A multiscale deep learning model is proposed for fluid flow in porous media. The method is formulated on hierarchical grid system, that is, a coarse grid and a fine grid. Deep learning network is used to train data on the coarse gird. Source term and permeability field is treated as input parameter and coarse-scale solution is treated as output parameter. We construct multiscale basis functions by solving local flow problems within coarse gridcells. Heterogeneity and interactions between matrix and fracture are captured by basis functions. Oversampling technique is applied to get more accurate small-scale details. Numerical experiments show that the multiscale deep learning model is promising for flow simulation in heterogeneous and fractured porous media.

Based on theory of fractal and assumption of continuity, a well test model for multiple fractured horizontal wells in fractal fractured shale gas reservoirs was established in which adsorption, desorption and cross flow between matrix and fractures are considered. Solution of the model was obtained with Laplace transform, point source solution and pressure drop superposition principles. Double logarithmic curves of dimensionless pressure changing over time were obtained. Characteristics of pressure of multiple fractured horizontal well in fractal fractured shale gas reservoirs and influence of parameters, such as fractal index and fractal dimension, on pressure and production curves were analyzed. It shows that the curves can be divided into 7 flow stages. Slope of unstable pressure well-test curves in the middle and late stages is greater with bigger fractal index or smaller fractal dimension.

Mimetic finite difference (MFD) method was applied to numerical simulation of non-Darcy flow in low permeability reservoirs. Principle of MFD method was described in details. And corresponding numerical formula of the non-Darcy flow model was developed. An IMPES scheme was used for solution of two-phase flow simulation. Several numerical examples are presented to demonstrate efficiency and applicability of the scheme.

A discrete fracture model is used to simulate water flooding in low permeability reservoirs considering nonlinear flow characteristic of matrix system.At first,we reduce dimension of fractures and propose a mathematical model.On this basis,unstructured grid and control volume finite difference method are applied to numerical computation considering complex geometry of fractures.Accuracy of simulator is validated through a simple example.At last,two water-injection tests are performed to analyze effect of different non-Darcy flow models.

A numerical method is developed to model nonlinear flow in heterogeneous and low-permeability reservoirs. In order to obtain pressure gradient of each gridcells precisely,we introduced two sets of physical quantities,cell face pressures and surface efflux for each gridcell. Principle of the method is elaborated based on single-phase fluid flow problem. Several numerical examples are presented to demonstrate efficiency and applicability of the scheme.

A locally conservative Galerkin (LCG) finite element method is proposed for two-phase flow simulations in heterogeneous porous media. The main idea of it is to use property of local conservation at steady state conditions to define a numerical flux at element boundaries. It provides a way to apply standard Ga/erkin finite element method in two-phase flow simulations in porous media. LCG method has all advantages of standard finite element method while explicitly conserving fluxes over each element. Several problems are solved to demonstrate accuracy of the method. All examples show that the formulation is accurate and robust, while CPU time is significantly less than mixed finite element method.

Based on a mathematical model of non-Darcy unsteady seepage flows in low-permeability porous media with moving boundary conditions,a differential equation of propagation velocity for moving boundary was deduced.It indicates that propagation velocity of a moving boundary is proportional to the second derivative of formation pressure with respect to radial distance on the moving boundary.And with Lagrange three-point interpolation formula,finite difference scheme of governing equation near a moving boundary was obtained.Exact position of a moving boundary is able to be tracked.Numerical results show that the front tracking method describes propagation behaviors of moving boundary of non.Darcy unsteady seepage flows in low-permeability porous media well.

With equivalent concept of single fracture,a discrete-fracture model is developed,in which macroscopic fractures are described explicitly as(n-1) dimensional geometry elements.This simple step greatly improves efficiency of numerical simulation.The model can really reflect impact of fractures on fluid flow through fractured reservoirs simultaneously.A fully coupling discrete-fracture mathematical model is implemented using Galerkin finite element method.Validity and accuracy of the model and numerical algorithm are demonstrated through several examples.Effect of fractures on water flooding in fractured reservoirs is investigated.It demonstrates that the discrete-fracture model is valid for fractured reservoirs,especially for those reservoirs in which macroscopic fractures exist.

Mathematical model with full tensorial permeability and mixed boundaries is presented to simulate single phase flow in heterogeneous reservoirs.A finite element method based on variational principle is proposed to solve the model.Fluid flows in homogeneous and heterogeneous reservoirs are simulated.It demonstrates that the finite element method is reliable and precise in determining flow behavior in heterogeneous reservoirs.It provides an important theoretical basis for detailed reservoir numerical simulation.

Considering factors as two-phase of oil and water, production history, areal heterogeneity of the reservoir, well-bore storativity and skin, we establish a mathematical well testing model with polymer flooding in producing period and a streamline model for unsteady well testing. Numerical solutions are gained by a stream-tube method. It shows that the pressure derivative curve moves upward with the increase of the ratio of oil/water viscosity. The effective permeability of formation decreases with an increase of production time. The effect of water-oil front cannot been reflected from the derivative curve in the water well drawdown log-log plot when a high permeable zone is distributed along the main streamline. The concave in derivative curve appears earlier with the increase of concentration of polymer injected.

A test well interpretation model consisting of matrix, fractures and vugs is presented in which the permeability of vugs decreases exponentially with pressure drop. A mathematical model which takes into account the effect of wellbore storage and skin factor is calculated in a fully-implicit finite-difference scheme. It is shown that the dimensionless permeability modulus causes a increase of pressure and its derivative. The interporosity-flow factor determines the time of the interporosity flow. The storativity-ratio influences the width and depth of the "concave" in the pressure derivative. The effect of the outer-boundary differs from that of a normal triple medium. The skin factor affects the whole pressure and the "heave" in the pressure derivative curve, while the dimensionless permeability modulus mainly affects the later pressure and pressure derivative.

This paper is concerned with the method,which approaches to analytical solution of the light field equation with the Gauss-Hermite mode in three-dimension free space.The number of modes needed for calculating has been eatimated.The method has the following advantages:(1)Simulation electron number is reduced by about two orders of magnitude in comparison with DM methed;(2)Electron equations and G-H modes can be computed parallely.

Higher-order accuracy schemes are established following PADE' approach.Based on the schemes,fast recursive algorithms for the simple equations.Can be obtained and satisfy the corresponding conservation law.The fast recursive algorithm in the reference ^[1] is a kind of specific form of the present recursive algorithms given here.