Zhaoqin Huang

Accurate multiscale finite element method for numerical simulation of two-phase flow in fractured media using discrete-fracture model

Numerical simulation in naturally fractured media is challenging because of the coexistence of porous media and fractures on multiple scales that need to be coupled. We present a new approach to reservoir simulation that gives accurate resolution of both large-scale and fine-scale flow patterns. Multiscale methods are suitable for this type of modeling, because it enables capturing the large scale behavior of the solution without solving all the small features. Dual-porosity models in view of their strength and simplicity can be mainly used for sugar-cube representation of fractured media. In such a representation, the transfer function between the fracture and the matrix block can be readily calculated for water-wet media. For a mixed-wet system, the evaluation of the transfer function becomes complicated due to the effect of gravity. In this work, we use a multiscale finite element method (MsFEM) for two-phase flow in fractured media using the discrete-fracture model. By combining MsFEM with the discrete-fracture model, we aim towards a numerical scheme that facilitates fractured reservoir simulation without upscaling. MsFEM uses a standard Darcy model to approximate the pressure and saturation on a coarse grid, whereas fine scale effects are captured through basis functions constructed by solving local flow problems using the discrete-fracture model. The accuracy and the robustness of MsFEM are shown through several examples. In the first example, we consider several small fractures in a matrix and then compare the results solved by the finite element method. Then, we use the MsFEM in more complex models. The results indicate that the MsFEM is a promising path toward direct simulation of highly resolution geomodels.

Effect of Large Capillary Pressure on Fluid Flow and Transport in Stress-sensitive Tight Oil Reservoirs

The pore sizes of unconventional reservoir rock, such as shale and tight rock, are on the order of nanometers. The thermodynamic properties of in-situ hydrocarbon mixtures in such small pores are significantly different from those of fluids in bulk size, primarily due to effect of large capillary pressure. For example, it has been recognized that the phase envelop shifts and bubble-point pressure is suppressed in tight and shale oil reservoirs. On the other hand, the stress-dependency is pronounced in low permeability rocks. It has been observed that pore sizes, especially the sizes of pore-throats, are subject to decrease due to rock deformation induced by the fluid depletion from over-pressurized tight and shale reservoirs. This reduction on pore spaces again affects the capillary pressure and therefore thermodynamic properties of reservoir fluids. Thus it is necessary to model the effect of stressdependent capillary pressure and rock deformation on tight and shale reservoirs. In this paper, we propose and develop a multiphase, multidimensional compositional reservoir model to capture the effect of large capillary pressure on flow and transport in stress-sensitive unconventional reservoirs. The vapor-liquid equilibrium (VLE) calculation is performed with Peng-Robinson Equation of State (EOS), including the impact of capillary pressure on phase behavior and thermodynamic properties. The fluid flow is fully coupled with geomechanical model, which is derived from the thermoporoelasticity theory; mean normal stress as the stress variable is solved simultaneously with mass conservation equations. The finite-volume based numerical method, integrated finite difference method, is used for space discretization for both mass conservation and stress equations. The formulations are solved fully implicitly to assure the stability. We use Eagle Ford tight oil formations as an example to demonstrate the effect of capillary pressure on VLE. It shows that the bubble-point pressure is suppressed within nano-pores, and fluid properties, such as oil density and viscosity, are influenced by the suppression due to more light components remained in liquid phase. In order to illustrate the effect of stress-dependent capillary pressure on tight oil flow and production, we perform numerical studies on Bakken tight oil reservoirs. The simulation results show that bubble-point suppression is exaggerated by effects of rock deformation, and capillary pressure on VLE also affects the reservoir pressure and effective stress. Therefore the interactive effects between capillary pressure and rock deformation are observed in numerical results. Finally, the production performance in the simulation examples demonstrates the large effect of large capillary pressure on estimated ultimate recovery (EUR) in stress-sensitive tight reservoirs.

Locally conservative Galerkin and finite volume methods for two-phase flow in porous media

In this paper we propose a locally conservative Galerkin (LCG) finite element method for two-phase flow simulations in heterogeneous porous media. The main idea of the method is to use the property of local conservation at steady state conditions in order to define a numerical flux at element boundaries. It provides a way to apply standard Galerkin finite element method in two-phase flow simulations in porous media. The LCG method has all the advantages of the standard finite element method while explicitly conserving fluxes over each element. All the examples presented show that the formulation employed is accurate and robust, while using less CPU time than finite volume method.

Assisted history matching for the inversion of fractures based on discrete fracture-matrix model with different combinations of inversion parameters

Accurate prediction of fracture distribution in fractured reservoirs is important in the development process. Considering that assisted history matching technology is an effective method for the inversion of reservoir parameters, the technology can also be applied for the inversion of fractures. Because applying assisted history matching technology for the inversion of fractures has an inherent defect of multiplicity of solution, it is therefore necessary to alleviate the multiplicity for the success of inversion. Although there are many factors affecting the multiplicity, the paper focuses on the study of the inversion results of different combinations of inversion parameters which are all representative parameters of fractures and determine the distribution of fractures. Firstly, we simulate the flow behavior in fractured media based on the discrete fracture matrix (DFM) module of Matlab Reservoir Simulation Toolbox (MRST) to explicitly describe the effect of fractures on flow behavior. Secondly, history matching objective function is established based on Bayesian theory and different kinds of representative parameters of fractures are chosen as inversion parameters. Thirdly, simultaneous perturbation stochastic approximation (SPSA) algorithm is adopted to minimize the objective function to achieve the inversion of fractures corresponding to different inversion parameters. Finally, theoretical cases verify that the inversion method is effective for the accurate prediction of fracture distribution and proper inversion parameters are crucial to the success of fracture inversion.

Flow of Particulate-Fluid Suspension in a Channel with Porous Walls

The problem of the dispersed particulate-fluid two-phase flow in a channel with permeable walls under the effect of the Beavers and Joseph slip boundary condition is concerned in this paper. The analytical solution has been derived for the longitude pressure difference, stream functions, and the velocity distribution with the perturbation method based on a small width to length ratio of the channel. The graphical results for pressure, velocity, and stream function are presented and the effects of geometrical coefficients, the slip parameter and the volume fraction density on the pressure variation, the streamline structure and the velocity distribution are evaluated numerically and discussed. It is shown that the sinusoidal channel, accompanied by a higher friction factor, has higher pressure drop than that of the parallel-plate channel under fully developed flow conditions due to the wall-induced curvature effect. The increment of the channel’s width to the length ratio will remarkably increase the flow rate because of the enlargement of the flow area in the channel. At low Reynolds number ranging from 0 to 65, the fluids move forward smoothly following the shape of the channel. Moreover, the slip boundary condition will notably increase the fluid velocity and the decrease of the slip parameter leads to the increment of the velocity magnitude across the channel. The fluid-phase axial velocity decreases with the increment of the volume fraction density.

A Multi-Porosity, Multi-Physics Model to Simulate Fluid Flow in Unconventional Reservoirs

Gas flow in shales is complicated by the highly heterogeneous and hierarchical rock structures (i.e., ranging from organic nanopores, inorganic nanopores, less permeable micro-fractures, more permeable macro-fractures, to hydraulic fractures). The dominant fluid flow mechanism varies in these different flow regimes, and properties of these rock structures are sensitive to stress changes with different levels. Although traditional single-porosity and double-porosity models can simulate certain time range of reservoir performance with acceptable accuracy, they are not generally applicable for the prediction of long-term performance and have limitations to improve our understandings of enhanced hydrocarbon recovery. In this paper, we present a multi-domain, multi-physics model, aiming to accurately simulate the fluid flow in shale gas reservoirs with more physics-based formulations. An idealized model has been developed for the purpose of studying the characteristic behavior of a fractured nanopore medium, which contains five regions: organic nanopores, inorganic nanopores, local micro-fractures, global natural fractures, and hydraulic fractures. Fluid flow governing equations in this model vary according to the different dominant fluid flow mechanisms in different regions. For example, the apparent permeability, which is the intrinsic permeability multiplied by a correction factor, is used to account for the gas slippage through nanopores of shale matrix; while the organic and inorganic nanopores in this matrix have different capacities for gas adsorption. On the other hand, for fluids flow in natural fractures and hydraulic fractures with high velocity, the non-Darcy flow model is used to capture the strong inertia when is comparable to viscous force. Numerical studies with practical interests are discussed. Several synthetic, but realistic test cases are simulated. Input parameters in these cases are evaluated using either the laboratory or theoretical work. Our results demonstrate that this model is able to capture the typical production behavior of unconventional reservoirs: a great initial peak, the sharp decline in the first few months, followed by a long flat production tail. A series of sensitivity analyses, which address the organic matter content, organic matter connectivity, natural fracture density, and hydraulic fracture spacing, will also be conducted.

A multiscale mixed finite element method with oversampling for modeling flow in fractured reservoirs using discrete fracture model

Abstract Fractures play a significant effect on the macro-scale flow, thus should be described exactly. Accurate modeling of flow in fractured media is usually done by discrete fracture model (DFM), as it provides a detailed representation of flow characteristic. However, considering the computational efficiency and accuracy, traditional numerical methods are not suitable for DFM. In this study, a multiscale mixed finite element method (MsMFEM) is proposed for detailed modeling of two-phase oil–water flow in fractured reservoirs. In MsMFEM, the velocity and pressure are first obtained on coarse grid. The interaction between the fractures and the matrix is captured through multiscale basis functions calculated numerically by solving DFM on the local fine grid. Through multiscale basis functions, this method can not only reach a high efficiency as upscaling technology, but also finally generate a more accurate and conservative velocity field on the full fine-scale grid. In our approach, oversampling technique is applied to get more accurate small-scale details. Triangular fine-scale grid is applied, making it possible to consider fractures in arbitrary directions. The validity of MsMFEM is proved through comparing experimental and numerical results. Comparisons of the multiscale solutions with the full fine-scale solutions indicate that the later one can be totally replaced by the former one. The results demonstrate that the multiscale technology is a promising method for multiscale flows in high-resolution fractured media.

On the interface boundary conditions for the Stokes-Darcy coupling problem

Recently, the Stokes-Darcy coupling flow problem has been taken much attention because of its importance in the fractured vuggy carbonate reservoirs. However, the descriptive equations for a free-flow region and a porous medium are different from each other due to the scale difference. Therefore, a suitable interface boundary conditions should be found to couple such two flows. To this end, a theoretical development is conducted based on the volume averaging up-scaling approach, which starts from the mathematical models on the micro pore-scale. After that, a novel set of interface conditions at the fluid-porous interface is developed. Such novel interface conditions are similar to the Beavers-Joseph conditions. However, a new coefficient has been introduced to modify the Darcy velocity. This coefficient is not only the function of the porosity and permeability of the porous medium, but also related to the structure characteristic of the interface transition zone. We have compared the theory with the experimental studies of Beavers-Joseph and our LDA experiment data. The results have shown that our new interface boundary conditions provide better agreement with the experimental data than the previous study.

Molecular Simulation of Displacement of Methane by Injection Gases in Shale.

Displacement methane (CH4) by injection gases is regarded as an effective way to exploit shale gas and sequestrate carbon dioxide (CO2). In our work, the displacement of CH4 by injection gases is studied by using the grand canonical Monte Carlo (GCMC) simulation. Then, we use molecular dynamics (MD) simulation to study the adsorption occurrence behavior of CH4 in different pore size. This shale model is composed of organic and inorganic material, which is an original and comprehensive simplification for the real shale composition. The results show that both the displacement amount of CH4 and sequestration amount of CO2 see an upward trend with the increase of pore size. The CO2 molecules can replace the adsorbed CH4 from the adsorption sites directly. On the contrary, when N2 molecules are injected into the slit pores, the partial pressure of CH4 would decrease. With the increase of the pores width, the adsorption occurrence transfers from single adsorption layer to four adsorption layers. It is expected that our work can reveal the mechanisms of adsorption and displacement of shale gas, which could provide a guidance and reference for displacement exploitation of shale gas and sequestration of CO2.

Hydraulic Fracturing Modeling and Its Extension to Reservoir Simulation Based on Extended Finite-Element Method (XFEM)

Abstract Hydraulic fracturing has become the most important technology in the development of shale gas and tight gas/oil reservoirs. The application of hydraulic fracturing is also essential for hot dry rock geothermal systems. It is important to understand how hydraulic fractures propagate in the formation and the effects of reservoirs heterogeneities on fracture propagation. In this chapter, a fully hydromechanical coupled model of hydraulic fracture propagation is presented, based on the extended finite-element method. Furthermore, its extension to reservoir simulation coupling with embedded discrete fracture model has also been developed and discussed.