Steady Unsaturated Flow Research Articles

Method for efficient calculating earth pressure of retaining wall considering plant transpiration

An accurate estimation of earth pressure on retaining walls is imperative to achieving its design. This paper presents an analytical method framework that considers the effect of plant transpiration relative to the traditional calculation approaches. Specifically, a closed-form solution for one-dimensional steady unsaturated flow considering plant transpiration is incorporated into a representation of effective stress to obtain the changes in matric suction, and effective stress. The representations are used to extend Hooke’s law and Rankine’s earth pressure theory to determine at-rest, active, and passive earth pressures. Subsequently, the analytical method is used in a series of analysis case studies on the influence of root architecture types, transpiration rates, and soil types on earth pressure, to reveal that it can rapidly obtain the earth pressure. Notably, the effect of plant transpiration on earth pressure is significant. Furthermore, it is found that soil types and transpiration rates have a larger influence than root architecture types. Collectively, the research not only reveals the effect of plant on earth pressure for retaining wall, but also provides a theoretical basis for further exploration of the contribution of plants to the stability of retaining wall.

Active earth pressure of 3D earth retaining structure subjected to rainfall infiltration

Rainfall infiltration is a key factor that determines the active earth pressure on an earth-retaining structure. Previous investigations on the stability of earth retaining structures were mainly conducted under the assumption that the backfills was dry, saturated, or subjected to a steady unsaturated flow, ignoring the time-varying characteristics of the rainfall infiltration process. Based on the upper bound theorem of limit analysis, this study conducted a stability analysis of a three-dimensional (3D) earth retaining structure (ERS) subjected to various rainfall infiltration patterns. An analytical model that was, capable of calculating the pore water pressure distribution due to rainfall infiltration, was employed to calculate the time-dependent pore water pressure change of unsaturated soil. A horizontal slice method was employed to take into account the change of effective unit weight of soil. Afterward, an analytical expression of the active earth pressure coefficient is derived from the energy balance equation. The effects of the 3D characteristics of ERSs and rainfall patterns on the active earth pressure coefficient and the failure pattern of ERSs were investigated. It was found that compared with plain strain conditions, the 3D characteristics of ERSs lead to a lower value for the active earth pressure and better stability. The rainfall patterns not only directly determined the stability and active earth pressure of an ERS, but also played a dominant role in critical failure pattern of the ERS.

Non-associated plasticity analysis of slope stability under steady unsaturated flow conditions

This study presents an analytical framework of limit analysis for the ultimate load of unsaturated soil slopes with the non-associated plasticity. The method of characteristics combined with boundary value problems is applied to build the slip line field that satisfies the stress boundary and the velocity boundary simultaneously. According to the proposed approach, a value of the ultimate load of slopes can be directly addressed, which is within the narrow range determined by the upper bound (UB) and the lower bound (LB) solutions. The non-associated Mohr-Coulomb criterion is presented to investigate the effect of the matric suction and the dilation on slope stability. The comparison with the existing methods is conducted to validate the presented solutions. The results indicate that the effect of the flow condition and the dilation on the ultimate load of four types of soil slopes can be estimated accurately by the proposed approach.

Bearing Capacity Analysis of Pile-Stabilized Slopes under Steady Unsaturated Flow Conditions

AbstractWhen analyzing slopes stabilized by piles, the soil has been commonly considered dry or saturated, whereas soils are usually unsaturated in practice. Suction stress is caused by positive an...

Stability study of slope reinforced with piles under steady unsaturated flow conditions

In this study, based on the log-spiral upper bound limit analysis method that integrated with the generalized effective stress, a new formula for calculating the lateral stabilizing force provided by a row of piles required to increase the slope safety factor to a desired value under vertical steady unsaturated flow was developed, and the most critical solutions were obtained by the optimization method. Moreover, the proposed method can be simplified to the case without piles for computing the stability number for a slope in the limit state of failure. By comparing the results of the proposed method to others for some examples, its feasibility was verified. Numerical study for the parametric effects was presented, indicating that limit lateral stabilizing force depends strongly on the soil types, the slope geometry, the seepage conditions and that in routine design without consideration of suction stress can lead to a less conservative result in clay and sand, but can obtain a relatively safe result in silt and loess. This paper gives some useful suggestions for engineering application and provides a useful method for further study.

Upper Bound Analysis for Collapse Failure of Shield Tunnel Face Excavated in Unsaturated Soils Considering Steady Vertical Flow

Natural soils are mostly in an unsaturated state and the corresponding mechanical properties differ significantly from a saturated one. In traditional stability analysis for a shield tunnel face, the soil mass is typically assumed to be dry or saturated for convenience of analysis. In this work, based on the upper bound theorem of classical plasticity theory and the log-spiral failure mechanism, face stability of the shield tunnel excavated in unsaturated soils under vertical steady flow is studied. The profile of shear strength is determined by virtues of the unified effective stress approach and the analytical solution of matric suction under unsaturated steady flow. On this basis, the analytical expression for the supporting pressure is deduced and the sequential quadratic programming is employed to search for the optimal upper bound solution. Through parametric analysis it is found the fitting parameters of SWCC and the groundwater level affect the supporting pressure notably. Besides, the relevant influence of vertical discharge is also significant when clayey soils are concerned.

Active earth pressure for soils with tension cracks under steady unsaturated flow conditions

The calculation of active earth pressure has commonly been performed under dry or saturated conditions. However, in practice soils are usually unsaturated. With the aim of estimating active earth pressure under steady unsaturated flow conditions, this paper describes a new method within the framework of the kinematic approach of limit analysis. A closed-form formula is adopted to estimate the suction stress of unsaturated soils subjected to vertical steady flow. A rotational failure mechanism composed of a log-spiral curve and a vertical crack is presented for calculations of the external work rate and internal energy dissipation rate. Pre-existing cracks as well as cracks that form as part of the failure mechanism are considered in the present work. The maximum depth of cracks is determined by requiring the vertical crack boundary to be stable. Explicit expressions about the thrust of active earth pressure for different crack types are derived from the work–energy balance equation, and the most critical solutions are obtained by optimizing the variables from the failure mechanism. Numerical results for different parameters are calculated and given in the form of graphs for practical use. A parametric study is performed to investigate the impact of different parameters on the active earth pressure.

Stability of 3D slope under steady unsaturated flow condition

Slope stability has commonly been analyzed by considering dry or saturated soils under two-dimensional (2D) plane-strain conditions. However, in practice, soils are often unsaturated, and many slope failures exhibit a three-dimensional (3D) feature. In this study, the kinematic limit analysis method is adopted to estimate the stability of slopes subjected to vertical unsaturated steady flow in the context of a 3D rotational failure mechanism. A closed-form representation is employed to express the shear strength behavior of unsaturated soils. According to the work-energy balance equation, the critical cohesion for a slope in the limit state is calculated and listed in the form of a stability number. An optimization scheme is used to search for the maximum stability number. By comparing with the previously published 2D and 3D solutions, the proposed method is validated. Charts of different parameters are plotted for parametric analysis and practical use in the design of slopes. It is shown that accounting for the 3D effect and the soil suction can lead to a less conservative result. This paper provides a useful method for assessing the combined effect of 3D boundary conditions and unsaturated flow conditions on the stability of slope and gives some useful suggestions for engineering application.

Analysis and Upscaling of Unsaturated Flow through Randomly Heterogeneous Soil

The spatial variability of soil hydraulic properties has to be considered to provide realistic predictions of unsaturated flow and transport at field scale. Steady unsaturated flow through randomly heterogeneous soils is analyzed and upscaled in this study. Numerical experiments of unsaturated flow have been performed on single realizations of unsaturated soil parameters generated by using threedimensional turning band random generator. In the first part of the study a linear relation for unsaturated hydraulic conductivity as a function of soil suction (.) given by K = Ksea. is used, with constant beta and saturated hydraulic conductivity (K-s) as perfectly correlated random space functions. In the first part of the study, linear conductivity relation given by Gardner is used, with Gardner's constant (beta) and saturated conductivity (Ks) as perfectly correlated random space functions. The steady flow fields generated are analyzed to explore the effect of variability of the product (a): the pore-size distribution parameter whose reciprocal represents the characteristic capillary length (h(cap)) and the correlation length (lambda Z) of the generated soil properties. A set of one-dimensional simulations are performed, with various values of the product beta lambda(Z) ratio of correlation length to capillary dispersion length (lambda(z)/ hcap = 1, >> 1 and e 1) used to compare the moisture distribution tendency. The resulting steady pressure fields are compared and analyzed. In some cases, the soil behaves as homogeneous, and in other cases as stratified. The second part of the study seeks to identify simple upscaling laws for block-scale nonlinear constitutive relationships under high tension, with the parameters of their measurement scale counterparts represented by the van Genuchten model. Random fields are generated to represent layering of perfect and imperfectly stratified soils, and numerical simulations are performed in flows that are perpendicular and parallel to stratification. Upscaling of hydraulic conductivity curve Kd.and moisture retention curve theta(Psi) is performed for perfectly stratified and imperfectly stratified soils, and anisotropy is also studied for both types of layering considered. It is demonstrated that the upscaled nonlinear curves can be represented by simple exponential relations and/ or by linear relations obtained by piecewise linearization. The linear relations obtained can be expressed in equivalent terms that can be used in analytical solutions to directly obtain effective hydraulic conductivity values. (C) 2016 American Society of Civil Engineers.

Failure analysis of an infinite unsaturated soil slope

Increases in rainfall and groundwater level rising can cause more frequent failures of unsaturated soil slopes, resulting in large-scale landslides. The aim of this paper is to propose a methodology for prediction of the failure of an infinite soil slope subject to steady unsaturated flow conditions. Probabilistic models for soil porosity, friction angle, matric suction and saturation are developed. The probability of slope failure for different groundwater levels and infiltration (or evaporation) intensities is investigated. Numerical results show that, for a slope with deep groundwater, the slope failure is mainly controlled by the rainfall infiltration, whereas for shallow groundwater, the slope failure is governed by the location of the groundwater level. A merit of the proposed methodology is that the fluctuation of the soil porosity along the depth of a slope is modelled as a Gaussian random field. The paper concludes that the proposed methodology can be used for prediction of slope failures under steady unsaturated flow conditions. Accurate prediction of slope failures can prevent catastrophic consequences of landslides.

Role of suction stress on service state behavior of geosynthetic-reinforced soil structures

Abstract Several design guidelines adopt limit state methods (e.g., earth pressure or limit equilibrium slope stability methods) to estimate the reinforcement loads for geosynthetic-reinforced soil structures (GRSSs). Field measurements usually reveal lower tensile loads in the reinforcements than that predicted by conventional design methods. Consequently, limit state methods have been criticized for being conservative or inaccurate. However, these lower-than-expected loads are primarily due to redundancy in design, attributed to several factors such as toe resistance, soil volumetric dilation, underestimation of soil shear strength, and the effect of suction stress. While these factors commonly contribute to the performance of the GRSSs, they are not accounted for in design procedures. This disregard is accredited to complexities and uncertainties associated with reliable quantification of these factors during the life span of the structure. By properly quantifying the role of suction stress, this study aims to quantitatively explain a part of the discrepancy which exists between the tensile loads in the reinforcement, which are estimated by working-condition design methods versus those by limit state design methods. A suction stress-based formulation is presented for calculating the active earth pressures coefficient ( K a ) of unsaturated backfill under unsaturated steady flow conditions. The formulation is derived by implementing an analytical solution of one-dimensional steady flow into a limit equilibrium-based effective stress analysis. The proposed formulation is used in conjunction with the earth pressure method to illustrate the effect of suction stress on predicted reinforcement loads. Two backfills, referred to as marginal and high quality backfill, along with three representative annual rain events are examined. The results are compared with two classical earth pressure methods as well as two empirical methods. The results show that the suction stress contribution can lead to significantly lower reinforcement loads than that predicted by classic earth pressure methods for backfill types. Empirical methods lead to lower loads in some cases and higher predicted loads in others. The proposed formulation provides a mechanics-based approach to explain a part of the discrepancy which exists between the measured tensile loads in the reinforcement in the field versus those by limit state design methods.

Active Earth Pressures for Unsaturated Retaining Structures

AbstractAlthough commonly ignored in classical earth pressure methods, precipitation-induced failures in retaining structures highlight a need for explicit consideration of negative pore pressures or matric suction. Changes in the degree of saturation in the backfill can significantly affect active earth pressures. This paper presents an analytical framework for calculating the thrust of active earth pressures under unsaturated steady flow conditions. The proposed method addresses limitations associated with the collapse mechanism and effective stress representation in alternative analytical methods. The formulation was derived by implementing an analytical solution for one-dimensional (1D) steady flow into a limit equilibrium–based effective stress analysis. A closed-form solution for the thrust was obtained by employing a log spiral surface, considering the effect of tension cracks, backslope inclination, and batter for c′-ϕ′ soils with variable backfill saturation. A unified effective stress representa...

A Note on the Apparent Conductivity of Stratified Porous Media in Unsaturated Steady Flow Above a Water Table

We aim at deriving the apparent unsaturated conductivity (AUC) K (ap), defined as the ratio between the mean flux and the mean head gradient in a stratified vadose zone above the water table. This is achieved for steady flow generated by a constant infiltrating flux applied at the soil surface. By adopting the first-order approximation in the two parameters of the conductivity curve, and under a few additional simplifying assumptions, we were able to analytically compute K (ap). It is shown that this latter varies between K (ap) ≡ K H (the harmonic mean) at the water table, and K (ap) ≡ K ef (the effective conductivity in gravitational flow) far above the water table. Profiles of the AUC are illustrated, and the impact of parameters values is discussed.

Capability of the Kd model to predict radionuclides behaviour and transport in unsaturated columns under steady flow conditions

In the frame of the IRSN Chernobyl Pilot Site Project, a waste trench containing hot particles was instrumented to study the fate of radioactive pollutants. Results have shown the formation of a 90 Sr plume in the downstream aquifer, mainly present in the Aeolian layer after 15 years. Laboratories experiments were performed to analyse the transport of 90 Sr and 137 Cs in the Aeolian sand in both saturated and unsaturated conditions. Batch experiments were performed to study the chemical equilibrium state of the soil/solution system under static conditions. Stirred flow-through reactor experiments were performed to study the kinetics and reversibility of sorption reactions at the surface of solid particles under dynamic conditions. Column experiments in the structured porous media were also performed in saturated and unsaturated flow conditions. Experimental data pointed out a non-linear, instantaneous and reversible sorption process of 90 Sr. 137 Cs exhibits a non-linear behaviour with a partial irreversible sorption process. A suitable cation-exchange model was used to describe the solute/soil reactions. The former model was coupled with transport models to describe behaviour of 90 Sr and 137 Cs in both saturated and unsaturated steady flow conditions. Main transport parameters were identified and discussed. An effective simple Kd type approach to describe radionuclides transport properties under unsaturated conditions was evaluated.

Two‐dimensional steady unsaturated flow through embedded elliptical layers

New analytic element solutions are presented for unsaturated, two‐dimensional steady flow in vertical planes that include nonoverlapping impermeable elliptical layers and elliptical inhomogeneities. The hydraulic conductivity, which is represented by an exponential function of the pressure head, differs between the inside and outside of an elliptical inhomogeneity; both the saturated hydraulic conductivity and water retention parameters are allowed to differ between the inside and outside. The Richards equation is transformed, through the Kirchhoff transformation and a second standard transformation, into the modified Helmholtz equation. Analytic element solutions are obtained through separation of variables in elliptical coordinates. The resulting equations for the Kirchhoff potential consist of infinite sums of products of exponentials and modified Mathieu functions. In practical applications the series are truncated but still fulfill the differential equation exactly; boundary conditions are met approximately but up to machine accuracy, provided that enough terms are used. The pressure head, saturation, and flow may be computed analytically at any point in the vadose zone. Examples are given of the shadowing effect of an impermeable elliptical layer in a uniform flow field and funnel‐type flow between two elliptical inhomogeneities. The presented solutions may be applied to study transport processes in vadose zones containing many impermeable elliptical layers or elliptical inhomogeneities.

Colloid Mobilization and Transport in Undisturbed Soil Columns. I. Pore Structure Characterization and Tritium Transport

While it is recognized that preferential flow may increase the transport of colloids, less is known about the actual influence of preferential flow on colloid mobilization in situ. Changes in pore structure upon soil exposure to drying and rewetting may additionally affect colloid mobilization. Information about the pore structure and the active flow volume, as well as the changes in these properties, are therefore important when investigating colloid mobilization. We investigate the pore structure characteristics and the transport of tritium (3H2O) during steady unsaturated flow conditions. A total of 54 soil columns sampled along a natural clay gradient representing six clay contents (12, 18, 24, 28, 37, and 43% clay) were equilibrated to three different initial matric potentials (IMP), ψ = −2.5, −100, and −15500 hPa. Pore structure characteristics were deduced from water retention characteristics and measurements of air‐filled porosity and air permeability. Tracer experiments were conducted at 1 mm h−1 and with a suction of 5 hPa. A mobile–immobile region model (MIM) and a three‐region model (2MIM) with two mobile and one immobile region were used for describing the breakthrough curves (BTCs). The 2MIM model was able to fit the data well and predicted the existence of two mobile flow regions, most pronounced at higher clay content. The 12% clay soil exhibited matrix‐dominated flow behavior, which is probably attributable to a large fraction of drained pores disconnecting the rapidly conducting flow system. Soils with ≥18% clay exhibited asymmetrical BTCs with early breakthrough and tailing and an increasing amount of immobile water, indicating preferential flow. Drying and rewetting, because of associated changes in the pore structure, significantly reduced the degree of preferential flow.

Colloid Mobilization and Transport in Undisturbed Soil Columns. I. Pore Structure Characterization and Tritium Transport

While it is recognized that preferential flow may increase the transport of colloids, less is known about the actual influence of preferential flow on colloid mobilization in situ. Changes in pore structure upon soil exposure to drying and rewetting may additionally affect colloid mobilization. Information about the pore structure and the active flow volume, as well as the changes in these properties, are therefore important when investigating colloid mobilization. We investigate the pore structure characteristics and the transport of tritium (3H2O) during steady unsaturated flow conditions. A total of 54 soil columns sampled along a natural clay gradient representing six clay contents (12, 18, 24, 28, 37, and 43% clay) were equilibrated to three different initial matric potentials (IMP), ψ = −2.5, −100, and −15500 hPa. Pore structure characteristics were deduced from water retention characteristics and measurements of air-filled porosity and air permeability. Tracer experiments were conducted at 1 mm h−1 and with a suction of 5 hPa. A mobile–immobile region model (MIM) and a three-region model (2MIM) with two mobile and one immobile region were used for describing the breakthrough curves (BTCs). The 2MIM model was able to fit the data well and predicted the existence of two mobile flow regions, most pronounced at higher clay content. The 12% clay soil exhibited matrix-dominated flow behavior, which is probably attributable to a large fraction of drained pores disconnecting the rapidly conducting flow system. Soils with ≥18% clay exhibited asymmetrical BTCs with early breakthrough and tailing and an increasing amount of immobile water, indicating preferential flow. Drying and rewetting, because of associated changes in the pore structure, significantly reduced the degree of preferential flow.

Unsaturated quasi-linear flow analysis in V-shaped domains

An explicit, essentially 2D solution for steady unsaturated seepage flow from infinity to a corner, with boundaries kept at constant suction, is obtained for the Kirchhoff potential by the method of separation of variables. A system of coordinates coinciding with the corner boundaries is selected. Distributions of the pressure, stream lines and velocities are derived. Non-existence of steady flows at certain corner orientations, deflection of the incident flow by slanted boundaries and inflection points on the stream lines close to the vertex are discussed. One-dimensional limit for zones far from the trough is examined. Refraction and further collimation of the upper 2D flow in the second underlying porous medium with implications to geotechnical capillary barriers is studied. A vadose zone originated accretion is matched with a saturated ‘wing’, which appears on a slanted bedrock. Mathematically, this matching is done by linking the Dupuit–Forchheimer and quasi-linear models. A tilted water table is shown to entrain the recharging moisture with curvilinear stream lines in the unsaturated zone.

Quasi‐linear steady infiltration from a sloping land surface

The present work derives a general analytical solution to the problem of steady unsaturated flow in hillslopes. The geometry of the domain consists of either a semi‐infinite medium or a finite area contained between two parallel sloping planes with a prescribed flux boundary condition at the top and at the bottom. The top boundary condition models rainfall infiltration while the bottom boundary condition models either an impermeable or a semipermeable underlying layer if the leakage flux is known a priori. Particular solutions for point, line, strip, and rectangle source configurations at the sloping land surface are presented. The numerical results for the strip and rectangle source solution show the effect of the slope, impermeable base, and soil type on the distribution of the pressure and flux in the domain. In particular, the three‐dimensional rectangular source solution displays the interaction between the slope, soil characteristics, and the point of maximum potential at which the saturation of the soil begins.

Removal of E. coli during intermittent filtration of wastewater effluent as affected by dosing rate and media type

Wastewater effluent dosing rates of 25 and 50 mm/day were intermittently applied in eight daily doses of 3.125 or 6.25 mm each, to 15-cm diameter 80 cm high columns packed with two types of Light Weight Aggregates (LWA) and one type of activated carbon aggregates. After three months of wastewater effluent application at 25 mm/day to stabilize the filter systems, Escherichia coli was spiked once each day onto the surface of the columns and wastewater effluent was applied at 25 mm/day for the months. The same procedure was repeated for effluent application rate of 50 mm/day. During operation, hydraulic behavior was monitored by moisture tensiometers located 5, 10, 20 and 40 cm below the filter surface as well as by radiotracer studies. Removal behavior was assessed by sampling and analysis of the column percolate and media within the column. The removal of E. coli was decreased as a result of increasing the dosing rate for all three media. In all media, the highest removal rates were observed in the upper part of the columns. Sorption head measurements showed that each effluent dose rapidly penetrates through the upper part of the filters, until a steady state, unsaturated flow was established in the lower sections. Different flow patterns were observed for the two dosing rates. For the dosing rate of 50 mm/day, the flow was penetrating faster, and to a deeper level before establishing steady unsaturated flow. Fast flow through the upper part of the filter, where the bacterial removal is most effective, may explain the significantly lower removal for the dosing rate of 50 mm/day. The dynamic behavior of the filter columns showed that most of the water movement took place right after dose application, during intermittent dosing. This indicates that dose size may be just as important for bacterial removal as the daily dosing rate.