This paper applies GFDM to analyze coupled two-phase flow and geomechanics for the first time, to our knowledge, is also the first meshless solver for two-phase fluid-solid coupling problems in porous media. Firstly, the implicit GFDM-based discrete schemes of the hyperbolic two-phase flow equation and elliptic stress equilibrium equation coupled by Biot's poroelastic theory are derived. Then the nonlinear solver based on Newton's method is used to solve the fluid pressure, water saturation, and displacement at each node simultaneously. Two technical details are found critical to the implementation of this method. The one is using different radii of the node influence domain to discretize the flow equation and the stress equilibrium equation respectively, and the other one is the selection criterion of the two radii of the node influence domain, which improves the implementation flexibility of the multi-physics coupling calculation, reduces the dissipation error of the calculation results of the convection-dominated water saturation distribution, and ensure a low computational cost. Several numerical test cases are implemented to analyze the computational efficiency and accuracy, and illustrate that GFDM can achieve good computational performance in case of regular domains, irregular domains, and different point clouds. Overall, this work may provide an important reference for building a GFDM-based general-purpose numerical simulator for multi-physics flow problems in porous media. 

Generalized finite difference method based meshless analysis for coupled two-phase porous flow and geomechanics


 In this paper, a meshless numerical modeling method named mesh-free discrete fracture model (MFDFM) of fractured reservoirs based on the newly developed extended finite volume method (EFVM) is proposed. First, matching and nonmatching point cloud generation algorithms are developed to discretize the reservoir domain with fracture networks, which avoid the gridding challenges of the reservoir domain in traditional mesh-based methods. Then, taking oil/water two-phase flow in fractured reservoirs as an example, MFDFM derives the EFVM discrete scheme of the governing equations, constructs various types of connections between matrix nodes and fracture nodes, and calculates the corresponding transmissibilities. Finally, the EFVM discrete scheme of the governing equations and the generalized finite difference discrete scheme of various boundary conditions form the global nonlinear equations, which do not increase the degree of nonlinearity compared with those in the traditional finite volume method (FVM)-based numerical simulator. The global equations can be solved by the existing nonlinear solver in the FVM-based reservoir numerical simulator by only adding the linear discrete equations of boundary conditions, which reduce the difficulty of forming a general purpose MFDFM-based fractured reservoir numerical simulator. Several numerical test cases are implemented to illustrate that the proposed MFDFM can achieve good computational performance under matching and nonmatching point clouds, and for heterogeneous reservoirs, complex fracture networks, complex boundary geometry, and complex boundary conditions, by comparing the computational results of MFDFM with embedded discrete fracture model (EDFM). Thus, MFDFM retains the computational performances of the traditional mesh-based methods and can avoid the difficulties of handling complex geometry and complex boundary conditions of the computational domain, which is the first meshless numerical framework to model fractured reservoirs in parallel with the mesh-based discrete fracture model (DFM) and EDFM.

/pdf/a-meshless-numerical-modeling-method-for-fractured-3u8lmd5h.pdf

A Meshless Numerical Modeling Method for Fractured Reservoirs Based on Extended Finite Volume Method

Machine learning techniques have garnered significant attention in various engineering disciplines due to their potential and benefits. Specifically, in reservoir numerical simulations, the core process revolves around solving the partial differential equations delineating oil, gas, and water flow dynamics in porous media. Discretizing these partial differential equations via numerical methods is one cornerstone of this simulation process. The synergy between traditional numerical methods and machine learning can enhance the precision of partial differential equation discretization. Moreover, machine learning algorithms can be employed to solve partial differential equations directly, yielding rapid convergence, heightened computational efficiency, and accuracies surpassing 95%. This manuscript offers an overview of the predominant numerical methods in reservoir simulations, focusing on integrating machine learning methodologies. The innovations in fusing deep learning techniques to solve reservoir partial differential equations are illuminated, coupled with a concise discussion of their inherent advantages and constraints. As machine learning continues to evolve, its conjunction with numerical methods is poised to be pivotal in addressing complex reservoir engineering challenges.

Progress and Challenges of Integrated Machine Learning and Traditional Numerical Algorithms: Taking Reservoir Numerical Simulation as an Example

This paper applied the space-time generalized finite difference scheme for solving the nonlinear dispersive shallow water waves as the modified Camassa−Holm equation, modified Degasperis-Procesi equation, Fornberg-Whitham equation, and its modified form. The proposed meshless numerical scheme was composed of the space-time generalized finite difference method, the two-step Newton-Raphson method, and the time-marching method. The space-time approach can treat the temporal derivative as one of the spatial derivatives. This numerical technique enables all the partial derivatives in the governing equation can be discretized by a spatial discretization method, and the mixed derivative can efficiently deal with using the proposed meshless numerical scheme. The space-time generalized finite difference method is advanced from the Taylor series expansion and the moving-least square method. The numerical discretization process is only related to functional data and weighting coefficients on the central node and its nearby nodes. Thus, the matrix system composed of nonlinear algebraic equations will be a sparse matrix and can be efficiently solved by the two-step Newton-Raphson method. Furthermore, the time-marching method was utilized to proceed with the space-time domain along the time axis. In this paper, several numerical examples were tested to verify the capability of the proposed space-time generalized finite difference scheme. 

Numerical Solutions of the Nonlinear Dispersive Shallow Water Wave Equations Based on the Space-Time Coupled Generalized Finite Difference Scheme

Abstract This study examines the Darcy–Brinkman model for 2-D magnetohydrodynamic fluid flow across porous stretching/shrinking surfaces. We consider water as a traditional base fluid in which the Al2O3, Silver (Ag), Titanium dioxide (TiO2), and Gold (Au) nanoparticles are submerged in preparation of a tetra-dusty nanofluid. The novelty of the present problem is to examine the influence of Boussinesq-Stokes suspension tetra dusty nanofluid flow with the Darcy–Brinkman model. The governing nonlinear partial differential equations are transferred to a set of ordinary differential equations through suitable similarity conversions. The methodology, the governing equations of fluid and dusty phases is transformed into nonlinear ODEs using similarity transformations. The findings demonstrate that the tetra nanofluid phase heat and mass transpiration performance is greater compared to the dusty phase. Developing tetra dusty nanofluid has significantly upsurged the heat transmission for manufacturing with numerous applications in engineering, biological, and physical sciences.

Darcy–Brinkman model for Boussinesq–Stokes suspension tetra dusty nanofluid flow over stretching/shrinking sheet with suction/injection

In this paper, a novel meshless method that can handle porous flow problems with singular source terms is developed by virtually constructing node control domains. By defining a connectable node cloud, this method uses the integral of the diffusion term and generalized finite-difference operators to derive overdetermined equations of the node control volumes. An empirical method of calculating reliable node control volumes and a triangulation-based method to determine the connectable point cloud are developed. The method focuses only on the volume of the node control domain rather than the specific shape, so the construction of node control domains is virtual and does not increase the computational cost. To our knowledge, this is the first study to construct node control volumes in the meshless framework, termed the node control domain-based meshless method (NCDMM), which can also be regarded as an extended finite-volume method (EFVM). Taking two-phase porous flow problems as an example, discrete NCDMM schemes satisfying local mass conservation are derived by integrating the generalized finite-difference schemes of governing equations on each node control domain. Finally, commonly used low-order finite-volume method (FVM)-based nonlinear solvers for various porous flow models can be directly employed in the proposed NCDMM, significantly facilitating general-purpose application. Theoretically, the proposed NCDMM has the advantages of previous meshless methods for discretizing computational domains with complex geometries, as well as the advantages of traditional low-order FVMs for stably handling a variety of porous flow problems with local mass conservation. Four numerical cases are implemented to test the computational accuracy, efficiency, convergence, and good adaptability to the calculation domain with complex geometry and various boundary conditions. 

/pdf/a-novel-meshless-method-based-on-the-virtual-construction-of-1n62wwo6.pdf

A novel meshless method based on the virtual construction of node control domains for porous flow problems

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A novel meshless method based on the virtual construction of node control domains for porous flow problems