Calculating phase diagrams involving solid solutions via non-linear equations, with examples using THERMOCALC

Phase diagrams involving solid solutions are calculated by solving sets of non-linear equations. In calculating P–T  projections and compatibility diagrams, the equations used for each equilibrium are the equilibrium relationships for an independent set of reactions between the end-members of the phases in the equilibrium. Invariant points and univariant lines in P–T  projections can be calculated directly, as can coordinates in compatibility diagrams. In calculating P–T  and T–x / P–x pseudosections – diagrams drawn for particular bulk compositions – the equilibrium relationship equations are augmented by mass balance equations. Lines in pseudosections, where the mode of one phase in the lower variance equilibrium is zero, and points, where the modes of two phases are zero, can then be calculated directly. The software, THERMOCALC, allows the calculation of these and a range of other types of phase diagram. Examples of phase diagrams and phase diagram movies, with instructions for their production, along with the THERMOCALC input and output files, and the Mathematica^TM functions for assembling them, are presented in this paper, partly in hard copy and partly on the JMG web sites (http://www.gly.bris.ac.uk/www/jmg/jmg.html, or equivalent Australian or USA sites).
     

Multibody approach for reactive transport modeling in discontinuous-heterogeneous porous media

Computational Geosciences - In the context of long-term degradation of porous media, the coupling between fracture mechanics and reactive transport is investigated. A reactive transport model in a...     

去耦合方法求解非均质地区地下水中多组分反应-运移模型

近些年来地下水中多组分反应-运移模型在地球科学及环境领域开始得到应用,但其求解较为复杂,为了提高计算效率,可以采用去耦合化方法处理,从而使模型求解得到简化。针对自然界中广泛存在的非均质地质体,提出该类条件下的去耦合化方法,即根据水-岩间、水溶组分间反应的不同,将整个研究区划分为若干子区域,获得对应的去耦合化矩阵。对化学场中各子区域间相邻边界进行设定,达到简化模型求解的目的。最后,以一维非均质介质中基于热力学平衡的反应-运移问题作为算例,基于以上方法进行求解,并与该算例经PHAST软件所示的结果较为一致。结果表明,基于去耦合化方法获得的各离子浓度随时间演变和沿空间分布特征与PHAST所示的结果较为一致,显示该方法在非均质区域模拟溶质运移等方面具有较好的适用性。     

A numerical study of fronts in random media using a reactive solute transport model

We simulate the random front solutions of a nonlinear solute transport equation with spatial random coefficients modeling inhomogeneous sorption sites in porous media. The nonlinear sorption function is chosen to be Langmuir type, and the random coefficients are two independent stationary processes with fast decay of correlations. The model equation is in conservation form, and the random fronts are similar to random viscous shocks. We find that the average front speed is given by an ensemble averaged explicit Rankine–Hugoniot relation, and the front position fluctuates about its mean. Our numerical calculations show that the standard deviation is of the order O( $\sqrt t $ ) for large time, and the front fluctuation scaled by $\sqrt t $ converges to a Gaussian random variable wih mean zero. We come up with a formal theory of front fluctuation, yielding an explicit expression of the root t normalized front standard deviation in terms of the random media statistics. The theory agrees remarkably with the numerically discovered empirical formula.     

A global approach to reactive transport: application to the MoMas benchmark

Water resource management involves numerical simulations in order to study contamination of groundwater by chemical species. Not only do the aqueous components move due to physical advection and dispersion processes, but they also react together and with fixed components. Therefore, the mass balance couples transport and chemistry, and reactive transport models are partial differential equations coupled with nonlinear algebraic equations. In this paper, we present a global method based on the method of lines and differential algebraic system (DAE) solvers. At each time step, nonlinear systems are solved by a Newton-LU method. We use this method to carry out numerical simulations for the reactive transport benchmark proposed by the MoMas research group. Although we study only 1D computations with a specific geochemical system, several difficulties arise. Numerical experiments show that our method can solve quite difficult problems, get accurate results and capture sharp fronts.     

A sequential partly iterative approach for multicomponent reactive transport with CORE2D

CORE^2D V4 is a finite element code for modeling partly or fully saturated water flow, heat transport, and multicomponent reactive solute transport under both local chemical equilibrium and kinetic conditions. It can handle coupled microbial processes and geochemical reactions such as acid–base, aqueous complexation, redox, mineral dissolution/precipitation, gas dissolution/exsolution, ion exchange, sorption via linear and nonlinear isotherms, and sorption via surface complexation. Hydraulic parameters may change due to mineral precipitation/dissolution reactions. Coupled transport and chemical equations are solved by using sequential iterative approaches. A sequential partly iterative approach (SPIA) is presented which improves the accuracy of the traditional sequential non-iterative approach (SNIA) and is more efficient than the general sequential iterative approach (SIA). While SNIA leads to a substantial saving of computing time, it introduces numerical errors which are especially large for cation exchange reactions. SPIA improves the efficiency of SIA because the iteration between transport and chemical equations is only performed in nodes with a large mass transfer between solid and liquid phases. The efficiency and accuracy of SPIA are compared to those of SIA and SNIA using synthetic examples and a case study of reactive transport through the Llobregat Delta aquitard in Spain. SPIA is found to be as accurate as SIA while requiring significantly less CPU time. In addition, SPIA is much more accurate than SNIA with only a minor increase in computing time. A further enhancement of the efficiency of SPIA is achieved by improving the efficiency of the Newton–Raphson method used for solving chemical equations. Such an improvement is obtained by working with increments of log concentrations and ignoring the terms of the Jacobian matrix containing derivatives of activity coefficients. A proof is given for the symmetry and non-singularity of the Jacobian matrix. Numerical analyses performed with synthetic examples confirm that these modifications improve the efficiency and convergence of the iterative algorithm. Changbing Yang is now at The University of Texas at Austin, USA.     

Two-dimensional reactive transport modeling of the alteration of a fractured limestone by hyperalkaline solutions at Maqarin (Jordan)

Two-dimensional reactive transport modeling of the Maqarin Eastern Springs site, a natural analogue for the alteration of a fractured limestone by high-pH Portland cement waters, has been performed using the CrunchFlow code. These 2D calculations included transport by advection–dispersion–diffusion along a single fracture and diffusion in the wall rock. Solute transport was coupled to mineral dissolution and precipitation. A limited sensitivity analysis evaluated the effect of different values of primary mineral surface areas, flow velocity and sulfate concentration of the inflowing high-pH solution.Major secondary minerals include ettringite–thaumasite, C–S–H/C–A–S–H and calcite. C–S–H/C–A–S–H precipitation is controlled by the dissolution of primary silicates. Ettringite precipitation is controlled by diffusion of sulfate and aluminum from the wall rock to the fracture, with aluminum provided by the dissolution of albite. Calcite precipitation is controlled by diffusion of carbonate from the wall rock. Extents of porosity sealing along the fracture and in the fracture-wall rock interface depend on assumptions regarding flow velocity and composition of the high-pH solution. The multiple episodes of fracture sealing and reactivation evidenced in the fracture infills were not included in the simulations. Results can qualitatively reproduce the reported decrease in porosity in the fractures and in the wall rock next to the fractures. Instances of porosity increase next to fractures caused by carbonate dissolution were not reproduced by the calculations.     

A semianalytical approach for solving first-order perturbation equations of dissolution-timescale reactive infiltration instability problems in fluid-saturated rocks

This paper presents a semianalytical approach for solving first-order perturbation (FOP) equations, which are used to describe dissolution-timescale reactive infiltration instability (RII) problems in fluid-saturated rocks. The proposed approach contains two parts because the chemical dissolution reaction divides the whole problem domain into two subdomains. In the first part, the interface-condition substitution strategy is used to derive the analytical expressions of purely mathematical solutions for the FOP equations in the upstream subdomain, where the dissolution chemical reaction is ceased and the FOP equations are weakly coupled. In the second part, the finite element method (FEM) is used to derive the analytical expressions of numerical solutions for the FOP equations in the downstream subdomain, where the dissolution chemical reaction needs to be considered and the FOP equations are strongly coupled so that it is impossible to derive purely mathematical solutions for them. Particular attention is paid to the development of the element-by-element forward marching strategy, which is associated with the use of the FEM for solving this new kind of scientific problem. The related analytical results demonstrated that (1) both the dynamic characteristic of a reactive infiltration system and the dimensionless wavenumber can have pronounced influences on the distribution of the FOP dimensionless acid concentration within the entire domain of the dissolution-timescale RII problems in fluid-saturated rocks and (2) the FOP dimensionless acid concentration distribution exhibits two significantly different patterns in the upstream and downstream subdomains of the dissolution-timescale RII system.     

A low-temperature kinetic study of the exsolution of pentlandite from the monosulfide solid solution using a refined Avrami method

A refined Avrami method, that assumes that the activation energy is a function of reaction extent y, was used to analyze the kinetics of the exsolution of pentlandite from mss/pyrrhotite (bulk composition, (Fe_0.77Ni_0.19)S) over the temperature range 473 to 573 K. The experimental results show the reaction rates vary from 1.6 × 10⁻⁵ to 5.0 × 10⁻⁷ s⁻¹ at 473 K and from 9.4 × 10⁻⁵ to 4.1 × 10⁻⁷ s⁻¹ at 573 K. Examination of exsolution textures indicated that the mechanism of exsolution did not change significantly over the temperature range investigated. The activation energy (E_a) decreases from 49.6 to 20.7 kJ mol⁻¹ over the course of the reaction. The decrease in E_a with y is related to the change in the dominant factor of pentlandite exsolution, from nucleation dominant at the beginning to metal ion diffusion dominant at the end. The classic Avrami method provides average values of kinetic parameters for the overall solid-state reaction while the refined Avrami method provides more a detailed indication of the variation of kinetic parameters over the course of the reaction. Previously published kinetic data for the exsolution of pentlandite from mss/pyrrhotite are reevaluated using the refined Avrami method.     

Mixing induced reactive transport in fractured crystalline rocks

In this paper the solute retention properties of crystalline fractured rocks due to mixing-induced geochemical reactions are studied. While fractured media exhibit paths of fast flow and transport and thus short residence times for conservative solutes, at the same time they promote mixing and dilution due to strong heterogeneity, which leads to sharp concentration contrasts. Enhanced mixing and dilution have a double effect that favors crystalline fractured media as a possible host medium for nuclear waste disposal. Firstly, peak radionuclide concentrations are attenuated and, secondly, mixing-induced precipitation reactions are enhanced significantly, which leads to radionuclide immobilization. An integrated framework is presented for the effective modeling of these flow, transport and reaction phenomena, and the interaction between them. In a simple case study, the enhanced dilution and precipitation potential of fractured crystalline rocks are systematically studied and quantified and contrasted it to retention and attenuation in an equivalent homogeneous formation.     

Activities of components in omphacitic solid solutions

Available experimental data on mixing of disordered C2/c clinopyroxenes in the system diopside-jadeite-hedenbergite-acmite are reviewed and evaluated. Because the methods used to determine jadeite activity suffer from severe uncertainty at high jadeite mol fractions, these data cannot be used to infer asymmetry in the jadeite-diopside or the jadeite-hedenbergite solid solutions. If the measurement uncertainties are taken into account, a single parameter (regular, or reciprocal energy) suffices to describe the mixing properties of these two solid solutions. It is argued that a two-site entropy of mixing satisfies the experiments and is consistent with the C2/c disordered nature of the solid solutions; the data in the range 600–1300° C are consistent with a temperature-independent interaction energy, implying no discernible excess entropy. The available experimental data imply W=26±2 kJ mol^–1 for jd-di, and W=25±3 kJ mol^–1 for jd-hd, solid-solutions. Landau theory for a tricritical phase transformation (C2/c-P2/n) is in good agreement with the calorimetrically determined disordering enthalpy, and may be used to derive a simple expression for the activities in ordered omphacite solid solutions. The derived activities of jadeite at 600° C in ordered omphacites are remarkably close to those reported previously for short-range ordered pyroxenes. A simple model is presented for determining the activities of end-members in the system jadeite-diopside-hedenbergite-acmite.     

Local structural heterogeneity in garnet solid solutions

Local heterogeneities in pyrope-almandine, almandine-grossular and pyrope-grossular solid solutions have been investigated using IR-powder absorption spectroscopy. Correlations of the wavenumber shifts and line broadening systematics with the thermodynamic mixing properties were found. Wavenumber shifts of the highest energy modes correlate closely with the Si-O bond distances and give an indirect view of the average distortions across the three solid solutions. They have a linear behaviour for Py-Alm, but show positive variations from linearity for Alm-Gr and Py-Gr systems. An effective line width (Δ_corr) of the absorption bands over a given wavenumber interval was obtained using the autocorrelation function. Line broadening is associated with local heterogeneities arising from cation substitution in the structure of samples at intermediate compositions. Non-linearities of the line broadening were found for Alm-Gr and Py-Gr and have a shape similar to the enthalpy of mixing, ΔH_mix. An empirical analysis was therefore carried out to compare ΔH_mix and Δ_corr quantitatively. Low-temperature far-IR spectra were recorded for the end-members pyrope, almandine and grossular and far-IR and mid-IR low-temperature spectra for Py₆₀Gr₄₀ in the temperature range 292–44 K. Softening of the lowest energy band with decreasing temperature was observed in the spectrum of pyrope and more enhanced in the spectrum of Py₆₀Gr₄₀. The same softening occurs by substitution of grossular component into pyrope. High energy modes of Py₆₀Gr₄₀ show the effect of saturation below 110–130 K, which correlates with the volume saturation at low temperature. This could provide an alternative explanation for the heat capacity anomaly found for Py-Gr solid solution at low-temperatures.     

A kinetic study of enargite dissolution in ammoniacal solutions

The kinetics of dissolution of natural enargite in ammoniacal solution has been investigated. Unlike other copper sulfide minerals, the dissolution of copper from enargite is very slow and only 60% of the copper is extracted after 24 hours. The rate of dissolution is favored by high temperatures and high oxygen pressures. The experimental data seem to fit an electrochemical mechanism.     

Removal of dexamethasone from aqueous solutions using modified clinoptilolite zeolite (equilibrium and kinetic)

The aim of this study was to remove a known pharmaceutics, dexamethasone, from an aqueous solution using clinoptilolite zeolite (CP). CP is a natural, versatile and inexpensive mineral, which has been investigated and applied in the last few decades. Herein, the experiments were carried out in the common conditions of a batch system in room temperature, and the effects of some parameters such as pH of the solution, initial concentration of dexamethasone, adsorbent dose and contact time were studied. Kinetic and isotherm of adsorption processes of dexamethasone on CP were surveyed in the current study. Results revealed that the maximum efficiency (78 %) occurred in pH = 4. The adsorption process followed a pseudo-second-order kinetic model as well as Freundlich and Sips isotherm models fitted with the experimental data well.