A combined method using Lattice Boltzmann Method(LBM)and Finite Volume Method(FVM)to simulate geothermal reservoirs in Enhanced Geothermal System(EGS)

With the development of industrial activities,global warming has accelerated due to excessive emission of CO_(2).Enhanced Geothermal System(EGS)utilizes deep geothermal heat for power generation.Although porous medium theory is commonly employed to model geothermal reservoirs in EGS,Hot Dry Rock(HDR)presents a challenge as it consists of impermeable granite with zero porosity,potentially distorting the physical interpretation.To address this,the Lattice Boltzmann Method(LBM)is employed to simulate CO_(2)flow within geothermal reservoirs and the Finite Volume Method(FVM)to solve the energy conservation equation for temperature distribution.This combined method of LBM and FVM is imple-mented using MATLAB.The results showed that the Reynolds numbers(Re)of 3,000 and 8,000 lead to higher heat extraction rates from geothermal reservoirs.However,higher Re values may accelerate thermal breakthrough,posing challenges to EGS operation.Meanwhile,non-equilibrium of density in fractures becomes more pronounced during the system's life cycle,with non-Darcy's law becoming significant at Re values of 3,000 and 8,000.Density stratification due to buoyancy effects significantly impacts temperature distribution within geothermal reservoirs,with buoyancy effects at Re=100 under gravitational influence being noteworthy.Larger Re values(3,000 and 8,000)induce stronger forced convection,leading to more uniform density distribution.The addition of proppant negatively affects heat transfer performance in geothermal reservoirs,especially in single fractures.Practical engineering considerations should determine the quantity of proppant through detailed numerical simulations.

NUMERICAL SIMULATION OF FORGING AND SUBSEQUENT HEAT TREATMENT OF A ROD BY A FINITE VOLUME METHOD

A method to simulate processes of forging and subsequent heat treatment of an axial symmetric rod is formulated in eulerian description and the feasibility is investigated. This method uses finite volume mushes for troching material deformation and an automatically refined facet surface to accurately trace the free surface of the deforming material.In the method,the deforming work piece flows through fixed finite volume meshes using eulerian formulation to describe the conservation laws,Fixed finite volume meshing is particularly suitable for large three-dimensional deformation such as forging because remeshing techniques are not required, which are commonly considered to be the main bottelencek in the ssimulations of large defromation by using the finite element method,By means of this finite volume method, an approach has been developed in the framework of 'metallo-thermo-mechanics' to simulate metallic structure, temperature and stress/strain coupled in the heat treatment process.In a first step of simulation, the heat treatment solver is limited in small deformation hypothesis,and un- coupled with forging. The material is considered as elastic-plastic and takes into account of strain, strain rate and temperature effects on the yield stress.Heat generation due to deformation,heat con- duction and thermal stress are considered.Temperature - dependent phase transformation,stress-in- duced phase transformation,latent heat,transformation stress and strain are included.These ap- proaches are implemented into the commerical commercial computer program MSC/SuperForge and a verification example with experimental date is given as comparison.

A Finite Volume Method with Unstructured Triangular Grids for Numerical Modeling of Tidal Current

The finite volume method （FVM） has many advantages in 2-D shallow water numerical simulation. In this study, the finite volume method is used with unstructured triangular grids to simulate the tidal currents. The Roe scheme is applied in the calculation of the intercell numerical flux, and the MUSCL method is introduced to improve its accuracy. The time integral is a two-step scheme of forecast and revision. For the verification of the present method, the Stoker＇s problem is calculated and the result is compared with the mathematically analytic solutions. The comparison indicates that the method is feasible. A sea area of a port is used as an example to test the method established here. The result shows that the present computational method is satisfactory, and it could be applied to the engineering fields.

High Accuracy Finite Volume Method for Solving Nonlinear Aeroacoustics Problems on Unstructured Meshes

The paper presents a finite volume numerical method universally applicable for solving both linear and nonlinear aeroacoustics problems on arbitrary unstructured meshes. It is based on the vertexcentered multi-parameter scheme offering up to the 6th accuracy order achieved on the Cartesian meshes. An adaptive dissipation is added for the numerical treatment of possible discontinuities. The scheme properties are studied on a series of test cases, its efficiency is demonstrated at simulating the noise suppression in resonance-type liners.

PERTURBATIONAL FINITE VOLUME METHOD FOR THE SOLUTION OF 2-D NAVIER-STOKES EQUATIONS ON UNSTRUCTURED AND STRUCTURED COLOCATED MESHES

Based on the first-order upwind and second-order central type of finite volume (UFV and CFV) scheme, upwind and central type of perturbation finite volume (UPFV and CPFV) schemes of the Navier-Stokes equations were developed. In PFV method, the mass fluxes of across the cell faces of the control volume (CV) were expanded into power series of the grid spacing and the coefficients of the power series were determined by means of the conservation equation itself. The UPFV and CPFV scheme respectively uses the same nodes and expressions as those of the normal first-order upwind and second-order central scheme, which is apt to programming. The results of numerical experiments about the flow in a lid-driven cavity and the problem of transport of a scalar quantity in a known velocity field show that compared to the first-order UFV and second-order CFV schemes, upwind PFV scheme is higher accuracy and resolution, especially better robustness. The numerical computation to flow in a lid-driven cavity shows that the under-relaxation factor can be arbitrarily selected ranging from (0.3) to (0.8) and convergence perform excellent with Reynolds number variation from 10~2 to 10~4.

Simulation of unconventional well tests with the finite volume method

The finite volume method has been successfully applied in several engineering fields and has shown outstanding performance in fluid dynamics simulation. In this paper, the general framework for the simulation ofnear-wellbore systems using the finite volume method is described. The mathematical model and the numerical model developed by the authors are presented and discussed. A radial geometry in the vertical plane was implemented so as to thoroughly describe near-wellbore phenomena. The model was then used to simulate injection tests in an oil reservoir through a horizontal well and proved very powerful to correctly reproduce the transient pressure behavior. The reason for this is the robustness of the method, which is independent of the gridding options because the discretization is performed in the physical space. The model is able to describe the phenomena taking place in the reservoir even in complex situations, i.e. in the presence of heterogeneities and permeability barriers, demonstrating the flexibility of the finite volume method when simulating non-conventional tests. The results are presented in comparison with those obtained with the finite difference numerical approach and with analytical methods, if possible.

A finite volume method for global electromagnetic induction forward modeling on collocated unstructured grids

Global electromagnetic induction provides an efficient way to probe the electrical conductivity in the Earth’s deep interior.Owing to the increasing geomagnetic data especially from high-accuracy geomagnetic satellites,inverting the Earth’s three-dimensional conductivity distribution on a global scale becomes attainable.A key requirement in the global conductivity inversion is to have a forward solver with high-accuracy and efficiency.In this study,a finite volume method for global electromagnetic induction forward modeling is developed based on unstructured grids.Arbitrary polyhedral grids are supported in our algorithms to obtain high geometric adaptability.We employ a cell-centered collocated variable arrangement which allows convenient discretization for complex geometries and straightforward implementation of multigrid technique.To validate the method,we test our code with two synthetic models and compare our finite volume results with an analytical solution and a finite element numerical solution.Good agreements are observed between our solution and other results,indicating acceptable accuracy of the proposed method.

An Unstructured Finite Volume Method for Impact Dynamics of a Thin Plate

The examination of an unstructured finite volume method for structural dynamics is assessed for simulations of systematic impact dynamics. A robust display dual-time stepping method is utilized to obtain time accurate solutions. The study of impact dynamics is a complex problem that should consider strength models and state equations to describe the mechanical behavior of materials. The current method has several features, l） Discrete equations of unstructured finite volume method naturally follow the conservation law. 2） Display dual-time stepping method is suitable for the analysis of impact dynamic problems of time accurate solutions. 3） The method did not produce grid distortion when large deformation appeared. The method is validated by the problem of impact dynamics of an elastic plate with initial conditions and material properties. The results validate the finite element numerical data

Dynamo equation solution using Finite Volume Method for midlatitude ionosphere

Ionosphere is the layer of atmosphere which plays an important role both in space based navigation,positioning and communication systems and HF signals. The structure of the electron density is a function of spatio-temporal variables. The electrodynamic medium is also influenced with earth’s magnetic field, atmospheric chemistry and plasma flow and diffusion under earth’s gravitation. Thus, the unified dynamo equation for the ionosphere is a second order partial differential equation for quasi-static electric potential with variable spatial coefficients. In this study, the inhomogeneous and anisotropic nature of ionosphere that can be formulated as a divergence equation is solved numerically using Finite Volume Method for the first time. The ionosphere and the operators are discretized for the midlatitude region and the solution domain is investigated for Dirichlet type boundary conditions that are built in into the diffusion equation. The analysis indicates that FVM can be a powerful tool in obtaining parametric electrostatic potential distribution in ionosphere.

Cubic Finite Volume Methods for Second Order Elliptic Equations with Variable Coefficients

In this paper, we present a finite volume framework for second order elliptic equations with variable coefficients based on cubic Hermite element. We prove the optimal H1 norm error estimates. A numerical example is given at the end to show the feasibility of the method.

Study on fluid-structure interaction in liquid oxygen feeding pipe systems using finite volume method

The fluid-structure interaction may occur in space launch vehicles,which would lead to bad performance of vehicles,damage equipments on vehicles,or even affect astronauts＇ health.In this paper,analysis on dynamic behavior of liquid oxygen （LOX） feeding pipe system in a large scale launch vehicle is performed,with the effect of fluid-structure interaction （FSI） taken into consideration.The pipe system is simplified as a planar FSI model with Poisson coupling and junction coupling.Numerical tests on pipes between the tank and the pump are solved by the finite volume method.Results show that restrictions weaken the interaction between axial and lateral vibrations.The reasonable results regarding frequencies and modes indicate that the FSI affects substantially the dynamic analysis,and thus highlight the usefulness of the proposed model.This study would provide a reference to the pipe test,as well as facilitate further studies on oscillation suppression.

Three-Dimensional Phase Field Simulations of Hysteresis and Butterfly Loops by the Finite Volume Method

Three-dimensional simulations of ferroelectric hysteresis and butterfly loops are carried out based on solving the time dependent Ginzburg-Landau equations using a finite volume method. The influence of externally mechanical loadings with a tensile strain and a compressive strain on the hysteresis and butterfly loops is studied numerically. Different from the traditional finite element and finite difference methods, the finite volume method is applicable to simulate the ferroelectric phase transitions and properties of ferroelectric materials even for more realistic and physical problems.

SIMULATION OF SOLUTE TRANSPORT USING A COUPLING MODEL BASED ON FINITE VOLUME METHOD IN FRACTURED ROCKS

The discrete version of solute transport equation for porous matrix depicted with the continuum model and the discrete fractured-network model are derived for fractured rocks with the Finite Volume Method （FVM）. The two models are coupled according to the continuity condition of hydraulic head and concentration and the conservation of flow flux and mass flux in the contact plane between porous matrix and fractures. Numerical results show that the simulated concentration of the coupling model based on the FVM agrees well with that from analytical solution, which demonstrates that the coupling model can effectively be applied to the simulation of solute transport in fractured rocks.

FINITE VOLUME METHOD FOR SIMULATION OF VISCOELASTIC FLOW THROUGH A EXPANSION CHANNEL

A finite volume method for the numerical solution of viscoelastic flows is given. The flow of a differential Upper-Convected Maxwell （UCM） fluid through an abrupt expansion has been chosen as a prototype example. The conservation and constitutive equations are solved using the Finite Volume Method （FVM） in a staggered grid with an upwind scheme for the viscoelastic stresses and a hybrid scheme for the velocities. An enhanced-in-speed pressure-correction algorithm is used and a method for handling the source term in the momentum equations is employed. Improved accuracy is achieved by a special discretization of the boundary conditions. Stable solutions are obtained for higher Weissenberg number （We）, further extending the range of simulations with the FVM. Numerical results show the viscoelasticity of polymer solutions is the main factor influencing the sweeo efficiency.

Integrating 1D and 2D hydrodynamic,sediment transport model for dam-break flow using finite volume method

The purpose of this study is to set up a dynamically linked 1D and 2D hydrodynamic and sediment transport models for dam break flow.The 1D-2D coupling model solves the generalized shallow water equations,the non-equilibrium sediment transport and bed change equations in a coupled fashion using an explicit finite volume method.It considers interactions among transient flow,strong sediment transport and rapid bed change by including bed change and variable flow density in the flow continuity and momentum equations.An unstructured Quadtree rectangular grid with local refinement is used in the 2D model.The intercell flux is computed by the HLL approximate Riemann solver with shock captured capability for computing the dry-to-wet interface for all models.The effects of pressure and gravity are included in source term in this coupling model which can simplify the computation and eliminate numerical imbalance between source and flux terms.The developed model has been tested against experimental and real-life case of dam-break flow over fix bed and movable bed.The results are compared with analytical solution and measured data with good agreement.The simulation results demonstrate that the coupling model is capable of calculating the flow,erosion and deposition for dam break flows in complicated natural domains.

INTERVAL FINITE VOLUME METHOD FOR UNCERTAINTY SIMULATION OF TWO-DIMENSIONAL RIVER WATER QUALITY

Under the interval uncertainties, by incorporating the discretization form offinite volume method and interval algebra theory, an Interval Finite Volume Method (IF-VM) wasdeveloped to solve water quality simulation issues for two-dimensional river when lacking effectivedata of flow velocity and flow quantity. The IFVM was practically applied to a segment of theXiangjiang River because the Project of Hunan Inland Waterway Multipurpose must be started workingafter the environmental impact assessment for it. The simulation results suggest that there existrather apparent pollution zones of BOD_5 downstream the Dongqiaogang discharger and that of CODdownstream Xiaoxiangjie discharger, but the pollution sources have no impact on the safety of thethree water plants located in this river segment. Although the developed IFVM is to be perfected, itis still a powerful tool under interval uncertainties for water environmental impact assessment,risk analysis, and water quality planning, etc. besides water quality simulation studied in thispaper.

Numerical Study of Heat and Moisture Transfer in Textile Materials by a Finite Volume Method

This paper focuses on the numerical study of heat and moisture transfer in clothing assemblies,based on a multi-component and multiphase flow model which includes heat/moisture convection and conduction/diffusion as well as phase change.A splitting semi-implicit finite volume method is proposed for solving a set of nonlinear convection-diffusion-reaction equations,in which the calculation of liquid water content absorbed by fiber is decoupled from the rest of the computation.The method maintains the conservation of air,vapor and heat flux(energy).Four types of clothing assemblies are investigated and comparison with experimental measurements are also presented.

NUMERICAL INVESTIGATION ON NON-NEWTONIAN FLOWS THROUGH DOUBLE CONSTRICTIONS BY AN UNSTRUCTURED FINITE VOLUME METHOD

This article presents a numerical investigation on a steady non-Newtonian flow through a two-dimensional channel with double constrictions. The power-law mode is employed in describing the non-Newtonian behavior of the flow. An unstructured finite volume method combined with a fractional-step projection method is developed for the discretization of incompressible equations governing the non-Newtonian flows. The important flow dynamics related with the arterial diseases, such as the wall shear stress and vortex generation, are also numerically studied in detail. Numerical results reveal that there are marked differences between Newtonian and non-Newtonian models.

TWO-GRID CHARACTERISTIC FINITE VOLUME METHODS FOR NONLINEAR PARABOLIC PROBLEMS＊

In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two- grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O（H2） or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = 0（I log hll/2H3）. These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods.

A NEW FINITE VOLUME METHOD FOR SOLVING SHALLOW WATER EQUATIONS

This paper presents a new numerical model using the finite volume method with arbitrary polygonal elements,It is suitable for simulating the shallow water problem with com- plex topography and boundary conditions.The calculated results show that this kind of method has the advantages of good stability,convenience of solving,easy programming,less,computing time and storage,high accuracy,ect.