H34E-01 INVITED
Upscaling Soil Hydraulic Properties Using Field-Scale Inverse Modeling and Bayesian Model Averaging
Parameter estimation through inverse modeling is a powerful method to derive effective values of the soil hydraulic parameters at various spatial scales. This approach circumvents many of the problems associated with conventional upscaling methods, but is typically carried out using a single conceptual mathematical model of the underlying vadose zone system, rejecting a-priori valid alternative conceptual models and underestimating uncertainty in the model itself. Methods based on Bayesian Model Averaging (BMA) have been proposed in the statistical literature as a means to explicitly account for conceptual model uncertainty. In this talk, we will highlight some of our recent work on a combined BMA and inverse modeling framework to estimate effective field-scale soil hydraulic properties, including their underlying uncertainty. We demonstrate our approach using observed tensiometric data from the Spydia experimental field site in New Zealand.
H34E-02
Relation Between Connected Patterns in Heterogeneous Soil Parameter Fields and Solute Transport Models in the Unsaturated Zone
Transport of solutes, such as agrochemicals, in the vadose zone is mostly described by an advection-dispersion equation, where the flow velocity of water in is described by the Richards equation. Soil is in reality highly heterogeneous, so the hydraulic parameters vary in space and their detailed structure is unknown. Heterogeneity of hydraulic soil parameters has a strong influence on flow and transport processes. As an example, it determines dispersion of solute concentration. As water and mass fluxes usually have to be predicted on length scales much larger than the typical length scales of heterogeneities, flow and transport models have to be upscaled to predict spatial averages of state variables (water content or solute concentration). Upscaled models for flow and transport in aquifers are quite well established. In the unsaturated zone, where variances of hydraulic parameters can be extremely high, assumptions such as smoothly varying, moderately heterogeneous hydraulic parameter fields can often not be made to derive upscaled models. Heterogeneity of soil is usually captured by modeling hydraulic parameters as correlated random fields. These fields are mostly directly or indirectly assumed to be multi-Gaussian. This implies that no information is used upon whether a certain parameter range is spatially connected or forms isolated clusters. However, connectivity has been found to have a strong influence on parameters of upscaled flow models, in particular if the variance of parameters is high. In this presentation, the influence of connected structures of heterogeneous hydraulic parameter fields on upscaled solute transport models in the vadose zone will be discussed. Upscaled models are derived using homogenization theory. The models are analyzed for different configurations of connected and isolated parameter ranges and for different parameter contrasts. Homogenization theory is based on an expansion of the flow- and transport equation in terms of the ratio between typical large length scale (for example the medium size) and typical small length scale (for example the length scale of a macroscopic representative elementary volume). By analyzing different parameter contrasts, quantified in terms of the expansion parameter, it can be demonstrated that, for example, the occurrence of non-equilibrium effects in the upscaled model depends crucially on the information about connectivity of different parameter ranges. Besides the type of upscaled model, also the effective model parameters depend on this type of information and can deviate significantly from effective parameters derived under the assumption that parameter fields are multi-Gaussian. The influence of connectivity of parameter fields on upscaled transport models in the vadose zone will be demonstrated with different multi-Gaussian and non-Gaussian test fields.
H34E-03
Upscaling Flow and Transport Through Block Permeability Inclusions by an Analytical Approach
Numerical simulation of flow and transport in natural porous media becomes very challenging when one takes into account the multi-scale heterogeneity of geological formations. Instead of obtaining the fine-scale solution numerically, which may be computationally expensive, another approach is to derive an up-scaled equation that provides the averaged solution and an approximation to the small scale dynamics. We present an analytical method to obtain the up-scaled coefficient when the heterogeneous coefficients (e.g., permeability) are periodic and rapidly oscillating and can be defined locally as step functions describing inclusions in a primary matrix. The new contribution involves deriving an analytical approximation for the solution of the periodic cell-problem, obtained by a two-scale asymptotic expansion of the governing heterogeneous equation. The analytical approximation also provides a lower bound of the generalized Voigt-Reiss' inequality. By defining a corrector to the approximation, the up-scaled coefficient is obtained as well as an analytical basis function that is used to derive the first-order term of the two-scale expansion. The known analytical results in the literature, such as the geometric average for a checkerboard geometry, are derived as particular cases and a comparison with some numerical results from the literature is presented. We demonstrate the convergence properties numerically, for linear and nonlinear problems of interest for flow in porous media. Further, we apply these results to special cases of transport and multi-scale diffusion, for a range of inclusion surface areas.
H34E-04 INVITED
Temporal Dynamics of the Spatial Variance of Sub-Grid Soil Moisture: A Look at Scaling Implications
Experimental efforts to define the dynamics of sub-grid spatial variance of soil moisture have led to contradictory results. Moreover, most reports of soil moisture variability range from qualitative to descriptively quantitative, and are unsupported by a theoretical framework for moisture variance dynamics. In this talk we present a conservation equation for the spatial variance of sub-grid root-zone soil moisture, based on first principles of statistical fluid mechanics. We arrive at a variance budget in which explicit covariances between moisture fields and land surface flux fields act to produce or destroy variance through time (according to the sign of the correlation between the flux and state fields). A series of examples are used to explore how simple forms of soil, vegetation, precipitation, topography, and initial moisture variability lead to evolving covariances between spatial fields of soil moisture and particular land surface fluxes, and how these covariances relate to the temporal trajectory of the spatial variance of soil moisture. We isolate a set of processes and conditions that demonstrate spatial variance production through time and a set that demonstrate variance destruction. Of particular interest is the tendency for transpiration and infiltration-runoff processes to either produce or destroy variance, depending on the background wetness regime. Field data are also employed and shown to demonstrate a temporal behavior of the spatial variance that is readily described by the proposed approach. The implications of this theory for multi-scale analysis of soil moisture variability is explored and discussed. Ultimately, this work should aid field data interpretation and, when supplemented with a closure model for the variance budget, lead to improved land surface flux predictability over coarse grids.
H34E-05
The average equilibrium capillary pressure-saturation relationship in two-phase flow in porous media
We investigate the upscaling of capillary pressure-saturation form Darcy scale to larger scales. Often when simulations are performed to investigate flow processes in reservoirs and aquifers, the simulated domain has a size on the scale of tens of meters to kilometers. Oil reservoirs and aquifers usually have complex structures, consisting of heterogeneities on different length scales. In an ideal situation, a numerical flow simulator would include all these variations. However, often, the numerical gridsize is much larger than the scale of medium heterogeneities. Therefore, it is desirable to obtain an uspcaled capillary pressure-saturation relationship that can be used to represent the capillary pressure-saturation relationship at the scale of the numerical grid. We have investigated several different averaging methods that can be used to obtain an upscaled capillary pressure- saturation relationship. The average of saturation over a certain volume follows directly from its definition. However, for pressure it is not that obvious and several ways of obtaining an average pressure are investigated. We introduce a number of different averaging operators and performing numerical simulations of a static primary drainage experiment in a homogeneous domain with dimensions similar to a laboratory sample. Results show that a new definition of macroscale pressure, in which centroids of the two phases is taken into account, appears to be the most suitable. We have also shown that the way Pc-Sw relationship is measured results in a curve that is not an intrinsic property of the two-phase system, but is dependent on several factors, such as the applied boundary conditions.
H34E-06
Calibration of a Soil Water Uptake Model Using Model Ensemble and Prior Information in a Semiarid Environment Using Global and Local Search Methods
A common model used to simulate actual evapotranspiration in watershed scale hydrologic models is the Kristensen and Jensen model (e.g. Mike She or MODHMS models). While the Kristensen and Jensen model was originally developed for Nordic climates, it has been extensively used in other types of environments without specific calibration or testing of its performance in climates other than the one for which the model was developed. In semiarid watershed hydrology, evapotranspiration is the main output component of the mass balance and is critical for a correct description of the hydrologic processes during interstorm periods. In this work we calibrate and study the performance of the Kristensen and Jensen model in a semiarid rangeland environment in southwest Spain. For this, a full soil water atmosphere model was used to describe the water fluxes in a column of soil. The model describes variably saturated water flow in the soil using Richards' equation and the van Genuchten soil retention curves. The Kristensen and Jensen model is used to calculate direct evaporation and the water uptake by grass cover. Seven parameters are simultaneously calibrated. Two are for the van Genuchten retention curve and three for the Kristensen and Jensen model. Hydraulic conductivity is assumed to decay exponentially with depth. The decay exponent and the hydraulic conductivity at zero depth are the two remaining parameters to be calibrated. Given the large set of free parameters involved, the calibration set up involves two sources of information: soil moisture measurements at four different depths in the soil column and an auxiliary simple linear model relating maximum daily temperatures and average soil moisture; and two sources of prior information: field capacity measured on soil cores and the maximum dry weight biomass when the soil is fully covered by grass. A global search method (SCE-UA) is used to locate the global minimum in the allowed parameter error space and a local search gradient based algorithm (Levenberg-Marquardt) is used to refine the search from the global solution and to obtain information in the vicinity of the minima. The results indicate that correct parameters of the soil retention curve are more critical for a proper simulation of the water uptake than are Kristensen and Jensen model parameters. Furthermore the calibrated van Genuchten parameters differ from the suggested values for silt-loam soils and they force a steeper effective retention curve and effective field capacities values that are lower than those measured in cores.
H34E-07 INVITED
Soil Moisture Variability and Mean Soil Moisture: A Stochastic Hydraulic Perspective
Soil moisture is a key variable in understanding water and energy fluxes in terrestrial systems. The characterization of soil moisture variability is one of the major challenges in hydrological sciences today. Especially the relationship between soil moisture variability and mean soil water content is receiving considerable attention as it plays an important in upscaling and downscaling of soil moisture fields and in the parameterization of terrestrial and climate models. We show that the relationship between mean moisture content and its standard deviation can be predicted by stochastic analysis of unsaturated Brooks-Corey flow in heterogeneous soils. Based on a sensitivity analysis, it is found that parameters of the moisture retention characteristic and their spatial variability determine to a large extent the shape of the soil moisture variance-mean water content function. Predicting this function for eleven textural classes we found that the standard deviation of soil moisture peaked between 0.17 and 0.23 for most textural classes. Differing values were found for the more sandy soils. The simulated range of soil moisture agrees with field findings reported in the literature. It was found that pore-size distribution of soils is the primary parameter controlling the maximum value of the soil moisture standard deviation. We demonstrate the potential of inversely estimating soil hydraulic parameters and their statistics from soil moisture data using a case study with generated functions.