Cu (II) removal intensification using Fe<Subscript>3</Subscript>O<Subscript>4</Subscript> nanoparticles under inert gas and magnetic field in a microchannel

In the present study, Cu (II) ions removal from aqueous solution was intensified by exciting magnetic nanoparticles under inert gas, magnetic field and combination of these two mixing methods in a T-type microchannel. The flow patterns and liquid–liquid two-phase mass transfer were studied in three different magnet distances from mixing channel (3, 6 and 10 mm) and also in the presence of different inert gas flow rates (1, 3 and 5 mL/min). Depending on the mixing method and the flow rate of both phases, several distinct flow patterns were observed including slugs, droplet, parallel and dispersed flows. The performances of mixing techniques for mass transfer enhancement based on relative removal efficiency ratio (λ) and mass transfer coefficient ratio (γ) were compared with simple layout (without nanoparticles, magnetic field and inert gas). The results showed that simultaneous using of inert gas and magnetic field can drive the nanoparticles as mixer. Liquid–liquid mass transfer with 27–62% enhancement in E and 235–285% in K _L a compared with plain one was observed.     

Short Communication: A numerical method for diffusive transport with moving boundaries and discontinuous material properties

A method for the numerical simulation of diffusive transport with moving boundaries is developed and tested. The variable domain is mapped onto a fixed region, which introduces a term of convective form to the transformed governing equation. The resulting convection/diffusion equation is solved by a finite-difference method. An ‘Immersed Interface’ Method (IIM) is introduced in order to retain second-order accuracy near discontinuities in material properties, where the solution is not smooth. The method performs well in benchmark calculations against an analytical solution. The IIM scheme is capable of treating a strong discontinuity in the gradient, and it is readily extended to two or three dimensions. The methods are illustrated through a calculation for the temperature profile in a growing continental ice sheet, in which the thermal properties are discontinuous at the rock/ice interface. © 1997 by John Wiley & Sons, Ltd.     

EXPERIMENTAL STUDIES OF MAGMA MIXING

The paper is devoted to experiments on mixing of natural melts of different compositions at 1300-1850° C and 1-12 kbars. Two series of experiments were carried out: one involving gravity-driven convective mixing and one involving diffusive mixing. The results demonstrate the effectiveness of mixing of contrasting magmas in the course of relative motion. Less viscous mafic melt transforms into andesitic much more easily than viscous silicic melt. The latter tends to “dissolve” into the mafic melt. Diffusive runs revealed selective behavior of alkalies and other components due to diffusion. Uphill diffusion of alkalies may cause double-diffusive convection in intercoupled melts. Diffusive interaction of two contrasting melts is explained as a multistage chemical reaction following the principle of acid-base interaction of components in silicate melts.     

Fully coupled generalised hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part II: numerical scheme and numerical results

We consider the modeling and simulation of compositional two-phase flow in a porous medium, where one phase is allowed to vanish or appear. The modeling of Marchand et al. (in review) leads to a nonlinear system of two conservation equations. Each conservation equation contains several nonlinear diffusion terms, which in general cannot be written as a function of the gradients of the two principal unknowns. Also the diffusion coefficients are not necessarily explicit local functions of them. For the generalised mixed finite elements approximation, Lagrange multipliers associated to each principal unknown are introduced, the sum of the diffusive fluxes of each component is explicitly eliminated and the static condensation leads to a “global” nonlinear system of equations only in the Lagrange multipliers also including complementarity conditions to cope with vanishing or appearing phases. After time discretisation, this system can be solved at each time step using a semi-smooth Newton method. The static condensation involves “local” nonlinear systems of equations associated to each element, solved also by a semismooth Newton method. The algorithm is successfully applied to 1D and 2D examples of water–hydrogen flow involving gas phase appearance and disappearance.     

Discontinuous Galerkin method for two-component liquid–gas porous media flows

We consider two-component (typically, water and hydrogen) compressible liquid–gas porous media flows including mass exchange between phases possibly leading to gas-phase (dis)appearance, as motivated by hydrogen production in underground repositories of radioactive waste. Following recent work by Bourgeat, Jurak, and Smaï, we formulate the governing equations in terms of liquid pressure and dissolved hydrogen density as main unknowns, leading mathematically to a nonlinear elliptic–parabolic system of partial differential equations, in which the equations degenerate when the gas phase disappears. We develop a discontinuous Galerkin method for space discretization, combined with a backward Euler scheme for time discretization and an incomplete Newton method for linearization. Numerical examples deal with gas-phase (dis)appearance, ill-prepared initial conditions, and heterogeneous problem with different rock types.     

A modified rock physics model for analysis of seismic signatures of low gas-saturated rocks

In seismic applications, the bulk modulus of porous media saturated with liquid and gas phases is often estimated using Gassmann's fluid substitution formula, in which the effective bulk modulus of the two-phase fluid is the Reuss average of the gas and liquid bulk moduli. This averaging procedure, referred to as Wood's approximation, holds if the liquid and gas phases are homogeneously distributed within the pore space down to sizes well below the seismic wavelength and if the phase transfer processes between liquid and gas domains induced by the pressure variations of the seismic wave are negligible over the timescale of the wave period. Using existing theoretical results and low-frequency acoustic measurements in bubbly liquids, we argue that the latter assumption of “frozen” phases, valid for large enough frequencies, is likely to fail in the seismic frequency range where lower effective bulk modulus and velocity, together with dispersion and attenuation effects, are expected. We provide a simple method, which extends to reservoir fluids a classical result by Landau and Lifshitz valid for pure fluids, to compute the effective bulk modulus of thermodynamically equilibrated liquid and gas phases. For low gas saturation, this modulus is significantly lower than its Wood's counterpart, especially at the crossing of bubble point conditions. A seismic reflector associated to a phase transition between a monophasic and a two-phase fluid thus will appear. We discuss the consequences of these results for various seismic applications including fizz water discrimination and hydrocarbon reservoir depletion and CO₂ geological storage monitoring.     

Modeling compositional compressible two-phase flow in porous media by the concept of the global pressure

We derive a new formulation for the compositional compressible two-phase flow in porous media. We consider a liquid–gas system with two components: water and hydrogen. The formulation considers gravity, capillary effects, and diffusivity of each component. The main feature of this formulation is the introduction of the global pressure variable that partially decouples the system equations. To formulate the final system, and in order to avoid primary unknowns changing between one-phase and two-phase zones, a second persistent variable is introduced: the total hydrogen mass density. The derived system is written in terms of the global pressure and the total hydrogen mass density. The system is capable of modeling the flows in both one and two-phase zones with no changes of the primary unknowns. The mathematical structure is well defined: the system consists of two nonlinear parabolic equations, the global pressure equation, and the total hydrogen mass density equation. The derived formulation is fully equivalent to the original one. Numerical simulations show ability of this new formulation to model efficiently the phase appearance and disappearance.     

煤层气井有杆排采泵筒煤粉流动特征

煤层气井煤粉颗粒在有杆排采泵筒内固液两相流中的流动特征是埋泵、卡泵和凡尔漏现象的重要因素。基于泵筒中液体携煤粉的运动学和动力学分析,建立了泵筒中液体流动和煤粉运移的计算模型,并依据仿真分析得到不同排液量和煤粉粒径时,煤粉在泵筒内的运移特征。结果表明,泵筒中的煤粉运移与液体流动特征相近,煤粉运移速度正负的分界点随排液量的增大而逐渐扩大,煤粉排出量也随之不断提升。泵筒中液体携煤粉在泵筒入口附近发生湍流,并在固定阀阀孔两侧由于涡流而发生煤粉沉淀,而在泵筒内部固液两相流动则变为层流运动。两相流进泵速度较低或煤粉粒径较大时,泵入口附近开始出现煤粉沉淀,煤粉运移速度损失较小。该研究首次系统分析煤层气井泵筒内煤粉流动特征,为防煤粉有杆泵的设计及其排采作业方法提供了重要依据。      

An empirical theory for gravitationally unstable flow in porous media

In this paper, we follow a similar procedure as proposed by Koval (SPE J 3(2):145–154, 1963) to analytically model CO₂ transfer between the overriding carbon dioxide layer and the brine layer below it. We show that a very thin diffusive layer on top separates the interface from a gravitationally unstable convective flow layer below it. Flow in the gravitationally unstable layer is described by the theory of Koval, a theory that is widely used and which describes miscible displacement as a pseudo two-phase flow problem. The pseudo two-phase flow problem provides the average concentration of CO₂ in the brine as a function of distance. We find that downstream of the diffusive layer, the solution of the convective part of the model, is a rarefaction solution that starts at the saturation corresponding to the highest value of the fractional-flow function. The model uses two free parameters, viz., a dilution factor and a gravity fingering index. A comparison of the Koval model with the horizontally averaged concentrations obtained from 2-D numerical simulations provides a correlation for the two parameters with the Rayleigh number. The obtained scaling relations can be used in numerical simulators to account for the density-driven natural convection, which cannot be currently captured because the grid cells are typically orders of magnitude larger than the wavelength of the initial fingers. The method can be applied both for storage of greenhouse gases in aquifers and for EOR processes using carbon dioxide or other solvents.     

Numerical modeling of three‐phase dissolution of underground cavities using a diffuse interface model

Natural evaporite dissolution in the subsurface can lead to cavities having critical dimensions in the sense of mechanical stability. Geomechanical effects may be significant for people and infrastructures because the underground dissolution may lead to subsidence or collapse (sinkholes). The knowledge of the cavity evolution in space and time is thus crucial in many cases. In this paper, we describe the use of a local nonequilibrium diffuse interface model for solving dissolution problems involving multimoving interfaces within three phases, that is, solid–liquid–gas as found in superficial aquifers and karsts. This paper generalizes developments achieved in the fluid–solid case, that is, the saturated case [1]. On one hand, a local nonequilibrium dissolution porous medium theory allows to describe the solid–liquid interface as a diffuse layer characterized by the evolution of a phase indicator (e.g., porosity). On the other hand, the liquid–gas interface evolution is computed using a classical porous medium two‐phase flow model involving a phase saturation, that is, generalized Darcy's laws. Such a diffuse interface model formulation is suitable for the implementation of a finite element or finite volume numerical model on a fixed grid without an explicit treatment of the interface movement. A numerical model has been implemented using a finite volume formulation with adaptive meshing (e.g., adaptive mesh refinement), which improves significantly the computational efficiency and accuracy because fine gridding may be attached to the dissolution front. Finally, some examples of three‐phase dissolution problems including density effects are also provided to illustrate the interest of the proposed theoretical and numerical framework. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical Reliability for Mixed Methods Applied to Flow Problems in Porous Media

This paper is devoted to the numerical reliability and time requirements of the Mixed Finite Element (MFE) and Mixed-Hybrid Finite Element (MHFE) methods. The behavior of these methods is investigated under the influence of two factors: the mesh discretization and the medium heterogeneity. We show that, unlike the MFE, the MHFE suffers with the presence of badly shaped discretized elements. Thereat, a numerical reliability analyzing software (Aquarels) is used to detect the instability of a matrix-inversion code generated automatically by a symbolic manipulator. We also show that the spectral condition number of the algebraic systems furnished by both methods in heterogeneous media grows up linearly according to the smoothness of the hydraulic conductivity. Furthermore, it is found that the MHFE could accumulate numerical errors if large jumps in the tensor of conductivity take place. Finally, we compare running-times for both algorithms by giving various numerical experiments.     

How do granitoid magmas mix with each other? Insights from textures,trace element and Sr–Nd isotopic composition of apatite and titanite from the Matok pluton (South Africa)

In plutonic systems, magma mixing is often modelled by mass balance based on whole-rock geochemistry. However, magma mixing is a chaotic process and chemical equilibration is controlled by non-linear diffusive–advective processes unresolved by the study of bulk samples. Here we present textural observations, LA-(MC-)ICP-MS trace element and Sr–Nd isotopic data of accessory apatites and titanites from a hybrid granodiorite of the Neoarchean Matok pluton (South Africa), collected in a zone of conspicuous mixing between mafic and felsic magmas. Apatite grains mostly show a pronounced zoning in CL images, corresponding to abrupt changes in REE and HFSE concentrations recording their transfer through compositionally different melt domains during mixing. These grains crystallized early, at temperatures of 950–1000 °C. Titanite grains crystallized at temperatures of 820–900 °C (Zr-in-sphene thermometry). They show limited intra-grain chemical variations but huge inter-grain compositional scatter in REE and HFSE, pinpointing crystallization within a crystal mush, from isolated melt pockets having different composition from one another owing to incomplete chemical homogenization and variable Rayleigh fractionation. These chemical–textural characteristics, in combination with partitioning models and Polytopic Vector Analysis, point to “self-mixing” between co-genetic dioritic and granodioritic/granitic magmas. Both resulted from differentiation of mantle-derived mafic melts, showing that mixing does not necessarily involve magmas from contrasted (crust vs. mantle) sources. Systematic variations in εNd _t (?4.5 to ?2.5) and ⁸⁷Sr/⁸⁶Sr_(i) (0.703–0.707) of titanite and apatite grains/domains crystallized from the two magmas point to an isotopically inhomogeneous mantle source, which is not resolved by bulk-rock isotopic data. Interaction between the two magmas must have occurred at relatively high temperatures (ca. 900°C) so that their viscosity contrast remained low, allowing efficient mechanical mixing. Despite this, chemical homogenization was incomplete, as recorded by diffusive fractionation between REE–HFSE and Sr. Modelling thereof reveals that chemical exchange between the liquid phases of the two mixed magmas did not last more than a few tens to hundreds of years. The chemical equilibration between mixed magmas thus strongly depends on the considered elements and observational length scales.     

The use of a mixed scheme: mixed hybrid finite elements method/finite volumes (MHFE/FV), for the modeling of contaminants transport in unsaturated porous mediums

This work is intended for the development of a numerical method to simulate flows and solute transport in multiphasical porous medium taking into consideration the interaction of solid/solute. More precisely, the studied problem is modeled by a coupled system composed of an elliptical equation (for the flow) and an equation convection–diffusion–reaction (for the transfer). Numerical simulations were realistic for two-dimensional problems confirming the stability and efficiency of the combined scheme in the characterization of a pollutant transport through an unsaturated zone of an industrial site.     

On persistent primary variables for numerical modeling of gas migration in a nuclear waste repository

Numerical simulation of gas migration driven by compressible two-phase partially miscible flow in porous media is of major importance for safety assessment of deep geological repositories for long-lived high-level nuclear waste. We present modeling of compositional liquid and gas flow for numerical simulations of hydrogen migration in deep geological radioactive waste repository based on persistent primary variables. Two-phase flow is considered, with incompressible liquid and compressible gas, which includes capillary effects, gas dissolution, and diffusivity. After discussing briefly the existing approaches to deal with phase appearance and disappearance problem, including a persistent set of variables already considered in a previous paper (Bourgeat et al., Comput Geosci 13(1):29–42, 2009), we focus on a new variant of the primary variables: dissolved hydrogen mass concentration and liquid pressure. This choice leads to a unique and consistent formulation in liquid saturated and unsaturated regions, which is well adapted to heterogeneous media. We use this new set of variable for numerical simulations and show computational evidences of its adequacy to simulate gas phase appearance and disappearance in different but typical situations for gas migration in an underground radioactive waste repository.     

Analysis of Numerical Algorithms for Computing Rapid Momentum Transfers between the Gas and Dust in Simulations of Circumstellar Disks

Approaches used in modern numerical simulations of the dynamics of dust and gas in circumstellar disks are tested. The gas and dust are treated like interpenetrating continuous media that can exchange momentum. A stiff coupling between the gas and dust phases is typical for such disks, with the dust stopping time much less than the characteristic dynamical time scale. This imposes high demands on the methods used to simulate the dust dynamics. A grid-based, piecewise-parabolic method is used as the basic algorithm for solving the gas-dynamical equations. Numerical solutions obtained using various methods to compute the momentum exchanges are presented for the case of monodisperse dust. Numerical solutions are obtained for shock tube problem and the propagation of sound waves in a gas–dust medium. The studied methods are compared in terms of their ability to model media with (a) an arbitrary (short or long) dust stopping time, and (b) an arbitrary dust concentration in the gas (varying the dust to gas mass ratio from 0.01 to 1). A method for computing the momentum exchange with infinite-order accuracy in time is identified, which makes it possible to satisfy the conditions (a) and (b) with minimal computational costs. A first-order method that shows similar results in the test computations is also presented. It is shown that the proposed first-order method for monodisperse dust can be extended to a regime when the dust is polydisperse; i.e., a regime represented by several fractions with different stopping times. Formulas for computing the gas and dust velocities for polydisperse dust with each fraction exchanging momentum with the gas are presented.     

Reaction of zoning of garnet

Compositional zoning of garnet in metamorphic or igneous rocks preserves evidence of the equilibration history of the sample and can be interpreted in terms of a growth-fractionation, diffusion-exchange, or diffusion-reaction model. Diffusion zoning is usually assumed to result from exchange reactions between garnet and other phases as the partitioning coefficient varies in response to changing environmental conditions, primarily temperature. However, in many natural environments where garnet grew originally in divariant equilibrium with other phases, changing conditions can promote continuous or “divariant” reactions and consequent compositional shifts of phases that can be much greater in some systems showing these reactions than those related to the small changes of partitioning. Diffusional zoning related to overstepping of these continuous reactions must be related to incongruent reaction and necessitates formulation of a kinetic diffusion-reaction model involving moving phase boundaries as well as solid-state diffusion. Three samples containing zoned garnets from the metamorphic aureole around the Ronda ultramafic intrusion in southern Spain are used to illustrate two possible models of diffusion-reaction processes. The examples are particularly informative because the reactions are demonstrably irreversible and evidence of the reaction system is preserved. Partitioning data indicates that compositions of product phases are not in equilibrium with the original garnet and do not vary with extent of reaction; therefore, exchange reactions with garnet were not possible and garnet changed composition only by incongruent reaction. After a small amount of reaction, Mg/Fe of the rim composition approaches a value apparently in equilibrium with product phases, but the garnets are zoned inward to the original garnet composition preserved in the interior. Grossularite content is approximately constant and spessartite content variable but small, thus, the rim composition of pyrope or almandine is assumed to be fixed by the external reaction process and is taken as a boundary condition in the following models. The zoning profile of pyrope or almandine component between the fixed rim and core compositions (assumed to extend to ∞) is described in semiinfinite, half-space models appropriate for large garnets with narrow rims. The first model corresponds to a reaction system in which all garnet compositions are metastable (case 1) and zoning depends on the independent variables of the diffusion constant, velocity of the interface between garnet and matrix, and time. The second model, corresponding to systems in which the initial garnet composition is metastable but an equilibrium composition is stable (case 2), depends on the independent variables diffusion constant, time, and a function of reaction compositions. In case 1 the consumption velocity is assumed constant and a steady state zoning profile is reached at large time, whereas, in case 2, the velocity decreases with the concentration gradient and steady state is not possible. The models were tested using a reaction time estimated from cooling models of the aureole, mass of garnet consumed, determined petrographically, and phase compositions. The two cases are somewhat independent in that different parameters are independent variables. The estimate of the diffusion constant of 10^?18±2 cm²/sec (assumed to be a mutual or binary coefficient for almandine and pyrope) is considered reasonable for the temperature range of reaction (probably 600–900° C), and the two models are consistent considering the probable error and possible real temperature differences. It is obvious that details of the metamorphic reaction system must be known to successfully apply diffusion models. Kinetic models, involving consumption or growth of the phase as well as diffusion are probably necessary when dealing with natural rocks. Several possible and interesting complications, such as cross coupling between components, can be investigated if more data were available. Experimental determination of diffusion constants allow natural reaction rates to be estimated by this method. Diffusion zoning is an important consideration that could increase the efficiency of experimentation with chemically recalcitrant phases.     

利用时间分裂的错格伪谱法模拟地震波在基于改进BISQ模型的双相介质中的传播

改进的BISQ(Biot-Squirt)模型中各参数具有明确的物理意义和可实现性,在不引入特征喷流长度的情况下可将Biot流动和喷射流动两种力学机制有机地结合起来;而高精度的地震波场数值模拟技术是研究双相介质地震波传播规律的重要手段。本文从本构方程、动力学方程和动力学达西定律出发,推导了基于改进BISQ模型的双相各向同性介质的一阶速度--应力方程组;采用时间分裂错格伪谱法求该方程组的数值解,模拟半空间及层状双相介质中的地震波场。数值模拟结果表明:①与传统方法相比,时间分裂错格伪谱法波场数值模拟的精度更高,压制网格频散效果更好;②在非黏滞相界情况下,慢纵波呈传播性,而在黏滞相界情况下,慢纵波呈扩散性,以静态模式出现在震源位置;③双相介质分界面处,各类波型复杂的反射透射规律可由数值模拟结果清晰展现。     

地幔对流的数值模拟方法

地幔对流的数值模拟研究为理解地壳运动、地幔内部的性质和地球的动力学机制提供了手段。笔者回顾了地幔对流数值模拟的发展,总结了各种常用的描述地幔对流的数学模型,边界条件以及数值解法。分析了级数展开方法,有限差分方法,有限元方法和多重网格方法的性质和特点,针对某些方法编写了二维地幔对流的计算机软件,使用了实际的地质参数,无穷大的Ｐｒａｎｄｔｌ数,在１０４到１０７之间的Ｒａｙｌｅｉｇｈ数,并给出了数值结果和地质解释。     

Identification of diffusion parameters in a nonlinear convection–diffusion equation using the augmented Lagrangian method

Numerical identification of diffusion parameters in a nonlinear convection–diffusion equation is studied. This partial differential equation arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. The forward problem is discretized with the finite difference method, and the identification problem is formulated as a constrained minimization problem. We utilize the augmented Lagrangian method and transform the minimization problem into a coupled system of nonlinear algebraic equations, which is solved efficiently with the nonlinear conjugate gradient method. Numerical experiments are presented and discussed. This work was partially supported by the Research Council of Norway (NFR), under grant 128224/431.     

Arbitrary Lagrangian–Eulerian method for large‐strain consolidation problems

In this paper, an arbitrary Lagrangian–Eulerian (ALE) method is generalized to solve consolidation problems involving large deformation. Special issues such as pore‐water pressure convection, permeability and void ratio updates due to rotation and convection, mesh refinement and equilibrium checks are discussed. A simple and effective mesh refinement scheme is presented for the ALE method. The ALE method as well as an updated‐Lagrangian method is then used to solve some classical consolidation problems involving large deformations with different constitutive laws. The results clearly show the advantage and efficiency of the ALE method for these examples. Copyright © 2007 John Wiley & Sons, Ltd.