RICHARDS equation

The Richards equation is widely used in research to estimate water flow in both saturated and unsaturated soils. It is also used in software proposed for use in evaluation of ET landfill covers. 

Theoretical estimation of water flow in unsaturated soils is difficult and complex. The derivation of the versions of the Richards equation commonly solved in modern models required several assumptions. In addition, it is difficult to accurately estimate likely field values for unsaturated soil hydraulic conductivity on the scale of a complete ET cover. Nevertheless, the Richards equation provides useful estimates of flow of water within the soil where adequate estimates of soil hydraulic conductivity are available. 

Payer80 states that the UNSAT-H model was developed to assess the water dynamics of arid sites and, in particular, estimate recharge fluxes for scenarios pertinent to waste disposal facilities. It addresses soil-water infiltration, redistribution, evaporation, plant transpiration, deep drainage, and soil heat flow as one-dimensional processes. The UNSAT-H model simulates water flow using the Richards equation, water vapor diffusion using Fick s law, and sensible heat flow using the Fourier equation. 

UNSAT-H uses the Richards equation, Fick s law, and the Fourier equation to estimate the flow of soil-water, vapor, and heat. This may be the strongest part of the model because these are the most rigorous, currently known, theoretical methods for estimating these parameters. 

UNSAT-H does not address the effects of soil density on plant growth and water balance. Disadvantages caused by the computational methods used to estimate soil water flow include the following (1) the model requires the user to choose from several submodels to solve the Richards equation this choice should be made by a person with training in advanced soil physics and (2) the model requires the input of several soil parameters that are difficult to estimate for the completed cover soil. 

The HYDRUS computer model was developed by the Agricultural Research Service of the USDA to estimate water flow in unsaturated soils that support plant growth81 It was developed as a onedimensional model, and then modified to allow solution of two-dimensional problems.82 HYDRUS employs the Richards equation to solve water flow in unsaturated soil however, it uses different solution methods from those used in UNSAT-H. It also requires extensive data input. The available windows version simplifies data entry and model operation. 

Table 25.3 compares the characteristics of these four models.14 UNSAT-H and HYDRUS are the most widely known Richards equation models that use modem soil physics principles to estimate water movement within the soil profile. HELP and EPIC are widely known engineering models. 

Richards equation, vapor flow, etc. f Requires independent user estimates for input data. 

Richards equation (Richards 1931), based on a mass conservation balance, together with Eq. 9.1, can be used to describe the transient flow of water through a partially saturated porous medium. In one dimension (vertically), Richards equation is given as... 

In this equation r) is the volume of water per unit volume of solid, t is time, F is the Darcy flux of water relative to the solid, and m is a space like coordinate defined in terms of the distribution of the solid phase. Substitution of Darcy s law in this equation results in equations similar to the Richards equation and, at least in 1-dimension, these can be solved like their non-swelling analogues. This equation implicitly deals with volume change that might accompany change in water content. 

Convection (also known as advection) is the vector, which results from the DARCY or the RICHARDS equations. It describes the flow velocity or the flow distance for a certain time t. In general convection has the major influence on... 

For one-dimensional horizontal water movement in homogeneous soil, Richards equation reduces to... 

It is also possible (at least, in a formal way) to apply a fractional version of Richards equation to simulate one-dimensional water transport in horizontal columns. We were able to fit the FADE to data on horizontal water infiltration (data not shown). However, the parameter a had to be set to values greater than two to fit the experimental data. This range of a is theoretically unjustified (Benson et al., 1999 Meerschaert et al., 1999). This example serves as a reminder about the danger of drawing analogies between water and solute transport in soils, since the underlying physical processes are different, Particles of soil water moving faster than others are affected by the structure of pore surfaces and move in films rather than in bulk volume by convection. One possible way to model the water transport is to use the diffusivity model proposed by Jumarie (1992) ... 

In this chapter, we use the results of numerical infiltration experiments in dual porosity media performed with a three-dimensional lattice-gas model to characterize preferential flow as response to rainfall intensity. From the temporal and spatial evolution of the water content during infiltration and drainage, we evaluate the adequacy of a kinematic wave approximation to describe the flow. We also discuss the conceptual basis of the asymptotic kinematic approach to Richards equation in comparison with the macropore kinematic equation. 

The Richards equation is one of the most popular formalism used to describe water flow in chemical transport models ... 

Conventionally used Darcy s law and Richards equation (2 order partial differential equations) are exanq>les of a system with a positive feedback, but witiiout a negative feedback conqmnent. Negative and positive feedback mechanisms are taken into account by using tiie difference-differential equation for soil-moisture balance (75) and the Kuramoto-Sivashinsky equation 74) (see below). 

The water balance of landfill covers also has been simulated using the unsaturated flow approach in the form of a one-dimensional, finite-difference computer program, UNSAT-H, that solves a modified version of Richards equation that governs unsaturated flow through porous media (Khire el al. 1997). In this case, surface runoff is determined as the difference between precipitation and infiltration, where infiltration is based on the saturated and unsaturated hydraulic conductivities of the soils constituting the cover. Thus, unlike the empirical approach in the HELP model, the determination of surface runoff in UNSAT-H is based directly on the physical properties of the soil profile. 

Without delay measure the extinction of the solution against a cell containing 90% acetone (Note i) at 7500, 6650, 6450, 6300, and 4800 A (Note /). If the Richards equations are to be used for carotenoids (see below) a further measurement at 5100 A is required, and if the scor/unesco equations are used the measurement at 6650 A should be replaced by one at 6630 A. Record the extinction values to the nearest 0.001 unit in the range 0-0.4 and the nearest 0.005 for extinctions exceeding about 0.4. Correct the extinctions at each wavelength by the procedure described in Section G below. 

Flow in porous media is difficult to be accurately modeled quantitatively. Richards equation can give good results, but needs constitutive relations. These are usually empirically based and require extensive calibration. The parameters needed in the calibration are amongst others capillary pressure and pressure gradient, volumetric flow, liquid content, irreducible liquid content, and temperature. In practice it is usually too demanding to measure all these parameters. 

Structural Reliability Estimation for Seismic Loading, Table 1 Parameters of Richard equation for M-0 curves... 

Flow is more difficult to model in the unsaturated zone because hydraulic conductivity changes as water content changes. In unsaturated flow models, the Darcy-Richards equation, which is a more general form of Darcy s law, is used. For isotropic porous media in three dimensions. 

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