A Subgrid-Scale Stabilized Finite Element Method for Multicomponent Reactive Transport through Porous Media

Standard Galerkin finite element methods (GFEM) lack stability in solving advection-dominated solute transport in porous media. They usually require prohibitively fine grids and extremely small time steps to solve for advection-dominated problems. The algebraic subgrid-scale stabilized (ASGS) finite element method has been proved to overcome such problems for single-species reactive transport. Its potential for dealing with multicomponent reactive transport has not yet been explored. Here we present a numerical formulation of ASGS for steady and transient multicomponent reactive transport. Subgrid-scale transport equations are solved first by using an ASGS approximation and their solutions are substituted back into the grid-scale equations. A sequential iteration approach (SIA) is used to solve for coupled transport and chemical equations. Coupling of ASGS and SIA, ASGS+SIA, has been implemented in a reactive transport code, CORE^2D V4, and verified for conservative solute transport. ASGS+SIA has been tested for a wide range of 1-D transient multicomponent reactive transport problems involving different types of chemical reactions such as: (1) Kinetically controlled aqueous species degradation, (2) Kinetic mineral dissolution, (3) Serial-parallel decay networks, and (4) Cation exchange and pyrite oxidation at local equilibrium. ASGS+SIA always provides accurate solutions and therefore offers an efficient option to solve for advection-dominated multicomponent reactive transport problems.     

Conservative monotone streamline upwind formulation using simplex elements

A streamline upwind formulation is presented for the treatment of the advection terms in the general transport equation. The formulation is monotone and conservative and is based on the discontinuous nature of the advection mechanism. The results of there benchmark test cases for the full range of flow Peclet numbers are presented. The new formulation is shown to accurately model the advection phenomenon with significantly smaller numerical diffusion than the existing methods. The results are also free of all spatial oscillations. Considerable savings in computer storage and execution time have been achieved by employing the three-noded triangular element for which exact integrations exist. The formulation is straightforward and can be readily incorporated into any finite element code using the conventional Galerkin approach.     

Effects of Clay Dispersion on Aquifer Storage and Recovery in Coastal Aquifers

Cyclic injection, storage, and withdrawal of freshwater in brackish aquifers is a form of aquifer storage and recovery (ASR) that can beneficially supplement water supplies in coastal areas. A 1970s field experiment in Norfolk, Virginia, showed that clay dispersion in the unconsolidated sedimentary aquifer occurred because of cation exchange on clay minerals as freshwater displaced brackish formation water. Migration of interstitial clay particles clogged pores, reduced permeability, and decreased recovery efficiency, but a calcium preflush was found to reduce clay dispersion and lead to a higher recovery efficiency. Column experiments were performed in this study to quantify the relations between permeability changes and clay mineralogy, clay content, and initial water salinity. The results of these experiments indicate that dispersion of montmorillonite clay is a primary contributor to formation damage. The reduction in permeability by clay dispersion may be expressed as a linear function of chloride content. Incorporating these simple functions into a radial, cross-sectional, variable-density, ground-water flow and transport model yielded a satisfactory simulation of the Norfolk field test – and represented an improvement over the model that ignored changes in permeability. This type of model offers a useful planning and design tool for ASR operations in coastal clastic aquifer systems.     

Simulation of Cement/Clay Interactions: Feedback on the Increasing Complexity of Modelling Strategies

During the last decade, numerous studies have focused on long-term predictive reactive transport modelling of cement/clay interactions. These simulations have been performed using modelling strategies of growing complexity, e.g. (i) taking more minerals into account, (ii) considering the effect of dissolution/precipitation kinetics versus thermodynamic equilibrium, (iii) refining the spatial discretisation of the models, etc. The present study reviews these simulations in order to identify the main factors affecting numerical results (e.g. mass transport, mesh, selected phases). Simulations are reproduced here with a consistent set of data and input parameters arranged with increasing order of complexity. Only such a standardised approach can allow a proper comparison of numerical results. Modelled reaction pathways (i.e. mineralogical transformations) appear to be independent from the chosen modelling assumptions. Irrespective of the simulated case or the underlying hypotheses, the geochemical transformations remain located very close to the cement/clay interface.     

Modelling Low-Salinity Oil Recovery Mechanisms Using an Ion Dissociation Model

Experimental evidence shows that injecting low-salinity water during the oil recovery process can lead to an increase in the amount of oil recovered. Numerous mechanisms have been proposed to explain this effect, and, in recent years, two which have gained notable support are multicomponent ionic exchange (MIE) and pH increase. Both mechanisms involve ion exchange reactions within the thin film of water separating the oil in a reservoir from the clay minerals on the surface of the reservoir rock. Since the reactions occur on the molecular scale, an upscaled model is required in order to accurately determine the dominant mechanism using centimetre-scale experiments. In this paper, we develop the first stages of this upscaling process by modelling the pore-scale motion of an oil slug through a clay pore throat. We use a law-of-mass-action approach to model the exchange reactions occurring on the oil–water and clay–water interfaces in order to derive expressions for the surface charges as functions of the salinity. By balancing the disjoining pressure in the water film with the capillary pressure across the oil–water interface, we derive an expression for the salinity-dependent film thickness. We compare the two mechanisms by modifying an existing model for the velocity of an oil slug through a pore throat. Numerical results show that the velocity increases as the salinity decreases. The percentage increase is larger for the MIE mechanism, suggesting that MIE may be the dominant causal mechanism; however, this will vary depending on the particular clay and oil being studied.

     

Experiments on transport of hydrophobic particles and gas bubbles in porous media

Adhesion of hydrophobic colloids (clay minerals) on the surface of bubbles of air and the transport of the composite units formed by bubbles and mineral particles were observed in a glass micro model.When a clay mineral suspension flowed in a porous medium that contained bubbles of air trapped in small pores, particles accumulated preferentially on the upstream portion of the bubbles, and quasi-stable bubble-mineral particle units were formed. With an increase in the flow velocity, the particles moved along the interface between the bubble and the liquid and accumulated on the downstream portion of the bubbles. A large stress could mobilize the units which, occasionally, accumulated in larger voids.The mechanism suggested is adhesion of the particles on the surface of the bubble due to compression of their diffuse electrical double layer. The adsorbed particles can be moved by shear stresses which act in the region of water molecules between the well-organized layers of water on the surfaces of the bubble and the clay particles. A large enough shear stress causes the bubbles to become more streamlined, allowing them to move in the channel system. If in contact, the common lamina of the bubbles can withdraw and rupture.Bubbles transport from 20 to 50 times more particles than can be transported by average suspension.     

龙游石窟岩面水岩作用研究

在龙游石窟砂岩胶结物的粘土矿物类型及特性研究的基础上,对窟内的积水、洞室顶部的裂隙渗水、地表水及衢江水等水样进行化学分析,结果表明石窟砂岩的矿物组成及构造, 决定了洞室渗水的水质,是造成水质不同的主要原因。对石窟洞室砂岩试样分别浸沉在上述水及蒸馏水中,并定期对浸沉水样进行化学分析,研究了石窟砂岩与水的相互作用机理,结果表明：由大气降水入渗而形成的渗透水对石窟的影响主要是渗透过程中发生的水化、溶脱、氧化等反应,从而导致洞室砂岩的矿物组成及构造产生变化,同时也直接影响了渗透水的水质。因此,粘土矿物与水之间有着相互制约,相互影响的作用,但是岩石中所含粘土矿物特性是使砂岩发生风化的主要原因。     

Deformation and Flow Driven by Osmotic Processes in Porous Materials: Application to Bituminised Waste Materials

This paper presents a coupled Chemo-Hydro-Mechanical (CHM) analysis of the behaviour of leached Bituminized Waste materials (BW). Under geological disposal conditions the main factor that affects the long-term behaviour of this kind of materials is water uptake. First, the long-term behavior of BW in contact with water has been studied. A formulation has been proposed for the analysis of deformation induced by dissolution of salts in porous media in contact with water. The equations include the effect of coupled transport phenomena and the formulation has been included as an extension in the coupled THM program CODE_BRIGHT. The impact of osmotic forces on the swelling of the material has been investigated by simulating water uptake swelling tests under confined conditions. The numerical analysis has proven to be able to furnish a satisfactory representation of the main observed patterns of the behaviour. A sensitivity analysis has also been carried out to examine the effect of various key parameters.     

Application of a level set method for simulation of droplet collisions

A level set technique for interface tracking is presented, both for the continuum surface force formulation and the ghost fluid method approach. A projection method is used to solve incompressible Navier–Stokes equations that are coupled to a transport equation for the level set function, defined as the algebraic distance to interface. Results are presented for head-on droplet collisions in coalescence and reflexive regimes with a 2D axi-symmetric code, and for an off-center droplet collision in a separation regime for a large impact parameter with 3D code. Simulations provided realistic and various droplet collision behaviors and they correspond to experimental observations.     

Theoretical Analysis of Anion Exclusion and Diffusive Transport Through Platy-Clay Soils

Diffusive transport through geosynthetic clay liners and engineered compacted clay landfill liners is the primary mechanism for mass transport from well-engineered modern landfills. For this reason, accurate estimates of diffusion coefficients for clay soils are essential for the design of engineered liner systems. A long-standing theoretical problem is the role of anion exclusion on the estimation of diffusion coefficients for ionic solutes migrating through charged porous media. This paper describes the steady-state solution of a fully coupled set of transport equations modeling ion movement through a permanently charged platy-clay soil. The microscale analysis takes into account the actual diffusion coefficient for each ion species, ion-pairing (as required by electroneutrality of the solution), as well as anion exclusion and cation inclusion ,arising from the permanent charge on clay particles. To render the problem tractable, the theoretical analysis focuses on an extremely small two-dimensional unit cell in an ideal, saturated, two-phase porous medium. The analysis presented here is limited to a particular geometrical example, but this example is sufficiently general for characteristic behaviours of systems of this kind to be identified. Most importantly, new insight is gained into the mechanism of ion migration through a charged platy-clay soil. The numerical results obtained from this study show that the identification of macroscopic transport quantities such as effective diffusion coefficients and membrane potentials from diffusion cell tests using standard diffusion theory only hold for a specific system. While ion exclusion behaviours are often referred to in the literature, as far as the authors are aware there has been no previous detailed microscale analysis of their role in steady-state diffusion through a charged platy-clay soil.     

A finite element formulation for laminar flow of a fluid with microstructure

A finite element formulation for the steady laminar flow of an incompressible fluid with microstructure has been developed. The particular fluids considered are commonly known as micropolar fluids, in which case suspended particulate microstructures are modelled by an ‘extended’ continuum formulation. The particle microspin is a new kinematic variable which is independent of the classical vorticity vector and thereby allows relative rotation between particles and the surrounding fluid. This formulation also gives rise to couple stresses in addition to classical force or traction stresses. The finite element formulation utilizes a variational approach and imposes conservation of mass through a penalty function. A general boundary condition for microspin has been incorporated whereby microspin at a solid boundary is constrained to be proportional to the fluid vorticity. The proportionality constant in this case can vary from zero to unity. Sample solutions are presented for fully developed flow through a straight tube and compared with an analytical solution. Results are also generated for flow through a constricted tube and compared with a Newtonian fluid solution.     

Non-linear finite element analysis of flexible risers in presence of buoyancy force and seabed interaction boundary condition

A non-linear finite element static analysis for flexible risers with large displacements is presented using a four-node pipe elbow element with 24 degrees of freedom. A pipe–soil interaction model is used to identify seabed boundary condition. The effects of buoyancy force as well as steady-state current loading are considered in the finite element solution for riser structures response. In fact considering buoyancy force among with current loading and seabed interaction boundary condition in this paper leads to a particular formulation. The resulting formulation has been implemented in a finite element code which is subsequently used to model and analyze some typical riser configurations. The results of some sample solutions are given to illustrate the accuracy and capability of the formulation.     

Finite element analysis of the piezocone test in cohesive soils using an elastoplastic–viscoplastic model and updated Lagrangian formulation

In this research, the finite element analysis of piezocone penetration has been conducted using the elastoplastic–viscoplastic bounding surface model in the updated Lagrangian reference frame. A finite element formulation has been performed considering the viscoplastic contribution of the model and the theory of mixtures has been incorporated to explain the behavior of the soil. The formulated model has been implemented into a finite element program, EPVPCS-S (elastoplastic–viscoplastic coupled system-soil), to analyze the mechanism of piezocone penetration. The results of the finite element analysis have been compared and investigated with the experimental results from the piezocone penetration and dissipation tests conducted using LSU/CALCHAS (Louisiana State University Calibration Chamber System).     

Macroscopical non-linear material model for ferroelectric materials inside a hybrid finite element formulation

A new approach for modeling hysteretic non-linear ferroelectric ceramics is presented, based on a fully ferroelectric/ferroelastic coupled macroscopic material model. The material behavior is described by a set of yield functions and the history dependence is stored in internal state variables representing the remanent polarization and the remanent strain. For the solution of the electromechanical coupled boundary value problem, a hybrid finite element formulation is used. Inside this formulation the electric displacement is available as nodal quantity (i.e. degree of freedom) which is used instead of the electric field to determine the evolution of remanent polarization. This involves naturally the electromechanical coupling. A highly efficient integration technique of the constitutive equations, defining a system of ordinary differential equations, is obtained by a customized return mapping algorithm. Due to some simplifications of the algorithm, an analytical solution can be calculated. The automatic differentiation technique is used to obtain the consistent tangent operator. Altogether this has been implemented into the finite element code FEAP via a user element. Extensive verification tests are performed in this work to evaluate the behavior of the material model under pure electrical and mechanical as well as coupled and multi-axial loading conditions.     

A Three-Scale Model of pH-Dependent Flows and Ion Transport with Equilibrium Adsorption in Kaolinite Clays: II Effective-Medium Behavior

In part I (Lima et al., Transp Porous Media, 2009), a three-scale model governing the movement of an aqueous saline solution containing four monovalent species (Na⁺, H⁺, Cl^?, OH^?) in kaolinite clays was derived. Unlike purely macroscopic approaches, the novelty of the formulation relied on the double averaging of the nanoscopic electro- chemistry of particle/electrolyte solution interface ruled by the electrical double layer coupled with protonation/deprotonation reactions. The passage from the nano to the micro (pore)-scale gave rise to ion-sorbed concentrations and slip velocity at the solid/fluid interface which are coupled with the microscopic Stokes problem and Nernst–Planck equations governing the hydrodynamics and ion transport in the micropores. Application of a formal homogenization procedure led to macroscopic governing equations with effective electro-chemical parameters, such as retardation coefficients, electro-osmotic permeability, and electric conductivity. In this study, we reconstruct the constitutive laws of the macroscopic coefficients by solving the nano and microscopic closure problems. New generalized isotherms for Na⁺ and H⁺ ? OH^? sorption are build-up based on a perturbation approach and the limitations of classical Freundlich isotherm for modeling ion sorption at the solid/fluid interface are discussed. The macroscopic governing equations are discretized by the finite volume method and numerical simulations of a transient electroosmosis experiment for desalination of a clay sample by electrokinetics are presented.     

A multi‐moment transport model on cubed‐sphere grid

A conservative, single‐cell‐based semi‐Lagrangian transport model is proposed in this paper. Using multi‐moment concept, an additional moment, i.e. volume‐integrated average (VIA), is treated as the model variable besides the point value (PV) updated in the traditional semi‐Lagrangian schemes. A quadratic interpolation function is constructed based on local degrees of freedom defined within each single cell. The PV moment is advanced by the semi‐Lagrangian formulation, whereas the VIA moment is updated by a finite volume formulation to rigorously ensure the numerical conservation. The numerical fluxes are computed from the PV moments defined along the boundary edges of the control volume. The scheme is extended to the spherical geometry through the application of the cubed‐sphere grid that eliminates the polar singularity in the conventional longitude/latitude coordinates by using the quasi‐uniform grid spacing covering the whole sphere. The single‐cell‐based scheme is well suited for the treatment of the connections between different patches. A simple quasi‐monotone limiter to the PV moment is applied to suppress non‐physical oscillations. The proposed scheme has been validated via representative benchmark tests and the performance is competitive to other existing transport schemes. Copyright © 2010 John Wiley & Sons, Ltd.