-415 (2003)
© 2003 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH PAPER
Reevaluation of the Multistep Outflow Method for Determining Unsaturated Hydraulic Conductivity
Haruyuki Fujimaki*,a and
Mitsuhiro Inoueb
a Institute of Agricultural and Forest Engineering, University of Tsukuba, 1-1-1 Tennohdai, Tsukuba, Ibaraki 305-8572, Japan
b Arid Land Research Center, Tottori University, 1390 Hamasaka, Tottori 680-0001, Japan
* Corresponding author (fujimaki{at}sakura.cc.tsukuba.ac.jp).
Received 10 January 2003.
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ABSTRACT
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The multistep outflow (MSO) method is widely used for determining the unsaturated hydraulic conductivity, which is essential for accurate prediction of water flow in soils. However, the reliability of this method has yet to be adequately verified through comparisons with independently measured conductivities of soils having different textures and structures. We conducted MSO experiments with three soils of different textures to test the reliability of the conventional MSO. Three methods were used to determine the hydraulic conductivities, K(
), from identical experiments: an optimization method with readings of a tensiometer (OM), a direct method using one tensiometer (DM1), and a direct method using two tensiometers (DM2). Outflow data were also used in the three methods. Additionally, steady-state downward flow experiments using two tensiometers were performed for comparison. For two of the three soils, OM and DM1 clearly underestimated K() determined by SDFM. On the other hand, DM2 gave good agreement with the SDFM data. These results and the discrepancies between the fitted and observed tensiometer readings near the bottom imply the existence of a hydraulic resistance at the soil–porous plate interface. The hydraulic resistance might be caused by pore plugging with fine particles (colloids) transported to the ceramic plate after stepwise changes in pressure. We suggest that an additional tensiometer be installed near the bottom of the sample and be used for checking the optimized K() with the direct method (DM2).
Abbreviations: DM1, direct method using one tensiometer • DM2, direct method using two tensiometers • MSO, multistep outflow • OM, optimization method • SDFM, steady-state downward flow method
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INTRODUCTION
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ACCURATE DETERMINATION of soil hydraulic properties (i.e., the soil water retention and hydraulic conductivity functions) is a prerequisite for predicting water and solute movement in soils. Many methods for determining hydraulic properties have been developed during the past 40 yr or more (Bruce and Klute, 1956; Gardner and Miklich, 1962). In the last two decades, several inverse methods based on parameter optimization for soil hydraulic functions have been developed to make the measurement of hydraulic properties faster and less laborious. The MSO method in combination with inversion is now being widely used (e.g., Vereecken et al., 1997; Weerts et al., 1999).
In spite of the extensive use of the method, to our knowledge the reliability of the MSO method has not been sufficiently evaluated through comparisons with independently measured unsaturated hydraulic conductivities of a large variety of soil types. In particular, the hydraulic resistance at the interface between the soil sample and the bottom porous plate has been assumed to be negligible, although pore plugging may be more common than assumed, as stated by Gee et al. (2002). The effect of hydraulic resistances at the soil–plate interface may be more important for determining the hydraulic conductivity than for measurement of retention data. Eching et al. (1994) compared the MSO results with a direct method and an evaporation method. However, this study may not have been fully adequate for an accurate comparison, since they did not include hydraulic resistances at the soil–plate interface in their analysis. Also, only one soil was used in the comparisons with the evaporation method of Wendroth et al. (1993). On the other hand, Hollenbeck and Jensen (1998) reported several experimental limitations of MSO for sand. Their results suggest the need for more tests on the MSO method for a wide range of soil textures.
The objective of this study was to assess the accuracy of the conventional MSO method by comparing results with independently measured unsaturated hydraulic conductivities for three soils. Hydraulic conductivity curves based on an inverse parameter estimation analysis of a MSO experiment are compared with a direct method using readings of tensiometers inserted at two different depths, as well as hydraulic conductivity data from a steady-state downward flow method (SDFM; Klute and Dirksen, 1986).
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MATERIALS AND METHODS
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Soil Samples
Experiments were performed for Masa loamy sand, Yazu sandy loam, and Tohaku loam. All of the soils were taken from Tottori Prefecture, Japan. Results of the particle-size analyses, bulk densities, and saturated hydraulic conductivities are listed in Table 1. All of the experiments described below were conducted at 25 ± 1°C. Saturated hydraulic conductivities were measured using the falling head method. In this study we used disturbed soil samples to enable comparisons with other methods.
Multistep Outflow Experiment
Each soil was well leached, air dried, sieved through a 2-mm screen, and packed to predetermined bulk densities in 0.05-m-i.d. and 0.051-m-high stainless-steel cylinders assembled in pressure cells (Sankeirika Corp, Tokyo) with 0.005-m-thick, 10-m air-entry ceramic plates (B01M3, Soil Moisture Corp., Santa Barbara, CA) at the bottom (Fig. 1). To pack the soils as uniformly as possible, we divided the samples into five masses equivalent to 0.01 m (or 0.011 m) for each sample. We then confirmed whether the increment of 0.01 m (or 0.011 m) was achieved for each incremental filling and tapping. This resulted in about 0.002 m excess over the top of the column wall, but shrinkage during the initial wetting reduced the height by about 0.002 m. The hydraulic conductivity and thickness of the ceramic plate were 1.25 x 10-5 m s-1 and 0.005 m, respectively. To flush the air bubbles from underneath the porous plate, the cell had an extra outlet at the bottom.
Before packing, two tensiometers were placed vertically as shown in Fig. 1. The vertical centers of the porous cup were placed at 0.026 and 0.046 m below the soil surface. The lengths of the porous cups were 0.01 m. To avoid a convergent flow just below the upper porous cup (which would violate the assumption of one dimensionality), a "dummy" plastic probe with the same diameter (0.006 m) as the tensiometer tubing was attached below the tensiometer. An epoxy canopy of 1-mm thickness isolated the lower porous cup from the porous plate.
The soil samples were wetted from below by imposing a positive pressure of about 0.1 m at the bottom of the plate and allowed to saturate overnight. To prevent pore plugging due to microbial growth, a 1.2 mmol L-1 sodium azide (NaN3) solution, a well-known biocide widely used as a pesticide (Seki et al., 1998), was used as the soil solution. After near saturation, atmospheric pressure was applied at the bottom of the soil by placing the outlet of the needle at a height of 0.02 m below the bottom. Since surface tension at the outlet induces a pressure of about 0.02 m, the resultant pressure head at the bottom might be about 0 m. After equilibrium, the cell cover was attached and quick desorption was started by applying a pneumatic pressure of 0.5 m from the top. The bottom outlet was connected to a reservoir placed on a load cell. The reservoir and load cell were placed in a plastic box to minimize evaporative water loss, while a small opening at the cable outlet kept the inside at atmospheric pressure.
Both the outflow amount and the pressure heads were monitored automatically at 300-s intervals. When the outflow essentially ceased, the pneumatic pressure was increased to 2.0 m, as shown in Fig. 2. After the next hydraulic equilibrium was nearly attained, the pneumatic pressure was changed to 6.0 m. This was the final step. The sample was then removed from the cell to measure the final volumetric water content gravimetrically at the point when the outflow nearly ceased. These hydraulic equilibriums enabled an independent determination of the soil water retention curve, providing four "pressure plate" retention measurements at -0.026, -0.526, -2.026, and -6.026 m. Soil water pressure heads were calculated as the difference between the measured soil water pressure head and the applied pneumatic pressure.
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Fig. 2. Comparison of the observed and optimized outflow, and pressure head. The arrows in (a) indicate when the direct methods were performed. The pressure heads here indicate tensiometer readings or pressure heads at the bottom (i.e., atmospheric one) subtracted from the pneumatic pressure head.
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Inverse Parameter Estimation of the Multistep Outflow Experiment
The values of hydraulic parameters in hydraulic functions used in solving the water flow equation were optimized with the Levenberg–Marquardt maximum neighborhood method (Marquardt, 1963). A Pascal code by Press et al. (1989), which implements the Levenberg–Marquardt maximum neighborhood method, was used with some modification. The one-dimensional water balance equation combining the liquid and gaseous phases is given by
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where is the volumetric water content (m3 m-3), t is the time (s), q
is the liquid water flux (m s-1), qv is the water vapor flux, z is the depth (m), and Sw is the sink term, which is zero in this study. The liquid water flux is described by Darcy's Law:
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where K is the hydraulic conductivity (m s-1), and h is the pressure head (m). Because the pressure head never decreased to below -6.0 m in the experiment, and no temperature gradient was induced, the water vapor flux, qv, could be neglected. The soil hydraulic functions were assumed to be described by the widely used equations presented by van Genuchten (1980):
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where m = 1 - 1/n, subscripts "sat" and "r" denote the saturated and residual values, respectively, and 
(m-1), n, and l are empirical parameters.
The objective function to be minimized in the algorithm is
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where b is the vector of optimized parameters, j represents the different sets of measurements, N is the number of the measurement sets, nj are the number of measurements within a particular set, wj,i are weights associated with a particular measurement point. The measurements p*j,i are of type j at time ti, pj, i(b) are the corresponding model predictions using the parameters in b, and 
j are the variances of the measurements of data type j. The coefficients (nj, j)-1 are used as the weighting factors to adjust the numbers and the measurement scales. In this study, three measurement sets were included in O(b): the initial and final average water content in the soil core (j = 1), tensiometer readings at the center (j = 2), and the cumulative outflow (j = 3). The negative values of the imposed pneumatic pressure were applied as the lower boundary condition. This condition is essentially the same as was done by Eching et al. (1994).
The initial values of the average water contents were used as the initial estimates of sat. The initial r was set at one-half of the average at the final sampling, except for the Tohaku loam for which the soil r was fixed at zero. The initial estimates of and n were iteratively determined with the bisection method so that Eq. [3] satisfied the two values of the retention data ( at h = -0.53 and -2.03 m). The saturated hydraulic conductivities determined using the falling head method were used as the initial Ksat. The initial l was set at -1, which is the value suggested by Schaap and Leij (2000), found through an empirical method. Using this and other available information for setting the initial parameters may promote a quick convergence of the optimization.
The numerical solutions of Eq. [1] and [2] were obtained using a finite difference scheme with modified Picards iteration (Celia and Bouloutas, 1990). The scheme of space discretization and solution was similar to that of the SWAP code (van Dam et al., 1997). Space increments were set following a nearly geometric progression so that the topmost and bottommost increments were 0.005 and 0.001 m, respectively. The calculated pressure heads at z = 0.026 m were obtained with linear interpolations of values at adjacent elements. The time increments were controlled automatically so that the number of iterations in each time step would be around 5, with an upper limit of 0.05 h. The initial conditions were that of hydraulic equilibrium with -0.051 m at the soil surface. The hydraulic resistance of the porous plate was incorporated.
Direct Estimation for the Multistep Outflow Experiment
Unsaturated hydraulic conductivity can also be estimated directly from an inversion of Darcy's Law:
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where the subscripts represent the tensiometer depth level. Several direct methods have been suggested (Eching et al., 1994; Wildenschild et al., 2001). Since the method by Wildenschild et al. (2001) requires a specially designed location for the tensiometers, it could not be applied in this experiment. Also, since the method by Eching et al. (1994) assumes the nonexistence of resistance at the soil–plate interface, the output from this method would not be reliable. Therefore, in addition to Eching's method, we performed a newly designed direct method.
Eching et al. (1994) calculated the hydraulic conductivity at the center of the core. They estimated the water flux at the center as one-half that of the outflow flux. This implies that the water flux was distributed linearly with depth. On the other hand, the hydraulic gradients were given by a linear approximation between the top (z1 = 0) and the bottom (z2 = L). They approximated the pressure heads at the top (h1) with tensiometer readings at the center. Those at the bottom (h2) were derived from the saturated hydraulic conductivity of the porous plate (Kp) and the outflow flux (qbot):
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where hptop is the pressure head at the upper boundary of the porous plate, and Lp is the thickness of the porous plate (0.005 m). They assumed that h2 equals hptop and that Kp does not change with time. We also applied this method to our experiment, calling it the direct method with one-tensiometer readings (DM1).
For the second direct method, we calculated the hydraulic conductivity at the center (z = 0.036 m) between two tensiometers. We also assumed that water flux linearly distributes with depth: 3.6/5.1 = 71% of the outflow rate was used as the flux at the center. On the other hand, the tensiometer pair provided the hydraulic gradient with a linear approximation. Note that a linear interpolation gives an accurate gradient at the center if the distribution is parabolic. Through our numerical experiments, this assumption of linearity of flux was found to be valid in practice, except for just after hbot changed. On the other hand, at a late stage in each step when the outflow is about to stop, the head difference of the tensiometers is so small that an error in output or pressure transducer can result in a large error. Thus, direct estimations were performed when the cumulative outflow amount in each step reached one-half of the final outflow for the step, as indicated by the arrows at 18.3 and 25 h in Fig. 2a. Additionally, data from the first step were eliminated because the hydraulic gradients were too small to allow errors in the tensiometer readings to be neglected. Figure 3 shows an example of the simulated flux profiles for Masa loamy sand. These profiles result from numerical solutions with optimized parameter values (listed below) at the moments when the direct K data were calculated. Linearly approximated profiles (broken lines) are in fair agreement with the simulated ones. We used Eq. [3], with parameters obtained from a curve fitting of the retention data obtained from near-equilibrium outflow data to convert from pressure head to water content.
Steady-State Downward Flow Experiment
Hydraulic conductivities at higher water contents from saturation to a pressure head of -2.0 m were measured with a SDFM (Klute and Dirksen 1986). The differences in bulk density compared with the soils used for the MSO experiments were <0.02. Tensiometers were inserted horizontally into a 0.25-m-high soil column at depths of 0.10 and 0.20 m. The use of the same core as the multistep outflow experiment might have enabled more reliable comparison. However, since the tensiometers were inserted vertically in the outflow experiment and the dimensions are rather small for SDFM, the same core could not be used.
Sodium azide (1.2 mmol L-1) solution was applied with a peristaltic pump from the top while negative pressure was applied through a porous plate at the bottom. Hydraulic conductivities at the average pressure heads between the pair of tensiometers were obtained under nearly unit hydraulic gradient at successive steady-state conditions during drainage from saturation. Pressure gradients were obtained by linear approximation of the tensiometer readings. The retention curves determined with the multistep outflow experiments were used to convert from pressure head to water content.
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RESULTS AND DISCUSSION
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Optimized Numerical Solutions
Measured and fitted outflow and tensiometer readings are shown in Fig. 2. Excellent agreements between the measured data and the numerical solutions were obtained for tensiometer readings at z = 0.026 m for both methods, except for the third step of the Yazu sandy loam. Also, the fitted outflow amounts agreed well with the observations. However, the fitted tensiometer readings at z = 0.046 m underestimated, while those at z = 0.026 m matched the measurements, except for the third step of the Yazu sandy loam. These discrepancies are obvious for the second stages of the Masa loamy sand, the second and third stages of the Yazu sandy loam, and the third stage of the Tohaku loam. Note that the tensiometer readings at z = 0.046 m are not included in the objective function. For the third step of the Yazu sandy loam, the simulated differences in tensiometer readings between z = 0.026 and 0.046 m were larger than the measured differences. These results imply that the hydraulic conductivity functions are underestimated, since, with the "optimized" hydraulic conductivities, greater hydraulic gradients than the measured ones were required in the simulation to drive nearly the same flux as the observation.
Soil Water Retention Curve
In Fig. 4, inversely determined retention curves and fitted retention curves for retention data from near-equilibrium outflow data are shown. For Tohaku loam, the r was fixed at zero, since the nonrestricted fitting gave a negative value. The parameter values are listed in Table 2. The inversely determined curves and the fitted curves both agreed well with the retention data for all three soils, indicating the reliability of both the inversely determined and the fitted curves.
Unsaturated Hydraulic Conductivity
Hydraulic conductivities, K(), determined from the four methods described above, are given in Fig. 5. The optimized parameter values are presented in Table 2. Standard errors of l are also listed. Standard errors of the other parameters are <1% of the parameter values. The optimized K() (OM) are smaller than data from the SDFM, except for the case of the near saturation of the Tohaku loam. For around K = 10-7 m s-1, which is the practical lower limit of the SDFM, underestimation is apparent for the Yazu loamy sand and the Tohaku loam. The small standard errors of the parameters also confirm the underestimation. Hydraulic conductivities from the DM1 readings gave a similar value to the OM curve. On the other hand, the data from the DM2 readings are consistently greater than data obtained from the DM1, and are in good agreement with the independently measured hydraulic conductivities (SDFM).
In addition to the discrepancy between the observed and the fitted tensiometer readings at z = 0.046 m, these results imply the existence of hydraulic resistance at the soil–porous plate interface. We did not use the 0.01 M CaC12 solution that is commonly used to minimize clay dispersion for either MSO or the SDFM. However, clay dispersion cannot account for the discrepancy between the DM2 and DM1 relative to the SDFM and optimized K() data from the OM. Thus, the K() determined from the OM and the DM1 might be "bulk hydraulic conductivities" that include the effect of the hydraulic resistance in the soil–porous plate interface. Eching et al. (1994) verified the reliability of the OM through comparison with data from the DM1. However, since the DM1 does not distinguish the hydraulic resistance in the soil–porous plate interface from the "bulk" hydraulic conductivity without its verification, it cannot be used as the reference for the accuracy of OM. Granted that our results do not necessarily indicate that the optimized and directly calculated hydraulic conductivities by Eching et al. (1994) are underestimated, they reveal an uncertainty in the reliability of the hydraulic conductivities from the OM and DM1. Discrepancies at near saturation for the Masa loamy sand and the Yazu sandy loam might not be due to hydraulic resistance in the soil–porous plate interface, since the relative contribution of the data in the first step in the objective function would be small due to the smaller possible differences in tensiometer readings (i.e., <0.5 m) and the shorter duration to attain near equilibrium. Also, smaller conductivities of air at near saturation would give smaller bulk K() than K() from the steady-state method that is not affected by the air conductivity. However, if this was the main reason, it would contradict the greater K() at near saturation for the Tohaku loam. Thus the former, the small sensitivities of the data at near saturation, would account for the underestimation.
Hydraulic Resistance in the Soil–Porous Plate Interface
As mentioned by Gee et al. (2002), soil shrinkage and pore plugging with inorganic (soil) colloids or microbial slime can increase hydraulic resistance in the soil–porous plate interface. None of the three soils showed significant shrinkage. In general, the effect of soil shrinkage on hydraulic resistance would be limited, since gravity ensures contact between the soil and the porous plate. In this study, a 1.2 mmol L-1 NaN3 solution would have prevented microbial growth, and the short duration of the experiment would have limited the microbial growth, even if NaN3 had not been used. Therefore, pore plugging with inorganic (soil) colloids might be the main cause of the hydraulic resistance in the experiment. In multistep outflow experiments, stepwise changes of pressure head at the lower boundary would drive flow that is fast enough to induce colloid transport to the porous surface. Additionally, the multiple steps would promote subsequent movement and accumulation of colloid on the porous surface. Also, since the soil samples were disturbed and the outflow experiments were their first desaturation, the pore structure might have been unstable, and the movement of unsettled fine particles could have induced the plugging on the plate. If this is the case, the effect of plugging may be minor for undisturbed soil samples. Still, we cannot presume that movement of fine particles never occurs, since MSO may induce a rapid unsaturated flow that the soil (tilled soil in particular) has never experienced. A saturation–desaturation pretreatment would reduce additional plugging, but then we could not use the independently measured bulk conductance of the porous plate and subsequent tubing. Further, pretreatment makes the method more laborious and time-consuming. On the other hand, the DM2 and SDFM do not require such pretreatment.
For the Masa loamy sand, its low clay content would have limited colloid transport, and thereby result in good agreement between data from the DFFM and DM2 and data from the OM and DM1.
Continuous and slow changes in pneumatic pressure would minimize colloid transport, but would lower the uniqueness of the optimized parameters.
It is possible that the properties of the two soils were unique and that conventional methods are applicable to most soils. However, since the mobility of soil colloids would be more the rule than the exception, the assumption of zero hydraulic resistance in the soil–porous plate interface should be questioned. Our results do not doom the OM to failure, but call for researchers to take care. We suggest that, as we did, an additional tensiometer be installed near the bottom to verify the optimized K() with the direct method using the additional tensiometer (DM2). If this results in disagreement, one should employ the data from DM2 since it is valid whatever the value of the hydraulic resistance in the soil–porous plate interface. Researchers seem to prefer inverse results that are ready to be applied to simulations. However, it needs to be remembered that the results of MSO are applicable to only a specific range of water content. For the low-pressure range where MSO cannot be applied with a reasonable time, we should measure K() using another method, such as the instantaneous profile method of Mehta et al. (1994) or the sorptivity method of Dirksen (1979). Also, since MSO is not suitable for determining the saturated hydraulic conductivity, Ksat, we should measure Ksat independently. With the discrete data from such methods, we can determine K() by curve-fitting the data, covering nearly the whole range of water contents.
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CONCLUSION
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We conducted multistep outflow experiments using three soils having different textures to test the reliability of the conventional multistep outflow method. To determine hydraulic conductivity, K(), from identical experiments using sieved and packed soil samples, three methods were applied: an optimization method with one tensiometer reading (OM), a direct method with one tensiometer reading (DM1), and a direct method with two tensiometers (DM2). The two former methods are conventional ones, while the third is newly presented here. Additionally, steady-state downward flow experiments using two tensiometers (SDFM) were performed for comparison, although the applicable range is narrower than for MSO, and different cores are used than those for the MSO.
For at least two of the three soils, conventional methods (OM and DM1) underestimated K(). On the other hand, DM2 gave good agreement with data from the SDFM. These results and the discrepancies between the fitted and observed tensiometer readings near the bottom imply the existence of hydraulic resistance in the soil–porous plate interface. The hydraulic resistance might be caused mainly by pore plugging with fine particles as a result of colloid transport by fast flows just after the stepwise change of pressure head at the bottom.
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REFERENCES
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ABSTRACT
INTRODUCTION
