Transient Unsaturated Flow Research Articles

A Dimension-Reduced Line-Element Method to Model Unsaturated Seepage Flow in Porous Media

Contrary to the continuum hypothesis, which averages water flow across the entire domain, including both grains and pores, the line-element model concentrates unsaturated flow in the pore space in the intermediate region of horizontal and vertical channels. The flux equivalent principle is used to deduce the equivalent unsaturated hydraulic conductivity, the flow velocity and the continuity equations. It is found that the relative hydraulic conductivities derived from the line-element model and the continuum model are identical. The continuity equations in the two models are also similar, except that the coefficient in the water content term is half that in the line-element model. Thus, the unsaturated flow problem in porous media is transformed into a one-dimensional problem. A dimension-reduced finite line-element method is proposed that includes a complementary algorithm for Signorini’s-type boundary conditions involving the seepage-face boundary and the infiltration boundary. The validity of the proposed model is then proved by good agreement with analytical, experimental and simulated results for one-dimensional infiltration in a vertical soil column, unsaturated flow in a sand flume with drainage tunnels, and transient unsaturated flow water-table recharge in a soil slab, respectively. In general, the proposed method has good computational efficiency, especially for smaller mesh sizes and short time intervals.

Modelling of Hydraulic Dynamics in Sludge Treatment Reed Beds with Moving Boundary Condition

The conventional method of simulation using fixed mesh method (FMM) of discretization is a well-known and trusted procedure in modelling hydraulic dynamics. However, new ideas of innovation in modelling should be advanced. The moving mesh method (MMM) has been considered as a novel approach in modelling hydraulic dynamics after depending on the existing simulation model for decades. The MMM is capable of describing the moving boundary condition of an actual wetland system due to water ponding. An idealized model should be able to simulate the actual hydraulic flows through the system with the corresponding porosity. Hence, a combination of MMM and FMM (MM-FMM) of discretization for hydraulic dynamics is studied in this project to model the flux with respect to water ponding scenario in a sludge treatment reed bed and unsaturated transient flow within the bed. Such method has evidently proved to simulate the actual hydraulic flows in contrast to conventional method. The application of MMM limits the maximum flux to keep within its saturated conductivity, thus reduces the effect of flow overprediction. Subsequently, the simulated results for hydraulic head and moisture content can be predicted for actual condition of different cases according to their respective fluxes.

Foundation bearing capacity estimation on unsaturated soil slope under transient flow condition using slip line method

In this paper, an analytical method is proposed for estimating the bearing capacity of soil within the framework of slip line theory. The method incorporates a closed-form solution taking account the transient unsaturated vertical flow conditions. The proposed framework is validated for both level and sloping ground scenarios considering saturated and unsaturated soil conditions. More specifically, the bearing capacity of a strip foundation located on an unsaturated soil slope under the influence of different rainfall infiltration conditions is evaluated. A comprehensive parametric study is also conducted taking account the influence of the rainfall influx, rainfall duration, slope angle, foundation setback distance, soil type, and the depth of the ground water table. In addition, the contribution of matric suction towards the bearing capacity under transient flow conditions is evaluated. Results of this study provide a rigorous understanding of the performance of foundations on sloping ground extending the mechanics of unsaturated soils.

3D stability analysis of tunnel face with influence of unsaturated transient flow

The analysis of tunnel face stability is commonly implemented with the assumption that the soil is dry or fully saturated. However, many practical cases occur in partially saturated soils, and the unsaturated component of the shear strength warrants consideration for a more realistic solution. This paper extends the current understanding of the tunnel face stability, which, for the first time, incorporates the contributions of unsaturated transient flow into the analytical framework of the three-dimensional (3D) tunnel face stability. A closed-form describing the spatio-temporal variability of saturation, suction stress, and apparent cohesion was derived and introduced into the Mohr-Coulomb criterion to consider the unsaturated transient flow. Within the framework of the kinematic approach of limit analysis, the work rate balance equation including the unsaturated component was established based on a 3D horn-like failure mechanism. A parameter analysis concerning the hypothetical fine sand, silts, and clay was performed to study some parameters of interest. The results show that unsaturated transient flows play a positive role in reducing the required face pressure. The magnitude of the influence is controlled by factors including soil types, thickness of the soil layer, and infiltration time. Compared with neglecting the unsaturated effect, the reduction of normalized critical face pressure caused by transient flows can reach 16.22% in fine sands, 24.98% in silts, and 85.62% in clays. The results suggest that the contribution of unsaturated transient flow to clays, particularly in the early infiltration stage, should be considered. Findings of this work can provide an insight to quantitively capture the influence of the unsaturated transient infiltration and the resulting transient changes of the 3D tunnel face stability.

Stochastic Foundation to Solving Transient Unsaturated Flow Problems Using a Fractional Dispersion Term

Abstract Mathematical modeling of unsaturated water seepage through soils uses the core concepts of continuum mechanics. More precisely, it combines the continuum mass balance equation and Darcy–Bu...

Three-dimensional active earth pressure under transient unsaturated flow conditions

The active earth pressure evaluation was typically conducted assuming dry backfills and/or two-dimensional collapse mechanisms. In practice, retained slope collapse usually shows a three-dimensional (3D) feature and soils are usually unsaturated. The present work develops a framework for evaluating the 3D active earth pressure under transient unsaturated flow conditions. The framework is implemented by using the kinematic approach of limit analysis, in conjunction with a closed-form solution for the transient unsaturated flow. Based on a 3D horn-like failure mechanism, the active earth pressure coefficient is determined. Results for three hypothetical soils of fine sand, silt, and clay are calculated. It emerges that the inclusion of 3D effects and unsaturated transient flow leads to a smaller active earth pressure coefficient. The 3D effects are mainly influenced by the width-to-height ratio of walls; several certain cases analyzed in this paper show that the difference in the active earth pressure coefficient with and without considering 3D effects can be as much as about 27% when the width-to-height ratio equates to 2, whereas this difference decreases to about 10% and 5% when the ratio reaches to 5 and 10. The contribution of transient flow is mainly affected by the soil types; considering the unsaturated flow effect in clay can approximately reduce the active earth pressure of 67%, whereas this difference corresponds to 34% and 22% in silt and fine sand. The obtained results can help to quantitatively understand changes in the active earth pressure due to 3D effects and the presence of transient infiltration.

The use of hyperbolic and Asaoka’s methods to estimate the moisture diffusivity and the hydraulic conductivity of unsaturated soils from tests to determine the SWRC

The equation governing the unsaturated transient flow in a soil sample when subjected to suction (ψ) at its base and that of the classical 1-D consolidation are exactly alike (equation of diffusion). The former can be arrived at by Richards’ equation, being in this case the moisture diffusivity (D), instead of the coefficient of consolidation (cv), the governing parameter. D need not be constant, but rather a function of the volumetric water content, D=D(θ), and is defined as the ratio of the hydraulic conductivity, k(θ), over the specific water capacity, C(θ)=dθ/dψ, i.e., the slope of the SWRC. The hyperbolic method has been used for several geotechnical purposes and, most importantly, as an alternative to Asaoka’s method for predicting the final settlement and cv of soft soils undergoing consolidation, improved by preloading. This paper shows that both methods prove to be very useful as a means of obtaining D(θ) and k(θ) at a certain range of θ, provided that a reduced number of water contents at known elapsed times are determined over the medium stage of this transient flow. It is addressed in the paper both by theoretical grounds and on the light of experimental data of 4 soils.

Modelling the Unsteady Flow of Water into a Partly Saturated Soil

Abstract Flows in unsaturated medium are frequent in the field of civil engineering and more particularly in geotechnics. The study undertaken here tries to solve the unsaturated transient flow equation in porous media using the finite element method. Numerical solution of a finite element discretization is used along with an implicit integration scheme of the time stepping. A functional one that makes it possible to find the distribution of the hydraulic load has been proposed and a calculation program has been developed. The results obtained by this program called TFAP (Transient Flow Analysis Program) are compared to other previous work in the subject. The authors show the importance of this study through two real examples. Liakopoulos conducted several experiments on the water drainage through a vertical column filled with Del Monte sand. One of these experiments was chosen to perform a simulation by the model. The results of the calculation are compared with the experimental data as well.

Earth pressure profiles in unsaturated soils under transient flow

Assessing the effect of heavy precipitations on earthen structures warrants studying earth pressure profiles in unsaturated soils under transient flow. This paper presents an analytical framework to determine the changes in at-rest, active, and passive earth pressures of unsaturated soils due to transient infiltration. A closed-from solution for one-dimensional transient unsaturated flow is incorporated into a suction stress-based representation of effective stress to obtain the tempo-spatial changes in matric suction, suction stress, and effective stress. The profiles are used to extend Hooke's law and Rankine's earth pressure theory, leading to the determination of unsaturated at-rest, active, and passive earth pressures at different depths and times. The analytical framework is used in a set of parametric study for three hypothetical soils of clay, silt, and fine sand. The results reveal the nonlinear characteristics of earth pressure profiles during transient infiltration for all three soils, whereas a linear trend is generally seen under steady-state flow conditions. Further, the depth of tension crack under transient flow is different than the one resulted from steady-state analysis. A transition from tension to compression stress is seen near the soil surface. This observation can be particularly important for fine-grained soils, implying that the depth of tension zone is affected by suction stress changes. Findings of this study can contribute toward a better understanding of service state behavior and forensic studies of earthen structures under heavy precipitations.

New 3D Analytical Solution for Modeling Transient Unsaturated Flow Due to Wetting and Drying

New 3D Analytical Solution for Modeling Transient Unsaturated Flow Due to Wetting and Drying

Transient unsaturated flow in the drainage layer of a highway: solution and drainage performance

In order to quickly remove water entering pavement layers so as to mitigate water-induced damage, drainage layers are widely used in highway pavements. This paper introduces a 1-D mathematical model in order to quantify the transient behaviour of the groundwater flow in the drainage layer of highway pavements. The model takes into account the capillary effect and the nonlinear relationship between hydraulic conductivity and saturation. The analytical solution is obtained and validated against the 2-D Richards equation-based model. Then, sensitivity analysis is carried out and the results show that the drainage time is reduced and drainage efficiency is enhanced by capillarity. The drainage capacity (the total water drained from the drainage layer) depends on the water retention curve (e.g. αG in the Gardner equation). In view of material selection, a relatively low value of αG helps improve the drainage capacity of the drainage layer, whereas a suitable amount of fine particles also leads to a better drainage performance. Furthermore, the drainage material with hydraulic properties is recommended in terms of drainage performance improvement and strength requirement. The analytical solution, which is simple and easy to use, provides guidance for assessing the capillary effects on the effectiveness and efficiency of subsurface drainage systems of a highway.

Efficient Approach to Solving Transient Unsaturated Flow Problems. II: Numerical Solutions

Efficient Approach to Solving Transient Unsaturated Flow Problems. II: Numerical Solutions

Efficient Approach to Solving Transient Unsaturated Flow Problems. I: Analytical Solutions

AbstractAlthough the finite-difference method (FDM) has been commonly used to numerically solve Richard’s equation, numerical difficulties are often encountered, even for comparatively simple probl...

Head-based isogeometric analysis of transient flow in unsaturated soils

An isogeometric analysis (IGA) is introduced to obtain a head-based solution to Richards equation for unsaturated flow in porous media. IGA uses Non-Uniform Rational B-Spline (NURBS) as shape functions, which provide a higher level of inter-element continuity in comparison with Lagrange shape functions. The semi-discrete nonlinear algebraic equations are solved using a combination of implicit backward-Euler time-integration and Newton-Raphson scheme. The time-step size is adaptively controlled based on the rate of changes in the pore pressure. The results from the proposed formulation are compared and verified against an analytical solution for one-dimensional transient unsaturated flow in a homogenous soil column. The proposed method is then applied to four more complex problems including two-dimensional unsaturated flow in a two-layered soil and a semi-circular furrow. The test cases in two-layered soil system involve sharp variations in the pressure gradient at the intersection of the two media, where the pore water pressure abruptly changes. It is shown that the proposed head-based IGA is able to properly simulate changes in pore pressure at the soils interface using fewer degrees of freedom and higher orders of approximation in comparison with the conventional finite element method.

储存-释放型腾发覆盖层的水文地质过程:现场土柱实验与数值模拟结果的滞后效应对比

Non-hysteretic numerical codes are often used to design store-and-release (SR) cover systems. Hysteretic and non-hysteretic predictions were compared using field measurements with instrumented column tests, consisting of fine-grained SR material (phosphate limestone waste) with high hysteretic behavior (hysteretic ratio ≈11) placed over a capillary break layer. Transient unsaturated water flow was then predicted using a one-dimensional code (HYDRUS-1D) under semi-arid climatic conditions over 1 year. Non-hysteretic simulation based on the main wetting curve of the SR material and hysteretic simulation were validated with the measured data. The compared results showed better agreement between measured and predicted values for non-hysteretic simulation during wet periods, whereas the hysteretic scenario showed better agreement with field measurements during dry periods. The influence of hysteresis effects on the complex transient unsaturated water flow of the tested scenarios and conditions is considered minor.

Estimating flow and transport parameters in the unsaturated zone with pore water stable isotopes

Abstract. Determining the soil hydraulic properties is a prerequisite to physically model transient water flow and solute transport in the vadose zone. Estimating these properties by inverse modelling techniques has become more common within the last 2 decades. While these inverse approaches usually fit simulations to hydrometric data, we expanded the methodology by using independent information about the stable isotope composition of the soil pore water depth profile as a single or additional optimization target. To demonstrate the potential and limits of this approach, we compared the results of three inverse modelling strategies where the fitting targets were (a) pore water isotope concentrations, (b) a combination of pore water isotope concentrations and soil moisture time series, and (c) a two-step approach using first soil moisture data to determine water flow parameters and then the pore water stable isotope concentrations to estimate the solute transport parameters. The analyses were conducted at three study sites with different soil properties and vegetation. The transient unsaturated water flow was simulated by solving the Richards equation numerically with the finite-element code of HYDRUS-1D. The transport of deuterium was simulated with the advection-dispersion equation, and a modified version of HYDRUS was used, allowing deuterium loss during evaporation. The Mualem–van Genuchten and the longitudinal dispersivity parameters were determined for two major soil horizons at each site. The results show that approach (a), using only the pore water isotope content, cannot substitute hydrometric information to derive parameter sets that reflect the observed soil moisture dynamics but gives comparable results when the parameter space is constrained by pedotransfer functions. Approaches (b) and (c), using both the isotope profiles and the soil moisture time series, resulted in good simulation results with regard to the Kling–Gupta efficiency and good parameter identifiability. However, approach (b) has the advantage that it considers the isotope data not only for the solute transport parameters but also for water flow and root water uptake, and thus increases parameter realism. Approaches (b) and (c) both outcompeted simulations run with parameters derived from pedotransfer functions, which did not result in an acceptable representation of the soil moisture dynamics and pore water stable isotope composition. Overall, parameters based on this new approach that includes isotope data lead to similar model performances regarding the water balance and soil moisture dynamics and better parameter identifiability than the conventional inverse model approaches limited to hydrometric fitting targets. If only data from isotope profiles in combination with textural information is available, the results are still satisfactory. This method has the additional advantage that it will not only allow us to estimate water balance and response times but also site-specific time variant transit times or solute breakthrough within the soil profile.

Influence of river level fluctuations and climate on riverbank stability

Riverbank collapse is a natural part of the evolution of rivers. An unprecedented period of dry conditions and low flows between 2005 and 2010 led to more than 162 reported riverbank collapse-related incidents along the Lower River Murray, in South Australia (downstream of Lock 1 at Blanchetown to Wellington). On 4 February, 2009 a 60×20m (70,000m3) section of riverbank, near Long Island Marina, Murray Bridge, collapsed into the river, taking with it three unoccupied vehicles and several trees. This paper aims to: (i) model the Long Island Marina riverbank collapse incident in both 2D and 3D; (ii) examine the influence and sensitivity of river level fluctuations and climatic factors on riverbank stability; and (iii) determine the dominant triggers affecting collapse. The analysis has been undertaken using an integration of the limit equilibrium method, transient unsaturated flow modeling and digital elevation model and high resolution aerial images from a Geographic Information System. The paper demonstrates the efficacy of this framework and the accuracy of the predictions. It also reveals that river fluctuation, rather than climatic influences, dominates riverbank collapse in the Lower River Murray.

Analysis of excess water impact on the structural performance of flexible pavements

Excess water in pavement foundations is one of the major factors contributing to pavements deterioration. In recent years, a number of research studies have been carried out to understand the water movement into pavements and to assess its detrimental effects on the mechanical response of the foundations. However, these studies do not quantify the impact of suction reduction or pore pressure buildup in the foundations on the pavement response. In this paper, coupled finite element analyses are carried out to investigate the effect of the excess water in the granular foundations on the structural performance of flexible pavements. The coupled analyses simulate critical features governing the foundations' hydromechanical response including the transient unsaturated flow and porous nonlinear behaviour of the foundations under moving wheel loads. Furthermore, a parametric study that examines the influences of various loading and foundation parameters on the performance of pavements subjected to excessive moisture scenarios is carried out. The numerical analysis results obtained in this paper are qualitatively in line with the empirical observations reported in the literature.

Interplay between resident and infiltrating water: Estimates from transient water flow and solute transport

► Interplay between resident (‘old’) and infiltrating (‘new’) water is investigated. ► Transient unsaturated water flow and solute transport measurements are presented. ► Piston-like displacement of old water is followed by mixing and entrainment. ► The flow regimes are governed by initial water content and soil particle grain size. This study investigates the interplay between resident (‘old’) water and incoming (‘new’) water in homogeneous, partially saturated sand undergoing infiltration, using a series of laboratory column experiments. It was hypothesized that old water pockets, defined both spatially and temporally, may be established during infiltration. This in turn, may affect the flow pattern of the infiltrating new water and/or contravene current geochemical reactions. Sands with three different particle size distributions and three initial water contents were employed. The upper end of each column was irrigated with water containing a conservative tracer, at three different flow rates, while free-drainage conditions were employed at the lower end. Analysis of the resulting 27 infiltration events was based on the Richards equation to describe fluid flow. Subsequently, the mobile-immobile model (MIM) was employed to describe the solute transport; the measurements were reproduced satisfactorily by these models. Results were further analyzed using mass balance considerations. Two regimes were identified: an initial piston-like mechanism that displaces old water, followed by a slow mixing/entrainment of the remaining old water. The relative contributions of these regimes appear to depend on the initial water content and the average grain size. In some cases, up to one-third of the old water fraction remained in the system following five flowthrough pore volumes. Comparison between the measured fractions of old water remaining in the system at the end of each infiltration event (i.e., after five flowthrough pore volumes) and the immobile water content (optimized from the MIM) relative to the initial water content, exhibited a significant linear correlation, with values about threefold higher for the optimized immobile fraction.

Inverse optimization of hydraulic, solute transport, and cation exchange parameters using HP1 and UCODE to simulate cation exchange

Reactive transport modeling is a powerful tool to evaluate systems with complex geochemical relations. However, parameters are not always directly measurable. This study represents one of the first attempts to obtain hydrologic, transport and geochemical parameters from an experimental dataset involving transient unsaturated water flow and solute transport, using an automatic inverse optimization (or calibration) algorithm. The data come from previously published, controlled laboratory experiments on the transport of major cations (Na, K, Mg, Ca) during water absorption into horizontal soil columns that were terminated at different times. Experimental data consisted of the depth profiles of water contents (θ), Cl concentrations, and total aqueous and sorbed concentrations of major cations. The dataset was used to optimize several parameters using the reactive transport model, HP1 and the generic optimization code, UCODE. Although the soil hydraulic and solute transport parameters were also optimized, the study focused mainly on the geochemical parameters because the soil columns were constructed from disturbed soil. The cation exchange capacity and the cation exchange coefficients for two exchange models (Gapon and Rothmund–Kornfeld) were optimized. The results suggest that both calibrated models satisfactorily described the experimental data, although the Rothmund–Kornfeld model fit was slightly better. However, information content and surface response analyses indicated that parameters of the Gapon model are well identifiable, whereas those of the Rothmund–Kornfeld model were strongly correlated. The calibrated geochemical parameters were validated using an independent dataset. In agreement with the identifiability analysis, the Gapon approach was better than the Rothmund–Kornfeld model at calculating the observed concentrations of major cations in the soil solution and on the exchange sites.