Solving Flow Equations Research Articles

Predictive modeling of flow characteristics in supersonic separators using machine learning

In the pursuit of cleaner energy sources, pre-processing fuel streams is crucial before distribution. Challenges arise during this process, leading to operational issues and diminished transmission efficiency. Among different technologies, supersonic separators are the ones that separate undesired components from the main gas stream more effectively. Precise forecasting of nozzle flow rate and shockwave location at a low cost can markedly enhance the efficiency of manufacturing and rating. In this research, the one-dimensional method has been developed to solve flow equations simultaneously by implementing equations of state. CFD has been used for the verification of the developed model, and the effect of different equations of state is investigated. Results show that the one-dimensional model significantly reduces the calculation time. Different EOSs result in up to 2.0 % and 3.5 % variation in the prediction of shock location and the mass flow rate, respectively.A machine learning methodology is employed to predict the location of shockwaves and the choked flow rate at an even lower cost. A dataset of 160 entries, generated using Peng Robinson's EOS, serves as the basis for this analysis. Prediction tasks are executed using Random Forest, k-nearest Neighbors, and Multilayer Perceptron models. While all three models demonstrate proficiency in data interpolation with R2 scores up to 0.9962 and 0.9951 for the shockwave location and mass flow rate, respectively, discerning their performance in extrapolation appears contingent on the specific case. In the case of extrapolation, Multilayer Perceptron and k-Nearest Neighbors show better match to one-dimensional results.

The weighted multi-scale connections networks for macrodispersivity estimation

Macrodispersivity is critical for predicting solute behaviors with dispersive transport models. Conventional methods of estimating macrodispersivity usually need to solve flow equations and are time-consuming. Convolutional neural networks (CNN) have recently been proven capable of efficiently mapping the hydraulic conductivity field and macrodispersivity. However, the mapping accuracy still needs further improvement. In this paper, we present a new network shortcut connection style called weighted multi-scale connections (WMC) for convolutional neural networks to improve mapping accuracy. We provide empirical evidence showing that the WMC can improve the performance of CNN in macrodispersivity estimation by implementing the WMC in CNNs (CNN without short-cut connections, ResNet, and DenseNet), and evaluating them on datasets of macrodispersivity estimation. For the CNN without short-cut connections, the WMC can improve the estimating R2 by at least 3% on three datasets of conductivity fields. For ResNet18, the WMC improved the estimated R2 by an average of 2.5% on all three datasets. For ResNet34, the WMC improved the estimated R2 by an average of 5.6%. For ResNet50, the WMC improved the estimated R2 by an average of 16%. For ResNet101, the WMC improved the estimating R2 by an average of 30%. For DenseNets, the improved estimated R2 ranges from 0.5% to 5%. The WMC can strengthen feature propagation of different sizes and alleviate the vanishing-gradient issue. Moreover, it can be implemented to any CNN with down-sampling layers or blocks.

Effects of Annular-finned Tube Deformation on Thermal Stresses in a Heat Exchanger

Nowadays, along with the developments in the industry, many machines and structures are exposed to very high-temperature environments, which has caused various types of thermal load in them. These conditions have led to focused studies on thermal stresses in structures. Heat exchangers are among the structures in which the temperature difference causes thermal stress. In this paper, “finite volume” and “finite element” methods are used to solve flow equations and thermal stress equations in solids, respectively. In addition, the effect of changes in the shape of the annular finned pipe on the thermal stresses in the heat exchanger is investigated. The study results on changing the pipe shape from circular to elliptical show that by changing the pipe shape, the thermal stress in the fin can be significantly reduced; for instance, in the cases reviewed in this paper, the thermal stress has decreased by 9%. Furthermore, the results of thermal stress show that the Maximum Effective Stress point, both before and after changing the shape, is still located at the fin’s base. So, according to the results of this study, examining the effect of pipe shape changing on thermal stress is an imperative measure when designing heat exchangers.

Accurate Two-Phase Flow Simulation in Faulted Reservoirs by Combining Two-Point Flux Approximation and Mimetic Finite Difference Methods

Summary Faulted reservoirs are commonly modeled by corner-point grids (CPGs). Because the two-point flux approximation (TPFA) method is not consistent on non-K-orthogonal grids, multi-phase flow simulation using TPFA on CPGs may have significant discretization errors if grids are not K-orthogonal. We developed a novel method to improve the simulation accuracy where the faults are modeled by polyhedral cells, and mimetic finite difference (MFD) methods are used to solve flow equations. We use a cut-cell approach to build the mesh for faulted reservoirs. A regular K-orthogonal grid is first constructed, and then cells are divided where fault planes are present. Most cells remain K-orthogonal while irregular non-K-orthogonal polyhedral cells can be formed with multiple cell divisions. We investigated three spatial discretization methods for solving the pressure equation on general polyhedral grids, including the TPFA, MFD, and TPFA-MFD hybrid methods. In the TPFA-MFD hybrid method, the MFD method is only applied to the part of the domain with severe grid non-K-orthogonality, while the TPFA method is applied to the rest of the domain. We compare flux accuracy between TPFA and MFD methods by solving a single-phase flow problem. The reference solution is obtained on a rectangular grid, while the same problem is solved by TPFA and MFD methods on a grid with non-K-orthogonal cells near a fault. Fluxes computed using TPFA exhibit larger errors in the vicinity of the fault, while fluxes computed using MFD are still as accurate as the reference solution. We also compare saturation accuracy for two-phase (oil and water) flow in faulted reservoirs when the pressure equation is solved by different discretization methods. Compared with the reference saturation solution, saturation exhibits non-physical errors near the fault when the pressure equation is solved by the TPFA method. Because the MFD method yields accurate fluxes over general polyhedral grids, the resulting saturation solutions agree with reference saturation solutions with an enhanced accuracy when the pressure equation is solved by the MFD method. Based on the results of our simulation studies, we observe that the accuracy of the TPFA-MFD hybrid method is very close to the accuracy of the MFD method, while the TPFA-MFD hybrid method is computationally cheaper than the MFD method.

A Flownet-Based Method for History Matching and Production Prediction of Shale or Tight Reservoirs with Fracturing Treatment

Summary Based on a newly developed physics-based data-driven model FlowNet, this paper presents an effective method for history matching and production prediction of fractured shale or tight reservoirs without any prior information about fracture geometry. In this method, four types of well nodes including fracturing cluster nodes, fracture nodes, stimulated reservoir volume (SRV) nodes, and matrix nodes are allocated in the reservoir. Then, the reservoir model is simplified as a flow network composed of some 1D connection elements between these nodes. Some grids are divided on each connection element, and the grids on the same connection element are of equal width and permeability. Subsequently, a fully implicit nonlinear solver is used to solve flow equations in this FlowNet grid system to obtain pressure, phase saturation, and production rates, etc. Efficient history-matching procedure based on the FlowNet model of the fractured reservoir is used to determine the parameters of connection elements, and then fast production prediction can be conducted. Five numerical examples including single-well depletion, waterflooding development with natural fractures, multiple-well interference, three-phase flow, and an actual waterflooding field case validate that this presented FlowNet-based method can achieve good history matching and production prediction for various flow problems in shale or tight reservoirs with fracturing treatment, and the history-matched transmissibility and volume of connection elements can reflect the existence of high-conductivity fractures.

Analysis of the impacts of relative permeability and mobility ratio on heterogeneity loss error during upscaling of geological models

The detailed geological fine grids are upscaled to create reliably sized simulation coarse models to solve flow equations in a more efficient way. Any upscaling process results in a loss of accuracy, along with an increase of errors. Numerical dispersion, heterogeneity loss, and connectivity misrepresentation are responsible for the upscaling errors. Recognizing the source of each error, and the behavior of influential factors through upscaling process could provide an optimum level of upscaling and an evaluation of upscaling methods’ accuracy. Despite the importance of upscaling error, little attention has been paid to this subject. This paper represents a rigorous analysis of the heterogeneity loss behavior associated with the relative permeability contrast and the mobility ratio under a waterflooding process. For this purpose, heterogeneous fine grid models are constructed by the fractional Brownian motion process. The models are upscaled by three upscaling factors. The models achieved are implemented to eliminate the impact of numerical error among upscaling errors in order to focus strictly on heterogeneity loss. Water–oil displacement simulation is then performed on fine and corresponding refined upscaled models at three different ratios of relative permeabilities and mobility ratios. In the next stage, the relation between flow performance error and heterogeneity loss is investigated by the heterogeneity loss plot. The slope of this plot provides the reservoir engineer an insight to evaluate the performance of upscaling methods and the behavior of the influential factors on upscaling errors. Moreover, by using the heterogeneity loss plot for each ratio, a limit of coarsening is presented. Based on the results, the heterogeneity loss error is affected more by the mobility ratio contrast than the relative permeability difference. Also, it is demonstrated that water-wet reservoirs with light oil are more sensitive to the level of upscaling.

Correction to: Constraint energy minimizing generalized multiscale finite element method in the mixed formulation

This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast.

Development of a transient non-isothermal two-phase flow model for gas kick simulation in HTHP deep well drilling

The simulation of gas/liquid two-phase flow, considering the heat transfer between the annulus fluid and the surrounding environment, is of significance in predicting temperature and pressure distributions after a gas kick in HTHP deep well drilling. This paper presents the development of a transient non-isothermal two-phase flow model for the dynamic simulation of multiphase flow in the wellbore after a gas kick. The drift-flux model is used to describe gas/liquid two-phase flow, and multiple transient energy conservation equations are used for predicting temperature profiles of fluids in the drillpipe, drillpipe, the fluid in the annulus, casing string, and formation. As for numerical scheme of this strongly coupled model, the advection upstream splitting model (AUSMV) hybrid scheme is adapted for solving flow equations, while the finite difference approach is adapted for simultaneously solving energy conservation equations of the wellbore-formation system. Physical properties of gas and liquid phases are updated at each timestep. Predicted temperature and pressure distributions are validated against the field data. Flow behaviors predicted by the models with and without the heat transfer effect are compared. The effects of some major parameters (reservoir pressure, choke pressure, geothermal gradient, liquid mass flow rate) on temperature, pressure, and gas fraction distributions along the wellbore are investigated.

Constraint energy minimizing generalized multiscale finite element method in the mixed formulation

This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass conservative on the fine grid. Our main goal is to obtain first-order convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that can not be localized, in general. This is built on our previous work on the Generalized Multiscale Finite Element Method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrast-dependent) eigenvalues are selected. These basis functions represent the high-conductivity channels (which connect the boundaries of a coarse block). Next, we solve local problems to construct multiscale basis functions for the velocity field. These local problems are formulated in the oversampled domain taking into account some constraints with respect to auxiliary spaces. The latter allows fast spatial decay of local solutions and, thus, allows taking smaller oversampled regions. The number of basis functions depends on small eigenvalues of the local spectral problems. Moreover, multiscale pressure basis functions are needed in constructing the velocity space. Our multiscale spaces have a minimal dimension, which is needed to avoid contrast-dependence in the convergence. The method's convergence requires an oversampling of several layers. We present an analysis of our approach. Our numerical results confirm that the convergence rate is first order with respect to the mesh size and independent of the contrast.

Robust Finite Element Approaches to Systemic Circulation Using the Locally Conservative Galerkin (LCG) Method

In this paper novel finite element algorithms for robustly solving flow equations of a systemic circulation are presented and compared. A novel, Locally Conservative Galerkin (LCG) method is developed by adopting semi- and fully- implicit time discretisations to address the behavior of blood flow in the human circulatory system. These techniques are efficient for nonlinear system of equations used in blood flow models and achieve rapid convergence. Several comparisons are made between the methods to demonstrate the validity and stability of the current numerical models to solve the blood flow characteristics in a human body.

Current noise of the interacting resonant level model

We study the zero-frequency current noise of the interacting resonant level model for arbitrary bias voltages using a functional renormalization group approach. For this we extend the existing nonequilibrium scheme by deriving and solving flow equations for the current-vertex functions. On-resonance artificial divergences of the latter found in lowest-order perturbation theory in the two-particle interaction are consistently removed. Away from resonance they are shifted to higher orders. This allows us to gain a comprehensive picture of the current noise in the scaling limit. At high bias voltages, the current noise exhibits a universal power-law decay, whose exponent is to leading order in the interaction identical to that of the current. The effective charge on resonance is analyzed in detail employing properties of the vertex correction. We find that it is only modified to second or higher order in the two-particle interaction.

On the use of Darcy's law and invasion‐percolation approaches for modeling large‐scale geologic carbon sequestration

AbstractMost large‐scale flow and transport simulations for geologic carbon sequestration (GCS) applications are carried out using simulators that solve flow equations arising from Darcy's law. Recently, the computational advantages of invasion‐percolation (IP) modeling approaches have been presented. We show that both the Darcy's‐law‐ and the gravity‐capillary balance solved by IP approaches can be derived from the same multiphase continuum momentum equation. More specifically, Darcy's law arises from assuming creeping flow with no viscous momentum transfer to stationary solid grains, while it is assumed in the IP approach that gravity and capillarity are the dominant driving forces in a quasi‐static two‐phase (or more) system. There is a long history of use of Darcy's law for large‐scale GCS simulation. However, simulations based on Darcy's law commonly include significant numerical dispersion as users employ large grid blocks to keep run times practical. In contrast, the computational simplicity of IP approaches allows large‐scale models to honor fine‐scale hydrostratigraphic details of the storage formation which makes these IP models suitable for analyzing the impact of small‐scale heterogeneities on flow. However, the lack of time‐dependence in the IP models is a significant disadvantage, while the ability of Darcy's law to simulate a range of flows from single‐phase‐ and pressure‐gradient‐driven flows to buoyant multiphase gravity‐capillary flow is a significant advantage. We believe on balance that Darcy's law simulations should be the preferred approach to large‐scale GCS simulations. © 2015 Society of Chemical Industry and John Wiley & Sons, Ltd

Annular Flow Characteristics of Pseudoplastic Fluids

In this paper, the problem of axial annular flow of non-Newtonian fluids is examined. By utilizing the slot analogy, a Fanning friction factor—Reynolds number relationship for a power law fluid was developed and presented. Good agreement over the entire range of flow regimes was obtained between model predictions and experimental data. The advantage of the proposed approach is that it eliminates the need to determine the dimensionless radial position of zero shear stress required to solve flow equations. Practical applications of this work include processes in the petroleum and chemical industries in which annular flow of non-Newtonian fluids is a common occurrence.

Hybrid Vortex Method for the Aerodynamic Analysis of Wind Turbine

The hybrid vortex method, in which vortex panel method is combined with the viscous-vortex particle method (HPVP), was established to model the wind turbine aerodynamic and relevant numerical procedure program was developed to solve flow equations. The panel method was used to calculate the blade surface vortex sheets and the vortex particle method was employed to simulate the blade wake vortices. As a result of numerical calculations on the flow over a wind turbine, the HPVP method shows significant advantages in accuracy and less computation resource consuming. The validation of the aerodynamic parameters against Phase VI wind turbine experimental data is performed, which shows reasonable agreement.

A semi-parabolic wake model for large offshore wind farms based on the open source CFD solver OpenFOAM

Wake effect represents one of the main sources of energy loss and uncertainty when designing offshore wind farms. Traditionally analytical models have been used to optimize and estimate power deficits. However these models have shown to underestimate wake effect and consequently overestimate output power (1, 2). This means that analytical models can be very helpful at optimizing preliminary layouts but not as accurate as needed for an ultimate fine design. Different techniques can be found in the literature to study wind turbine wakes that include simplified kinematic models and more advanced field models, that solve flow equations with different turbulence closure schemes. See the review papers of Crespo et al. (3), Vermeer et al. (4), and Sanderse et al. (5). Purely elliptic Computational Fluid Dynamics (CFD) models based on the actuator disk technique have been developed during the last years (6-8). They consider wind turbine rotor as a disk where a distribution of axial forces act over the incoming air. It is a fair approach but it can still be computationally expensive for big wind farms in an operative mode. With this technique still active, an alternative approach inspired on the parabolic wake models (9, 10) is proposed. Wind turbine rotors continue to be represented as actuator disks but now the domain is split into subdomains containing one or more wind turbines. The output of each subdomain is mapped onto the input boundary of the next one until the end of the domain is reached, getting a considerable decrease on computational time, by a factor of order 10. As the model is based on the open source CFD solver OpenFOAM, it can be parallelized to speed-up convergence. The near wake is calculated so no initial wind speed deficit profiles have to be supposed as in totally parabolic models and alternative turbulence models, such as the anisotropic Reynolds Stress Model (RSM) can be used. Traditional problems of elliptic models related to the estimation of the reference wind speed at each rotor position are mitigated due to the semi-parabolic algorithm. The model has been validated at the ECN test farm and at the offshore Horns Rev wind farm with significant results and also have been compared to other wake models.

Analytic continuation of functional renormalization group equations

Abstract Functional renormalization group equations are analytically continued from imaginary Matsubara frequencies to the real frequency axis. On the example of a scalar field with $ \mathcal{O} $ (N) symmetry we discuss the analytic structure of the flowing action and show how it is possible to derive and solve flow equations for real-time properties such as propagator residues and particle decay widths. The formalism conserves space-time symmetries such as Lorentz or Galilei invariance and allows for improved, self-consistent approximations in terms of derivative expansions in Minkowski space.

Study on adjoint-based optimization method for multi-stage turbomachinery

Adjoint-based optimization method is a hotspot in turbomachinery. First, this paper presents principles of adjoint method from Lagrange multiplier viewpoint. Second, combining a continuous route with thin layer RANS equations, we formulate adjoint equations and anti-physical boundary conditions. Due to the multi-stage environment in turbomachinery, an adjoint interrow mixing method is introduced. Numerical techniques of solving flow equations and adjoint equations are almost the same, and once they are converged respectively, the gradients of an objective function to design variables can be calculated using complex method efficiently. Third, integrating a shape perturbation parameterization and a simple steepest descent method, a frame of adjoint-based aerodynamic shape optimization for multi-stage turbomachinery is constructed. At last, an inverse design of an annular cascade is employed to validate the above approach, and adjoint field of an Aachen 1.5 stage turbine demonstrates the conservation and areflexia of the adjoint interrow mixing method. Then a direct redesign of a 1+1 counter-rotating turbine aiming to increase efficiency and apply constraints to mass flow rate and pressure ratio is taken.

An efficient parallel mixed method for flow simulations in heterogeneous geological media

The permeability of a 3D geological fracture network is determined by triangulating the fractures and solving the 2D Darcy's equation in each fracture. Here, the numerical modelling aims to simulate a great number of networks made up of a great number of fractures i.e. from 103 to 106 fractures. Parallel computing allows us to solve very large linear systems improving the realism of simulations. Several algorithms to simulating fluid flow are proposed for the cases of significant matrix permeability. In the case of a weak permeability matrix, the flow is focused in the fractures having a strong permeability and fluids percolate through networks of interconnected fractures. In this paper, we present a complete parallel algorithm for solving flow equations in fracture networks. We consider an imprevious matrix. The different parts of the algorithm are detailed. Numerical examples using the mixed finite element (MFE) method for various fracture networks illustrate the efficiency and robustness of the proposed algorithm. To the best of our knowledge, results for parellel simulation of fluid flow in discrete-fractured media with impervious matrix using the MFE method are the first to appear in the literature.

The shallow water equations in Lagrangian coordinates

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. We apply numerical methods traditionally used to solve differential equations in Eulerian coordinates, to solve the shallow water equations in Lagrangian coordinates. The difficulty with solving in Lagrangian coordinates is that the transformation from Eulerian coordinates results in solving a highly nonlinear partial differential equation. The non-linearity is mainly due to the Jacobian of the coordinate transformation, which is a precise record of how the particles are rotated and stretched. The inverse Jacobian must be calculated, thus Lagrangian coordinates cannot be used in instances where the Jacobian vanishes. For linear (spatial) flows we give an explicit formula for the Jacobian and describe the two situations where the Lagrangian shallow water equations cannot be used because either the Jacobian vanishes or the shallow water assumption is violated. We also prove that linear (in space) steady state solutions of the Lagrangian shallow water equations have Jacobian equal to one. In the situations where the shallow water equations can be solved in Lagrangian coordinates, accurate numerical solutions are found with finite differences, the Chebyshev pseudospectral method, and the fourth order Runge–Kutta method. The numerical results shown here emphasize the need for high order temporal approximations for long time integrations.

High resolution conjugate filters for the simulation of flows

This paper proposes a Hermite-kernel realization of the conjugate filter oscillation reduction (CFOR) scheme for the simulation of fluid flows. The Hermite kernel is constructed by using the discrete singular convolution (DSC) algorithm, which provides a systematic generation of low-pass filter and its conjugate high-pass filters. The high-pass filters are utilized for approximating spatial derivatives in solving flow equations, while the conjugate low-pass filter is activated to eliminate spurious oscillations accumulated during the time evolution of a flow. As both low-pass and high-pass filters are derived from the Hermite kernel, they have similar regularity, time-frequency localization, effective frequency band and compact support. Fourier analysis indicates that the CFOR-Hermite scheme yields a nearly optimal resolution and has a better approximation to the ideal low-pass filter than previous CFOR schemes. Thus, it has better potential for resolving natural high frequency oscillations from a shock. Extensive one- and two-dimensional numerical examples, including both incompressible and compressible flows, with or without shocks, are employed to explore the utility, test the resolution, and examine the stability of the present CFOR-Hermite scheme. Small ratio of point-per-wavelength (PPW) is achieved in advancing a wavepacket and resolving a shock/entropy wave interaction. The present results for the advection of an isentropic vortex compare very favorably to those in the literature.