Changing ideas in hydrology — The case of physically-based models

Prophecy, reality and uncertainty in distributed hydrological modelling

Steady unsaturated flow with vertical mean infiltration through unbounded heterogeneous porous media is analyzed using a perturbation approximation of the stochastic flow equation which is solved by spectral representation techniques. The hydraulic conductivity K is related to the capillary pressure head ψ by K = Ks exp (−αψ), where Ks is the saturated conductivity, and α is a soil parameter. A general formulation is presented for the case with Ks and α represented as statistically homogeneous spatial random fields. In part 1, solutions are developed assuming α is constant and representing Ks variability by one-dimensional and three-dimensional isotropic random fields. Results are obtained for head variances and covariance functions, effective hydraulic conductivities, variances of the unsaturated hydraulic conductivity, flux variances, and variance of pressure gradient. When the parameter α is relatively large, corresponding to coarse textured soils, the head variance decreases and all of the results demonstrate a trend toward gravitationally dominated one-dimensional vertical flow. The effective conductivity is dependent on the correlation scale of ln Ks and the mean hydraulic gradient.

Stochastic Analysis of Unsaturated Flow in Heterogeneous Soils: 1. Statistically Isotropic Media

In this paper we examine how the nature of spatial variability affects hydrologic response over a range of scales using five field studies as examples. The nature of variability was characterized as either stochastic, when random, or deterministic, when due to known, nonrandom sources. We have emphasized how that characterization may change with the scale of hydrologic model. The five field examples, along with corresponding sources of variability, were (1) infiltration and surface runoff affected by shrub canopy, (2) groundwater recharge affected by soil depth, (3) groundwater recharge and streamflow affected by small-scale topography, (4) frozen soil runoff affected by elevation, and (5) snowfall distribution affected by large-scale topography. In each example there was a scale, the deterministic length scale, over which the hydrologic response was strongly dependent upon the specific, location-dependent ecosystem properties. Smaller-scale variability may be represented as either stochastic or homogeneous with nonspatial data. In addition, changes in scale or location sometimes resulted in the introduction of larger-scale sources of variability that subsume smaller-scale sources. Thus recognition of the nature and sources of variability can reduce data requirements by focusing on important sources of variability and using nonspatial data to characterize variability at scales smaller than the deterministic length scale. All the sources of variability described are present in the same watershed and affect hydrologic response simultaneously. Physically based models should therefore utilize both spatial and stochastic data where scale appropriate. Other implications for physically based modeling are that modeling algorithms should reflect larger-scale variability which generally has greater impact and that model and measurement grids should be consistent with the nature of variability.

Scale and the Nature of Spatial Variability: Field Examples Having Implications for Hydrologic Modeling

Models of water flow in the upper soil layer of spatially variable fields are developed. Field variability is assumed to take place in the horizontal plane. The saturated hydraulic conductivity is assumed to be a random variable of lognormal distribution, and head suction and moisture content are related to it by simple analytical relationships. The aim of the study is to derive the expectation and variances of the moisture content, suction heads, hydraulic conductivity, and water flux as functions of depth and time for infiltration and redistribution. Toward this aim a simplified solution of vertical flow in a homogeneous column, based on the concept of moving front, is developed. The statistical procedure for using this solution in a spatially variable field is outlined.

Unsaturated flow in spatially variable fields: 1. Derivation of models of infiltration and redistribution

Present-day soil-water physics enables useful quantitative predictions in the laboratory and in simple field situations. Difficulties, however, frequently arise for areas of appreciable size in the field. Known and unknown heterogeneities, on many scales, may vitiate predictions based on theory for homogeneous, or very simple heterogeneous, systems. Two types of heterogeneity are distinguished, deterministic and stochastic. The first often demands an extension of established analyses and may involve important phenomena absent from the analogous homogeneous problem. Stochastic heterogeneity may involve many scales and is imperfectly known. The statistical properties may be stationary, but in more complicated cases, randomness may be embedded in (either known or unknown) systematic trends. Some aspects of unsaturated and generally unsteady flow in heterogeneous systems are reviewed: the mathematical nature of the flow equation; the concept of scale-heterogeneity; analytical and quasi-analytical solutions. The enormity of the total problem of unsaturated unsteady flows in stochastic heterogeneous systems is illustrated through a dialectic of eight successive stages of simplification. The concept of the autocorrelation function governing λ, the internal characteristic length, is introduced; and the problem posed in terms involving the distribution and autocorrelation functions of λ, the reduced potential and conductivity functions, and the initial and boundary conditions as the data, from which it is required to establish distribution functions of various descriptors of the flow. The solution to a grossly simplified example of horizontal absorption is presented. Mean apparent sorptivity decreases rapidly to about one fifth of the mean (and about half the minimum) sorptivity of the component soils. Variation about the mean is very great but decreases as absorption proceeds. The example epitomizes the failure of additivity of properties in stochastic heterogeneous media, which arises because spatial textural changes in either sense tend to reduce unsaturated flow rates.

J. R. Philip

Papers

Field Heterogeneity: Some Basic Issues