Numerical Simulation Study using the Explicit Finite Difference Method for Petroleum Reservoir

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.

Fluid flow in porous media is the key scientific problem in the development of oil and gas reservoirs. The traditional mechanics of fluid flow in porous media which based on the continuum hypothesis and Darcy′s law plays an important role in developing conventional oil and gas resources. In recent years, unconventional reservoirs are drawing more and more attention all over the world, therefore the development theory and technology, especially the corresponding flow mechanisms have become the hot research issues. The unconventional reservoirs exhibit distinct multiscale characteristics, even with six orders of magnitude difference. In addition, the application of massive multi-stage hydraulic fracturing can induce strong stress interactions. Therefore, the traditional theory of fluid flow in porous media cannot accurately describe the flow characteristics in unconventional reservoirs. In essence, the development of unconventional oil and gas resources involves multiphase fluids (e.g. oil, water and gas) flow in multi-scale porous media with multi-field coupling and various flow patterns. Therefore, the concept of modern system of multiphase flow in porous media is proposed, which means multiphase fluids flowing in multi-scale porous media with multi-field coupling and various flow patterns. The research status and development tendency are reviewed from the aspects of: (1) micro- and nanoscale oil and gas flow simulation; (2) upscaling for reservoir simulation, (3) macroscale flow simulation of unconventional oil and gas reservoirs; (4) simulation of flow in large scale fractured and vuggy carbonate reservoirs and (5) physical simulation of hydrocarbon transport in porous media. More specifically, in nanoscale the density functional theory and molecular simulation method can be used to study the interfacial phenomena to understand the hydrocarbon transport behavior in nanopores and provide key parameters for mesoscale flow simulation. The current study of nanoscale simulation mainly focuses on developing more realistic molecular structure model to represent the heterogeneous shale samples. Microscale simulation methods involve pore network model, lattice Boltzmann method, direct simulation of Navies-Stokes equation, level-set method and smoothed particle hydrodynamics, etc. Digital core and pore network model are the fundamental research platforms. Various methods can be used to reconstruct digital cores with multiscale pore structures and mineral compositions. The complex physicochemical phenomena namely adsorption/desorption, wettability change and boundary effect should be considered in the microscale flow simulations and extensive works have been done in microscale gas flow simulations. The future work on microscale simulation should focus on the multiphase flow mechanisms with multi-field coupling. The multiscale characteristics of unconventional reservoirs indicate the necessity of upscaling process to introduce the microscale flow mechanisms to macroscale. Homogenization theory and volume averaging method are the main upscaling approaches. Current upscaling methods are mostly based on the periodic boundary condition and are unreliable to be used in complex oil and gas reservoirs, which needs further study. In addition, more research needs to be conducted on the upscaling from molecular scale to mesoscale. In macroscale simulations of unconventional oil and gas reservoirs, the fluid-structure interaction should be considered and high efficiency numerical algorithm needs to be established. For large scale fractured and vuggy carbonate reservoirs, the non-Darcy flow characteristics and different flow regimes in vugs and fractures should be taken into account during flow simulation. Physical simulations of hydrocarbon transport in porous media are conducted at two scales: macroscale, nano- and microscale. Macroscale physical simulations aim at monitoring the dynamic saturation and pressure fields change under the realistic reservoir conditions. Nano- and microscale physical simulations are mainly applied to study the fluid transport mechanisms in single pore or throat. In summary, the proposed theory of multiphase fluids flowing in multi-scale porous media with multi-field coupling and various flow patterns can be applied to study the fluid flow problems in unconventional oil and gas industries.

A mixed pressure-velocity formulation to model flow in heterogeneous porous media with physics-informed neural networks

Pressure transient behavior in an infinite reservoir with horizontal well: Non-Newtonian fluid flow in porous media

Analytical study of fluid flow modeling by diffusivity equation including the quadratic pressure gradient term

1 - Modelling fluid flow in saturated porous media and at interfaces

Understanding fluid flow in porous media is essential with complex and multiphase fluid flow. We demonstrate that high-resolution in-line density measurements are a valuable tool in this regard. An in-line densitometer is used in fluid flow in porous media applications to quantify fluid production and obtain quantitative and qualitative information such as breakthrough times, emulsion/foam generation, and steam condensation. In order to determine the potential applications for in-line densitometry for fluid flow in porous media, a series of sand pack floods were performed with a densitometer placed at the outlet of a sand pack. All fluids passed through the measurement cell at experiential temperatures and pressures. An algorithm was developed and applied to the density data to provide a quantitative determination of oil and water production. The second series of tests were performed at high temperature and pressure, with a densitometer placed at the inlet and outlet of a sand pack, for steam applications. In both series of experiments, data acquisition was collected at 1 hertz and the analyzed density data was compared to results from the conventional effluent analysis, including Dean-Stark, toluene separations, magnetic susceptibility measurement, and flash calculations where applicable. The high-resolution monitoring of effluent from a flow experiment through porous media in a system with two phases of known densities enables two-phase production to be accurately quantified in the case of both light and heavy oil. The frequency of measurements results in a high-resolution history of breakthrough times and fluid behavior. In the case of monitoring steam injection processes, reliable laboratory tests show that in-line density measurements enable the determination of steam quality at the inlet and outlet of a sand pack and qualitative determination of steam condensation monitoring The use of in-line densitometry provides insight on the monitoring of complex fluid flow in porous media, which typical bulk effluent analysis is not able to do. The ability to measure produced fluids at high resolution and extreme temperatures reduces mass balance error associated with the effluent collection and broadens our understanding of complex fluid flow in porous media.

Summary In recent years, Physics Informed Neural Networks (PINNs) have generated considerable interest in the scientific computing community as an alternative, or potential contender, to traditional numerical discretization methods such as finite- difference, volume, and element methods, for solving partial differential equations (PDE). In PINNs, the governing equations are incorporated into a loss function as a regularization term to guide the neural network so that the solution respects the underlying physics, hence the name 'physics informed'. In this work, we investigate the performance of PINNs in solving the highly anisotropic diffusion equation that models fluid flow in subsurface porous media. Several levels of permeability anisotropy are tested. The results show that PINNs have excellent performance when the solution is smooth regardless of the strength of permeability anisotropy. However, PINNs struggle to give adequate results when the solution has large gradients, for example, when the solution is induced by a concentrated source term. The problem is exacerbated by higher levels of permeability anisotropy. Our results highlight a few limitations in the current implementation of physics-informed neural networks for fluid flow in porous media and show that we still have ways to go before it can compete with traditional numerical methods.

Food Insecurity During Long-term Recovery From Hurricane Katrina: A Longitudinal Analysis

Numerical study of inertial effects on permeability of porous media utilizing the Lattice Boltzmann Method

A finite-element scheme has been formulated, which is capable of solving transient analysis of both conductive and convective heat transfers due to fluid flow in porous media. The model also includes the latent heat effect to consider the phase change aspect of a frozen medium. To test the validity of the model, it was applied to six cases for which analytical solutions are available. The test cases cover (i) single-phase fluid flow through porous media, (ii) radial conduction with and without phase change, (iii) conductive and convective heat transfer in an aquifer, and (iv) two-phase immiscible flow in porous media. In all these cases, good agreement with analytical solutions are observed validating the computational scheme. This computational scheme should be useful in solving frozen ground problems, thermal stimulation technique for natural gas recovery from hydrates, and single-phase and two-phase convective heat transfer problems in enhanced oil recovery scheme in petroleum engineering.

A model to describe microscopic fluid flow in porous media is proposed. It offers the potential to predict relative permeability curves from microscopic rock and fluid properties. Conventional descriptions of fluid flow in porous media are based on Darcy's equations which require relative permeabilities and capillary pressure curves to be measured on macroscopic porous material. Prior to 1980, these macroscopically measured parameters could not be quantitatively related to microscopic rock properties. Recently, percolation theory has been used to relate macroscopic rock properties to relative permeabilities in the limit of capillary flow. Diffusion Limited Aggregation has been shown to simulate microscopic flow in the limit of infinite mobility ratios. The model described in this paper is a new approach to simulating flow through porous media on a microscopic scale. It is based on a variation of diffusion limited aggregation. The model can match coreflood average saturation profiles and production histories as predicted by Darcy's equations while generating saturation distributions resembling viscous fingering. The model can also simulate the limiting cases of infinite mobility ratio and zero flow rates as previously modeled by diffusion limited aggregation and percolation theory. With some simplifying assumptions, differential equations very similar to Darcy's equations can be derived from the microscopic interpretation of fluid behavior in porous media used in this model.

The equation for fluid flow in porous media is analyzed in this paper with the aid of Lie symmetry method (LSM) and invariant subspace method (ISM). Infinitesimal generators, the entire geometric fields of the vectors and the symmetry groups of the equation being considered are given. One-dimensional optimal systems of sub-algebra are reported with corresponding reduced nonlinear ordinary differential equations. By means of ISM, we determine the exact solutions and invariant subspaces (ISs) for the equation under consideration. Moreover, with the aid of the new theorem of conservation, we establish the conservation laws (CLs) for the governing equation. The construction of the conserved vectors reveals the integrability and existence of soliton solutions of the equation for fluid flow in porous media.

An ELLAM-MFEM Solution Technique for Compressible Fluid Flows in Porous Media with Point Sources and Sinks

Analysis of transient pressure response near a horizontal well has been based largely upon the analysis of mathematical solutions for the diffusivity equation. The diffusivity equation satisfies the principle of mass conservation, but does not consider the principle of equilibrium. No deformation properties of reservoir are included in the diffusivity equation. However, in some deformable reservoirs, the stress and deformation caused by injection or production could be so significant that it might affect the pressure response. This paper presents analyses of transient pressure response near a horizontal well using a coupled diffusion-deformation method. The results are compared with those obtained from the single diffusivity equation. Implications on practical applications such as well testing are addressed. Introduction Reliable information about in situ reservoir conditions is important in many phases of petroleum engineering. Much of the information can be obtained from well testing by measuring the variations in wellbore pressure that result from changes in the operating condition of the well. For example, pressure transients measured at an injecting well (under constant injecting rate) after it is shut in can be used to estimate the permeability of the reservoir. Well testing theory is based largely upon the solutions for the diffusivity equation, the differential equation for fluid flow in porous medium. Traditionally, it is assumed that the permeability, porosity and compressibility of fluid are constant, and pressure gradient is small(1,2). In fact, the change in fluid pressure is related to the change in the volume of fluid and reservoir rock. In order to obtain a better insight into the reservoir properties, two mechanisms should be considered in the interaction between the interstitial fluid and the porous rock:an increase of pore pressure induces a dilation of the rock; andcompression of the rock causes a rise of pore pressure, if the fluid is prevented from escaping the pore network. On the other hand, if pore pressure induced by compression of the rock is allowed to dissipate through diffusive fluid mass transport, further deformation of the rock progressively takes place. The earliest theory to account for this coupled diffusion-deformation mechanism was developed by Biot(3) who proposed a theory for three-dimensional consolidation of soils. Terazghi(4) developed a model for one-dimensional consolidation of soils. Rice and Cleary(5) linked the theory of poroelasticity to the concepts in rock mechanics with the full account of fluid and solid grain compressibility's. The objective of this paper is to study the effects of coupled diffusion-deformation mechanisms and compressibility's of fluid and grain on the pressure response near a horizontal well. Numerical cases in oil sand and sandstone reservoirs are generated to investigate the above effects, using a coupled numerical simulator. Then, the results are compared with those from the well testing method. Analysis Methods Single Diffusion Equation (Well Testing Analysis Technique) In 1967, Matthews and Russell(2) published their monograph dealing with well testing and analysis, which became a standard reference for many petroleum engineers. The basic fluid diffusion equation is a combination of the law of conservation of mass, Darcy's law, and state equation.