Integral Finite Difference Method Research Articles

A fast time integral finite difference method for a space-time fractional FitzHugh-Nagumo monodomain model in irregular domains

This work aims at proposing a fast time integral (FTI) method for the space-time fractional Fitzhugh-Nagumo (FHN) monodomain model in irregular domains, which is commonly used to characterize the transmembrane potential of the heart. In order to reduce the storage and the algorithm complexity due to the geometric configuration of the heart, the derivation of the sum of the exponentials (cf. [1]) in FTI method is improved by the integral transformation, integral truncation and Gauss-Legendre quadrature. Such strategy is applied to approximate the Caputo fractional derivative. The number of the exponentials, reduced in the FTI method, is related to the calculation efficiency. The CPU time of the FHN monodomain model using FTI method is reduced by an order of magnitude. Moreover, a second-order discrete method to the Riesz space fractional derivative adopts for the spatial discretization, then an implicit-explicit scheme is derived for the nonlinear FHN model under the finite difference method. Numerical results are reported to demonstrate the convergence behaviors, robustness and high efficiency of the proposed method.

Real-Time Simulation of Hydraulic Fracturing Using a Combined Integrated Finite Difference and Discontinuous Displacement Method: Numerical Algorithm and Field Applications

Real-time simulation of hydraulic fracturing operations is of critical importance to the field-scale stimulation applications. In this paper, we present an efficient yet reasonably accurate program for the numerical modeling of dynamic fractures. Our program, named as FracCSM, is based on combined Integrated Finite Difference (IFD) method and Discontinuous Displacement Method (DDM). FracCSM simulates the initiation and propagation of hydraulic fractures with DDM and mass/heat transport inside fractures by IFD. The frictional loss within the wellbore is also taken into consideration. In this way, we are able to model the propped height and length of the fractures subject to the stress interference effect. Moreover, FracCSM captures the stress shadow effect of multi-stage fractures. To facilitate the monitoring and decision making during the hydraulic fracturing process, we have developed a general framework that supports real-time simulation of fracture propagation. Our developed program demonstrates sound accuracy in comparison with existing simulators. The novelty of this work is the combined simulation algorithm to simulate the multiphysical process during hydraulic fracturing operations. We will demonstrate the program structure as well as the field applications of FracCSM to the real-time simulation of hydraulic fracturing operations in Sulige tight sandstone reservoir.

Numerical modeling of gas production from methane hydrate deposits using low-frequency electrical heating assisted depressurization method

With the increasing global demand for clean energy, methane hydrate has been regarded as a potential alternative energy source in future due to its huge resources and high energy density. In this study, a mathematical model is established to evaluate the production performance of low-frequency electrical heating assisted depressurization (LF-EHAD) method for hydrate deposits recovery. Within the framework of this model, the integral finite difference method and Newton-Raphson iteration are employed for solution, which is then validated by the experiment data. Using this model, the energy recovery behaviors with LF-EHAD from Shenhu hydrate deposits are investigated and then compared with those performed by hot water flooding and depressurization. Simulation results show that hydrate dissociation and gas production are significantly enhanced as a result of sufficient heat supplied by the electrical heating, validating the potential of the LF-EHAD for hydrate deposits recovery. After 2000 days of production, the cumulative gas production of LF-EHAD and hot water injection are 2.37 times and 1.47 times of that conducted by the depressurization method. Moreover, the heat consumption used to increase the formation temperature in the hydrate dissociation region under LF-EHAD is much less than that of hot water flooding, which results in a higher energy efficiency ratio (EER) of 9.65 for LF-EHAD, while the EER of hot water flooding is 4.42. Overall, LF-EHAD shows better performance than hot water flooding with the same amount of heat input because of higher gas production and energy utilization efficiency, but lower water production.

An embedded 3D fracture modeling approach for simulating fracture-dominated fluid flow and heat transfer in geothermal reservoirs

In this paper, we describe an efficient modeling approach, named embedded discrete fracture method (EDFM), for incorporating arbitrary 3D, discrete fractures, such as hydraulic fractures or faults, into modeling fracture-dominated fluid flow and heat transfer in fractured geothermal reservoirs. This technique allows 3D discrete fractures to be discretized independently from surrounding rock volume and inserted explicitly into a primary fracture/matrix grid, generated without including 3D discrete fractures in prior. An effective computational algorithm is developed to discretize these 3D discrete fractures and construct local connections between 3D fractures and fracture/matrix grid blocks representing the surrounding rock volume. The constructed gridding information on 3D fractures is then added to the primary grid. This embedded fracture modeling approach can be directly implemented into a developed geothermal reservoir simulator via the integral finite difference (IFD) method or with TOUGH2 technology. This embedded fracture modeling approach is very promising and computationally efficient to handle realistic 3D discrete fractures with complicated geometries, connections, and spatial distributions. Compared with other fracture modeling approaches, it avoids cumbersome 3D unstructured, local refining procedures, and increases computational efficiency by simplifying Jacobian matrix size and sparsity, while maintaining enough accuracy. Several numeral simulations are presented to demonstrate the utility and robustness of the proposed technique. Our numerical experiments show that this approach captures all the key patterns about fluid flow and heat transfer dominated by fractures in these cases. Thus, this approach is readily available to the simulation of fractured geothermal reservoirs with both artificial and natural fractures.

Robust implementations of the 3D-EDFM algorithm for reservoir simulation with complicated hydraulic fractures

The geometry of the subsurface hydraulic fracture is complicated which deviates from the conventional single pan shape. It can be in multi-wing shape by the formation heterogeneity and/or 3D fracture networks caused by the pre-exising natural fractures. Effective and flexible handling of hydraulic fractures is crucial for the reservoir simulation of unconventional reservoirs. In this paper, we discuss the robust implementations of the 3D-EDFM algorithm, which is applicable to numerical simulation of fluid flow in unconventional reservoirs with complicated, arbitrary 3D hydraulic fracture networks.Key features of this 3D-EDFM methodology consist of vertex-based geometric calculations, and polygons with three to six vertices from box-plane intersections can be captured with this algorithm. The 3D-EDFM has been implemented in reservoir simulators via the integral finite difference (IFD) method. Its numerical scheme and implementation are verified by comparisons with analytical solutions and other established numerical solutions. This method not only provides an effective approach of handling realistic hydraulic fractures, but also significantly improves computational efficiency, while keeping sufficient accuracy. For some extreme cases that the transient state flow lasts long (i.e., large-size matrix blocks with extremely low permeability), our numerical analysis indicates only one level of local grid refinement (LGR) is needed to retain the required accuracy. We also provide an analytical-solution based method to optimize this LGR grid size if needed.This enhanced method is able to handle multiple 3D hydraulic fractures with arbitrary strikes and dip angles, shapes, curvatures, conductivities, and connections. Therefore, the 3D-EDFM provides great flexibility to handle hydraulic-fracture input data, interpreted from geomechanics modelings as well as microseismic data, for reservoir simulation. We present illustrative application examples for three unconventional reservoir engineering problems of interest in practice: (1) a single fracture with a complicated shape due to multi-layer geological features, (2) multiple isolated fractures with curvatures due to the stress shadow effect, and (3) multiple fractures with complicated orientations caused by natural fracture interactions. The 3D-EDFM is demonstrated to capture all the key flow patterns dominated by the discrete fractures in these three cases.

A tool for pathline creation using TOUGH simulation results and fully unstructured 3D Voronoi grids

Pathline or streamline computation is used today to improve the understanding of fluid flow behavior in groundwater modelling. In particular, pathline computation on unstructured grids requires an accurate velocity interpolation method when the flow velocity field is given by numerical simulators only at the faces of the grid blocks; as for example in the case of the Integral Finite Difference method is used for spatial discretization. In the case of 3D Voronoi unstructured grids, a corner velocity interpolation (CVI) method allows continuous velocity field interpolation while preserving the mass conservation principle. We have developed a computer code, named TOUGH2Path, able to compute the velocity field based on the CVI method and generalized barycentric coordinates for 3D Voronoi polytope grid blocks tailored for the TOUGH family of codes. Results for the classical quarter five spot problem show that the coded algorithm correctly reproduces the pathlines computed with semi-analytical methods.

Stress-dependent permeability of organic-rich shale reservoirs: Impacts of stress changes and matrix shrinkage

Effective stress is gradually increased and permeability is gradually reduced during reservoir depletion caused by hydrocarbon production, while the organic-rich matrix might experience a shrinkage process that will boost the permeability. The main objective of this paper was to develop a mathematical simulator coupling gas flow process, geomechanics effects, and matrix shrinkage in order to evaluate their influences on reservoir permeability and production performance of organic-rich shale reservoirs. The mesh was divided into three different continuums: organic matter, non-organic matter, and natural fractures. Matrix shrinkage was only considered for organic matter because of gas desorption process, and the stress-dependent permeability was considered for all three sub-pore media. The flow and stress-equilibrium equations were solved by the fixed-stress sequential method, where the flow equations are solved first, followed by the mechanics equations. The displacements are solved for each grid node by finite element method, and the grid pressure is solved by the integral finite difference method. Different stress-dependent correlations are chosen to separately apply to these three sub-pore media. Based on the correlations, the porosity and permeability are updated at end of each time step. A synthetic reservoir model was built, where the permeability change and the cumulative gas production is calculated at each time step. Besides, the sensitivity analyses were investigated for Total Organic Carbon (TOC), matrix permeability, Young's modulus, Poisson's ratio, and bottom hole pressure.The coupled model was validated by comparing with the analytical solution of Mandel's consolidation problem. Numerical results show that the permeability and cumulative production is significantly reduced when the stress-dependent permeability is considered. The matrix shrinkage on organic matter could provide an obvious rebound on cumulative production at the late producing stage, because the permeability is boosted by the media shrinkage at low pressure. However, the production enhancement is highly related to bottom hole pressure and Total Organic Carbon. Due to large permeability decline, a higher TOC does not necessarily bring a higher cumulative production. When the stress-dependent permeability is considered, a larger matrix permeability encounters a bigger loss of production rate. A higher cumulative production is predicted from a lower either Young's modulus or Poisson's ratio. A large bottom hole pressure could offset certain production loss caused by the permeability decline. Overall, the triple-porosity coupled simulator can quantitatively interpret the impacts of matrix shrinkage and geomechanics on permeability change and gas production performance for organic-rich shale reservoirs. That provides more realistic production performance evaluation and economic assessment when the stress-dependent permeability needs to be considered. Additionally, this coupled flow-geomechanics model is also available to simulate the permeability change when water injection is performed to maintain reservoir pressure and prevent the permeability decline.

Dynamic Performances of the Herringbone-Grooved Gas Bearing

Aiming at studying the dynamic performances of herringbone-grooved gas bearings, a model is built to analysis the relationship between herringbone-grooved gas bearings parameters and its performances such as friction torque, stiffness, stamping, etc. In this model, the hydrodynamic effect of compressible fluid and the shunting effect of herringbone groove are taken into account, and the partial derivative method which based on local integral finite difference method is used to calculate the modal at the same time. The result shows that when the average gap and the length-diameter ratio are decreasing, the rotate speed and eccentricity ratio will be increasing and the hydrodynamic effect of the gas bearing will be enhanced. Increasing groove angle, decreasing width-ridge ratio or aggrandizing length-groove ratio can improve its load capacity. The parameters such as rotate speed and eccentricity ratio are the most important factors which affect the dynamic characteristics of herringbone-groove gas bearing greatly. Then the analysis method is verified through an experiment systematically. In the experiment, the moment balance method is used. The experiment result shows that the result of model is correct s and reliable.

3D Voronoi grid dedicated software for modeling gas migration in deep layered sedimentary formations with TOUGH2-TMGAS

As is known, a full three-dimensional (3D) unstructured grid permits a great degree of flexibility when performing accurate numerical reservoir simulations. However, when the Integral Finite Difference Method (IFDM) is used for spatial discretization, constraints (arising from the required orthogonality between the segment connecting the blocks nodes and the interface area between blocks) pose difficulties in the creation of grids with irregular shaped blocks. The full 3D Voronoi approach guarantees the respect of IFDM constraints and allows generation of grids conforming to geological formations and structural objects and at the same time higher grid resolution in volumes of interest. In this work, we present dedicated pre- and post-processing gridding software tools for the TOUGH family of numerical reservoir simulators, developed by the Geothermal Research Group of the DICAM Department, University of Bologna. VORO2MESH is a new software coded in C++, based on the voro++ library, allowing computation of the 3D Voronoi tessellation for a given domain and the creation of a ready to use TOUGH2 MESH file. If a set of geological surfaces is available, the software can directly generate the set of Voronoi seed points used for tessellation. In order to reduce the number of connections and so to decrease computation time, VORO2MESH can produce a mixed grid with regular blocks (orthogonal prisms) and irregular blocks (polyhedron Voronoi blocks) at the point of contact between different geological formations. In order to visualize 3D Voronoi grids together with the results of numerical simulations, the functionality of the TOUGH2Viewer post-processor has been extended. We describe an application of VORO2MESH and TOUGH2Viewer to validate the two tools. The case study deals with the simulation of the migration of gases in deep layered sedimentary formations at basin scale using TOUGH2-TMGAS. A comparison between the simulation performances of unstructured and structured grids is presented.

Simulation-assisted gas tracer test study of Tarim Yaha gas-condensate reservoir in China

To enhance oil recovery and optimize a production plan, injected water or gas fluid should be monitored and tracked, which can be realized by gas tracer technology. Tracer test technology has become an effective method for evaluating the flow condition of reservoirs and estimate reservoir properties. Gas tracers, such as chemical and radioactive materials, have been used to identify well-to-well connectivity, reservoir heterogeneity, high-permeability pay zones, sweep efficiency, and residual oil saturation. Moreover, tracer tests have been used to evaluate well interference among multiple hydraulically fractured wells and qualitatively predict possible fracture geometries. Therefore, the accurate quantification of tracer transport is of great importance. However, theoretical study of gas tracers in a condensate gas reservoir has not been thoroughly conducted to date. Current tracer transport models mostly focus on the gas tracer flow in oil reservoirs and cannot be directly applied to condensate gas reservoirs. To the best of our knowledge, the application of a gas tracer to recycle gas injection in gas condensate reservoirs has not been thoroughly studied. In this paper, we propose a mathematical model to quantitatively describe the gas tracer transport mechanism in condensate gas reservoirs. Our mathematical model is based on the mass conservation law and is numerically solved via the Integral Finite Difference method. The results predicted by our model have been compared to and verified by real data collected from the field. The model has been used to optimize the recovery of the Yaha Condensate Gas Field in the Tarim Basin, China. With the assistance of our tracer transport model, the pressure drawdown speed of the Yaha field has been successfully slowed from 0.8 MPa/year to 0.5 MPa/year. Generally, this study not only provides key insights into the transport mechanism of gas tracers but also presents the application of numerical tools to guide the production of gas condensate reservoirs. The technique introduced in this study could be potentially extended to condensate oil production in other fields.

Simulation of Coupled Thermal/Hydrological/Mechanical Phenomena in Porous Media

SummaryFor processes such as production from low-permeability reservoirs and storage in subsurface formations, reservoir flow and the reservoir stress field are coupled and affect one another. This paper presents a thermal/hydrological/mechanical (THM) reservoir simulator that is applicable to modeling such processes. The fluid- and heat-flow portion of our simulator is for general multiphase, multicomponent, multiporosity systems. The geomechanical portion consists of an equation for mean stress, derived from linear elastic theory for a thermo-poroelastic system, and equations for stress-tensor components that depend on mean stress and other variables. The integral finite-difference method is used to solve these equations. The mean-stress and reservoir-flow variables are solved implicitly, and the remaining stress-tensor components are solved explicitly. Our simulator is verified by use of analytical solutions for stress- and strain-tensor components and is compared with published results.

Modelling geothermal conditions in part of the Szczecin Trough – the Chociwel area

Abstract The Chociwel region is part of the Szczecin Trough and constitutes the northeastern segment of the extended Szczecin-Gorzów Synclinorium. Lower Jurassic reservoirs of high permeability of up to 1145 mD can discharge geothermal waters with a rate exceeding 250 m3/h and temperatures reach over 90°C in the lowermost part of the reservoirs. These conditions provide an opportunity to generate electricity from heat accumulated in geothermal waters using binary ORC (Organic Rankine Cycle) systems. A numerical model of the natural state and exploitation conditions was created for the Chociwel area with the use of TOUGH2 geothermal simulator (i.e., integral finite-difference method). An analysis of geological and hydrogeothermal data indicates that the best conditions are found to the southeast of the town of Chociwel, where the bottom part of the reservoir reaches 3 km below ground. This would require drilling two new wells, namely one production and one injection. Simulated production with a flow rate of 275 m3/h, a temperature of 89°C at the wellhead, 30°C injection temperature and wells being 1.2 km separated from each other leads to a small temperature drop and moderate requirements for pumping power over a 50 years’ time span. The ORC binary system can produce at maximum 592.5 kW gross power with the R227ea found as the most suitable working fluid. Geothermal brine leaving the ORC system with a temperature c. 53°C can be used for other purposes, namely mushroom growing, balneology, swimming pools, soil warming, de-icing, fish farming and for heat pumps.

Stability Evaluation of Highway Near Mined-Out Region

The stability evaluation of highway near a goaf is a key problem in highway construction projects. The goal of this research is to analyze the surface deformation of the highway No.108 in Baoshui area. Both Probability Integral Method (PIM) and Finite Difference Method (FDM) are used and compared to deal with this problem. The main aspects of this research consist of surface subsidence calculation and prediction of surface residual deformation. The results calculated by probability integral method and finite difference method are mainly in agreement,and they are in accord with surface deformation observation results of some sections. So these two kinds of method are proven to be effective in dealing this problem. Moreover, the surface residual deformation was predicted by using numerical simulation based on FLAC3D. To ensure the highway No.108 security, backfill grouting should be carried out in the areas where collapse pits and ground fissure are found near the highway.

A novel fully-coupled flow and geomechanics model in enhanced geothermal reservoirs

Abstract The geomechanical behavior of porous media has become increasingly important in stress-sensitive reservoirs. This paper presents a novel fully-coupled fluid flow-geomechanical model (TOUGH2-EGS). The fluid flow portion of our model is based on the general-purpose numerical simulator TOUGH2-EOS3. The geomechanical portion is developed from linear elastic theory for a thermo-poro-elastic system using the Navier equation. Fluid flow and geomechanics are fully coupled, and the integral finite-difference method is used to solve the flow and stress equations. In addition, porosity and permeability depend on effective stress and correlations describing that dependence are incorporated into the simulator. TOUGH2-EGS is verified against analytical solutions for temperature-induced deformation and pressure-induced flow and deformation. Finally the model is applied to analyze pressure and temperature changes and deformation at The Geysers geothermal field. The results demonstrate that the model can be used for field-scale reservoir simulation with fluid flow and geomechanical effects.

Modeling CO2 miscible flooding for enhanced oil recovery

The injection of fuel-generated CO2 into oil reservoirs will lead to two benefits in both enhanced oil recovery (EOR) and the reduction in atmospheric emission of CO2. To get an insight into CO2 miscible flooding performance in oil reservoirs, a multi-compositional non-isothermal CO2 miscible flooding mathematical model is developed. The convection and diffusion of CO2-hydrocarbon mixtures in multiphase fluids in reservoirs, mass transfer between CO2 and crude, and formation damages caused by asphaltene precipitation are fully considered in the model. The governing equations are discretized in space using the integral finite difference method. The Newton-Raphson iterative technique was used to solve the nonlinear equation systems of mass and energy conservation. A numerical simulator, in which regular grids and irregular grids are optional, was developed for predicting CO2 miscible flooding processes. Two examples of one-dimensional (1D) regular and three-dimensional (3D) rectangle and polygonal grids are designed to demonstrate the functions of the simulator. Experimental data validate the developed simulator by comparison with ID simulation results. The applications of the simulator indicate that it is feasible for predicting CO2 flooding in oil reservoirs for EOR.

Multiphase flow dynamics during CO2 disposal into saline aquifers

Injection of CO2 into saline aquifers is described by mass conservation equations for the three components water, salt (NaCl), and CO2. The equations are discretized using an integral finite difference method, and are solved using methods developed in geothermal and petroleum reservoir engineering. Phase change processes are treated through switching of primary thermodynamic variables. A realistic treatment of PVT (fluid) properties is given which includes salinity and fugacity effects for partitioning of CO2 between gaseous and aqueous phases. Chemical reactions and mechanical stress effects are neglected. Numerical simulations are presented for injection of CO2 into a brine aquifer, and for loss of CO2 from storage through discharge along a fault zone. It is found that simulated pressures are much more sensitive to space discretization effects than are phase saturations. CO2 discharge along a fault is a self-enhancing process whose flow rates can increase over time by more than an order of magnitude, suggesting that reliable containment of CO2 will require multiple barriers.

Stream tube integration method for analysing subsurface fluid flow

A stream tube integration method is introduced to solve transient subsurface fluid flow problems. The method combines a geometry-embedded form of Darcy's Law and the notion of location of average. Two types of problems, transient radial flow to a well of finite radius in an areally infinite aquifer and in a double porosity system, are solved by the stream tube integration method and the integral finite difference method. Results of the solutions show that the stream tube integration method, with fixed coarse mesh, are more accurate and better behaved than the integral finite difference method, with fine mesh. The fixed mesh stream tube integration method is readily extended to the moving mesh method. With much coarse mesh, the moving mesh technique can obtain the same accurate results as the fixed mesh stream tube integration method. It is suggested that the stream tube integration method is a viable way to state, solve, interpret and verify numerical solutions. The method provides efficient computation and improved accuracy for analysing subsurface fluid flow. © 1998 John Wiley & Sons, Ltd.

A multiple species transport model with sequential decay chain interactions in heterogeneous subsurface environments

The spatial and temporal distribution of solutes in groundwater is controlled by several physical and chemical processes. Among the chemical processes, sequential degradation phenomena play an important role in determining the fate of radioactive materials and certain types of organic compounds. We present a numerical model designed to evaluate the simultaneous transport and kinetically controlled sequential degradation (straight and branched chains) of several dissolved components in groundwater systems. The model utilizes a two‐step quasi‐linearization algorithm to solve the equations of chemical transport and transformation. The transport equations are solved explicitly using the integral finite difference method. The chemical transformation equations are solved using an implicit finite difference (in time) algorithm for each volume element in the discretized flow domain. Although this algorithm is designed to solve problems involving first‐order kinetics, it may be modified in certain instances to accommodate rate mechanisms other than first order. The chemical transformation module and the transport module are coupled via a source/sink term in the transport equation. This combination results in a numerical code that is computationally efficient. We have found that the model yields solutions which are in excellent agreement with available analytical solutions. Solution of a test problem based on the sequential degradation of the pesticide aldicarb demonstrates that the model can provide useful insights into the fate of solutes subject to certain degradation regimes in heterogeneous groundwater systems. Although the illustrative examples presented are one dimensional, the model itself is capable of handling two‐ and three‐dimensional problems. In addition, the modular structure of the model is built upon user‐specified chemical reactions. This allows for flexibility in problem definition, which may accommodate both reversible and irreversible reactions.

Temperature field modelling along the Northern Segment of the European Geotraverse and the Danish Transition Zone

A two-dimensional numerical solution of the heat conduction equation was used to calculate the crustal temperature field along two profiles of the Northern Segment of the European Geotraverse: the FENNOLORA profile (Baltic Shield) and profile 2 of the EUGENO-S Project (Danish Transition Zone). The solution of the 2-D steady-state equation of heat conduction was solved with the integrated finite-difference method. The crustal structure was discretized by a rectangular grid. Seismic refraction results were used to calculate the corresponding distribution of heat generation. We applied empirical equations linking seismic velocity with heat generation to convert the characteristic seismic velocity into heat production. In the upper crust, the temperature field shows no significant lateral variation. The lower crust, on the other hand, is characterized by lateral and vertical variations of the temperature field mainly at tectonic transition zones (i.e. Precambrian to Phanerozoic crustal domains). In the Baltic Shield, temperatures tend to decrease towards the north with increasing age of the shield. As expected, relatively low Moho temperatures were obtained beneath the Baltic Shield (300–500°C), whereas a maximum of 900 to 1000°C at the Moho was found to correspond to high surface heat flow in southern Sweden. In the geologically complex Danish Transition Zone, the calculated Moho temperatures reached values between 550° C in the Shield region and 650°C under the Danish Basin. The variation of mantle heat flow reflects the pattern of surface heat flow. In the Fennoscandian part of the shield, the Moho heat flow is 5–40 mW/m 2, in the Danish part 25–30 mW/m 2, and below the Danish Basin up to 45 mW/m 2. With the present data set the model shows a remarkable coincidence between high Q 0, the temperature field, and areas of major tectonic change. Sensitivity analysis shows that the surface heat flow and the heat generation in the upper crust are the most critical thermal parameters. Due to their relatively limited range of variation the heat production in the lower crust, the thermal conductivity and corrections applied to the seismic velocities play only a secondary role on the two-dimensional temperature field. The differences between 1-D and 2-D temperature models are most pronounced in the lower crust, where significant lateral temperature variations are seen.

General Purpose Compositional Model

Abstract A direct sequential method has been developed to simulate isothermal compositional systems. The solution technique is the same as that of the implicit pressure, explicit saturation (IMPES) method: one pressure is treated implicitly and (instead of the phase saturation) the component masses/moles are treated explicitly. A "volume balance" equation is used to obtain the pressure equation. A weighted sum of the conservation equations is used to eliminate the nonlinear saturation/concentration terms from the accumulation term of the pressure equation. The partial mass/mole volumes are used as "constants" to weight the mass/mole conservation equations. The method handles uniformly a range of cases from the simplified compositional (i.e., black-oil) models to the most complicated multiphase compositional models of incompressible and compressible fluid systems. The numerical solution is based on the integrated finite-difference method that allows one- (ID), two- (2D), and three-dimensional (3D) grids of regular or irregular volume elements to be handled with the same ease. The mathematical model makes it possible to develop modular versatile computer realizations; thus the model is highly suitable as a basis for general-purpose models.