Interface‐capturing finite element technique for transient two‐phase flow

A 6M digital twin for modeling and simulation in subsurface reservoirs

Development of a non-linear porous media flow relationship for oscillatory unsteady flow : K. R. Hall, G. M. Smith & D. J. Turcke, Journal of Coastal Research, 10(1), 1994, pp 158–169

Delta is an object-oriented code architecture based on the finite element method which enables simulation of a wide range of engineering mechanics problems in a parallel processing environment. Written in C{sup ++}, Delta is a natural framework for algorithm development and for research involving coupling of mechanics from different Engineering Science disciplines. To enhance flexibility and encourage code reuse, the architecture provides a clean separation of the major aspects of finite element programming. Spatial discretization, temporal discretization, and the solution of linear and nonlinear systems of equations are each implemented separately, independent from the governing field equations. Other attractive features of the Delta architecture include support for constitutive models with internal variables, reusable ``matrix-free`` equation solvers, and support for region-to-region variations in the governing equations and the active degrees of freedom. A demonstration code built from the Delta architecture has been used in two-dimensional and three-dimensional simulations involving dynamic and quasi-static solid mechanics, transient and steady heat transport, and flow in porous media.

Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell's equations in linear Lorentz media

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

In this research study, we conducted a coupled thermal-stress-fluid flow numerical model on the UTF Phase A SAGD project in order to investigate the fundamental geomechanical behaviour involved in the SAGD process, and to gain insight into the reservoir response to temperature and pore pressure changes. The numerical simulation is carried out by using a self-developed coupled finite element model which incorporates our proposed strain-induced permeability model. The obtained simulation results were compared with the measured data. Introduction Thermal recovery processes involve coupling between heat transfer, multiphase flow and stress/deformation, which has become an increasingly important subject in the petroleum field(1). Particularly, the coupling is crucial in problems such as borehole stability, hydraulic fracturing and injection/production induced deformation of the ground surface during the thermal recovery process in heavy oil or oil sand reservoirs. Numerical modelling of the coupled processes is historically carried out in the areas of geomechanics modelling and reservoir simulation. The former is to compute the stress-strain behaviour; therefore, the deformation. The latter is to essentially model the multiphase flow and heat transfer in porous media. Each of these disciplines simplifies the part of the problem that is not of primary interest. These approaches are unacceptable in situations where the coupling is strong and the changes of porosity and permeability cannot be accounted for by rock compressibility alone. Gutierrez and Lewis(2) extend Biot's theory to multiphase fluid flow in deformable porous media. Based on their formulation, they conclude that the coupling between the geomechanics and the multiphase flow occurs simultaneously. Thus, fully coupled system equations of deformations, multiphase flow and heat transfer should be solved simultaneously. Development of such kinds of fully coupled geomechanics-multiphase flow-heat transfer simulators needs tremendous effort, since the existing FEM geomechanics codes and FDM reservoir simulators cannot be used. Settari and Mourits(3) present an approach to couple the stress-strain behaviour to multiphase flow-heat transfer using porosity as a coupling parameter. The geomechanics module and the thermal reservoir simulator are used in a staggered manner. Pore pressure and temperature changes are calculated from the thermal reservoir simulator and transferred to the geomechanics module. The stress and the displacement changes are then calculated in the geomechanics simulation. An iterative algorithm is used to ensure that the porosity calculated from the geomechanics module is the same as that from the thermal reservoir simulator. The staggered technique employed to solve the coupled system equations allows for the use of the existing geomechanics codes in conjunction with a standard reservoir simulator. Currently, most of the commercial coupled geomechanics-multiphase flow-heat transfer simulators are developed in this way. The disadvantage of these kinds of coupled simulators is that the thermal reservoir module, usually developed using the finite difference method (FDM), cannot accommodate the full permeability tensor, since they adopt the standard discretization scheme, such as 5-spot for 2D problems and 7-spot for 3D problems. In this paper, the development of a coupled geomechanics-multiphase flow-heat transfer simulator using the finite element method (FEM) is described with the use of Galerkin's least squares (GLS) technique(4) to stabilize the saturation equation.

This work is about examples of applications of group classifications for continuous (differential equations) and discrete models of flow in porous media. Difference schemes and discrete dynamical systems are considered as discrete models. Group classifications has been carried out on the basis of continuous groups (Lie groups) of transformations in n-dimensional Euclidean spaces. The example of explicit difference scheme for Buckley-Leverett equation has been demonstrated where the correspondence to one of classes of the obtained earlier classifications of discrete dynamical system has been shown. The example of fractal capillary network on the basis of Sierpinski triangle has been demonstrated for the group classification of discrete dynamical systems. The approach of continuous symmetry absence or presence proof has been demonstrated using classifications of discrete dynamical systems. Difference schemes with continuous symmetry groups for partial differential equations of parabolic type, which correspond to the equation of gas flow in porous media and Rapoport-Leas equation (generalized Buckley-Leverett equation), are considered. Examples of difference schemes for these cases are given and related to classes of the obtained earlier classifications. Applications of continuous symmetry groups for numerical solutions generations are demonstrated for these cases.

A linearized energy–conservative finite element method for the nonlinear Schrödinger equation with wave operator

In this paper, a stabilized finite element technique, actualized by streamline upwind Petrov-Galerkin (SUPG) stabilized method and three-step finite element method (FEM), for large eddy simulation (LES) is developed to predict the wind flow with high Reynolds numbers. Weak form of LES motion equation is combined with the SUPG stabilized term for the spatial finite element discretization. An explicit three-step scheme is implemented for the temporal discretization. For the numerical example of 2D wind flow over a square rib at Re=4.2×105, the Smagorinsky’s subgrid-scale (SSGS) model, the DSGS model, and the DSGS model with Cabot near-wall model are applied, and their results are analyzed and compared with experimental results. Furthermore, numerical examples of 3D wind flow around a surface-mounted cube with different Reynolds numbers are performed using DSGS model with Cabot near-wall model based on the present stabilized method to study the wind field and compared with experimental and numerical results. Finally, vortex structures for wind flow around a surface-mounted cube are studied by present numerical method. Stable and satisfactory results are obtained, which are consistent with most of the measurements even under coarse mesh.

A finite element method is practical and applicable for many fields including for geotechnical engineering structures. Seepage through earth dam is difficult to analyze especially dams with multiple zones. The problem becomes complex if it requires deformation analysis taking into account seismic conditions. Therefore, finite element is the best tool for analyzing seepage flow in an earth fill dam. The main objective of the paper is to simulate the seepage flow through an earth fill dam. Hassan Kanosh Dam which is located in the north eastern part of Iraq is chosen as a case study. There are three periods in the life of a dam which may be critical from the standpoint of shear failure and which must be analyzed; during construction condition, full reservoir condition, and rapid drawdown condition. The program SIGMA/W which is a finite element software coupled with SEEP/W, (another GEO-SLOPE software product) to analyze the dam. The stability analysis performed showed that all parts of the dam are safe within the prescribed rang of factors of safety for the possible loading and operation cases. It is recommended, however, that close control of the quality of the works during construction must be maintained to realize the shearing strengths necessary to fulfill the assumed strength and even more. It is recommended to check routinely the foundation conditions during excavation to realize the same end results.

A sequential fully implicit multi-scale meshless multi-point flux method (MS-MMPFA) for nonlinear hyperbolic partial differential equations of fluid flow in heterogeneous porous media is described in this paper. The method extends the recently proposed the meshless multi-point flux approximation (MMPFA) for general fluid flow in porous media [Lukyanov, “Meshless Upscaling Method and its Application to a Fluid Flow in Porous Media”, Proceeding ECMOR XII, 2010] by utilizing advantages of the existing multi-scale finite volume (MSFV) schemes. The MMPFA is based on a gradient approximation commonly used in meshless method and combined with the mixed corrections which ensure linear completeness. In corrected meshless method, the domain boundaries and field variables at the boundaries are approximated with the default accuracy of the method. The MMPFA method was successfully tested for a number of problems where it was clearly shown that the MMPFA gives a good agreement with analytical solutions for a given number of particles. However, the level of detail and range of property variability included in reservoir characterization models leads to a large number of particles to be considered in MMPFA method. In this paper this problem is resolved using a sequential fully implicit MS-MMPFA method. The results are presented, discussed.

In this research study, we conducted a coupled thermalstress-fluid flow numerical modeling on the UTF Phase A SAGD project in order to investigate the fundamental geomechanical behavior involved in the SAGD process and to gain insight into the reservoir response to temperature and pore pressure changes. The numerical simulation is carried out using a self-developed coupled finite element model which incorporates our proposed strain-induced permeability model. The obtained simulation results were compared with the measured data. Introduction Thermal recovery processes involve coupling between heat transfer, multiphase flow and stress/deformation, which has become an increasingly important subject in the petroleum field(1). Particularly, the coupling is crucial in problems such as borehole stability, hydraulic fracturing and injection/production induced deformation of the ground surface during the thermal recovery processes in the heavy oil or oil sand reservoirs. Numerical modeling of the coupled processes is historically carried out in the areas of geomechanics modeling and the reservoir simulation. The former is to compute the stress-strain behavior, therefore the deformation; the latter is to essentially model the multiphase flow and heat transfer in porous media. Each of these disciplines simplifies the part of the problem that is not of primary interest. These approaches are unacceptable in situations where the coupling is strong and the changes of porosity and permeability cannot be accounted by rock compressibility alone. Gutierrez and Lewis(2) extend Biot's theory to multiphase fluid flow in deformable porous media. Based on their formulation, they conclude that the coupling between the geomechanics and the multiphase flow occurs simultaneously. Thus, fully coupled system equations of deformations, multiphase flow and heat transfer should be solved simultaneously. Development of such kinds of fully coupled geomechanics-multiphase flow-heat transfer simulators needs tremendous effort, since the existing FEM geomechanics codes and the FDM reservoir simulators cannot be used. Settari and Mourits(3) present an approach to couple the stress-strain behavior to multiphase flow, heat transfer using porosity as a coupling parameter. The geomechanics module and the thermal reservoir simulator are used in a staggered manner. Pore pressure and temperature changes are calculated from the thermal reservoir simulator and transferred to the geomechanics module. The stress and the displacement changes are then calculated in the geomechanics simulation. An iterative algorithm is used to ensure that the porosity calculated from the geomechanics module is the same as that from the thermal reservoir simulator. The staggered technique employed to solve the coupled system equations allows for the use of the existing geomechanics codes in conjunction with a standard reservoir simulator. Currently, most of the commercial coupled geomechanics-multiphase flow-heat transfer simulators are developed in this way. The disadvantage of these kinds of coupled simulators is that the thermal reservoir module, usually developed using finite difference method (FDM), cannot accommodate the full permeability tensor, since they adopt the standard discretization scheme such as 5-spot for 2-D problems and 7-spot for 3-D problems. In this paper, the development of a coupled geomechanicsmultiphase flow-heat transfer simulator using finite element method (FEM) is described with the use of the Galerkin's least squares (GLS) technique(4) to stabilized the saturation equation.

This work presents the development of an explicit/implicit particle finite element method (PFEM) for the 2D modeling of metal cutting processes. The purpose is to study the efficiency of implicit and explicit time integration schemes in terms of precision, accuracy and computing time. The formulation for implicit and explicit time marching schemes is developed, and a detailed study on the explicit solution steps is presented. The PFEM remeshing procedures for insertion and removal of particles have been improved to model the multiple scales of time and/or space of the solution. The detection and treatment of the rigid tool contact are presented for both, implicit and explicit schemes. The performance of explicit/implicit integration is studied with a set of different two-dimensional orthogonal cutting tests of AISI 4340 steel at cutting speeds ranging from 1 m/s up to 30 m/s. It was shown that if the correct selection of the time integration scheme is made, the computing time can decrease up to 40 times. It allows us to affirm that the computing time of the PFEM simulations can be excessive due to the used time marching scheme independently of the meshing process. As a practical result, a set of recommendations to select the time integration schemes for a given cutting speed are given. This is intended to minimize one of the negative constraints pointed out by the industry when using metal cutting simulators.

Using implicit time discretization methods combined with space discretization by finite differences for solving the diffusion equation in VLSI process simulation-leads to large systems of equations that have to be solved at every time-step. For this purpose, a multigrid (MG) algorithm was constructed. Spatial discretization in cartesian coordinates is done in the (non-transformed) physical domain. It is demonstrated that the method of discretization, relaxation and residual transfer used here combined with a nonlinear variant of the MG-method (Full Approximation Scheme (FAS)) yields convergence rates of less than or about 0.1, even for extremely large time steps, Further it is estimated that the amount of computational work is about ten percent of that for a single grid method using an explicit time discretization scheme for the problem at hand.

Spherical fuel elements are distributed randomly in the pebble bed reactor core and helium flow through the pebble bed to remove nuclear reaction heat. Pebble bed reactor core is usually treated as a uniform porous media flow in thermal hydraulic research. However the porosity distribution is nonuniform and the porosity near the wall increase sharply. A new random model is developed in this paper to investigate thermal hydraulic characteristics of pebble bed reactor core. Porosity assumption is based on porosity measurement of other research. Porosity simulation is divided into three parts according to the distance from wall. In the center of core, porosity is assumed to obey normal distribution, where average porosity is from the experimental relation based on statistical results. The mean and standard deviation of porosity distribution near the wall will increase because of the wall effect, where the distance from the wall is about three times of fuel ball’s diameter. The third part is zone from three times to five times of ball’s diameter departed from the wall. The wall effect of this zone is between center and the wall zone. Based on above assumption, a random porosity simulation is completed to apply in this research. COMSOL Multiphysics 3.5a software is used in this research. COMSOL Multiphysics are a calculation platform using proven Finite Elements Methods (FEM). In this research, Brinkman equation for porous media flow is applied in the simulation. Non-thermal Balance model is used in local heat transfer research between gas and pebble bed. A geometry model is built to simulate HTR-10. Temperature profile of variant porosity is gained from stationary analysis and comparison with uniform porosity is also discussed in the paper. For transient analysis, four cases simulation is carried out in the research. Case 1 and 2 simulate heat transfer phenomena with forced cooling system and with passive cooling system after reactor shut down. Way-Wigner-curve is used in Case 1 and Case 2 to simulate decay heat in the calculation. Case 3 and Case 4 simulate ATWS phenomena with natural convection and without natural convection system when blower is trip off in normal operation. Simulation results also are compared with some ATWS experiments and some discussion is done in the paper. From the results, it can be seen that random porosity will affect temperature distribution near the wall and make outlet temperature non-uniform greatly. The maximum temperature of variant porosity is much greater than the maximum temperature of uniform porosity at the same condition. Transient analyses of variant model show that passive cooling system can remove residual heat even in accident conditions when the blower trip off whether reactor shut down or not and the analyses results correspond substantially with experimental results. In general, variant porosity should be considered in the thermal hydraulic research of pebble bed reactor core. Variant porosity model can provide good prediction of heat transfer phenomena than uniform porosity model. Especially it can explain some transient analysis results.