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.

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.

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.

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.

Construction and analysis of the quadratic finite volume methods on tetrahedral meshes

We construct and analyze a family of quadratic finite volume method(FVM)schemes over tetrahedral meshes.In order to prove the stability and the error estimate,we propose the minimum V-angle condition on tetrahedral meshes,and the surface and volume orthogonal conditions on dual meshes.Through the technique of element analysis,the local stability is equivalent to a positive definiteness of a 9 × 9 element matrix,which is difficult to analyze directly or even numerically.With the help of the surface orthogonal condition and the congruent transformation,this element matrix is reduced into a block diagonal matrix,and then we carry out the stability result under the minimum V-angle condition.It is worth mentioning that the minimum V-angle condition of the tetrahedral case is very different from a simple extension of the minimum angle condition for triangular meshes,while it is also convenient to use in practice.Based on the stability,we prove the optimal H^(1) and L^(2) error estimates,respectively,where the orthogonal conditions play an important role in ensuring the optimal L^(2) convergence rate.Numerical experiments are presented to illustrate our theoretical results.

On the Convergence of a Crank-Nicolson Fitted Finite Volume Method for Pricing European Options under Regime-Switching Kou’s Jump-Diffusion Models

In this paper,we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou’s jumpdiffusion model which is governed by a system of partial integro-differential equations(PIDEs).We show that this scheme is consistent,stable and monotone as the mesh sizes in space and time approach zero,hence it ensures the convergence to the solution of continuous problem.Finally,numerical experiments are performed to demonstrate the efficiency,accuracy and robustness of the proposed method.

Interface Finite Volume Method with Low Calculating Amount for Step Size Varying Electro-Quasistatic Field Problem

In this paper,a novel numerical method called interface finite volume method(I-FVM)for calculation of step-varying Electro-quasistatic(EQS)field is proposed.First,the principle of I-FVM is derived.Then,with numerical example of double layers parallel plate structure under step voltage which has an analytical solution,effectiveness and correctness of the I-FVM are verified.It can be found the calculating time of I-FVM is only 30%of normal FVM without decreasing accuracy during the whole calculating process.Furthermore,an engineering example about the electric field of DBC(Direct Bonding Copper)structure in a high voltage IGBT device is given.It can be found that accuracy of the I-FVM is the same as normal FVM,while time cost of I-FVM is only 20.8%of normal FVM.At last,the I-FVM is extended to one dimension based on the two-direction tri-diagonal matrix algorithm(TDMA)method given in this paper which can save processing of LU decomposition compared to one-dimensional traditional TDMA.In conclusion,the novel method called I-FVM proposed in this paper can decrease calculating amount for a step size varying electro-quasistatic field calculation problem.It may be a good method for large-scale EQS field calculation.

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.

Gas kinetic flux solver based finite volume weighted essentially non-oscillatory scheme for inviscid compressible flows

A high-order gas kinetic flux solver(GKFS)is presented for simulating inviscid compressible flows.The weighted essentially non-oscillatory(WENO)scheme on a uniform mesh in the finite volume formulation is combined with the circular function-based GKFS(C-GKFS)to capture more details of the flow fields with fewer grids.Different from most of the current GKFSs,which are constructed based on the Maxwellian distribution function or its equivalent form,the C-GKFS simplifies the Maxwellian distribution function into the circular function,which ensures that the Euler or Navier-Stokes equations can be recovered correctly.This improves the efficiency of the GKFS and reduces its complexity to facilitate the practical application of engineering.Several benchmark cases are simulated,and good agreement can be obtained in comparison with the references,which demonstrates that the high-order C-GKFS can achieve the desired accuracy.

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.

A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries

In Li and Ren(Int.J.Numer.Methods Fluids 70:742–763,2012),a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain,in which the high-order numerical accuracy and the oscillations-free property can be achieved.In this paper,the method is extended to solve steady state problems imposed in a curved physical domain.The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations,and a geometrical multigrid method to solve the derived linear system.To achieve high-order non-oscillatory numerical solutions,the classical k-exact reconstruction with k=3 and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables.The non-uniform rational B-splines(NURBS)curve is used to provide an exact or a high-order representation of the curved wall boundary.Furthermore,an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state.A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.