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Heterogeneity is a natural characteristic of aquifers, which has a profound influence on groundwater flow and solute transport. However, the heterogeneity in an aquifer is difficult to characterize and its impact on groundwater response to Earth tides remains insufficiently explored. A common heterogeneity in thick aquifers is the decrease in porosity and hydraulic conductivity with depth. In the present study, we present an analytical model that includes this particular heterogeneity in both the unsaturated and saturated zones to explore its impact on the tidal response. Our results reveal that, as the hydraulic conductivity decays with depth, the amplitude ratio of the tidal response increases and its phase shift decreases, and the impact of the capillary zone on tidal responses is primarily restricted to the shallow region of the aquifer. Neglecting the decay of conductivity in the previous models would result in underestimation of the amplitude ratio and overestimation of the phase shift. The solutions are applied to the field data to effectively estimate the decay in conductivity with depth. This study highlights the importance of considering aquifer heterogeneity in the analysis of the response of saturated and unsaturated flow to Earth tides. Key Points Decay in conductivity with depth causes an increase in amplitude ratio and a decrease in phase shift Influence of capillary zones on tidal responses is restricted to shallow aquifers due to decrease in conductivity with depth Homogeneous solutions estimate mean conductivity well with smaller decay rates of conductivity, but underestimate with larger decay rates

Solute transport in unsaturated conditions is important in various applications and natural environments, such as groundwater flow in the vadose zone. Studies of unsaturated solute transport show complex characteristics (e.g. non-Fickian transport) due to larger variations in the pore-scale velocities compared to transport in saturated conditions. However, the physical processes at the pore scale are still not completely understood because direct three-dimensional observations at the pore scale are very limited. In this study, single-phase and two-phase solute transport was directly characterized by performing tracer injection experiments in a sintered glass and Bentheimer sandstone sample. These experiments were imaged by continuous scanning with fast laboratory-based micro-computed tomography. The network-scale flow velocities and transport properties were characterized by using the pore-based transient concentration fields to determine the tracer’s arrival time and filling duration in every pore. Important measures for dispersion (the scalar dissipation rate and filling duration) were determined and indicated a wide range in pore-scale velocities and the existence of stagnant and flowing pores for the unsaturated experiments. Furthermore, we performed the first quantification of the mass transfer coefficient between stagnant and flowing pores on three-dimensional experimental data. We also calculated the tortuosity directly from the interstitial velocity and the pore-based velocity. This was found to be 13% higher in unsaturated conditions compared to saturated conditions. Our results indicate that pore-scale structural heterogeneity increases the differences between saturated and unsaturated solute transport. This study thus provides further insight into pore-scale spreading and mixing of dissolved substances in unsaturated porous media.

In this study, an analytical model for the three-dimensional (3D) dynamic stability analysis of vegetation-rooted slopes is first developed under steady-state unsaturated flow conditions. Root reinforcement, defined as the increase in the soil shear strength produced by the mechanical and hydrological effects of vegetation roots, is included in the proposed analytical model. By combining the modified pseudo-dynamic approach (MPDA) and the kinematic theory of limit analysis to the 3D discretized failure model, the most critical failure surface and the corresponding factor of safety ( FS ) are derived to examine the stability of vegetation-rooted slopes with the aid of the optimization algorithm of particle swarm. The proposed approach is verified by comparing with published analytical solutions and numerical results. A series of parametric analysis are then conducted to examine the influence of seismic-related parameters, vegetation properties, possible surcharge and slope geometry parameters on the slope stability. Finally, a comparison between the slope stability under different root architectures is provided and discussed. The results show that, for these selected cases, the stability of vegetation-rooted slopes is significantly improved by approximately 45% compared to bare soil slopes, and the divergences of reinforcement effects between different root architectures can be negligible.

The soil slopes in nature are normally unsaturated, heterogeneous, and even carry cracks. In order to assess the stability of slope with crack under steady unsaturated flow and non-uniform conditions, this work proposes a novel discretization-based method to generate the rotational failure mechanism in the context of the kinematic limit analysis. A point-to-point strategy is used to generate the potential failure surface of the failure mechanism. The failure surface consists of a series of log-spiral segments instead of linear segments employed in previous studies. Two kinds of cracks—open cracks and formation cracks—are considered in the stability analysis. The maximum depth of the vertical crack is modified by considering the effect of the unsaturated properties of soils. According to the work–energy balance equation, the explicit expression about the slope factor safety for different crack types is obtained, which is formulated as a multivariate nonlinear optimization problem optimized by an intelligent optimization algorithm. Numerical results for different unsaturated parameters and non-uniform distribution of soil strength are calculated and presented in the form of graphs for potential use in practical engineering. Then, a sensitivity analysis is conducted to find more insights into the effect of unsaturation and heterogeneity on the crack slopes.

This work uses a mixture theory approach to describe kinematically constrained flows through porous media using an adequate constitutive relation for pressure that preserves the problem hyperbolicity even when the flow becomes saturated. This feature allows using the same mathematical tool for handling unsaturated and saturated flows. The mechanical model can represent the saturated–unsaturated transition and vice-versa. The constitutive relation for pressure is a continuous and differentiable function of saturation: an increasing function with a strictly convex, increasing, and positive first derivative. This significant characteristic permits the fluid to establish a tiny controlled supersaturation of the porous matrix. The associated Riemann problem’s complete solution is addressed in detail, with explicit expressions for the Riemann invariants. Glimm’s semi-analytical scheme advances from a given instant to a subsequent one, employing the associated Riemann problem solution for each two consecutive time steps. The simulations employ a variation in Glimm’s scheme, which uses the mean of four independent sequences for each considered time, ensuring computational solutions with reliable positions of rarefaction and shock waves. The results permit verifying this significant characteristic.

For the sustainable restoration of wet farm land degraded by the climate change-induced rise of ground water level (GWL) and soil salinity etc., the sheet pipe system is one of the most useful technologies which reduces cultivation obstacles due to the poor drainage by controlling the rapid drainage function and enabling farmers to produce profitable crops. This system is characterized mainly as a perforated polyethylene rolled-band sheet 180 mm in width and 1 mm thick which is drawn into the subsurface layer while transforming a drainage pipe with φ = 50 mm. The major advantage of this system is that since the sheet pipe is installed without trenching, the disturbance of land is minimized and the construction period can be shortened to about 1/4 (which reduces the cost approximately by 50%). In this study, by using the sheet pipe installed miniature-type model soil box, the drainage capacity of the sheet pipe was confirmed as being the same as the pipe-shaped standard drainage pipes. Based on the observations of the saturated–unsaturated flow and the maximum lowering rate of GWL was predicted. Finally, at the farm land wherein the free board of the adjoining canal was limited, the effectiveness of the sheet-pipe system was confirmed.

We provide a unified theory, within the framework of the multi-phase Darcy description, on gravity current, interfacial and unsaturated flows in a vertically heterogeneous porous layer, which finds applications in many geophysical, environmental and industrial contexts. Based on the assumption of vertical gravitational-capillary equilibrium, a theoretical model is presented to describe the time evolution of the saturation field and the interface shape, imposing a general formula for the vertical distribution of intrinsic permeability, porosity and capillary entry pressure. Example calculations are then provided in the Cartesian configuration to illustrate potential implications of the theory, imposing power-law distribution of vertical heterogeneity. Seven dimensionless parameters are identified, which arise from the standard Darcy description of multi-phase flow and measure the influence of vertical heterogeneity, viscosity ratio, and the competition between gravitational and capillary forces. Four asymptotic regimes are recognised, representing unconfined unsaturated flows, confined unsaturated flows, unconfined interfacial flows and confined interfacial flows. The influence of heterogeneity is then discussed in the two unsaturated flow regimes based on the evolution of the interface shape, frontal location, saturation distribution, and the time transition between unconfined and confined self-similar flows.

Saturated/unsaturated pore water flow induced by rainwater infiltration in a soil column composed of a mixture of Toyoura sand and a small amount of clay (kaolin minerals) and the rinsing rate (mass transfer) of dissolved NaCl accumulated in the pore system from previous road salt application were investigated by experiments and simulations. Experiments were conducted with variable kaolin minerals mass contents (mixing ratios) in the soil columns. Measured saturated hydraulic conductivity ( K s ) diminished with increased clay contents, i.e., K s =0.00771, 0.00560, 0.00536, 0.00519, and 0.00314 cm s −1 , for clay contents = 0.2, 0.5, 1, 2, and 5%, respectively. Experimental NaCl concentrations in the effluent from the bottom of the soil columns were about constant for times t ≈ 800, 1200, 1300, 1400, and 3400 s from the beginning of a rinsing experiment for the clay contents = 0.2, 0.5, 1, 2, and 5%, respectively. These NaCl concentrations then decreased with time quickly, and finally, approached zero. The presented model can reproduce experimental time variations of NaCl concentration in the effluent from the soil column reliably. Simulated salt mass left in the soil column with time also matches the experimental results for the clay contents = 0.2 and 0.5%. An inconsistency between simulated and experimental salt mass left in the soil columns becomes more significant as the clay content increases. These results suggest that the soil–water retention curve for the pure Toyoura sand can be applied to the soil column composed of kaolin minerals/Toyoura sand mixture when the clay content is small, i.e., less than 1%. Prediction of rinsing process becomes more difficult with increased clay content. However, the time required to remove saline water from the soil column to less than 1% of its initial value simulated by the model agrees closely with experimental results of 1000, 1500, 1700, 2100, and 5400 s, respectively.

Study of the saturated‐unsaturated flow in porous media is of interest in many branches of science and engineering. Among the various numerical simulation methods available, the finite difference method is advantageous because it offers simplicity of discretization. This method has been widely used for simulating saturated‐unsaturated flows. However, the simulation of geometrically complex flow domains requires the use of high‐resolution grids in conventional finite difference models because conventional finite difference discretization assumes an orthogonal coordinate system. This makes a finite difference model computationally less efficient than other numerical models that can treat nonorthogonal grids, such as the finite element model and finite volume model. To overcome this disadvantage, we use a coordinate transformation method and develop a multidimensional finite difference model for simulating saturated‐unsaturated flows; this model can treat nonorthogonal grids. The cross‐derivative terms derived by the coordinate transformation method are evaluated explicitly for ease of coding. Therefore, a 7 point stencil is used for implicit terms in the iterative calculation, as in the case of conventional finite difference models with an orthogonal grid. We assess the performance of the proposed model by carrying out test simulations. We then compare the simulation results with dense grid solutions in order to evaluate the numerical accuracy of the proposed model. To examine the performance of the proposed model, we draw a comparison between the simulation results obtained using the proposed model and the results obtained by using (1) a model in which all terms are considered fully implicitly, (2) a finite element model, and (3) a conventional finite difference model with a high‐resolution orthogonal grid.

The unsaturated flow problem is of important applied background and its mixed finite element (MFE) method can be used to simultaneously calculate both water content and flux in soil, which is the most ideal calculation method. Nonetheless, it includes many unknowns. Thereby, herein we will employ the proper orthogonal decomposition (POD) to lower the dimension of unknown solution coefficient vectors in the MFE method for the unsaturated flow problem. Thus, we first examine the MFE method for the unsaturated flow problem and the existence and convergence of the classical MFE solutions. We then take advantage of the initial L MFE solution coefficient vectors to generate a set of POD basis vectors and utilize the most POD basis vectors to create the preserving precision MFE reduced-dimension (PPMFERD) format. Under the circumstances, the PPMFERD format has the same basis functions as the classical MFE format so that it can maintain the same accuracy as the classical MFE format, but it only includes a few unknowns, so it greatly lightens the calculating load, retards the accumulation of computing errors, saves CPU runtime, and improves the accuracy of the real-time calculation in the computational process. Next, we employ the analysis of matrices to demonstrate the existence and convergence of the PPMFERD solutions such that the theoretical analysis becomes very simple and elegant. Finally, we take advantage of some numerical simulations to check on the correctness of the PPMFERD method. It shows that the PPMFERD method is effective and feasible for simulating both water content and flux in unsaturated flow soil.