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.

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.

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.

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.

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.

A hybrid finite volume/finite element method for incompressible generalized Newtonian fluid flows on unstructured triangular meshes

This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow （Power-Law model）. The collocated （i.e. non-staggered） arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.

A Coastal Ocean Model of Semi-Implicit Finite Volume Unstructured Grid

A two-dimensional coastal ocean model based on unstructured C-grid is built, in which the momentum equation is discretized on the faces of each cell, and the continuity equation is discretized on the cell. The model is discretized by semi-implicit finite volume method, in that the free surface is semi-implicit and the bottom friction is implicit, thereby removing stability limitations associated with the surface gravity wave and friction. The remaining terms in the momentum equations are discretized explicitly by integral finite volume method and second-order Adams-Bashforth method. Tidal flow in the polar quadrant with known analytic solution is employed to test the proposed model. Finally, the performance of the present model to simulate tidal flow in a geometrically complex domain is examined by simulation of tidal currents in the Pearl River Estuary.

An Accurate Multimoment Constrained Finite Volume Transport Model on Yin-Yang Grids

A global transport model is proposed in which a multimoment constrained finite volume （MCV） scheme is applied to a Yin-Yang overset grid. The MCV scheme defines 16 degrees of freedom （DOFs） within each element to build a 2D cubic reconstruction polynomial. The time evolution equations for DOFs are derived from constraint conditions on moments of line-integrated averages （LIA）, point values （PV）, and values of first-order derivatives （DV）. The Yin-Yang grid eliminates polar singularities and results in a quasi-uniform mesh. A limiting projection is designed to remove nonphysical oscillations around discontinuities. Our model was tested against widely used benchmarks; the competitive results reveal that the model is accurate and promising for developing general circulation models.

A Hybrid Finite-Volume and Finite Difference Scheme for Depth- Integrated Non-Hydrostatic Model

A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into hydrostatic and non-hydrostatic parts. The first part is solved by using the finite volume conservative discretization method, whilst the latter is considered by solving discretized Poisson-type equations with the finite difference method. The second-order accuracy, both in time and space, of the finite volume scheme is achieved by using an explicit predictor-correction step and linear construction of variable state in cells. The fluxes across the cell faces are computed in a Godunov-based manner by using MUSTA scheme. Slope and flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purpose. Wave breaking is treated as a shock by switching off the non-hydrostatic pressure in the steep wave front locally. The model deals with moving wet/dry front in a simple way. Numerical experiments are conducted to verify the proposed model.

A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

Characteristic-based finite volume scheme for 1D Euler equations

In this paper, a high-order finite-volume scheme is presented for the one- dimensional scalar and inviscid Euler conservation laws. The Simpson＇s quadrature rule is used to achieve high-order accuracy in time. To get the point value of the Simpson＇s quadrature, the characteristic theory is used to obtain the positions of the grid points at each sub-time stage along the characteristic curves, and the third-order and fifth-order central weighted essentially non-oscillatory （CWENO） reconstruction is adopted to estimate the cell point values. Several standard one-dimensional examples are used to verify the high-order accuracy, convergence and capability of capturing shock.

A TVD-WAF-based hybrid finite volume and finite difference scheme for nonlinearly dispersive wave equations

A total variation diminishing-weighted average flux （TVD-WAF）-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer （HLL） Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth- order monotone upstream-centered scheme for conservation laws （MUSCL）. The time marching scheme based on the third-order TVD Runge- Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.

A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

After discretization by the finite volume method,the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure.The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix.Moreover,we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations,which is unconditionally convergent for any positive constant.Meanwhile,the iteration methods introduce a new preconditioner for Krylov subspace methods.Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner,compared with the existing approaches.