Function Of Pressure Head Research Articles

Analytical solution for steady vertical flux through unsaturated soils based on van Genuchten-Mualem model

Various analytical solutions for steady-state vertical flux through unsaturated soils exist in the literature. These solutions often assume specific hydraulic conductivity forms such as exponential form, rational power form, or power-law form such as the Brooks-Corey model. The van Genuchten-Mualem model is widely used in practical applications of flow through unsaturated soils. It is among the most complicated hydraulic conductivity forms for which no analytical solutions are available. This paper deals with the development of an analytical solution for steady-state vertical flux using van Genuchten-Mualem model. The analytical solution is obtained using Maclaurin series expansions and the multinomial theorem. The general form expresses the depth as a function of pressure head. The analytical solution can be used to model both infiltration (positive flux rate) and evaporation (negative flux rate). The derived analytical solution is compared with numerical results and shows excellent agreement. It can also be used to identify the hydraulic parameters analytically, eliminating the need of a numerical inverse procedure. As far as the author’s knowledge, the proposed solution is the first analytical solution dealing with van Genuchten-Mualem model, which is directly derived from the flow equation without any predefined mathematical form of the pressure head.

Simple functions for describing soil water retention and the unsaturated hydraulic conductivity from saturation to complete dryness

Many applications require accurate soil hydraulic functions that can describe the dynamics of water in variably saturated systems from full saturation to complete dryness. Current functions considering zero water content at the dry end involve the summation of capillary and adsorption functions. This study proposes relatively simple yet complete soil hydraulic functions from full saturation to oven dryness that can be incorporated into existing models. The proposed water retention model is constructed by scaling the saturation function of Fredlund and Xing (1994) to describe capillary and adsorptive water retention as well as zero water content at oven dryness. To describe the hydraulic conductivity due to capillary and film flow, the model of Wang et al. (2018) is used. We further incorporated hysteresis in the water retention function of Wang et al. (2018). Testing on 241 hydraulic property data sets covering a wide range of soil textures from sands to clays, shows good agreement with laboratory observations, especially for coarse-textured soils. The proposed model accounts for the following processes: hysteretic capillary and adsorptive water retention, hysteretic hydraulic conductivity as a function of pressure head, non-hysteretic hydraulic conductivity as a function of water content, the effects of entrapped air, and closed scanning curves. This paper successfully derives the parameters of the main wetting curve from the main drying curve. Using the proposed functions for soil water flow modelling should lead to improved dynamic flow predictions, especially for drying soil conditions.

An effective excess charge model to describe hysteresis effects on streaming potential

Streaming potentials are produced by the coupling between the water flow and the electrical current generated by the drag of electrical charges within the pore water of the media. This electrokinetic coupling is strongly influenced by the hydraulic properties that control groundwater flow (permeability, saturation and pressure head). Under unsaturated conditions, hydrogeologic studies have widely established that the relationships of permeability and saturation with pressure head are different for drainage and imbibition experiments. The hysteresis phenomenon present on these properties produces a hysteretic behaviour on the streaming potential which has been recently observed in experimental data. Hysteresis can be explained by the presence of irregularities in the pore geometry of the media which affects the water flow and, therefore, the excess charge density that is effectively dragged by the flow. In this study, we present a physically-based analytical model to describe the hysteresis phenomenon in the estimates of the effective excess charge density. Under the assumptions of a porous medium represented by a bundle of tortuous capillary tubes with throats and a fractal pore size distribution, hysteretic curves are obtained for the effective excess charge density as a function of pressure head using a flux averaging technique. These analytical expressions are closed-form and depend on the medium petrophysical and chemical properties. The predictions of the proposed model are consistent with laboratory data from drainage-imbibition experiments. These results open up exciting possibilities for studies involving water movement and processes in the vadose zone.

On Similarity of Seismo Magnetic Power Density and Pressure Head Fractal Dimension for Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia

The quality and assessment of a reservoir can be documented in details by the application of seismo magnetic power density. This research aims to calculate fractal dimension from the relationship among seismo magnetic power density, maximum seismo magnetic power density and wetting phase saturation and to approve it by the fractal dimension derived from the relationship among inverse pressure head * pressure head and wetting phase saturation. Two equations for calculating the fractal dimensions have been employed. The first one describes the functional relationship between wetting phase saturation, seismo magnetic power density, maximum seismo magnetic power density and fractal dimension. The second equation implies to the wetting phase saturation as a function of pressure head and the fractal dimension. Two procedures for obtaining the fractal dimension have been utilized. The first procedure was done by plotting the logarithm of the ratio between seismo magnetic power density and maximum seismo magnetic power density versus logarithm wetting phase saturation. The slope of the first procedure = 3- Df (fractal dimension). The second procedure for obtaining the fractal dimension was determined by plotting the logarithm (inverse of pressure head and pressure head) versus the logarithm of wetting phase saturation. The slope of the second procedure = Df -3. On the basis of the obtained results of the fabricated stratigraphic column and the attained values of the fractal dimension, the sandstones of the Shajara reservoirs of the Shajara Formation were divided here into three units

On Similarity of Seismo Magentic Field and Pressure Head Fractal Dimension for Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia

The quality and assessment of a reservoir can be documented in details by the application of seismo magnetic field. This research aims to calculate fractal dimension from the relationship among seismo magnetic field, maximum seismo magnetic field and wetting phase saturation and to approve it by the fractal dimension derived from the relationship among inverse pressure head * pressure head and wetting phase saturation. Two equations for calculating the fractal dimensions have been employed. The first one describes the functional relationship between wetting phase saturation, seismo magnetic field, maximum seismo magnetic field and fractal dimension. The second equation implies to the wetting phase saturation as a function of pressure head and the fractal dimension. Two procedures for obtaining the fractal dimension have been utilized. The first procedure was done by plotting the logarithm of the ratio between seismo magnetic field and maximum seismo magnetic field versus logarithm wetting phase saturation. The slope of the first procedure = 3- Df (fractal dimension). The second procedure for obtaining the fractal dimension was determined by plotting the logarithm (inverse of pressure head and pressure head) versus the logarithm of wetting phase saturation. The slope of the second procedure = Df -3. On the basis of the obtained results of the fabricated stratigraphic column and the attained values of the fractal dimension, the sandstones of the Shajara reservoirs of the Shajara Formation were divided here into three units.

On Similarity of Seismo Magentic Field and Pressure Head Fractal Dimension for Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia

The quality and assessment of a reservoir can be documented in details by the application of seismo magnetic field. This research aims to calculate fractal dimension from the relationship among seismo magnetic field, maximum seismo magnetic field and wetting phase saturation and to approve it by the fractal dimension derived from the relationship among inverse pressure head * pressure head and wetting phase saturation. Two equations for calculating the fractal dimensions have been employed. The first one describes the functional relationship between wetting phase saturation, seismo magnetic field, maximum seismo magnetic field and fractal dimension. The second equation implies to the wetting phase saturation as a function of pressure head and the fractal dimension. Two procedures for obtaining the fractal dimension have been utilized. The first procedure was done by plotting the logarithm of the ratio between seismo magnetic field and maximum seismo magnetic field versus logarithm wetting phase saturation. The slope of the first procedure = 3- Df (fractal dimension). The second procedure for obtaining the fractal dimension was determined by plotting the logarithm (inverse of pressure head and pressure head) versus the logarithm of wetting phase saturation. The slope of the second procedure = Df -3. On the basis of the obtained results of the fabricated stratigraphic column and the attained values of the fractal dimension, the sandstones of the Shajara reservoirs of the Shajara Formation were divided here into three units.

Head Loss in Thin-Walled Drip Tapes with Continuous Labyrinth.

Thin-walled drip tapes with continuous labyrinth have been used for irrigation of vegetables and other short-cycle crops, especially due to their low cost. The continuous labyrinths welded into the pipe inner wall affect the head loss along such emitting pipes. In addition, the flow cross section of thin-walled pipes may change due to the effects of the operating pressure, which also has consequences for the head loss. The objective of this work was to investigate experimentally the friction factor and the head loss on thin-walled drip tapes with continuous labyrinths operated under various pressures. Two models of commercial thin-walled drip tapes with continuous labyrinths were evaluated. Nonperforated samples were used to determine the head-loss equations. The equations were adjusted as a function of flow rate and pressure head at the pipe inlet. Alternatively, the diameter in the Darcy–Weisbach equation was adjusted as a function of the pressure head by a power-law model. The possibility of using a mean diameter in the Darcy–Weisbach equation was also analyzed. Experimental investigation indicated that the friction factor in the Darcy–Weisbach equation can be accurately described using a power-law model, like the Blasius equation, but characterized by a coefficient a=0.3442 for the Turbo Tape and a=0.3225 for the Silver Tape. The obtained values of a are larger than those generally used and available in the literature. The influence of the operating pressure on the pipe diameter can be neglected for the purpose of calculating the head loss. The two approaches, considering the variation of the diameter with the pressure head and considering an optimum average diameter for the calculation of head loss by the Darcy–Weisbach equation, produce similar results, allowing accurate prediction of head loss. Evaluating the proposed mathematical models, 95% of predictions presented relative errors of head loss smaller than 5%. For the Turbo Tape, the optimum diameter for the purpose of calculating the head loss is 16.01 mm, which is very close to the value indicated by its manufacturer (15.9 mm). For the Silver Drip, the optimum diameter is 15.71 mm, while the manufacturer gives a value of 16.22 mm, which produces considerable error in the calculation of head loss.

Head loss in laser-perforated thin-walled polyethylene pipes for irrigation

ABSTRACT For reducing fixed and operational costs in pressurized irrigation systems, thin-walled polyethylene pipes with laser-perforated orifices are manufactured to operate under low pressure (up to 100 kPa). Hydraulic characterization of these materials is essential for designing irrigation systems. Considering the material elasticity and the thin wall thickness (about 200 μm), the internal diameter of these pipes may vary according to the operating pressure, resulting in changes of head losses. The purpose of this study was to analyze the head loss in flexible pipes with laser-perforated orifices, and to estimate the maximum length of laterals based on criteria of water distribution uniformity. Non-perforated pipe samples were tested to obtain equations of friction loss. Equations were fitted as a function of flow rate and pressure head at the pipe inlet, and, alternatively, the Darcy-Weisbach equation was modified considering the diameter expressed as a power-law function of pressure head. The equation of head loss as a function of flow rate and pressure head provided proper estimations and considered effects related to changes in the diameter of plastic pipes due to variations in the pressure head. The Darcy-Weisbach equation can be employed for estimating head loss in flexible pipes, whose diameter varies due to pressure, but the diameter must be calculated as a function of the pressure head at the lateral inlet.

Hydraulic properties of typical salt-affected soils in Jiangsu Province, China

Every year about 1,500 ha of land is reclaimed from the sea along the coastline of Jiangsu Province, China. It is important to characterize the hydraulic properties of this reclaimed land to be able to predict and manage salt and water movement for amelioration of these saline soils. In this paper, we report hydraulic properties of these salt-affected soils. The pressure-plate method, constant head method, the crust method and Klute's method were used in this study. The saturated hydraulic conductivities of the soils ranged from 128.66 to 141.26 cm/day and decreased with increasing soil depth. The unsaturated hydraulic conductivities followed an exponential function of pressure head. The soil water retention curves were similar for three soil layers in the soil. The saturated water content, field capacity and wilting point decreased with increasing soil depth. Plant available water contents of the three layers in the soil profile were 0.21, 0.20 and 0.19 cm3/cm3, respectively. The unsaturated soil water diffusivity of the studied soils ranged from 0.07 to 10.46 cm2/min, and was related to the water content via an exponential relationship.

Estimating finite difference block equivalent hydraulic conductivity for numerically solving the Richards' equation

AbstractA method is proposed for calculating the equivalent hydraulic conductivity (EHC) within a finite difference block (FDB). Application of the constant‐flux assumption of Darcy's Law, the EHC equals to the integration of effective hydraulic conductivity (Kw) as a function of pressure head (hw) divided by the head difference at the ends of the FDB. Error analysis show that the constant‐flux (CF) EHC estimates are better than those computed by the commonly used arithmetic‐mean (AM), geometric‐mean (GM), and harmonic‐mean (HM) techniques. CF EHC results are even more superior at larger interblock head difference situations.Simulations of water infiltration experiments show that simulations using the CF EHC or AM or GM weighting technique have only slight difference while applying the Neumann type boundary condition at the ground surface. In case of the Dirichlet type boundary condition, however, the CF EHC is superior to the other two in correctly estimating the depth of infiltration while enlarging the grid size. Therefore, it is recommended to adopt the CF EHC with a larger grid size to the more stable and more efficient results. Copyright © 2007 John Wiley & Sons, Ltd.

A conceptual model for the description of oxidation and transport processes in sulphidic waste rock dumps

Basic mathematical equations which describe the processes of sulphide oxidation and gas, heat and water transport in waste rock dumps are presented and discussed. The governing equations account for gas and water flow, vapourisation and condensation with latent heat effects, heat transport and mass balance. Gas, water and solid phases are assumed to be in local thermal equilibrium at all times. Air is approximated as an ideal three-component gas. Different semi-empirical relationships between physical values are used: Darcy's law for fluid flow, ideal gas law, the Van Genuchten formula for the relationship between degree of water saturation and pressure head, Mualem's formula for the relative hydraulic conductivity as a function of pressure head, etc. Some important global quantities, such as the fraction of sulphide sulphur oxidised and the global oxidation rate, are defined and considered as functions of time.

Quick Estimation of Hydraulic Conductivity in Unsaturated Sandy Loam Soil

Laboratory experiments were conductedfor determining hydraulic conductivity duringinfiltration in an unsaturated sandy loam soil, usingboth steady state and equilibrium methods. A constanthead Guelph permeameter and a volumetric pressureplate extractor were used. Based on two ponded heightsin the permeameter, the parameters of Gardner'sequation expressing the unsaturated hydraulicconductivity as a function of pressure head (i.e. thesaturated hydraulic conductivity Ks and theexponent α), were estimated simultaneously.Furthermore, it was found that the parameter α,could also be predicted from the soil-water retentioncurve based on equilibrium data obtained from theextractor. This indicated that, for the soil typestudied, one-ponded height in the permeameter methodcould be sufficient for the determination of theexponent α, provided that the soil-water retentioncurve is known.

Numerical investigation of steady liquid water flow in a variably saturated fracture network

Numerical simulation was used to study steady liquid water movement in a 5‐m by 5‐m vertical section containing a hypothetical fracture network under conditions of variable saturation. The fracture network was assumed to be embedded within an impermeable rock matrix. Three variations of a network were considered. The “mixed” network consisted of two fracture sets, a subvertical set containing five 125 μm average aperture fractures and a subhorizontal set containing four 25 μm average aperture fractures. The other two networks had identical fracture orientation and contained either all 125 μm or all 25 μm average aperture fractures. The TOUGH simulator was used to calculate the total steady liquid water flux through the network, the flux through individual fracture segments, and the pressure head at each fracture segment. A unit hydraulic gradient was imposed on the network by applying fixed pressure head boundaries (ranging from −0.25 to 0.0 m of water) of equal value to the top and bottom. Saturation and permeability versus pressure head relations for the two sets of fractures were determined with the VSFRAC model, which assumed that aperture was variable within an individual fracture. Results showed that the spatial distributions of pressure head and flux within the network, as well as the location of the dominant pathways, depended strongly on the prescribed boundary pressure head. For the mixed network, both pressure head and flux tended to become more spatially uniform when the boundary pressure head approached the pressure head at which the permeability thickness products of the large‐ and small‐aperture fractures are equal (the crossover pressure head). These results imply that for systems similar to the one considered here, interpretation of actual measurements of pressure head and flux may be quite complex, and that representation of variably saturated fracture networks as an equivalent continuum may be more valid for some ranges in pressure head than for others. Equivalent permeability as a function of pressure head was calculated for the fracture network, illustrating how information collected on individual fractures may be used to estimate the flow properties of rock at larger scales.

MACROPORE CHARACTERIZATION BY INDIRECT METHODS

Macropores are important in preferential flow; therefore, function and continuity of macropores are often more important than visual observation. The objective of this research was to compare several indirect methods of determining macropore volume. Ponded and tension infiltration measurements were made in situ at the soil surface and 0.25 m deep in a Waukegan silt loam (fine-silty over sandy, mixed, mesic Typic Hapludoll), and air-filled porosity at each pressure head was calculated from conductivity determinations. Undisturbed, unconfined samples were taken from the 0.25− to 0.35-m depth. Air-filled porosity as a function of pressure head was determined on these samples using wetting and draining equilibrium curves in rotated cores (to even out the gravitational gradient), measured/calculated after steady-state lab conductivity, and calculated from ped size distribution (assuming a ped shape factor and ped size/crack size relation, with known bulk density and ped density). Water adsorption/desorption in rotated cores in only four out of 20 samples displayed an “air-entry” pressure head, and no samples displayed a “water-entry” pressure head (that is the head at which no more water enters pores during equilibrium soil water adsorption). All methods tested estimated air-filled porosity in the same range. Air-filled porosity determined from ped-size distribution and from ponded and tension infiltration measurements was much less tedious than from rotated core measurements.

Modeling the Hydraulic Properties of a Multiregion Soil

AbstractNumerical techniques for modeling transport of solutes in multiple porosity systems have recently been developed to better represent the physical attributes of structured soils. Multiregion flow modeling requires characterization of the water content (θ) and hydraulic conductivity (K) as a function of pressure head (h) for each region of the system. A technique is presented for determining the θ(h) and K(h) relationships for a three‐region soil system based on a typical water retention data base. The demarcation between Region 1 (macropore) and Region 2 (mesopore) was assumed to occur at h = −10 cm. The effect of this assumption was evaluated against criteria of h = −2 and −25 cm, with h = −10 cm giving the best results. The θ(h) function was described as a Fermi function for Region 1 and combined with the van Genuchten model for describing Regions 2 and 3 (micropore). Continuity in θ(h) and K(θ) between regions was achieved and an excellent agreement between predicted and measured values was obtained for each region. However, the derivative of the θ(h) and K(θ) functions were occasionally not continuous through the junction points. The Fermi function based on measured and assumed parameters gave erroneous predictions of K(θ) when a large (25‐cm) range in h was used for Region 1. A modified Fermi function was developed that gave an excellent prediction of K(θ) based on assumed parameters.

A continuum model for water movement in an unsaturated fractured rock mass

The movement of fluids in a fractured, porous medium has been the subject of considerable study. This paper presents a continuum model that may be used to evaluate the isothermal movement of water in an unsaturated, fractured, porous medium under slowly changing conditions. This continuum model was developed for use in evaluating the unsaturated zone at the Yucca Mountain site as a potential repository for high‐level nuclear waste. Thus its development has been influenced by the conditions thought to be present at Yucca Mountain. A macroscopic approach and a microscopic approach are used to develop a continuum model to evaluate water movement in a fractured rock mass. Both approaches assume that the pressure head in the fractures and the matrix are identical in a plane perpendicular to flow. Both approaches lead to a single‐flow equation for a fractured rock mass. The two approaches are used to calculate unsaturated hydrologic properties, i.e., relative permeability and saturation as a function of pressure head, for several types of tuff underlying Yucca Mountain, using the best available hydrologic data for the matrix and the fractures. Rock mass properties calculated by both approaches are similar.