Extended multistep outflow method for the accurate determination of soil hydraulic properties near water saturation

Transcription

1 WATER RESOURCES RESEARCH, VOL. 47, W08526, doi: /2011wr010632, 2011 Extended multistep outflow method for the accurate determination of soil hydraulic properties near water saturation W. Durner 1 and S. C. Iden 1 Received 3 March 2011; revised 3 June 2011; accepted 24 June 2011; published 23 August [1] Inverse modeling of multistep outflow (MSO) experiments is an established and fast method to determine unsaturated hydraulic properties of soils. A disadvantage of the method is its low sensitivity with respect to the hydraulic conductivity function at saturation, which makes the respective estimation very uncertain. Thus, the use of independently measured values for the saturated hydraulic conductivity, K s, is generally recommended. This involves disadvantages, namely, the effort to conduct additional experiments and the general problems associated with the combination of data from different experimental sources. To overcome this problem, we propose to combine percolation and outflow in one experiment. This extended multistep outflow experiment (XMSO) starts with a completely water-saturated soil column, on top of which a small amount of water is ponding. The first experimental phase is a saturated percolation under falling-head conditions. After ponding ceases, the experiment continues as a standard MSO experiment with an unsaturated drainage process. The XMSO experiment is evaluated by inverse modeling, using measurements of cumulative outflow and tensiometric pressure head data. Our analysis of synthetic and real data demonstrates that XMSO yields very accurate estimates of saturated and near-saturated hydraulic properties, even for soils with structured pore systems. Furthermore, saturated hydraulic conductivities of the supporting porous plate and the soil can be simultaneously determined with great accuracy. We conclude that the XMSO experimental design solves the problem of the identification of near-saturated hydraulic properties of soil samples and reduces the estimation uncertainty of K s to a minimum. Citation: Durner, W., and S. C. Iden (2011), Extended multistep outflow method for the accurate determination of soil hydraulic properties near water saturation, Water Resour. Res., 47, W08526, doi: /2011wr Introduction [2] Multistep outflow (MSO) experiments evaluated by inverse modeling are an efficient way to determine simultaneously the water retention and hydraulic conductivity functions of soils [Eching et al., 1994; van Dam et al., 1994; Durner et al., 1999; Hopmans et al., 2002]. With the availability of fast computers and suitable numerical codes, the method is increasingly used and has become a standard method to determine hydraulic properties of soils [e.g., Puhlmann et al., 2009; Weihermüller et al., 2009; Figueras and Gribb, 2009; Laloy et al., 2010] and to study the phenomenon of nonequilibrium water flow in variably saturated porous media [O Carroll et al., 2010]. [3] A notorious problem of the MSO method is that it provides only limited information on the soil hydraulic properties in the water content range near and at saturation. As a consequence, the estimated soil hydraulic functions are prone to relatively large uncertainties in the corresponding pressure head range [Vrugt et al., 2001; Vrugt and Bouten, 1 Institut für Geoökologie, Technische Universität Braunschweig, Braunschweig, Germany. Copyright 2011 by the American Geophysical Union /11/2011WR ]. Although water fluxes and pressure head changes in a soil sample are nonlinearly related during the course of an MSO experiment, it is justified to state that the soil water retention function is primarily determined by the amounts of water withdrawn from the sample during the various applied pressure steps at the sample s boundary, i.e., by the average specific water capacity within the respective pressure head range. In contrast, the hydraulic conductivity is inferred from the flux density, i.e., the time scale of hydraulic equilibration after a pressure change. Close to saturation, most soils have a very small water capacity. Consequently, the matric potential in the soil sample will decrease very quickly as a response to the pressure drop at the system boundary. Correspondingly, the pressure head region near saturation is left very quickly and possibly even before significant outflow can be observed. Because of the absence of significant soil water flux in the beginning of the MSO experiment, it is impossible to determine the hydraulic conductivity function with satisfying accuracy during this phase of the experiment. [4] In order to identify the soil hydraulic functions, they are expressed as mathematical equations containing a limited number of unknown parameters. The determination of the soil hydraulic functions is then achieved by the estimation of the parameters using appropriate experimental information. W of13

2 In most parameter estimation applications in soil hydrology the number of degrees of freedom is minimized by coupling the soil water retention function to the hydraulic conductivity function by means of capillary bundle models, such as those of Childs and Collis-George [1950], Burdine [1953], or Mualem [1976]. The uncertainty of the estimated soil hydraulic properties is inferred from the uncertainty in the estimated parameters, either by means of the first-order, second-moment method [Durner et al., 2008] or Monte Carlo based sampling procedures [Vrugt and Bouten, 2002]. Capillary bundle models usually predict a relative hydraulic conductivity function K r (dimensionless) from the soil water retention function; thus, the former must be scaled by a factor to derive the hydraulic conductivity function. The factor most frequently chosen for scaling is the saturated hydraulic conductivity, K s [LT 1 ]. Since K s scales the conductivity function over the entire pressure head range from saturation to oven dryness, the conductivity function is sensitive to K s in the entire pressure head range and not only at water saturation, the state to which the parameter K s refers both terminologically and physically (see Figure 1, left). Because of this increased sensitivity, the uncertainty of K s will be greatly underestimated whenever parameter estimation is performed using capillary bundle models. With respect to the MSO method, this seemingly small uncertainty in K s is opposed to the fact that all parameters, including K s, are primarily determined by the experimental data obtained from that part of the experiment where significant outflow takes place. Since this phase corresponds to pressure heads smaller than the air entry value of the porous medium, it does not relate to saturated conditions. [5] As a consequence, a reliable determination of hydraulic conductivity close to saturation by inverse evaluation of standard MSO experiments is impossible. This is confirmed by comparisons of directly measured and MSO-estimated K s values, which can show extreme differences (see Vereecken et al. [1997, Figures 4a and 4b] for an intriguing example). Additional challenges originate from the fact that hydraulic conductivities tend to increase markedly when moving from slightly unsaturated to water-saturated conditions [Schaap and van Genuchten, 2005]. The precise estimation of hydraulic functions near saturation requires thus an independent measurement of K s, which is often obtained from different soil samples. Since K s is subject to considerable spatial variability [Wilson et al., 1989;Ünlü et al., 1990; Iqbal et al., 2005], the use of a fixed K s value, obtained from a measurement conducted on a different sample, can be a source of error in the interpretation and inversion of MSO data. For the pressure head range near water saturation, the reliability of conductivity estimates relies further on the validity of two assumptions: (1) the shape of the soil water retention function is matched perfectly by the selected model of the hydraulic properties, and (2) the underlying capillary bundle model for predicting the conductivity functions is correct. Durner [1994] analyzed the first assumption and found that water retention characteristics of undisturbed soils are rarely correctly described by the most frequently applied unimodal retention functions of Brooks and Corey [1964] and van Genuchten [1980]. Furthermore, he showed that even seemingly small deviations between the true retention characteristic and fitted functions near saturation lead to disproportionately large differences in the conductivity functions predicted by capillary models. In order to overcome these difficulties, Bitterlich et al. [2004] and Iden and Durner [2007] introduced the freeform estimation approach, which estimates both hydraulic functions independently and minimizes parameterization errors. As illustrated in Figure 1, a further advantage of the method is that K s does not scale the entire function K(h) but only exerts an influence on K(h) at pressure heads corresponding to conditions close to and at water saturation. [6] The aim of this paper is to present a new methodology, called the extended multistep outflow method (XMSO), which yields correct estimates of the unsaturated hydraulic properties and precise and consistent estimates of the hydraulic conductivity near water saturation from one single experiment. The experiment starts as a falling head experiment with saturated percolation and is continued as Figure 1. Schematic illustration of the influence of a change in the saturated hydraulic conductivity K s on the unsaturated hydraulic conductivity function K(h). (left) A capillary bundle model is used to compute K(h) from the soil water retention curve; that is, K s scales the entire function. (right) In the free form parameterization a change in K s exerts an influence only on the saturated and near-saturated conductivity. The function is obtained by cubic Hermite interpolation between the nodes (shown as dots). Note that K s is assigned to a pressure head of 1 cm here for convenience. 2of13

3 a standard MSO experiment. We first analyze the outflow and pressure head response of virtual soil columns by a sensitivity analysis. On the basis of inverse simulations with the free-form approach of Iden and Durner [2007] using synthetic data perturbed with noise, we show that the XMSO method not only improves greatly the identifiability of the hydraulic conductivity function near saturation but furthermore enables the exact determination of the initial and boundary conditions and the determination of the hydraulic conductivity of the porous plate used to support the soil material during the experiment. 2. Materials and Methods 2.1. Experiment Setup [7] The extended multistep outflow experiment is a combination of a saturated percolation experiment and a subsequent multistep outflow experiment. Since both experimental parts are conducted in one experimental run, we treat XMSO as a single experiment, combining the advantages of both techniques. The experimental setup, shown in Figure 2, follows the common design for outflow experiments [e.g., Hopmans et al., 2002]. A column of height L s [L], which contains the soil sample under investigation, is mounted on a porous plate of height L p [L], which is water permeable and air impermeable in the range of pressure heads occurring throughout the experiment. This plate is connected to a pressure reservoir, which allows us to impose a prescribed negative pressure head, h b [L], to the water phase at the lower boundary of the plate. This pressure is regulated by a computer-controlled magnetic valve connected to an air pressure reservoir that is applied to the top of the burette. At the top of the soil column, a second column of polymethyl methacrylate is mounted and connected in a water-tight manner to the casing of the soil column. The cumulative outflow per unit area across the lower boundary of the column, Q(t) [L], is recorded in high temporal resolution using a pressure transducer, and pressure heads at one or more depths, h i (t) [L], within the column are recorded using tensiometers. These measurements provide the experimental data for inverse modeling Conducting XMSO [8] To fully saturate the soil with water, the column is mounted onto the porous plate in an unsaturated state and then slowly saturated from the bottom. This is achieved by applying a gradually increasing pressure to the water phase at the lower boundary. After reaching full saturation, the pressure at the bottom is further increased, which results in a ponding of water on the soil surface. The saturation process is stopped when a ponding height h t (t ¼ 0) in the range of a few centimeters is reached. We assign this value to t ¼ 0 since this state corresponds to the beginning of the XMSO experiment. Reaching such a ponding requires the wall height of the column casing to exceed the soil surface by some centimeters. If the spatial coordinate z [L] is defined positive upward and the bottom of the plate is selected as reference height z ¼ 0 cm, the initial condition for the experiment is given by a hydrostatic pressure head distribution in the soil: hðz; t ¼ 0Þ ¼L s þ L p þ h t ðt ¼ 0Þ z: It is straightforward to use this initial condition for an in situ offset correction of the electronic pressure transducers for the pressure regulation at the lower boundary and the installed tensiometers because the initial ponding height can be determined with very high precision, as will be shown at the end of section The reference value for the initial pressure head at the bottom is given by equation (1) ð1þ Figure 2. Schematic of the experimental setup for the outflow experiment. 3of13

4 Figure 3. (left) Lower boundary condition and pressure head within the column at 1.8 cm from the top during the extended multistep outflow experiment (XMSO) experiment. (right) Cumulative infiltration and cumulative outflow. The experimental phases shown are (1) hydrostatic equilibrium during ponding (0 < t < t 0 ), (2) saturated percolation (t 0 < t t 1 ), (3) the first drainage phase without a change of the lower boundary condition (t 1 < t t 2 ), and (4) successive drainage after pressure decrease at lower boundary (t > t 2 ). The times t 0 and t 2 are known by the external experimental control, whereas t 1 is derived from the analysis of the tensiometer data. by setting z ¼ 0. Correspondingly, the reference initial pressure head of a tensiometer installed at a depth z i measured from the bottom can be computed by setting z ¼ z i in equation (1). [9] Figure 3 illustrates the temporal evolution of the boundary conditions, the pressure head within the column, and the fluxes across the column boundaries. The extended outflow experiment starts at time t 0 by lowering the pressure head at the lower boundary to h b ¼ 0 cm. This causes a sudden pressure change in the entire soil column and initiates instantaneously a saturated percolation. As long as surface ponding remains, i.e., h t (t) > 0 cm, the flux density across the lower boundary, q(t) [LT 1 ], will be slightly higher than the saturated conductivity, K s, since the gradient in the hydraulic head is slightly larger than unity. Furthermore, the flux density decreases exponentially with time as in a classic falling-head experiment [Kutilek and Nielsen, 1994]. [10] During the percolation phase, the tensiometer in the column will record a smoothly and continuously decreasing pressure head. When the ponding height reaches zero (we denote the corresponding time as t 1 ), the saturated percolation process turns into a drainage process. This is indicated by a pronounced change in the slope of the tensiometer readings. This change is abrupt and independent of the position of the tensiometer or the column height since at full saturation the specific water capacity, defined as the partial derivative of volumetric soil water content,, with respect to pressure head, C is zero everywhere in the column, which corresponds to an infinite hydraulic diffusivity and thus an almost-instantaneous propagation of a pressure signal through the medium. If the soil is homogeneous and if one neglects the resistance of the porous plate at the bottom, the pressure head at any position in the system will be zero at that moment. The fact that the end of the ponding can be exactly identified from the tensiometer readings is a crucial point in the XMSO experimental design. It allows us not only to exactly set the beginning of the MSO phase in the inverse simulation but also to precisely recalculate the initial ponding height because during saturated percolation the upper infiltration rate, q t (t), is equal to the drainage rate, q b (t) (Figure 3, right), and, accordingly, the initial ponding height is equal to the cumulative drainage at t 1. After hydrostatic conditions are reached, the pressure head at the lower boundary can be further reduced in steps; that is, the XMSO proceeds as a regular MSO experiment [Hopmans et al., 2002]) Measurements on Real Soils [11] We investigated an undisturbed soil column of 7.2 cm height and an inner diameter of 9.4 cm, resulting in a sample volume of 500 cm 3. The column was sampled from a depth of cm from the A l horizon of a Luvisol at an agricultural field near Braunschweig-Völkenrode in Germany. The field was fallow at the time of sampling. The soil s texture is 64% sand, 30% silt, and 6% clay. The experimental setup followed the XMSO procedure as described in section At the lower boundary, a 0.7 cm thick sintered glass plate was installed. The soil was saturated by infiltrating water from the bottom within 48 h until a ponding height of 2.6 cm was reached. The pressure heads applied at the lower boundary were þ10.5, 0, 10, 20, 30, 40, 60, 80, and 100 cm. After the XMSO experiment, the sample was weighed, dried at 105 C for 24 h, and weighed again in order to obtain the water content at the end of the experiment. The saturated water content was calculated from the final water content and the registered total cumulative outflow minus the initial ponding height. The accuracy of the pressure regulation, expressed as standard deviation, was hb ¼ 0:1 cm. The pressure head in the soil column was measured at a distance z 1 ¼ 6.1 cm from the bottom of the porous plate. The accuracy of the tensiometer readings was h ¼ 0:5 cm. Cumulative outflow was sampled in a burette and recorded with a precision of Q ¼ 0:01 cm. Temporal resolution for the outflow and tensiometer readings was one per second. For use in the modeling, the data were thinned out Determination of Saturated Conductivities [12] A point of concern in practical applications of the MSO method is to properly consider the resistance of the porous plate that is installed below the soil sample to 4of13

5 support it. Usually, the plate resistance is measured independently before or after the experiment and is then included as a fixed parameter in the numerical inversion. However, the disadvantage caused by this is that the value does not necessarily match the effective resistance during the experiment since an additional resistance might be caused by contact problems between the soil sample and the plate and the plate resistance might be altered by fine soil particles that may clog its pores. [13] A great advantage of the XMSO design is that it enables us to determine the hydraulic conductivity of the plate from the pressure head measurements during the percolation phase. A joint determination of the saturated hydraulic conductivity of the soil K s and the hydraulic conductivity of the porous plate K p can be achieved by two different methods. [14] First, the estimation of K s and K p can be included in the inverse evaluation of the entire XMSO experiment. This method is straightforward and probably the most convenient for the user. The disadvantage of this approach is that it does not in all cases lead to correct estimates of the conductivities because sufficient weight must be assigned to the data reflecting the possibly short time period corresponding to the saturated percolation phase. If the number of data in this time period is too small to ensure a sufficient contribution to the objective function, higher weights could be assigned to the respective data in the objective function. However, the specific values of the weights are not straightforward to assign, and there is a risk that such weights could be selected subjectively and not on a sound statistical basis. [15] As an alternative to the determination by inverse modeling of the entire XMSO experiment, the data corresponding to the falling-head phase can be separated from the entire data set and evaluated by parameter estimation using weights in the objective function that are in agreement with maximum likelihood theory. An additional advantage of this method is that the governing equations can be solved analytically, which makes them insensitive to numerical errors. We will outline this procedure in the following. [16] We denote the difference in the hydraulic head, i.e., the sum of pressure and gravitational head, between the top and the bottom of the soil-plate system by HðtÞ [L]. For the falling-head experiment, HðtÞ will decrease exponentially with time [Kutilek and Nielsen, 1994]: HðtÞ ¼H 0 exp K eff L t ; ð2þ where L [L] is the total length, i.e., the sum of L s and L p, t [T] is the time from the initiation of the experiment, when the pressure at the bottom is reduced to zero, K eff [LT 1 ]is the effective hydraulic conductivity of the soil-plate system, given by L K eff ¼ ; L s =K s þ L p =K p and H 0 [L] is the initial difference in hydraulic head between top and bottom, given by H 0 ¼ L s þ L p þ h t ðt ¼ 0Þ: ð3þ ð4þ Since the change in HðtÞ is equal to the outflow rate, the cumulative outflow [L] can be calculated as QðtÞ ¼H 0 HðtÞ: The flux density [L T 1 ] through the column is given by Darcy s law: ð5þ HðtÞ qðtþ ¼ K eff ; ð6þ L and the pressure head [L] at the interface between the porous plate and the soil is given by [Jury and Horton, 2004] qðtþ h p ðtþ ¼ L p þ 1 : ð7þ K p For a homogeneous soil, the pressure head distribution in the column during saturated flow is linear. The pressure head at the position of the tensiometer, z i [L], measured from the bottom of the plate is thus calculated as h i ðtþ ¼h p ðtþþ h tðtþ h p ðtþ L s z i L p : ð8þ Figure 4 shows the results of a sensitivity analysis using equations (2) (8). Results of numerical simulations using Hydrus-1D [Simunek et al., 2008] are included in Figure 4 to illustrate their correspondence to the analytical solution and to extrapolate the time series of the pressure head to the subsequent drainage phase with unsaturated conditions, Figure 4. Results of a sensitivity analysis for the hydraulic conductivity of the porous plate K p using equations (2) (8). The soil column is 7.2 cm long, the porous plate is 0.8 cm thick, and the initial ponding height is 2 cm. The saturated hydraulic conductivity of the soil K s is 100 cm d 1, and pressure head is calculated for a depth of 1.8 cm measured from the top of the column. The XMSO experiment starts at time t 0 ¼ 1 with a saturated percolation. After ponding ceases, saturated flow changes to unsaturated flow and must be simulated by a numerical solution of the Richards equation. 5of13

6 which cannot be calculated by the analytical solution. The reader is referred to section 2.3 for more details on the numerical simulations. It becomes evident that the time series of the pressure head is sensitive with respect to K p. This enables the determination of K p by inverse modeling using the cumulative outflow and pressure head data of the saturated percolation phase of the XMSO experiment. Without the pressure head measurements this is not possible because the outflow signal is sensitive only to K eff, resulting in a perfect correlation of K s and K p. The details on parameter estimation are described in section Numerical Modeling Simulating XMSO Using the Richards Equation [17] Isothermal, variably saturated water flow in a rigid soil is simulated using the one-dimensional Richards þ 1 ; ð9þ where h [L] is pressure head, t [T] is time, z [L] is the vertical coordinate (positive upward), C(h) [L 1 ] is soil water capacity, and K(h) [LT 1 ] is the hydraulic conductivity function. We solved equation (9) numerically using the Hydrus-1D software code of Simunek et al. [2008]. For an XMSO experiment the initial condition is given by equation (1). The upper boundary condition consists of two stages. During saturated percolation, as long as water is ponding at the soil surface, a surface reservoir condition is used, given by [Simunek et al., ¼ þ 1 ; z ¼ L; ð10þ where K s is the effective saturated hydraulic conductivity of the soil and h t is the ponding height. After ponding ceases, the upper boundary condition is switched to a noflux condition. In Hydrus-1D this system-dependent boundary condition is simulated by using an atmospheric boundary condition with surface layer and specifying zero precipitation and zero potential evaporation. The lower boundary condition is identical to that of a classic MSO experiment and consists of a stepwise decrease of pressure head at the bottom of the column. The automatic time step control in Hydrus-1D was restricted by a maximum time step of 0.01 h to prevent time steps from becoming too large during the numerically less demanding saturated percolation part of the XMSO Parameterization of Soil Hydraulic Functions [18] The solution of the Richards equation (9) requires the unsaturated hydraulic properties ðhþ and K(h), which are both nonlinear functions of pressure head. We parameterized the soil hydraulic functions by the classic van Genuchten Mualem (VGM) model of van Genuchten [1980], the modified van Genuchten Mualem (MVGM) model of Vogel et al. [2001], and the bimodal van Genuchten Mualem (BVGM) model of Durner [1994] for which closed-form expressions for K(h) are derived by Priesack and Durner [2006]. The unimodal VGM model is given by the retention function ðhþ ¼ r þ ð s r Þ½1 þð h Š m ; ð11þ Þ n where s and r are the saturated and residual water contents [L L 1 ], respectively, [L 1 ] and n (dimensionless) are shape parameters, and m (dimensionless) is defined by m ¼ 1 1/n. After defining the effective saturation, S e (dimensionless), as S e ¼ð r Þ= ð s r Þ; ð12þ the unsaturated hydraulic conductivity function is expressed as KðS e Þ¼K s Se ½1 ð1 S1=m e Þ m Š 2 ; ð13þ where (dimensionless) is a shape parameter and the above constraint m ¼ 1 1/n must be satisfied. The MVGM model overcomes the difficulties of the VGM model with respect to the shape of the conductivity function near saturation for values of n smaller than 2 [Vogel et al., 2001] and is given by the retention function ðhþ ¼ r þ ð m r Þ½1 þð hþ n Š m h < h c s h h c ; ð14þ where h c is an explicit air entry value of the porous medium and the fictitious water content, m, is defined by because m ¼ r þ ð s r Þ½1 þð h c Þ n Š m ð15þ ðh c Þ ¼! s : ð16þ The conductivity function of the MVGM model is given by where KðhÞ ¼ K sk r ðhþ h < h c K s h h c ; ð17þ K r ðs e Þ¼Se 1 FðS e Þ 2 ; ð18þ 1 Fð1Þ FðS e Þ¼ m; 1 Se 1=m ð19þ S e ¼ s r m r S e : ð20þ For soils with a structured pore system, the BVGM model of Durner [1994] yields a better match to experimental data than the VGM model. This model is a linear superposition of two effective saturation functions according to the van Genuchten model, given by ðhþ ¼ r þ ð s r Þ X2 w i ½1 þð i hþ ni Š mi ; ð21þ i¼1 6of13

7 where the constraint m i ¼ 1 1/n i restricts the number of degrees of freedom and ensures the existence of a closedform equation for the conductivity function, which is given by ( ) X 2 KðhÞ ¼K s w i ½1 þð i hþ ni Š mi 0 P 2 i¼1 i¼1 n w i i 1 ð i hþ ni 1 ½1 þð i h P 2 i¼1 w i i Þ ni Š mi o12 C : A ð22þ 2.4. Generation of Synthetic Data Sets [19] To compare the XMSO method to the MSO technique and to assess the correctness and the uncertainty of the soil hydraulic properties determined by inverse modeling, synthetic data sets were generated for four types of soil textures. The models and parameters of the soil hydraulic functions used for generation are summarized in Table 1. All synthetic data were generated by numerical simulation with HYDRUS-1D using the initial and boundary conditions outlined in section For use in the inverse analysis, time series of cumulative outflow across the bottom of the column and pressure head at a depth of 1.8 cm from the top were disturbed by normally distributed random error with an expected value of zero and standard deviations of 0.01 and 0.5 cm, respectively. The number of synthetic data points was 810 in all cases, consisting of 405 pressure head and 405 cumulative outflow points. [20] We simulated XMSO and MSO experiments for 8 cm long columns. The finite element mesh was discretized by 50 equally spaced nodes. The mass balance error was smaller than 0.01% in call cases. The initial condition was specified as a vertical pressure head distribution in accordance with equation (1). In case of the MSO experiments, we distinguished two different initial conditions. In the first case, termed MSO-S, we started the experiments at full water saturation by specifying a hydrostatic distribution of pressure head with a positive pressure head of 8 cm at the bottom. In the second case, denoted MSO-U, we started the simulated MSO experiment under unsaturated hydrostatic conditions with a value of 0 cm at the bottom. Note that this corresponds to partly unsaturated conditions only if the air entry pressure of the porous medium is greater than 8 cm. The influence of the porous plate was neglected. This corresponds to a situation where the hydraulic conductivity of the plate is much higher than that of the soil and water flow is not slowed down by the plate. We selected this simplification to ensure a better comparability of the results and to keep the presentation concise Inverse Modeling Synthetic Data [21] The synthetic data sets were evaluated by inverse modeling using the free-form algorithm described in detail by Iden and Durner [2007]. The basic idea is to estimate values of ðhþ and log K(h) at nodes positioned on the pressure head axis and to obtain continuous smooth functions by cubic Hermite interpolation. The number of nodes r is increased by a multilevel routine, and the position of the nodes is defined prior to the numerical optimization on the basis of the estimated retention function of the previous estimation level using r 1 nodes. We modified the algorithm such that the saturated hydraulic conductivity K s could be treated as an additional free parameter. Therefore, given r nodes, the number of estimated parameters amounts to 2r þ 1, i.e., r water contents and r þ 1 conductivities. The saturated water content, s, is treated as a fixed parameter and is assumed to be determined independently. Since the free-form algorithm uses a logarithmic pressure head axis to parameterize the hydraulic functions, s and K s are assigned to a pressure head of 1 cm. An additional modification of the free-form algorithm was implemented to restrict the position of the node right next to this pressure head to 4 cm. This ensures that no nodes are positioned at pressure heads between saturation and a pressure head of Table 1. Models of the Soil Hydraulic Functions and Corresponding Parameter Values Used for Generation of Synthetic Data Sets a Unimodal Soils c Parameter b Units Sand e VGM Silt f MVGM Clay g MVGM Bimodal d Loam BVGM s cm 3 cm r cm 3 cm or 1 cm n or n w cm n h c cm K s cm h a VGM, van Genuchten Mualem model; MVGM, modified van Genuchten Mualem model; BVGM, bimodal van Genuchten Mualem model. For details on models, see section b As defined by van Genuchten [1980] and Durner [1994]. c Soils discussed by Durner and Flühler [2005]. d Bimodal soil as used by Zurmühl and Durner [1998]. e Parameters represent 92.5% sand, 5% silt, and 2.5% clay. f Parameters represent 10% sand, 80% silt, and 10% clay. g Parameters represent 5% sand, 20% silt, and 75% clay. 7of13

8 4 cm. The inverse problem is defined by minimizing the weighted least squares objective function: OðpÞ ¼ Xnd i¼1 2; w i y obs i y sim i ð23þ where n d is the number of data points, y i obs and y i sim are observed and model-simulated data, respectively, w i are the weights set to 10,000 for cumulative outflow and 4 for pressure head data, respectively, and p is the vector of estimated model parameters, which is of length 2r þ 1. The weights are set equal to the reciprocal of the variance of the measurement error for each data type. This is in agreement with maximum likelihood theory if measurement errors have zero mean, are mutually independent, and are normally distributed with a variance that is identical for each observation belonging to the same data type. [22] Since both MSO-S and XMSO experiments are started at full water saturation, the soil water retention curve can be estimated from the saturated water content s and the cumulative outflow data. In the case of the MSO-U, which is started from a hydrostatic pressure head distribution with a pressure head of zero at the bottom, the column is not necessarily saturated with water at the beginning, and the level of the soil water retention curve is undefined because only differences in water content are recorded by the cumulative flux measurements. Therefore, we included the final volumetric water content of the soil column as an additional point in the objective function for MSO-U and assigned a weight of 10 8 to it. This weight reflects a measurement error of 10 4 for the final water content. [23] We ran the free-form algorithm from r ¼ 1tor ¼ 10. For values of r 5, the objective function (23) was minimized by the shuffled complex evolution algorithm SCE-UA of Duan et al. [1992]. For values of r > 5, the minimization was achieved by the covariance matrix adaptation strategy CMA-ES described by Hansen et al. [2003]. The use of CMA-ES is more efficient than SCE-UA while leading to correct results, as will be shown in section 3. We initialized the CMA-ES strategy by computing an initial estimate of the parameter covariance matrix by a first-order approximation using the parameter estimates of the previous optimization level with r 1 nodes. This covariance matrix was then multiplied by a factor of 10 to further disperse the normal distribution used for generating candidate points. The optimization was terminated when the relative range of the objective function or the relative range of all estimated parameters was smaller than in the entire population. All inverse simulations were done on a Quadcore PC using a parallelized Fortran 90 inversion code developed using OpenMP [Chapman et al., 2008] Real Data [24] In the case of the real data, we estimated the hydraulic conductivities K p and K s by separating the data from the falling head part of the experiment and then minimizing the objective function given by equation (23). The data in the objective function were the measured cumulative outflow and pressure head data during saturated percolation. The simulated system response was calculated by the analytical solution given by equations (5) and (8). In this case the vector of estimated model parameters is p ¼ [K s K p ] T. Minimization of the objective function was achieved by the MatlabVR routine lsqnonlin. [25] We then applied the free-form algorithm with the saturated hydraulic conductivities of the soil and plate fixed to the values determined from the saturated percolation part. Therefore, the number of degrees of freedom was reduced by one to 2r. The standard error of K s was accounted for in the uncertainty analysis of the soil hydraulic functions (see section 2.5.3). To compare the properties obtained by the free-form approach with classic models of the soil hydraulic functions, we furthermore estimated parameters of the van Genuchten Mualem model using the SCE-UA optimization scheme. The estimated parameters were, n, r,and, while K s and K p were fixed as described above. The saturated volumetric water content, s, was calculated from the gravimetrically determined final water content, the total cumulative outflow during the experiment, and the initial ponding height Uncertainty Analysis [26] The uncertainty of the estimated parameters was quantified by the parameter covariance matrix, which was derived by the first-order, second-moment method. Since this method assumes linearity in the underlying model, the resulting covariance matrix is only approximate. It was calculated as [Seber and Wild, 2003] ^C^p ¼ Oð^pÞ S T w n d n S 1; w ð24þ p where n p is the number of estimated model parameters and S w is the weighted sensitivity matrix with elements p s ij ¼ i w i j ð25þ i.e., the matrix of the partial derivatives of the model predictions f i with respect to the parameters scaled by the square root of the weights contained in the objective function. The square roots of the main diagonal elements of ^C^p are the estimated standard errors of the parameters from which confidence intervals can be calculated using Student s t distribution. Confidence intervals for the estimated soil hydraulic functions were calculated by the first-order, second-moment method as well. In case of the soil water retention curve, we first calculated the variance of a predicted water content as Var ^i ¼ Xn X n j¼1 ^ ^ i c jk k ð26þ where c jk denotes the entries of the parameter covariance matrix ^C^p. From this variance, the confidence interval for the level ð1 p Þ is calculated as ^ i 6t nd n p 1 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p Var ^i ; ð27þ 2 where t nd n p 1 p =2 is the value of Student s t distribution. The conductivity curve K(h) can be treated analogously 8of13

9 by replacing the partial derivatives in equation (26) and the predicted value in equation (27). 3. Results and Discussion 3.1. Inverse Modeling of Synthetic Data [27] Figure 5 shows a comparison of the results for the MSO-S (Figure 5, left) and XMSO (Figure 5, right) experiments, performed for the synthetic soil Loam and evaluated by inverse modeling with the VGM and free-form parameterization. Since the loam data were generated with the bimodal van Genuchten Mualem model, the inversion using the van Genuchten Mualem model suffers from a slight model error in the soil hydraulic functions, which reflects a practical situation that is the rule rather than the exception [Laloy et al., 2010]. Figure 5 (top row) depicts the synthetic measurement data and the model fits, Figure 5 (middle) depicts the true and identified soil water retention curves, and Figure 5 (bottom) depicts the true and estimated unsaturated conductivity functions. [28] In Figure 5 (top left), we see a typical time series of MSO experimental data. The best fitted outflow data of the VGM model cannot adequately match the synthetic observations, which is due to the model error in the soil hydraulic functions, whereas the free-form algorithm (r ¼ 8) matches the observations well. In correspondence with this, the estimated VGM soil water retention curve shows slight discrepancies to the true function, whereas the free-form function coincides perfectly with the true function. As shown in Figure 5 (bottom left), both models fail to match the true K s. Although the free-form method minimizes the parameterization error, a discrepancy between the estimated and true K s is evident. This is in contrast to the unsaturated range, where the true K(h) function is identified correctly. For the Figure 5. Comparison of the inverse modeling results for (left) MSO-S (started at water saturation) and (right) XMSO for the synthetic data set Loam. (top) Generated and optimal model-predicted pressure head and cumulative outflow data. (middle) True and identified soil water retention functions. (bottom) True and identified unsaturated hydraulic conductivity functions. Inverse modeling was performed using the van Genuchten Mualem model (VGM, 5 degrees of freedom) and the free-form algorithm (FF). For the latter, the results using eight nodes (17 degrees of freedom) are shown. Points used for cubic Hermite interpolation are denoted as blue dots. 9of13

10 Table 2. Comparison of the Estimates and Standard Errors (SE) of the Decimal Logarithm of the Saturated Hydraulic Conductivity K s Using the Synthetic Data Loam and the MSO and XMSO Experimental Designs a MSO XMSO Estimate SE Estimate SE VGM FF a The true value is 1.0. K s values are in cm h 1. MSO, multistep outflow; XMSO, extended multistep outflow experiment; FF, free-form algorithm. VGM parameterization, the missing sensitivity near saturation and the model error jointly lead not only to a significant deviation from the estimated K s but also to a mismatch of the conductivity function in the moist range. [29] In cases where the data were both generated and inverted with the VGM model, the discrepancy in K s was almost zero (data not shown). This could lead to the wrong conclusion that the MSO experiment provides sufficient information for the determination of K s. This, however, is a misconception. The correct identification of K s is due to the fact that the K(h) function is sensitive to K s as a scaling parameter over the entire pressure head range. If the functional extrapolation near saturation is without any model error, an unbiased estimate results. In real soils, this is a lucky situation, which cannot easily be tested without additional efforts. As pointed out by Iden and Durner [2007] and shown by this example, the freeform method illustrates that the MSO design provides only very limited information on the hydraulic conductivity near and at saturation and helps to avoid such wrong conclusions. [30] Figure 5 (top right) shows the time series of XMSO experimental data and the model fits. The fully saturated percolation phase gives an initially steep outflow curve, until the initial ponding height of 2 cm has percolated through the sample. Afterward, data correspond to those of the MSO experiment. As for the MSO design, the match of the data by the simulation based on the VGM model is not perfect, and the identified retention curve (Figure 5, middle) shows slight deviations from the true function. It differs, furthermore, from the function that was obtained by the MSO experiment. As for the MSO design, the estimated free-form function matches the data perfectly and identifies the soil water retention curve without error. [31] The most interesting part of Figure 5 is the conductivity curve plot in Figure 5 (bottom right). The change from MSO to XMSO results in an unbiased estimate of the saturated hydraulic conductivity for both parameterizations, which is the paramount advantage of the XMSO method. Even an inadequate functional model for the constitutive relationships will no longer cause K s estimates to be far from the true values. Any error in the estimate of K s can be easily and unambiguously identified by comparing observed and fitted data from the saturated percolation. Thus, XMSO will guarantee unbiased estimates of K s if used with care. [32] Table 2 lists estimated saturated hydraulic conductivities and the respective standard errors for the two experiment types and both parameterizations. The uncertainty in K s is strongly reduced when switching from MSO to XMSO, reflecting the increased sensitivity of the model response with respect to K s. The reduction is more pronounced for the free-form parameterization than for the VGM model because the free-form model shows much higher uncertainty for the MSO design, reflecting the missing information content of the experimental data in this pressure head region. In the case of the VGM model, the uncertainty of K s is always relatively small because the parameter is conditioned on the entire data set. The fact that the true value of K s is not covered by the 95% confidence interval [1.787, 1.889] for the MSO experiment evaluated using the VGM model highlights the uselessness of such uncertainty measures for a wrong model of the hydraulic properties. Conversely, the 95% interval obtained by the free-form method [0.852, 1.566] is much wider and covers the true value of 1.0. [33] Figure 6 summarizes the results for all four synthetic data sets. We restrict the graphical presentation to the free-form inversions because only those reflect reliable uncertainty estimates of the soil hydraulic properties, as discussed. Depicted are the estimated hydraulic conductivity functions and their 95% confidence intervals obtained with the free-form algorithm with eight nodes (17 degrees of freedom). There is a decreasing uncertainty in K s when switching from the initially unsaturated MSO experiment (MSO-U) to the initially saturated one (MSO-S) and further to the XMSO method. The fit of the generated data was always perfect, and the corresponding soil water retention functions matched the true ones in all cases (thus not shown for the sake of brevity). [34] Evidently, both variants of the MSO experiments lead to biased estimates of K s and relatively large uncertainties, as summarized in Table 3. Conversely, the XMSO method estimates of K s are unbiased and highly accurate. This highlights the significant improvement achieved by the slight change in the experimental design when switching from MSO to XMSO Inverse Modeling of Real Data [35] Figure 7 shows the results for the real measurements. The saturated hydraulic conductivities K s and K p were estimated using the data from the saturated percolation phase and the analytical solution given by equations (5) and (8). The observed and fitted pressure head and cumulative outflow data are shown in Figure 7 (top left). The agreement is excellent. The estimated parameters are K s ¼ cm h 1, with a standard error of cm h 1, and K p ¼ cm h 1, with a standard error of cm h 1. Keeping the parameters K s and K p fixed at these values, the XMSO experiment was inversely simulated. Figure 7 (top right) shows the matches of the VGM and FF simulations with r ¼ 5 to the measured data. A further increase in the number of degrees of freedom did not lead to an improvement in the goodness of fit. Both models fit the data sufficiently well, with a slightly better match of the FF fit toward at the end of the experiment. Figure 7 (bottom) shows the resulting hydraulic functions with their 95% confidence bands. The retention curves (Figure 7, bottom left) of the two parameterizations resemble each other closely, and the confidence bands indicate a very small uncertainty of the estimates. The hydraulic conductivity functions 10 of 13

11 Figure 6. Unsaturated hydraulic conductivity functions estimated using the free-form parameterization for the synthetic data sets and the known true functions. The shaded areas visualize the 95% confidence band of the estimated functions. Points used for cubic Hermite interpolation are denoted as blue dots. differ more but fall together at saturation. The quality of determination is reflected by the small uncertainty of K(h) at saturation and the excellent match to the experimental data during saturated percolation (Figure 7, top). Thus, the newly developed XMSO technique is shown to be successful in a practical application. Table 3. Estimates and Standard Errors (SE) of the Decimal Logarithm of the Saturated Hydraulic Conductivity K s Using Synthetic Data Sets for Three Different Experimental Designs a Soil MSO-U MSO-S XMSO True Estimate SE Estimate SE Estimate SE Sand Silt Clay Loam a Inversions were carried out with the free-form algorithm. Values are in cm h 1. MSO-U, unsaturated initial condition; MSO-S, MSO starting at full water saturation. 4. Conclusions [36] We presented a new method for the determination of unsaturated hydraulic properties of porous media that combines a saturated percolation under falling-head conditions with a subsequent multistep outflow experiment (MSO). The method is called the extended multistep outflow method (XMSO). In comparison to the MSO method, the only changes to be made are an extension of the column to enable surface ponding of water and a different initial condition. These changes in the experimental design are easily implemented and seem to be minor, but as shown in this study, they have a strong positive impact on the estimates of the saturated hydraulic conductivity, K s. Furthermore, the new design enables very accurate in situ offset calibrations for the tensiometers and the pressure transducer used to control the pressure head at the lower boundary. It can also be used to identify accurately the beginning of the MSO phase, i.e., the unsaturated water flow process. An analysis of synthetic data for a broad range of soil textures showed that XMSO leads to unbiased estimates of K s with standard errors close to zero. This is in contrast to the 11 of 13

12 Figure 7. Results for the real soil sample. (top left) Measured and fitted pressure head and cumulative outflow data using the analytical solution. (top right) Entire data set of the XMSO experiment and inversely simulated data using the van Genuchten Mualem (VGM) model and the free-form algorithm (FF) with five nodes. (bottom left) Estimated soil water retention curves. (bottom right) Estimated hydraulic conductivity functions. The shaded areas denote 95% confidence bands of the estimated soil hydraulic functions. Points used for cubic Hermite interpolation are denoted as blue dots. usually conducted MSO experiments, for which estimates are biased and uncertain. The advantages of the XMSO method were confirmed by the evaluation of a real soil column experiment by inverse modeling with a combined analytical-numerical model. Since the XMSO method yields accurate estimates of soil hydraulic properties from water saturation down to pressure heads in the range of a few hundred centimeters in relatively little time, it is an important tool for vadose zone research and practical applications. In order to further extend the measurement range to lower soil water contents, the method can easily be combined with evaporation experiments, as shown by Schelle et al. [2011], and additional water content measurements in the dry range, for instance, by pressure plates or the dew point method [Bittelli and Flury, 2009]. References Bittelli, M., and M. Flury (2009), Errors in water retention curves determined with pressure plates, Soil Sci. Soc. Am. J., 73, , doi: /sssaj Bitterlich, S., W. Durner, S. C. Iden, and P. Knabner (2004), Inverse estimation of the unsaturated soil hydraulic properties from column outflow experiments using free-form parameterizations, Vadose Zone J., 3, Brooks, R. H., and A. T. Corey (1964), Hydraulic properties of porous media, Hydrol. Pap. 3, Colo. State Univ., Fort Collins. Burdine, N. T. (1953), Relative permeability calculations from pore size distribution data, Trans. Am. Inst. Min. Metall. Pet. Eng., 198, Chapman, B., G. Jost, and R. van der Pas (2008), Using OpenMP: Portable Shared Memory Parallel Programming, MIT Press, Cambridge, Mass. Childs, E. C., and G. N. Collis-George (1950), The permeability of porous materials, Proc. R. Soc. London, Ser. A, 201, Duan, Q., S. Sorooshian, and V. Gupta (1992), Effective and efficient global optimization for conceptual rainfall-runoff models, Water Resour. Res., 28, , doi: /91wr Durner, W. (1994), Hydraulic conductivity estimation for soils with heterogeneous pore structure, Water Resour. Res., 30, , doi: / 93WR Durner, W., and H. Flühler (2005), Soil hydraulic properties, in Encyclopedia of Hydrological Sciences, edited by M. G. Anderson and J. J. McDonnell, pp , John Wiley, Chichester, U. K. Durner, W., B. Schultze, and T. Zurmühl (1999), State-of-the-art in inverse modeling of inflow/outflow experiments, in Proceedings of the International Workshop on Characterization and Measurement of the Hydraulic Properties of Unsaturated Porous Media, edited by M. T. van Genuchten, F. J. Leij, and L. Wu, pp , Univ. of Calif., Riverside. Durner, W., U. Jansen, and S. C. Iden (2008), Effective hydraulic properties of layered soils at the lysimeter scale determined by inverse modelling, Eur. J. Soil Sci., 59, , doi: /j x. Eching, S. O., J. W. Hopmans, and O. Wendroth (1994), Unsaturated hydraulic conductivity from transient multistep outflow and soil water pressure data, Soil Sci. Soc. Am. J., 58, Figueras, J., and M. Gribb (2009), Design of a user-friendly automated multistep outflow apparatus, Vadose Zone J., 8, Hansen, N., S. D. Müller, and P. Koumoutsakos (2003), Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES), Evol. Comput., 11(1), pp Hopmans, J. W., J. Simunek, N. Romano, and W. Durner (2002), Simultaneous determination of water transmission and retention properties Inverse methods, in Methods of Soil Analysis, Part 4, Physical Methods, Soil. Sci. Soc. Am. Book Ser., vol. 5, edited by J. H. Dane and G. C. Topp, pp , Soil Soc. of Am., Madison, Wis. Iden, S. C., and W. Durner (2007), Free-form estimation of the unsaturated soil hydraulic properties by inverse modeling using global optimization, Water Resour. Res., 43, W07451, doi: /2006wr Iqbal, J., J. Thomasson, J. Jenkins, P. Owens, and F. Whisler (2005), Spatial variability analysis of soil physical properties of alluvial soils, Soil Sci. Soc. Am. J., 69(4), of 13