PHT3D‐UZF: A Reactive Transport Model for Variably‐Saturated Porous Media

A modified version of the MODFLOW/MT3DMS‐based reactive transport model PHT3D was developed to extend current reactive transport capabilities to the variably‐saturated component of the subsurface system and incorporate diffusive reactive transport of gaseous species. Referred to as PHT3D‐UZF, this code incorporates flux terms calculated by MODFLOW's unsaturated‐zone flow (UZF1) package. A volume‐averaged approach similar to the method used in UZF‐MT3DMS was adopted. The PHREEQC‐based computation of chemical processes within PHT3D‐UZF in combination with the analytical solution method of UZF1 allows for comprehensive reactive transport investigations (i.e., biogeochemical transformations) that jointly involve saturated and unsaturated zone processes. Intended for regional‐scale applications, UZF1 simulates downward‐only flux within the unsaturated zone. The model was tested by comparing simulation results with those of existing numerical models. The comparison was performed for several benchmark problems that cover a range of important hydrological and reactive transport processes. A 2D simulation scenario was defined to illustrate the geochemical evolution following dewatering in a sandy acid sulfate soil environment. Other potential applications include the simulation of biogeochemical processes in variably‐saturated systems that track the transport and fate of agricultural pollutants, nutrients, natural and xenobiotic organic compounds and micropollutants such as pharmaceuticals, as well as the evolution of isotope patterns.     

A multiphase flow and multispecies reactive transport model for DNAPL-involved Compound Specific Isotope Analysis

This study presents a multiphase flow and multispecies reactive transport model for the simultaneous simulation of NAPL and groundwater flow, dissolution, and reactive transport with isotope fractionation, which can be used for better interpretation of NAPL-involved Compound Specific Isotope Analysis in 3D heterogeneous hydrogeologic systems. The model was verified for NAPL-aqueous phase equilibrium partitioning, aqueous phase multi-chain and multi-component reactive transport, and aqueous phase multi-component transport with isotope fractionation. Several illustrative examples are presented to investigate the effect of DNAPL spill rates, degradation rate constants, and enrichment factors on the temporal and spatial distribution of the isotope signatures of chlorinated aliphatic hydrocarbon groundwater plumes. The results clearly indicate that isotope signatures can be significantly different when considering multiphase flow within the source zone. A series of simulations indicate that degradation and isotope enrichment compete with dissolution to determine the isotope signatures in the source zone: isotope ratios remain the same as those of the source if dissolution dominates the reaction, while heavy isotopes are enriched in reactants along groundwater plume flow paths when degradation becomes dominant. It is also shown that NAPL composition can change from that of the injected source due to the partitioning of components between the aqueous and NAPL phases even when degradation is not allowed in NAPL phase. The three-dimensional simulation is presented to mechanistically illustrate the complexities in determining and interpreting the isotopic signatures with evolving DNAPL source architecture.     

Evaluating MT3DMS for Heat Transport Simulation of Closed Geothermal Systems

Owing to the mathematical similarities between heat and mass transport, the multi-species transport model MT3DMS should be able to simulate heat transport if the effects of buoyancy and changes in viscosity are small. Although in several studies solute models have been successfully applied to simulate heat transport, these studies failed to provide any rigorous test of this approach. In the current study, we carefully evaluate simulations of a single borehole ground source heat pump (GSHP) system in three scenarios: a pure conduction situation, an intermediate case, and a convection-dominated case. Two evaluation approaches are employed: first, MT3DMS heat transport results are compared with analytical solutions. Second, simulations by MT3DMS, which is finite difference, are compared with those by the finite element code FEFLOW and the finite difference code SEAWAT. Both FEFLOW and SEAWAT are designed to simulate heat flow. For each comparison, the computed results are examined based on residual errors. MT3DMS and the analytical solutions compare satisfactorily. MT3DMS and SEAWAT results show very good agreement for all cases. MT3DMS and FEFLOW two-dimensional (2D) and three-dimensional (3D) results show good to very good agreement, except that in 3D there is somewhat deteriorated agreement close to the heat source where the difference in numerical methods is thought to influence the solution. The results suggest that MT3DMS can be successfully applied to simulate GSHP systems, and likely other systems with similar temperature ranges and gradients in saturated porous media.     

Modeling Variably Saturated Multispecies Reactive Groundwater Solute Transport with MODFLOW‐UZF and RT3D

A numerical model was developed that is capable of simulating multispecies reactive solute transport in variably saturated porous media. This model consists of a modified version of the reactive transport model RT3D (Reactive Transport in 3 Dimensions) that is linked to the Unsaturated‐Zone Flow (UZF1) package and MODFLOW. Referred to as UZF‐RT3D, the model is tested against published analytical benchmarks as well as other published contaminant transport models, including HYDRUS‐1D, VS2DT, and SUTRA, and the coupled flow and transport modeling system of CATHY and TRAN3D. Comparisons in one‐dimensional, two‐dimensional, and three‐dimensional variably saturated systems are explored. While several test cases are included to verify the correct implementation of variably saturated transport in UZF‐RT3D, other cases are included to demonstrate the usefulness of the code in terms of model run‐time and handling the reaction kinetics of multiple interacting species in variably saturated subsurface systems. As UZF1 relies on a kinematic‐wave approximation for unsaturated flow that neglects the diffusive terms in Richards equation, UZF‐RT3D can be used for large‐scale aquifer systems for which the UZF1 formulation is reasonable, that is, capillary‐pressure gradients can be neglected and soil parameters can be treated as homogeneous. Decreased model run‐time and the ability to include site‐specific chemical species and chemical reactions make UZF‐RT3D an attractive model for efficient simulation of multispecies reactive transport in variably saturated large‐scale subsurface systems.     

Implementation of Solute Transport in the Vadose Zone into the “HYDRUS Package for MODFLOW”

The “HYDRUS package for MODFLOW” is an existing MODFLOW package that allows MODFLOW to simultaneously evaluate transient water flow in both unsaturated and saturated zones. The package is based on incorporating parts of the HYDRUS-1D model (to simulate unsaturated water flow in the vadose zone) into MODFLOW (to simulate saturated groundwater flow). The coupled model is effective in addressing spatially variable saturated-unsaturated hydrological processes at the regional scale. However, one of the major limitations of this coupled model is that it does not have the capability to simulate solute transport along with water flow and therefore, the model cannot be employed for evaluating groundwater contamination. In this work, a modified unsaturated flow and transport package (modified HYDRUS package for MODFLOW and MT3DMS) has been developed and linked to the three-dimensional (3D) groundwater flow model MODFLOW and the 3D groundwater solute transport model MT3DMS. The new package can simulate, in addition to water flow in the vadose zone, also solute transport involving many biogeochemical processes and reactions, including first-order degradation, volatilization, linear or nonlinear sorption, one-site kinetic sorption, two-site sorption, and two-kinetic sites sorption. Due to complex interactions at the groundwater table, certain modifications of the pressure head (compared to the original coupling) and solute concentration profiles were incorporated into the modified HYDRUS package. The performance of the newly developed model is evaluated using HYDRUS (2D/3D), and the results indicate that the new model is effective in simulating the movement of water and contaminants in the saturated-unsaturated flow domains.     

The influence of dimensionality on simulations of mass recovery from nonuniform dense non-aqueous phase liquid (DNAPL) source zones

The influence of model dimensionality on predictions of mass recovery from dense non-aqueous phase liquid (DNAPL) source zones in nonuniform permeability fields was investigated using a modified version of the modular three-dimensional transport simulator (MT3DMS). Thirty-two initial two- (2D) and three-dimensional (3D) tetrachloroethene–DNAPL source zone architectures, taken from a recent modeling study, were used as initial conditions for this analysis. Commonly employed source zone metrics were analyzed to determine differences between 2D and 3D predictions: (i) down-gradient flux-averaged contaminant concentration, (ii) reductions in contaminant mass flux through a down-gradient boundary, (iii) source zone ganglia-to-pool (GTP) ratio, and (iv) time required to achieve a remediation objective. 3D flux-averaged contaminant concentrations were approximately 3.5 times lower than concentrations simulated in 2D. This difference was attributed to dilution of the contaminant concentrations down gradient of the source zone. Contaminant flux reduction predictions for a given mass recovery were generally 5% higher in 3D simulations than in 2D simulations. The GTP ratio declined over time as mass was recovered in both 2D and 3D simulations. Although the source longevity (i.e., time required to achieve 99.99% mass recovery) differed between individual 2D and 3D realizations, the mean source longevity for the 2D and 3D simulation ensembles was within 2%. 2D simulations tended to over-predict the time required to achieve lower mass recovery levels (e.g. 50% mass recovery) due to a smaller contaminated area exposed to uncontaminated water. These findings suggest that ensemble averages of 2D numerical simulations of DNAPL migration, entrapment, dissolution, and mass recovery in statistically homogenous, nonuniform media may provide reasonable approximations to average behavior obtained using simulations conducted in fully three-dimensional domains.     

Application of a mixing-ratios based formulation to model mixing-driven dissolution experiments

We address the question of how one can combine theoretical and numerical modeling approaches with limited measurements from laboratory flow cell experiments to realistically quantify salient features of complex mixing-driven multicomponent reactive transport problems in porous media. Flow cells are commonly used to examine processes affecting reactive transport through porous media, under controlled conditions. An advantage of flow cells is their suitability for relatively fast and reliable experiments, although measuring spatial distributions of a state variable within the cell is often difficult. In general, fluid is sampled only at the flow cell outlet, and concentration measurements are usually interpreted in terms of integrated reaction rates. In reactive transport problems, however, the spatial distribution of the reaction rates within the cell might be more important than the bulk integrated value. Recent advances in theoretical and numerical modeling of complex reactive transport problems [De Simoni M, Carrera J, Sanchez-Vila X, Guadagnini A. A procedure for the solution of multicomponent reactive transport problems. Water Resour Res 2005;41:W11410. doi: 10.1029/2005WR004056, De Simoni M, Sanchez-Vila X, Carrera J, Saaltink MW. A mixing ratios-based formulation for multicomponent reactive transport. Water Resour Res 2007;43:W07419. doi: 10.1029/2006WR005256] result in a methodology conducive to a simple exact expression for the space–time distribution of reaction rates in the presence of homogeneous or heterogeneous reactions in chemical equilibrium. The key points of the methodology are that a general reactive transport problem, involving a relatively high number of chemical species, can be formulated in terms of a set of decoupled partial differential equations, and the amount of reactants evolving into products depends on the rate at which solutions mix. The main objective of the current study is to show how this methodology can be used in conjunction with laboratory experiments to properly describe the key processes that occur in a complex, geochemically-active system under chemical equilibrium conditions. We model three CaCO₃ dissolution experiments reported in Singurindy et al. [Singurindy O, Berkowitz B, Lowell RP. Carbonate dissolution and precipitation in coastal environments: Laboratory analysis and theoretical consideration. Water Resour Res 2004;40:W04401. doi: 10.1029/2003WR002651, Singurindy O, Berkowitz B, Lowell RP. Correction to Carbonate dissolution and precipitation in coastal environments: laboratory analysis and theoretical consideration. Water Resour Res 2005;41:W11701. doi: 10.1029/2005WR004433], in which saltwater and freshwater were mixed in different proportions. The integrated reaction rate within the cell estimated from the experiments are modeled independently by means of (a) a state-of-the-art reactive transport code, and (b) the uncoupled methodology of [12, 13], both of which use dispersivity as a single, adjustable parameter. The good agreement between the results from both methodologies demonstrates the feasibility of using simple solutions to design and analyze laboratory experiments involving complex geochemical problems.     

CO2-induced dissolution of low permeability carbonates. Part II: Numerical modeling of experiments

We used the 3D continuum-scale reactive transport models to simulate eight core flood experiments for two different carbonate rocks. In these experiments the core samples were reacted with brines equilibrated with pCO₂ = 3, 2, 1, 0.5 MPa (Smith et al., 2013 [27]). The carbonate rocks were from specific Marly dolostone and Vuggy limestone flow units at the IEAGHG Weyburn-Midale CO₂ Monitoring and Storage Project in south-eastern Saskatchewan, Canada. Initial model porosity, permeability, mineral, and surface area distributions were constructed from micro tomography and microscopy characterization data. We constrained model reaction kinetics and porosity–permeability equations with the experimental data. The experimental data included time-dependent solution chemistry and differential pressure measured across the core, and the initial and final pore space and mineral distribution. Calibration of the model with the experimental data allowed investigation of effects of carbonate reactivity, flow velocity, effective permeability, and time on the development and consequences of stable and unstable dissolution fronts.The continuum scale model captured the evolution of distinct dissolution fronts that developed as a consequence of carbonate mineral dissolution and pore scale transport properties. The results show that initial heterogeneity and porosity contrast control the development of the dissolution fronts in these highly reactive systems. This finding is consistent with linear stability analysis and the known positive feedback between mineral dissolution and fluid flow in carbonate formations. Differences in the carbonate kinetic drivers resulting from the range of pCO₂ used in the experiments and the different proportions of more reactive calcite and less reactive dolomite contributed to the development of new pore space, but not to the type of dissolution fronts observed for the two different rock types. The development of the dissolution front was much more dependent on the physical heterogeneity of the carbonate rock. The observed stable dissolution fronts with small but visible dissolution fingers were a consequence of the clustering of a small percentage of larger pores in an otherwise homogeneous Marly dolostone. The observed wormholes in the heterogeneous Vuggy limestone initiated and developed in areas of greater porosity and permeability contrast, following pre-existing preferential flow paths.Model calibration of core flood experiments is one way to specifically constrain parameter input used for specific sites for larger scale simulations. Calibration of the governing rate equations and constants for Vuggy limestones showed that dissolution rate constants reasonably agree with published values. However the calcite dissolution rate constants fitted to the Marly dolostone experiments are much lower than those suggested by literature. The differences in fitted calcite rate constants between the two rock types reflect uncertainty associated with measured reactive surface area and appropriately scaling heterogeneous distribution of less abundant reactive minerals. Calibration of the power-law based porosity–permeability equations was sensitive to the overall heterogeneity of the cores. Stable dissolution fronts of the more homogeneous Marly dolostone could be fit with the exponent n = 3 consistent with the traditional Kozeny–Carman equation developed for porous sandstones. More impermeable and heterogeneous cores required larger n values (n = 6–8).     

A Spatially Periodic Solute Boundary for MT3DMS and PHT3D

The assumption of spatial repetition is commonly made when producing bedform scale models of the hyporheic zone. Two popular solute transport codes, MT3DMS and PHT3D, do not currently provide the necessary boundary condition required to simulate spatial periodicity in hyporheic zone transport problems. In this study, we develop a spatially periodic boundary (SPB) for solutes that is compatible with a SPB that was previously developed for MODFLOW to simulate the flow component of spatially periodic problems. The approach is ideal for simulating groundwater flow and transport patterns under repeating surface features, such as ripples or dunes on the bottom of a lake or stream. The appropriate block‐centered finite‐difference approach to implement the boundary is presented and the necessary source code modifications are discussed. The performance of the solute SPB, operating in conjunction with the groundwater flow SPB, is explored through comparison of a multi‐bedform hyporheic‐zone model with a single bedform variant. The new boundary conditions perform well in situations where both dispersive effects and lateral seepage flux in the underflow regime beneath the hyporheic zone are minimal.     

Imaging geochemical heterogeneities using inverse reactive transport modeling: An example relevant for characterizing arsenic mobilization and distribution

The spatial distribution of reactive minerals in the subsurface is often a primary factor controlling the fate and transport of contaminants in groundwater systems. However, direct measurement and estimation of heterogeneously distributed minerals are often costly and difficult to obtain. While previous studies have shown the utility of using hydrologic measurements combined with inverse modeling techniques for tomography of physical properties including hydraulic conductivity, these methods have seldom been used to image reactive geochemical heterogeneities. In this study, we focus on As-bearing reactive minerals as aquifer contaminants. We use synthetic applications to demonstrate the ability of inverse modeling techniques combined with mechanistic reactive transport models to image reactive mineral lenses in the subsurface and quantify estimation error using indirect, commonly measured groundwater parameters. Specifically, we simulate the mobilization of arsenic via kinetic oxidative dissolution of As-bearing pyrite due to dissolved oxygen in the ambient groundwater, which is an important mechanism for arsenic release in groundwater both under natural conditions and engineering applications such as managed aquifer recharge and recovery operations. The modeling investigation is carried out at various scales and considers different flow-through domains including (i) a 1D lab-scale column (80 cm), (ii) a 2D lab-scale setup (60 cm × 30 cm) and (iii) a 2D field-scale domain (20 m × 4 m). In these setups, synthetic dissolved oxygen data and forward reactive transport simulations are used to image the spatial distribution of As-bearing pyrite using the Principal Component Geostatistical Approach (PCGA) for inverse modeling.     

Approaches to modelling coupled flow and reaction in a 2D cementation experiment

Porosity evolution at reactive interfaces is a key process that governs the evolution and performances of many engineered systems that have important applications in earth and environmental sciences. This is the case, for example, at the interface between cement structures and clays in deep geological nuclear waste disposals. Although in a different transport regime, similar questions arise for permeable reactive barriers used for biogeochemical remediation in surface environments.The COMEDIE project aims at investigating the coupling between transport, hydrodynamics and chemistry when significant variations of porosity occur. The present work focuses on a numerical benchmark used as a design exercise for the future COMEDIE-2D experiment. The use of reactive transport simulation tools like Hytec and Crunch provides predictions of the physico-chemical evolutions that are expected during the future experiments in laboratory. Focus is given in this paper on the evolution during the simulated experiment of precipitate, permeability and porosity fields.A first case is considered in which the porosity is constant. Results obtained with Crunch and Hytec are in relatively good agreement. Differences are attributable to the models of reactive surface area taken into account for dissolution/precipitation processes. Crunch and Hytec simulations taking into account porosity variations are then presented and compared. Results given by the two codes are in qualitative agreement, with differences attributable in part to the models of reactive surface area for dissolution/precipitation processes. As a consequence, the localization of secondary precipitates predicted by Crunch leads to lower local porosities than for predictions obtained by Hytec and thus to a stronger coupling between flow and chemistry. This benchmark highlights the importance of the surface area model employed to describe systems in which strong porosity variations occur as a result of dissolution/precipitation. The simulation of highly non-linear reactive transport systems is also shown to be partly dependent on specific numerical approaches.     

A COMSOL-PHREEQC Coupled Python Framework for Reactive Transport Modeling in Soil and Groundwater

The CPqPy framework coupling COMSOL and PHREEQC based on Python was developed. This framework can achieve the simulation of diversified situations including multi-physics coupling and geochemical reactions of soil and groundwater. The multi-physics coupling models are calculated in COMSOL, whereas PHREEQC was applied to calculate the geochemical models through the Phreeqpy library in Python. Feasibility and accuracy of CPqPy were verified and applied to two cases, including a solute transport model considering equilibrium reaction and ion exchange as well as a reactive transport model of a variable saturation soil considering kinetic reaction. The results show a high degree of credibility of CPqPy. The framework has the advantages of strong portability, and it can be further used in conjunction with multiple Python calculation libraries, which greatly extends the application of the reactive transport model.     

Modeling axisymmetric flow and transport

Unmodified versions of common computer programs such as MODFLOW, MT3DMS, and SEAWAT that use Cartesian geometry can accurately simulate axially symmetric ground water flow and solute transport. Axisymmetric flow and transport are simulated by adjusting several input parameters to account for the increase in flow area with radial distance from the injection or extraction well. Logarithmic weighting of interblock transmissivity, a standard option in MODFLOW, can be used for axisymmetric models to represent the linear change in hydraulic conductance within a single finite-difference cell. Results from three test problems (ground water extraction, an aquifer push-pull test, and upconing of saline water into an extraction well) show good agreement with analytical solutions or with results from other numerical models designed specifically to simulate the axisymmetric geometry. Axisymmetric models are not commonly used but can offer an efficient alternative to full three-dimensional models, provided the assumption of axial symmetry can be justified. For the upconing problem, the axisymmetric model was more than 1000 times faster than an equivalent three-dimensional model. Computational gains with the axisymmetric models may be useful for quickly determining appropriate levels of grid resolution for three-dimensional models and for estimating aquifer parameters from field tests.     

大地电磁局部畸变异常特征三维数值模拟研究

局部畸变问题曾经困扰大地电磁资料反演解释几十年,大地电磁三维数值模拟技术的发展为剖析局部畸变特点和得到可靠的反演成像结果提供了技术基础。本文采用三维数值模拟成像方法对典型三维局部畸变模型进行模拟分析。三维数值模拟结果显示:电场分量垂直电性分界面的极化模式视电阻率曲线(对应二维情况下TM模式)在穿越低阻异常体界面时,曲线会先上移后下移,而在穿越高阻异常体界面时,曲线会先下移后上移,这与电性分界面处积累面电荷产生的二次电场有关。三维模型中XY模式、YX模式视电阻率和相位在三维异常体附近的水平变化是呈现近似垂向对称的,该现象与电场垂直跨越电性界面时视电阻率的变化规律是吻合的,当测线分别沿X方向和Y方向展布时,三维情况下的XY和YX模式分别对应二维情况下的TM模式。低阻小异常体对区域构造响应的畸变影响比高阻小异常体要严重。低阻小异常体对二维区域响应的两种极化模式视电阻率和相位都有非常明显的畸变影响,相比较而言对TE模式的畸变要大于TM模式,因此我们在做二维反演解释时,可优先考虑拟合TM模式数据。位于小异常体中心上方测点的三维畸变响应虽然与对应真实二维区域响应的差异比较大,但可以等效于某种二维模型响应,这种由局部畸变造成的假二维响应在实际野外数据的解释中是需要注意的。      

Fast transport simulation with an adaptive grid refinement

One of the main difficulties in transport modeling and calibration is the extraordinarily long computing times necessary for simulation runs. Improved execution time is a prerequisite for calibration in transport modeling. In this paper we investigate the problem of code acceleration using an adaptive grid refinement, neglecting subdomains, and devising a method by which the Courant condition can be ignored while maintaining accurate solutions. Grid refinement is based on dividing selected cells into regular subcells and including the balance equations of subcells in the equation system. The connection of coarse and refined cells satisfies the mass balance with an interpolation scheme that is implicitly included in the equation system. The refined subdomain can move with the average transport velocity of the subdomain. Very small time steps are required on a fine or a refined grid, because of the combined effect of the Courant and Peclet conditions. Therefore, we have developed a special upwind technique in small grid cells with high velocities (velocity suppression). We have neglected grid subdomains with very small concentration gradients (zero suppression). The resulting software, MODCALIF, is a three-dimensional, modularly constructed FORTRAN code. For convenience, the package names used by the well-known MODFLOW and MT3D computer programs are adopted, and the same input file structure and format is used, but the program presented here is separate and independent. Also, MODCALIF includes algorithms for variable density modeling and model calibration. The method is tested by comparison with an analytical solution, and illustrated by means of a two-dimensional theoretical example and three-dimensional simulations of the variable-density Cape Cod and SALTPOOL experiments. Crossing from fine to coarse grid produces numerical dispersion when the whole subdomain of interest is refined; however, we show that accurate solutions can be obtained using a fraction of the execution time required by uniformly fine-grid solutions.     

面向目标自适应有限元法的带地形三维大地电磁各向异性正演模拟

传统三维大地电磁各向异性模拟均是基于规则六面体网格,计算精度有限且较难拟合复杂地质条件.本文采用面向目标自适应非结构矢量有限元法,对三维大地电磁各向异性介质进行模拟.首先从电场双旋度方程出发,利用伽辽金方法建立变分方程;然后利用电流密度连续性条件构建适合大地电磁各向异性问题的加权后验误差估计方法,实现面向目标的网格自适应正演;最后通过典型算例分析各向异性对网格自适应和大地电磁响应的影响特征以及各向异性的识别方法.本文算法能够高精度地拟合起伏地表和任意各向异性介质,适用于分析复杂地电条件大地电磁响应特征,为提高大地电磁资料解释水平提供了理论基础.     

Analysis of observed and model mt fields based on three-dimensional mathematical modeling: Case study of the Verkhnee Penzhino-Korf profile

We propose a system for the analysis of magnetotelluric (MT) data, which makes use of the invariant characteristics of the impedance tensor such as the maximum and minimum induction curves and the phase tensor. We examine the coefficients of the appearance and normalization of principal values of the impedance tensor. By the case study for Koryakiya, it is shown that the three-dimensional (3D) mathematical modeling and the Wiese-Parkinson vectors allow one to correct the results of one-dimensional (1D) and two-dimensional (2D) inversion of MT curves. Comparison between model and observed data based on the 1D inversion of MTS curves provides a pictorial view of the distortions of MT curves and their sensitivity to the parameters of a geological cross section.     

复杂二维/三维大地电磁的有限单元法正演模拟策略

复杂二维和三维大地电磁模型的正演数值模拟具有一定的挑战性。对于复杂的二维和三维大地电磁正演问题,我们采用有限单元法进行求解。有限单元法最后形成一个线性方程组,系数矩阵是大型稀疏的带状对称复系数矩阵,并且其条件数远大于1,为严重病态矩阵,求解其对应方程组会遇到很多困难。不完全LU分解处理的Bi-CGSTAB迭代方法可用于该线性方程组的求解,并且具有速度快、精度高和稳定性好等优点;为了模拟无穷远边界及满足计算机的内存需求,在保证计算精度的情况下设计了非均匀网格剖分;在程序编制中,只存储有限元系数矩阵的非零元素,大大减少了正演计算的时间。通过对二维和三维模型电磁响应的计算,验证了算法的正确性。