Experimental and numerical investigation of droplet flow mechanisms at fracture intersections

Understanding unsaturated flow behaviors in fractured rocks is essential for various applications.A fundamental process in this regard is flow splitting at fracture intersections.However,the impact of geometrical properties of fracture intersections on flow splitting is still unclear.This work investigates the combined influence of geometry(intersection angle,fracture apertures,and inclination angle),liquid droplet length,inertia,and dynamic wetting properties on liquid splitting dynamics at fracture intersections.A theoretical model of liquid splitting is developed,considering the factors mentioned above,and numerically solved to predict the flow splitting behavior.The model is validated against carefullycontrolled visualized experiments.Our results reveal two distinct splitting behaviors,separated by a critical droplet length.These behaviors shift from a monotonic to a non-monotonic trend with decreasing inclination angle.A comprehensive analysis further clarifies the impacts of the key factors on the splitting ratio,which is defined as the percentage of liquid volume entering the branch fracture.The splitting ratio decreases with increasing inclination angle,indicating a decrease in the gravitational effect on the branch fracture,which is directly proportional to the intersection angle.A non-monotonic relationship exists between the splitting ratio and the aperture ratio of the branch fracture to the main fracture.The results show that as the intersection angle decreases,the splitting ratio increases.Additionally,the influence of dynamic contact angles decreases with increasing intersection angle.These findings enhance our understanding of the impact of geometry on flow dynamics at fracture intersections.The proposed model provides a foundation for simulating and predicting unsaturated flow in complex fractured networks.

The capillary pressure curves from upscaling interfacial and unsaturated flows in porous layers with vertical heterogeneity

We provide the capillary pressure curves p_(c)(s)as a function of the effective saturation s based on the theoretical framework of upscaling unsaturated flows in vertically heterogeneous porous layers proposed recently(Z.Zheng,Journal of Fluid Mechanics,950,A17,2022).Based on the assumption of vertical gravitational-capillary equilibrium,the saturation distribution and profile shape of the invading fluid can be obtained by solving a nonlinear integral-differential equation.The capillary pressure curves p_(c)(s)can then be constructed by systematically varying the injection rate.Together with the relative permeability curves k_(rn)(s)that are already obtained.One can now provide quick estimates on the overall behaviours of interfacial and unsaturated flows in vertically-heterogeneous porous layers.

Modeling unsaturated flow in fractured rocks with scaling relationships between hydraulic parameters

Modeling unsaturated flow in fractured rocks is essential in various subsurface engineering applications,but it remains a great challenge due to the difficulties in determining the unsaturated hydraulic properties of rocks that contain various scales of fractures.It is generally accepted that the van Genuchten(VG)model can be applied to fractured rocks,provided that the hydraulic parameters could be representatively determined.In this study,scaling relationships between the VG parameters(a and n)and hydraulic conductivity(K)across 8 orders of magnitude,from 10^(-10)m/s to 10^(-2)m/s,were proposed by statistical analysis of data obtained from 1416 soil samples.The correlations were then generalized to predict the upper bounds of VG parameters for fractured rocks from the K data that could be obtained more easily under field conditions,and were validated against a limited set of data from cores,fractures and fractured rocks available in the literature.The upper bound estimates significantly narrow the ranges of VG parameters,and the representative values of a and n for fractured rocks at the field scale can then be determined with confidence by inverse modeling using groundwater observations in saturated zones.The proposed methodology was applied to saturated-unsaturated flow modeling in the right-bank slope at the Baihetan dam site with a continuum approach,showing that most of the flow behaviors in fractured rocks in this complex hydrogeological condition could be properly reproduced.The proposed method overcomes difficulties in suction measurement in fractured rocks with strong heterogeneity,and provides a feasible way for modeling of saturated-unsaturated flow in fractured rocks with acceptable engineering accuracy.

Method of coupling 1-D unsaturated flow with 3-D saturated flow on large scale

A coupled unsaturated-saturated water flow numerical model was developed. The water flow in the unsaturated zone is considered the one-dimensional vertical flow, which changes in the horizontal direction according to the groundwater table and the atmospheric boundary conditions. The groundwater flow is treated as the three-dimensional water flow. The recharge flux to groundwater from soil water is considered the bottom flux for the numerical simulation in the unsaturated zone, and the upper flux for the groundwater simulation. It connects and unites the two separated water flow systems. The soil water equation is solved based on the assumed groundwater table.and the subsequent predicted recharge flux. Then, the groundwater equation is solved with the predicted recharge flux as the upper boundary condition. Iteration continues until the discrepancy between the assumed and calculated groundwater nodal heads have a certain accuracy. Illustrative examples with different water flow scenarios regar.ding the Dirichlet boundary condition, the Neumann boundary condition, the a.tmospheric boundary condition, and the source or sink term were calculated by the coupled model. The results are compared with those of other models, including Hydrus-lD, SWMS-2D, and FEFLOW, which demonstrate that the coupled model is effective and accurate and can significantly reduce the computational time for the large number of nodes in saturated-unsaturated water flow simulation.

Unsaturated flow and solute transport in a porous column using spherical ore particles

This paper dealt with the development of a two-dimensional （2D） mathematical model for column leaching and confirmed the important simulation parameters through experiment. The unsaturated state of the variably saturated flow column and the solute transport of copper ions were studied during leaching. The fluid flow problem was handled using the Richards equation on the premise of an ambient pressure column air, where the van Genuchten formulas were applied to define the nonlinear relationships of pressure head with the retention and permeability properties. The ore column permeability test gave a varied hydraulic conductivity, which was analyzed in the model. In the solute transport problem, the copper ion concentration was solved using the advection-diffusion-reaction equation whose reaction term was determined by the joint analysis of experimental copper leaching rate and the shrinking core model. Particle-and column-scale leaching tests were carried out to illustrate the difference and connection of copper extraction in both processes. This fluid flow and solute transport cou-pled model was determined through the finite element method using the numerical simulation software, COMSOL Multiphysics.

An Unsaturated Soil Water Flow Problemand Its Numerical Simulation

A numerical model for the unsaturated flow equation with moisture content as prognostic variable is established in order to simulate liquid moisture flow in an unsaturated zone with homogeneous soil, and different initial and boundary conditions. For an infiltration or evaporation problem, its numerical solution by using a finite difference method is very sensitive to its upper boundary condition and the related soil parameters, and using a traditional finite element method usually yields oscillatory non-physics profiles. However, we obtain a nonoscillatory solution and evade a non-physics solution for the problem by using the mass-lumped finite element method. This kind of boundary conditions is handled very well. Numerical simulations for certain soils show that the numerical scheme can be used in simulation of liquid moisture flow for infiltration, evaporation, re-distribution and their alternate appearances. It can be also applied to a high-resolution land surface model.

Unsaturated Water Transport of Cement-based Materials

The slice-weighing method was used to investigate the unsaturated water transport of different cement pastes. The experimental results show that a sharp wetting front existed during water transport, the transport can be described by a non-linear diffusion equation, and transport coefficient of different materials exhibit various rules with water content of materials. The addition of fly-ash decreases transport coefficient of cement pastes in all the various water contents, even changes the transport mechanism.

A thermo-hydro-mechano-chemical formulation for modeling water transport around a ventilated tunnel in an argillaceous rock

Argillaceous rocks are being considered as potential host rocks for deep geological disposal. For the research work in DECOVALEX-2011, 5 participant research teams performed simulations of a labora- tory drying test and a ventilation experiment for Mont Terri underground laboratory built in argillaceous rock formation. Our study starts with establishing a coupled thermo-hydro-mechano-chemical （THMC） processes model to simulate water transport in rock around the ventilated tunnel. Especially in this THMC formulation, a three-phase and two-constituent hydraulic model is introduced to simulate the processes which occur during tunnel ventilation, including desaturation/resaturation in the rock, phase change and air/rock interface, and to explore the Opalinus clay parameter set, It can be found that water content evolution is very sensitive to intrinsic permeability, relative permeability and capillary pressure in clay rock. Water loss from surrounding rock is sensitive to the change of permeability in clay which is induced by excavation damaged zone. Chemical solute transport in the rock near ventilation experiment tunnel is simulated based on the coupled THMC formulation. It can be estimated that chemical osmotic flow has little significance on water flow modeling. Comparisons between simulation results from 5 teams and experimental observations show good agreement. It increases the confidence in modeling and indi- cates that it is a good start for fully THMC understanding of the moisture transportation and mechanical behavior in argillaceous rock.

ALMOST ANALYTIC SOLUTIONS AND THEIR TESTS OF THE HORIZONTAL DIFFUSION EQUATION FOR THE MOVEMENT OF WATER IN UNSATURATED SOIL

This paper discusses the analytic solutions of the diffusion equation for the movement of water in unsaturated soil. Firstly, according to the experience relation between the diffusivity 'D(theta)' and water content in soil 'theta', and through variable substitution, the original diffusion equation is converted into the form easy to solve. Then, the variables separation method together with the Boltzmann transform method is used to find out the solution of the new partial differential equation. so the analytic function of 0(x, t) is obtained, which is verified by the experiment of water flow through unsaturated soil.

A relative permeability model for deformable soils and its impact on coupled unsaturated flow and elasto-plastic deformation processes.

Relative permeability is an indispensable property for characterizing the unsaturated flow and induced deformation in soils. The widely used Mualem model is inadequate for deformable soils because of its assumption of a rigid pore structure and the resultant unique dependence of the tortuosity factor on the volumetric water content. In this study, a unified relationship between the relative permeability and the effective degree of saturation was proposed for deformable soils by incorporating our newly developed water retention curve model into the original Mualem model, in which a new tortuosity factor was defined using the fractal dimension of flow paths and the mean radius of water-filled pores for representing the effect of pore structure variation. The modified deformation-dependent relative permeability model was verified using test data on five types of soils; the verification revealed a much better performance of the proposed model than the original model, which commonly overestimates the relative permeability of deformable soils. Finally, the proposed model was implemented in a coupled numerical model for examining the unsaturated flow and elastoplastic deformation processes in a soil slope induced by rain infiltration. The numerical results showed that the deformation-dependent nature of relative permeability has a remarkable effect on the elastoplastic deformation in the slope and that neglect of the deformation-dependent behavior of relative permeability causes overestimation of the depth of failure.

EFFECT OF A VERTICAL FRACTURE ON HORIZONTAL UNSAT-URATED SEEPAGE FLOW

The movement of water flow in unsaturated fractured rock was investigated with the sandstone sample through experimental research and numerical simulation.The experimental results show that the arriving time of wetting front is delayed by the fracture,resulting in the increase of water saturation in the domain on the upstream side of the fracture,which will locally enhance water flow through the matrix.The numerical simulation with the finite volume method captures effectively these characteristics.The comparison between simulated and observed travel time and arriving time of wetting front shows that their difference are very small and the simulated results are in agreement with the observed results,which implies calibrated parameters are reliable and effective.Then according to the calibrated parameters,fractured models were established to examine how the change of large fractured aperture would affect the arriving time of wetting fronts,pressure heads and water saturation on the upstream and downstream sides of fracture.

Inverse modeling approach for determining soil hydraulic properties as affected by application of cattle manure

Numerical codes are extensively used in the modeling of water and solute transport in the vadose zone.The application of these codes depends on knowledge of soil hydraulic properties such as soil water retention curve and hydraulic conductivity.Application of cattle manure to the soil can increase soil organic matter(SOM)contents.Increases in SOM associated with changes in the structure and adsorption properties of soil and,thus,their hydraulic properties.In this study the effect of cattle manure on soil hydraulic properties was investigated using inverse method.Applied inverse method was based on Levenberg-Marquart optimization algorithm to estimate hydraulic properties of soil in transient condition using C++programming language along with forward model(HydroGeoSphere)as a numerical code.Nine iron cylinders of 57 cm in inner diameter and about 40 cm in height were filled with Sandy clay loam soil of 30 cm in height.Cattle manure applied at 0,30,and 60 Mg/ha at three replications in a completely random design.One year after cattle manure application,saturated hydraulic conductivity,porosity,and water retention curve parameters(van Genuchten function,αandβ)were estimated using inverse method.Statistical analysis showed that the automatic calibration is sensitive toαmore than the other parameters.The results showed that porosity,saturated hydraulic conductivity,residual water content,αandβincreased significantly(P<0.05)with application 30 and 60 Mg/ha cattle manure.But there was no significant difference(P<0.05)inβbetween application of 30 and 60 Mg/ha cattle manure.The study also indicated thatαwas 25.0%and 50.0%higher andβwas 9.6%and 12.6%lower than control treatment in 30 and 60 Mg/ha treatments.In addition,application cattle manure showed positive effect on hydraulic parameters of soil.

Closed-form Solution for a Moving Boundary Problem

This paper considers a moving boundary problem with Neumann boundary conditions and a general initial value, which occurs in an unsaturated flow with extraction. The closed form solution for the moving boundary problem is obtained using a Laplace transform boost. This method has been successfully applied to solve moving boundary problems with Dirichelet boundary conditions, but not to the case with Neumann boundary conditions.