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1.
In this article, an approach for the efficient numerical solution of multi-species reactive transport problems in porous media is described. The objective of this approach is to reformulate the given system of partial and ordinary differential equations (PDEs, ODEs) and algebraic equations (AEs), describing local equilibrium, in such a way that the couplings and nonlinearities are concentrated in a rather small number of equations, leading to the decoupling of some linear partial differential equations from the nonlinear system. Thus, the system is handled in the spirit of a global implicit approach (one step method) avoiding operator splitting techniques, solved by Newton’s method as the basic algorithmic ingredient. The reduction of the problem size helps to limit the large computational costs of numerical simulations of such problems. If the model contains equilibrium precipitation-dissolution reactions of minerals, then these are considered as complementarity conditions and rewritten as semismooth equations, and the whole nonlinear system is solved by the semismooth Newton method.  相似文献   

2.
CORE2D 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.  相似文献   

3.
The focus of this work is efficient solution methods for mixed finite element models of variably saturated fluid flow through deformable porous media. In particular, we examine preconditioning techniques to accelerate the convergence of implicit Newton–Krylov solvers. We highlight an approach in which preconditioners are built from block-factorizations of the coupled system. The key result of the work is the identification of effective preconditioners for the various sub-problems that appear within the block decomposition. We use numerical examples drawn from both linear and nonlinear hydromechanical models to test the robustness and scalability of the proposed methods. Results demonstrate that an algebraic multigrid variant of the block preconditioner leads to mesh-independent convergence, good parallel efficiency, and insensitivity to the material parameters of the medium.  相似文献   

4.
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5.
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6.
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.  相似文献   

7.
A new parallel solution technique is developed for the fully implicit three‐dimensional two‐phase flow model. An expandedcell‐centered finite difference scheme which allows for a full permeability tensor is employed for the spatial discretization, and backwardEuler is used for the time discretization. The discrete systems are solved using a novel inexact Newton method that reuses the Krylov information generated by the GMRES linear iterative solver. Fast nonlinear convergence can be achieved by composing inexact Newton steps with quasi‐Newton steps restricted to the underlying Krylov subspace. Furthermore, robustness and efficiency are achieved with a line‐search backtracking globalization strategy for the nonlinear systems and a preconditioner for each coupled linear system to be solved. This inexact Newton method also makes use of forcing terms suggested by Eisenstat and Walker which prevent oversolving of the Jacobian systems. The preconditioner is a new two‐stage method which involves a decoupling strategy plus the separate solutions of both nonwetting‐phase pressure and saturation equations. Numerical results show that these nonlinear and linear solvers are very effective.  相似文献   

8.
Subsurface contamination problems of metals and radionuclides are ubiquitous. Metals and radionuclides may exist in the solute phase or may be bound to soil particles and interstitial portions of the geologic matrix. Accurate tools to reliably predict the migration and transformation of these metals and radionuclides in the subsurface environment enhance the ability of environmental scientists, engineers, and decision makers to analyze their impact and to evaluate the efficacy of alternative remediation techniques prior to incurring expense in the field. A mechanistic-based numerical model could provide such a tool. This paper communicates the development and verification of a mechanistically coupled fluid-flow thermal-reactive biogeochemical-transport model where both fast and slow reactions occur in porous and fractured media. Theoretical bases, numerical implementations, and numerical experiments using the model are described. A definition of the “rates” of fast/equilibrium reactions is presented to come up with a consistent set of governing equations. Two example problems are presented. The first one is a reactive transport problem which elucidates the non-isothermal effects on heterogeneous reactions. It also demonstrates that the rates of fast/equilibrium reactions are not necessarily greater than that of slow/kinetic reactions in the context of reactive transport. The second example focuses on a complicated but realistic advective–dispersive–reactive transport problem. This example exemplifies the need for innovative numerical algorithms to solve problems involving stiff geochemical reactions. It also demonstrates that rates of all fast/equilibrium reactions are finite and definite. Furthermore, it is noted that a species-versus-time curve cannot be used to characterize the rate of homogeneous fast/equilibrium reaction in a reactive transport system even if one and only one such reaction is responsible for the production of this species.  相似文献   

9.
The transport of chemically reactive solutes (e.g. surfactants, CO2 or dissolved minerals) is of fundamental importance to a wide range of applications in oil and gas reservoirs such as enhanced oil recovery and mineral scale formation. In this work, we investigate exponential time integrators, in conjunction with an upwind weighted finite volume discretisation in space, for the efficient and accurate simulation of advection–dispersion processes including non-linear chemical reactions in highly heterogeneous 3D oil reservoirs. We model sub-grid fluctuations in transport velocities and uncertainty in the reaction term by writing the advection–dispersion–reaction equation as a stochastic partial differential equation with multiplicative noise. The exponential integrators are based on the variation of constants solution and solve the linear system exactly. While this is at the expense of computing the exponential of the stiff matrix representing the finite volume discretisation, the use of real Léja point or the Krylov subspace technique to approximate the exponential makes these methods competitive compared to standard finite difference-based time integrators. For the deterministic system, we investigate two exponential time integrators, the second-order accurate exponential Euler midpoint (EEM) scheme and exponential time differencing of order one (ETD1). All our numerical examples demonstrate that our methods can compete in terms of efficiency and accuracy compared with standard first-order semi-implicit time integrators when solving (stochastic) partial differential equations that model mixing and chemical reactions in 3D heterogeneous porous media. Our results suggest that exponential time integrators such as the ETD1 and EEM schemes could be applied to typical 3D reservoir models comprising tens to hundreds of thousands unknowns.  相似文献   

10.
In this study we use two dimensional chemical patterns and numerical modeling to estimate the relative rates of chemical transport along interphase boundaries (ib) and through grain (s) interiors during retrograde Fe–Mg exchange between garnet and biotite at a garnet–biotite–quartz triple junction. We demonstrate that systematic variations in garnet–rim compositions, which are frequently observed along garnet–quartz interfaces, and deviations from concentric retrograde zoning patterns start to develop when chemical transport along the interphase boundaries becomes slow during cooling. The capacities for chemical transport along an interphase boundary depend on the product D ib K ib/s a, where D ib is the diffusion coefficient of the exchangeable components within the interphase boundary medium, K ib/s is the equilibrium partitioning coefficient between the cation exchange partners and the interphase boundary medium and a is the interphase boundary width. The model is applied to analyze the retrograde zoning patterns in garnets from the Mozambique belt (SE-Kenya), which cooled from 820°C at a rate of ca. 2°C/my. It is found that non-equilibrated compositions in garnet along garnet/quartz interphase boundaries started to develop below 700°C due to insufficient rates of chemical transport along these boundaries. The transport capacities of garnet/quartz interphase boundaries was estimated to have been between about 1E-23 cm3/s (575°C) and 1E-20 cm3/s (700°C) from modeling the observed X Fe pattern in garnet close to a garnet–quartz–biotite triple junction and relying on published data on the diffusivity of Fe2+ in garnet. Similar transport capacities are obtained; when the interphase boundary is assumed to be filled with a material that has the transport properties and chemical composition of a free melt in equilibrium with garnet, biotite and quartz at the respective conditions. In contrast, if the transport properties of the interphase boundary medium are related to the diffusivities and solubility of Fe2+ and FeOH+ within a free aqueous solution, chemical transport along the interphase boundaries would be much more efficient, and exchange equilibrium would have been maintained during the entire cooling history of the rocks. The observation of systematic deviations from local equilibrium along the garnet–quartz interphase boundaries leads us to exclude the presence of an aqueous fluid along the interphase boundary at any time during cooling.  相似文献   

11.
Numerical approximation based on different forms of the governing partial differential equation can lead to significantly different results for two-phase flow in porous media. Selecting the proper primary variables is a critical step in efficiently modeling the highly nonlinear problem of multiphase subsurface flow. A comparison of various forms of numerical approximations for two-phase flow equations is performed in this work. Three forms of equations including the pressure-based, mixed pressure–saturation and modified pressure–saturation are examined. Each of these three highly nonlinear formulations is approximated using finite difference method and is linearized using both Picard and Newton–Raphson linearization approaches. Model simulations for several test cases demonstrate that pressure based form provides better results compared to the pressure–saturation approach in terms of CPU_time and the number of iterations. The modification of pressure–saturation approach improves accuracy of the results. Also it is shown that the Newton–Raphson linearization approach performed better in comparison to the Picard iteration linearization approach with the exception for in the pressure–saturation form.  相似文献   

12.
A novel numerical method based on the finite element approach is established for the zero current method approach for calculating multi-species ionic diffusion. The proposed numerical method uses the direct calculation of the coupled set of equations in favor of the staggering approach. A one-step truly implicit time stepping scheme is adopted together with an implementation of a modified Newton–Raphson iteration scheme for search of equilibrium at each considered time step calculation. Results from the zero current case are compared with existing results from the solutions of the more general Gauss’ law method.  相似文献   

13.
We present a reservoir simulation framework for coupled thermal-compositional-mechanics processes. We use finite-volume methods to discretize the mass and energy conservation equations and finite-element methods for the mechanics problem. We use the first-order backward Euler for time. We solve the resulting set of nonlinear algebraic equations using fully implicit (FI) and sequential-implicit (SI) solution schemes. The FI approach is attractive for general-purpose simulation due to its unconditional stability. However, the FI method requires the development of a complex thermo-compositional-mechanics framework for the nonlinear problems of interest, and that includes the construction of the full Jacobian matrix for the coupled multi-physics discrete system of equations. On the other hand, SI-based solution schemes allow for relatively fast development because different simulation modules can be coupled more easily. The challenge with SI schemes is that the nonlinear convergence rate depends strongly on the coupling strength across the physical mechanisms and on the details of the sequential updating strategy across the different physics modules. The flexible automatic differentiation-based framework described here allows for detailed assessment of the robustness and computational efficiency of different coupling schemes for a wide range of multi-physics subsurface problems.  相似文献   

14.
Summary A forward model is proposed to reproduce the formation of garnet under conditions of sluggish diffusion transport in the matrix. Starting from a matrix consisting of chlorite and quartz, the amount of garnet growth and the chemical composition was calculated at each P–T increment in the system MnO–FeO–MgO–Al2O3–SiO2–H2O. Sluggish diffusion transport was introduced considering the local equilibrium between garnet surface and the matrix within a given diffusion distance (equilibration volume). Varying the diffusion distance, calculations were performed along the prograde P–T path of the Sambagawa metamorphic belt, Japan. The final size of the garnet grains was largely proportional to the diffusion distance. In contrast to the model without diffusion limitations, a shorter diffusion distance resulted in a rise of the Mg/(Mg + Fe) ratio in garnet before Mn approached zero. These results indicate that the chemical composition trend in zoned garnet from the Sambagawa belt is consistent with growth under sluggish material transport. The calculated amount of garnet growth increases dramatically with temperature. The amount of newly grown natural garnet in the Sambagawa metamorphic rocks was plotted against temperatures, where chemical compositions of garnet were calibrated against temperatures with the Gibbs’ method. This trend was also consistent with the modelled garnet behaviour.  相似文献   

15.
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.  相似文献   

16.
This paper presents a hydrogeochemical modeling code HYDROS, which combines the multi-component transport model with equilibrium speciation module MINTEQA2. The processes of adsorption, aqueous speciation and mineral precipitation/dissolution are represented in the model. The numerical model uses a sequential iterative approach for solving the solute transport and the equilibrium geochemistry modules. Further the transport part is solved using an operator split approach wherein a finite volume method is used for solving the advective equations while a classical finite difference method is employed for solving the dispersive equations. The model performance is evaluated by comparing it with MINTOX for a literature problem. HYDROS is then applied to the case study of the transfer of transition metals with organic colloids in the swamp groundwater system of the experimental Nsimi watershed, representative of the humid tropical ecosystem of the South Cameroon Plateau. Field observations at the site swamp system suggest that the carbon is mainly transferred as organic colloids (i.e., dissolved organic carbon) produced by the slow biodegradation of the swamp organic matter. Using HYDROS, the behaviour of Al(III) and Fe(III) elements in the base flow system is simulated during inter rain events of a short rainy season (May–June 1996). The elemental time-series for Al, Fe, Cl, pH compare well with the simulation results. The colloids are found to have a strong impact on the mobilization and transfer of Al(III) and Fe(III), which are considered to have low mobility in weathering environment.  相似文献   

17.
A new approach is proposed for incorporating solid solution reactions into mass conservation equations describing reaction paths in both closed and open systems. The method is applicable to problems involving advective, dispersive, and diffusive transport in a porous medium. By representing the continuously variable solid solution composition with a discrete set of stoichiometric solids that span composition space, combined with a kinetic formulation of their rates of reaction, a self-determining spatial and temporal evolution of the solid solution concentration and composition is obtained. It is demonstrated that equilibrium of an aqueous solution with a stoichiometric solid derived from a solid solution corresponds to equilibrium of the solid solution itself if and only if equilibrium of the stoichiometric solid is stable. One advantage of this approach is that it is unnecessary to introduce any additional compositional variables to represent the solid solution. Discretization may be over the entire range of composition space, or over some subset depending on the system. A major consequence of the kinetic discrete-composition solid solution representation is that modeling solid solutions is similar to modeling pure mineral phases with the exception of a weighting factor applied to reaction rates of stoichiometric solids corresponding to a common solid solution. With this approach, precipitation leads to a discrete zonation of the solid solution that approximates the continuous variation in composition expected for the actual solid solution. The approach is demonstrated for a hypothetical ideal and non-ideal binary solid solution AxB1−xC for a reaction path formulation and reactive transport involving advection and diffusion.  相似文献   

18.
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.  相似文献   

19.
We present an approach developed to compute chemical equilibrium and its corresponding reactive chemical transport when dominating precipitated species (DPS) exist. In computing chemical equilibrium, most models take the concentrations or activities of component species and precipitated species as the master variables. However, when the amount of a precipitated species is much larger than those of other species, small computational errors on this DPS concentration might introduce large errors on the concentrations of other species and would cause non‐mass‐conserved numerical results. To deal with the existence of DPS, we pick as master variables the concentration change, rather than the concentration, of DPS to compute chemical equilibrium. Since the concentration changes of DPS will no longer be much larger than the concentrations of other species in determining equilibrium, our approach is able to provide correct numerical results. We also employ the modified total analytical concentrations, rather than the total analytical concentrations, of aqueous components as the dependent variables in presenting and solving corresponding transport equations. Several examples are given to reveal the numerical problems associated with DPS and to demonstrate the success of our approach. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

20.
The application of a powerful evolutionary optimization technique for the estimation of intrinsic formation constants describing geologically relevant adsorption reactions at mineral surfaces is introduced. We illustrate the optimization power of a simple Genetic Algorithm (GA) for forward (aqueous chemical speciation calculations) and inverse (calibration of Surface Complexation Models, SCMs) modeling problems of varying degrees of complexity, including problems where conventional deterministic derivative-based root-finding techniques such as Newton–Raphson, implemented in popular programs such as FITEQL, fail to converge or yield poor data fits upon convergence. Subject to sound a priori physical–chemical constraints, adequate solution encoding schemes, and simple GA operators, the GA conducts an exhaustive probabilistic search in a broad solution space and finds a suitable solution regardless of the input values and without requiring sophisticated GA implementations (e.g., advanced GA operators, parallel genetic programming). The drawback of the GA approach is the large number of iterations that must be performed to obtain a satisfactory solution. Nevertheless, for computationally demanding problems, the efficiency of the optimization can be greatly improved by combining heuristic GA optimization with the Newton–Raphson approach to exploit the power of deterministic techniques after the evolutionary-driven set of potential solutions has reached a suitable level of numerical viability. Despite the computational requirements of the GA, its robustness, flexibility, and simplicity make it a very powerful, alternative tool for the calibration of SCMs, a critical step in the generation of a reliable thermodynamic database describing adsorption equilibria. The latter is fundamental to the forward modeling of the adsorption behavior of minerals and geologically based adsorbents in hydro-geological settings (e.g., aquifers, pore waters, water basins) and/or in engineered reactors (e.g., mining, hazardous waste disposal industries).  相似文献   

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