Water Flow Equations Research Articles

Numerical analysis of soil water uniformity under sprinkler irrigation

A two-dimensional finite difference numerical analysis of the equation for water flow in unsaturated soils was used to examine the uniformity of wetting of the root zone during sprinkler irrigation. This was modelled by infiltration that varied over the soil surface and by the subsequent redistribution phase after infiltration ceased. Two soils, a sand and a sandy loam were used in this investigation. A representative vertical plane of soil 1·0m wide was considered to receive precipitation with a uniformity coefficient of 60%. The uniformity coefficient of the nett addition of water to a root zone 500 mm deep was evaluated. During infiltration, the uniformity improved from 60% to 67% for the sand while for the sandy loam it improved from 60% to 73%, reaching a stable value as the root zone became wetted. Further increase in uniformity occurred after precipitation ceased, as a result of drainage from the root zone, reaching a uniformity in excess of 80% as drainage proceeded. The results show that the soil profile plays a significant role in improving the uniformity of irrigation, and must be considered when specifying uniformity criteria for the design of sprinkler irrigation systems.

Simulated effects of flow rate on phosphate retention in soils

Phosphate in waste water disposed in the field can be partially or wholly retained in soil, eliminating ground water pollution. The retention ability of soil is affected by many factors, such as soil pH, minerals, structure, water content, solute concentration, solution pH, etc. The flow rate of waste water within a soil profile may be an important factor as well. It is the purpose of this paper to simulate different flow patterns in soils and their effect on the renovation of waste water by soils. A system of partial differential equations, including water flow and phosphate transport equations with a P-soil reaction model, were solved simultaneously to simulate the saturated or unsaturated flow of phosphate solution and its reactions in soils. Euler implicit and general explicit finite difference methods were applied to solve these equations. The results indicate that phosphate retention by soils from a finite quantity of waste effluent could be increased by decreasing the flow rate. An on-and-off intermittent disposal of waste water would also increase the retention capacity when compared to continuous flooding of the same amount of effluent. The main cause of these effects is the difference of reaction time of phosphate with soils.

In‐Situ Determination of Soil Hydraulic Properties during Drainage

AbstractIn‐situ soil water retention and hydraulic conductivity relations were determined for two soils from knowledge of initial and boundary conditions and water content profiles during drainage. Empirical relations were assumed for each soil between volumetric water content (θ) and soil water pressure head (h), and between hydraulic conductivity (K) and θ. Assigning values to the parameters in these relations made it possible to solve the general water flow equation for the same initial and boundary conditions as encountered in the drainage experiments. Calculated water content profiles were compared with measured water content profiles at corresponding times. Differences in measured and calculated θ values provided information for changes to be made in the parameters. This procedure was repeated until a satisfactory agreement was found and the determined relations apparently represented the soils in question.

Comments on “An Adaptive Finite Difference Scheme for the One‐Dimensional Water Flow Equation”

Comments on “An Adaptive Finite Difference Scheme for the One‐Dimensional Water Flow Equation”

Reply to “Comments on An Adaptive Finite Difference Scheme for the One‐Dimensional Water Flow Equation”

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Ground water and geotechnical problems

This paper is based on the Author's lectures presented during his stay as a Visiting Professor at The City University, London, in November 1982. Starting from a general view of the main problems, it describes the driving forces in hydro-geomechanical processes, especially the forces induced by pressure changes in the underground water. The estimation of the stresses and strains in the ground (soil and rock) requires the solution of coupled systems of equations of soil and ground water flow and geotechnical strain or failure of the ground. The paper describes in detail problems of regional ground subsidence due to pumping in open-cast mine areas and problems of estimation of seepage forces in saturated or unsaturated loose rocks with anisotropic conductivities and in fissured hard rocks with conductivities characterised by a tensor. Possibilities of the solution of those problems with analog and numerical models are briefly discussed.

Effect of Mud Filtrate Invasion on Apparent Productivity in Drillstem Tests in Low-Permeability Gas Formations

Summary This paper shows that mud-filtrate invasion prior to drillstem testing in low-permeability gas formations can cause significant reduction of gas flow rate observed during the test. In addition, apparent gas permeabilities determined from test data can be much lower than the true values. These conclusions are supported by simulator results and field data. Introduction Interpretation of drillstem tests (DST's) in low-permeability gas formations is difficult. and this difficulty leads to uncertainty in identifying formations that, when stimulated, could become commercial completions. Operators have tong suspected that mud-filtrate invasion into these low-permeability formations could contribute to these uncertainties because of significant alterations in relative permeability and capillary pressure. These uncertainties led us to a study of the effects of mud-filtrate invasion on response in a DST; this paper reports the results of that study. In this investigation, we used a two-phase, two-dimensional, fully implicit reservoir simulator to model DST's. The input data for the simulator were obtained from special core analyses. openhole tops, and DST's from the Falher sandstone in western Canada. Model Description The two-dimensional, two-phase model used in this study has been described previously in the literature. The model simulates the flow of gas and water in the formation using a simultaneous determination of saturations and pressures from finite-difference approximations of the gas and water flow equations. Capillary pressure and relative permeability data are input into the model; the model is therefore capable of simulating imbibition, water blocking, and cleanup effects. These effects are potentially important in DST performance. To simulate a DST, uniform initial pressures and saturations were established in the model for all reservoir cells. To model the imbibition of filtrate into the formation prior to the DST, the pressure in the wellbore was set equal to (and maintained at) the hydrostatic mud pressure, and the water (filtrate) saturation was maintained at 100%. Water saturations and pressures around the wellbore increased during the imbibition period as filtrate moved into the reservoir. To model a mud filter cake, the permeabilities in the formation adjacent to the wellbore were reduced. Following the imbibition period, a flow period was simulated. During the flow period, the pressure in the wellbore was maintained at a fixed pressure and the wellbore volume was set equal to the value of a typical drillpipe open to flow. Following the flow period, a pressure buildup test was simulated, with the wellbore volume set equal to a typical volume below the shut-in valve in a DST tool assembly . JPT P. 299^

Conditions governing the use of approximations for the Saint-Vénant equations for shallow surface water flow

The solutions of the Saint-Vénant equations are compared with those of the kinematic, diffusion and gravity wave approximations, for a range of constant Froudé and kinematic wave numbers, with two different lower boundary conditions: (1) critical flow; and (2) zero depth gradient. For each lower boundary condition, zones are defined in the F 0, k-field in which either kinematic, diffusion or gravity wave solutions may be used to approximate the full Saint-Vénant solutions.

An Adaptive Finite Difference Scheme for The One‐Dimensional Water Flow Equation

AbstractThe pressure bead form of the general flow equation for water in a porous medium was numerically solved using a scheme that allowed both the time step (Δt) and the space increment (Δz) to be changed during the flow process. The mass balance equation was used as a check for the accuracy of the simulation. A fixed number of grid points in the space direction was continuously redistributed to allow for small Δz values in regions of large changes in pressure head gradients while allowing large Δz values in regions of small changes. The method, which is unconditionally stable, is demonstrated for an infiltration and an evaporation process.

Numerical studies of Jökulhlaups

The aim of this paper is to investigate Jökulhlaups: outbursts of ice-dammed lakes. The governing equations of unsteady water flow through straight, circular conduits, as derived by the authors in a previous paper, are compared with the equations of Nye, and for the steady state situation with Röthlisberger's theory. The dynamic theory is treated numerically by a finite-difference technique. Run-off simulations are illustrated for a model of the Grimsvötn Jökulhlaup, a periodic outburst of a subglacial lake beneath the Vatnajökull in Iceland. We study how various parameters, such as the friction coefficient of the conduit, and constants in the flow law of ice, influence the evolution of the outburst. In particular, we compute discharge hydrographs both incorporating and neglecting the rate of change of internal energy. It is shown that this term is not negligible and that the lake temperature, as a boundary condition, strongly influences the time-discharge relation and might explain the abrupt end of the outburst well before the lake has been emptied.

Effect of Field Variability in Soil Hydraulic Properties on Solutions of Unsaturated Water and Salt Flows

AbstractUsing actual field variability data and the stochastic approach, effects of spatial variability of soil hydraulic properties on the predictions of water and salt flow distributions, and of correlation structure, are evaluated. The governing water and salt flow equations are solved numerically using actual field data of soil hydraulic conductivity and differential water capacity. These data, which had been obtained previously from measurements made in 30 different locations in a field, were the same as those obtained from a Monte‐Carlo simulation that considered the correlations between the parameters of the hydraulic functions. Simulated values of pressure head (h), water fluxes (q), wetting front positions (Zf), and solute concentration (c) for each of the 30 locations in the field were used to estimate distribution types (normal or log‐normal) and its moments, for h, q, Zf, and c. The two‐point autocorrelation functions of pressure head, wetting front, and solute concentration have been calculated and used to estimate the integral scales of h, Zf, and c. Distances in the field in which the horizontal flow components were negligibly small compared with the vertical flow components were calculated and found to be in the order of 10 cm and larger. Field dispersion of solutes was calculated from the simulated salt distribution results. Values of field scale dispersivity were estimated to be larger than the pore scale dispersivity and to increase with soil depth. An “equivalent” uniform porous medium, which was defined at the soil surface, exists for large infiltration time when a unit hydraulic gradient may be assumed.

Application of Soil Water Flow Theory in Field Simulation

A numerical model based on a finite difference approximation of the basic differential equation for soil water flow is described. The equation includes a sink term representing moisture extraction by root systems, and allowance for evaporation from the soil surface is made too. Estimation of these terms is based on a potential transpiration calculated from Penman-Monteith's equation, on a potential evaporation calculated from Ritchie's equation and in both cases on relations to soil moisture status over various soil depths. The model simulations are compared with field observations of moisture content and soil moisture tension. Further, the model has been applied in evaluating how irrigation treatment effects the increases in evapotranspiration and deep percolation.

MODELING NITROGEN TRANSPORT AND TRANSFORMATIONS IN SOILS

ABSTRACT We developed a numerical model to simulate water and nitrogen transport and transformations through water-unsaturated, multilayered soil profiles. The nitrogen transformation processes considered were nitrification, denitrification, immobilization, mineralization, and ionic exchange of ammonium. Plant uptakes of water and nitrogen were also included. We used an explicit-implicit finite difference approximation method to solve the nitrogen transport and transformation equations simultaneously with the water flow equation. Model evaluation and sensitivity analysis for a wide range of values for the rate of nitrification, distribution coefficient for ammonium exchange, and rate of N uptake were investigated.

Modeling Water Flux on Strip-Mined Land

ABSTRACT USING numerical models based on the one dimen-sional water flow equation, we simulated infiltration and redistribution of water in strip mine spoils. The technique used appeared satisfactory for topsoiled pro-files. In nontopsoiled profiles better fit was obtained by assuming that the flux occurred through the larger chan-nels and was a function of gravity alone. Comparison with experimental results suggested that if such a model is postulated, allowance is needed for water retained on the surfaces of coarse fragments and by fines. The pro-posed model appears to describe reasonably well the general shapes of moisture profiles on disturbed land, but specific results appear at times to be in error. The er-ror may be due to movement of water through discrete channels within the profile.

A mixed numerical analytical method for groundwater flow simulation

When finite difference or finite element schemes are used to solve the equations of ground water flow, it is not possible to obtain an accurate representation of the flow field near singularities without a considerable refinement of the grid. This difficulty is readily overcome by the use of a mixed numerical analytic scheme which is proposed in this work. Functions which accurately represent the effect of the singularity near its center but die off away from it are subtracted from the solution. The difference is a smooth function which is adequately represented by existing schemes. In addition to wells which are discussed here, springs and extended singularities can also be handled. The scheme is mainly of interest when it is of importance to trace the flow of pollutants in an area where many singularities may be present.

Modeling soil water changes in a well‐structured, freely draining soil

Changes of soil water content were measured in three monolith lysimeters and compared with changes predicted by a soil water budgeting program and a program incorporating a finite element solution of the unsaturated soil water flow equation. Both models underestimated water uptake during periods of high evapotranspiration. While the physically based model gave accurate estimates of flow from a lysimeter in the absence of root water uptake, the empirical budget provided the more practical approach to long‐term water balance calculations.

Three-Dimensional Simulation of Geologic, Hydrodynamic, and Thermodynamic Development of a Sedimentary Basin--New Approach: ABSTRACT

An understanding of paleopressures and paleotemperatures can guide and limit the application of geologic, geochemical, hydrodynamic, and thermodynamic principles in our study of the evolution of a sedimentary basin. Although we have qualitative understanding of the influence of paleopressure and paleotemperature on the interaction of sedimentation, compaction, and subsidence, a quantitative evaluation of the system is needed to understand basin history. Accordingly, a three-dimensional, deterministic dynamic model was constructed to quantify mass and energy transport in sedimentary sequences of a basin. A new water flow equation was derived for a compacting porous medium under moving boundary conditions, and was coupled with the heat flow equation for the transfer of heat by conduction and forced convection (due to water movement). By varying heat flux, initial physical and thermal properties of sediments, paleobathymetric estimates and sedimentation rate, this model can compute pressure, temperature, and physical and thermal properties as a function of space and time. Studies with this model show that pressure and temperature are closely interrelated with the geologic developent of a basin. Changes in heat flux alter relations between pressure and physical and thermal properties, and depth. For example, all the dependent and independent variables being equal, a change in heat flux affects the thickness of sediments in such a way that a 1,300-m thick clay layer subjected to a geothermal gradient of 40°C/km will compact to 1,236 m for an increase of 5°C/km, but will expand to 1,400 m with a decrease of 5°C/km to 35°C/km. Consequently, plots of pressure and physical and thermal data against depth are altered. This dynamic model was successfully used to study a real basin. Pressures and physical and thermal data were computed with an error of less than 8%, and temperatures with an error of approximately 2°C, with respect to all other data. End_of_Article - Last_Page 806------------

Infiltration and drainage calculations using spatially scaled hydraulic properties

Infiltration and drainage results are calculated for spatially varying hydraulic properties. The soil water flow equation is solved in a form valid for scaled soil water characteristics and unsaturated hydraulic conductivities. The single solution can then be utilized for individual sites chosen randomly by using a scaling coefficient. This procedure reduces repeated tedious individual calculations to simple algebra or interpolations. Examples include infiltration using semi‐analytical forms and drainage based on a single finite difference solution. Monte Carlo simulations were used in most cases.

COMPUTER SIMULATION MODEL FOR PREDICTING SOIL WATER CONTENT PROFILES

A simulation model describing water movement through field-cropped soils was developed. The soil water flow equation, which included a root extraction term, was approximated using a finite difference technique. Hydraulic conductivity (K) and diffusivity (D) functions were calculated by using the Millinton-Quirk technique. The chosen boundary conditions, rainfall, evapotranspiration, and a fluctuating water table approximated those existing in the field. Submodels to calculate interception, evaporation, and transpiration were included in the model. Model predictions were in good agreement with field data from a fairly uniform soil. On a soil containing discontinuous clay varves the predictions were not satisfactory when the original calculated K and D functions were used. Predicted and measured water content profiles were in closer agreement when the K and D functions were modified within the limits of the measured moisture retention curve data. /Author/

A model for coupled heat and moisture transfer during soil freezing

The objective of this report is to present simulation analyses of heat and water flow during soil freezing. The analyses utilize a modified form of a model presented by Harlan. The model is tested by using experimental data reported by Dirksen and Miller and Jame and Norum. The basic elements of the model are the heat and water flow equations. An implicit finite-difference scheme is used to solve these two equations for the boundary conditions used in the experiments. The agreement between simulated and experimental results is illustrated for temperature, water, and ice contents.