Finite-Volume Discretisations for Flow in Fractured Porous Media

Over the last decades, finite-volume discretisations for flow in porous media have been extended to handle situations where fractures dominate the flow. Successful discretisations have been based on the discrete fracture-matrix models to yield mass conservative methods capable of explicitly incorporating the impact of fractures and their geometry. When combined with a hybrid-dimensional formulation, two central concerns are the restrictions arising from small cell sizes at fracture intersections and the coupling between fractures and matrix. Focusing on these aspects, we demonstrate how finite-volume methods can be efficiently extended to handle fractures, providing generalisations of previous work. We address the finite-volume methods applying a general hierarchical formulation, facilitating implementation with extensive code reuse and providing a natural framework for coupling of different subdomains. Furthermore, we demonstrate how a Schur complement technique may be used to obtain a robust and versatile method for fracture intersection cell elimination. We investigate the accuracy of the proposed elimination method through a series of numerical simulations in 3D and 2D. The simulations, performed on fractured domains containing permeability heterogeneity and anisotropy, also demonstrate the flexibility of the hierarchical framework.     

Fracture-Flow-Enhanced Matrix Diffusion in Solute Transport Through Fractured Porous Media

Over the past few decades, significant progress of assessing chemical transport in fractured rocks has been made in laboratory and field investigations as well as in mathematic modeling. In most of these studies, however, matrix diffusion on fracture–matrix surfaces is considered as a process of molecular diffusion only. Mathematical modeling based on this traditional concept often had problems in explaining or predicting tracer transport in fractured rock. In this article, we propose a new conceptual model of fracture-flow-enhanced matrix diffusion, which correlates with fracture-flow velocity. The proposed model incorporates an additional matrix-diffusion process, induced by rapid fluid flow along fractures. According to the boundary-layer theory, fracture-flow-enhanced matrix diffusion may dominate mass-transfer processes at fracture–matrix interfaces, where rapid flow occurs through fractures. The new conceptual model can be easily integrated with analytical solutions, as demonstrated in this article, and numerical models, as we foresee. The new conceptual model is preliminarily validated using laboratory experimental results from a series of tracer breakthrough tests with different velocities in a simple fracture system. Validating of the new model with field experiments in complicated fracture systems and numerical modeling will be explored in future research.     

Theoretical and Experimental Investigations of Miscible Displacement in Fractured Porous Media

In this paper a mathematical model for miscible displacement in fractured porous media is developed. The model takes into account mechanisms of mass transfer between fracture and matrix. The model is normalized by using the dimensionless parameters, which characterize the process, and the analytical solutions of the resulting system of equations are provided by utilizing the method of characteristics. For comparison the results of model with experimental results, laboratory displacement tests have been performed in fractured systems under miscible displacement. The porous media used were cylindrical Asmari cores from Iranian reservoirs containing an artificially vertical fracture. Normal heptane and kerosene were two miscible fluid used. There is very good agreement between experiments and model prediction.     

Contaminant Transport in Fractured Media with Sources in the Porous Domain

We study contaminant flow with sources in a fractured porous mediumconsisting of a single fracture bounded by a porous matrix. In the fracturewe assume convection, decay, surface adsorption to the interface, and lossto the porous matrix; in the porous matrix we include diffusion, decay,adsorption, and contaminant sources. The model leads to a nonhomogeneous,linear parabolic equation in a quarter-space with a differential equationfor an oblique boundary condition. Ultimately, we study the problemu _t = u _yy – u + f(x,y,t),x,y>0, t>0, u _t = –u _x + u _y – u on y = 0; u(0,0,t) =u₀(t), t>0,with zero initial data. Using Laplace transforms we obtain the Green'sfunction for the problem, and we determine how contaminant sources in theporous media are propagated in time.     

Pore-Level Observation of Free Gravity Drainage of Oil in Fractured Porous Media

This work presents results from two sets of experiments conducted to study, in pore level, the role of fracture aperture and tilt angle on the stability of liquid bridges and the shape of a front during free gravity drainage process. Glass micromodels of two different aperture sizes were used to monitor the mechanism of gravity drainage of air?Ccrude oil system, rotating around a bottom corner to create different tilting angles. Oil content within the matrix blocks was determined as a function of time using a series of images obtained during the experiments, from which net drainage rate from the upper and lower matrix blocks is calculated. Liquid bridges are more frequent but less stable at early time of drainage. The liquid bridges, which have widths as thin as 50 ??m, can resist instability to maintain continuity. Liquid bridges formed in stacks with higher tilt angles are more stable, enhancing oil drainage from the upper matrix block and causing higher recoveries. Quantitative analysis of the results shows that a wider fracture aperture increases the oil production rate, but reduces the ultimate recovery. Furthermore, stacks with higher tilt angles present larger ultimate recoveries and smaller production rates. The front geometry in the lower block deviates from linearity due to formation of liquid bridges in the middle fracture. The results of this work can be helpful to better understand the interaction between fractures and matrix blocks.     

Permeability of Porous Media from Simulated NMR Response

Nuclear Magnetic Resonance (NMR) is an increasingly popular well-logging tool in petroleum industry because it is the only tool that attempts to estimate formation permeability. In this paper, spatially correlated porous media are generated. Permeabilities of these media are computed by the lattice Boltzmann method. NMR relaxation responses are simulated by a random walk technique and formation factors are computed by solving a Laplacian equation. The testing of commonly used NMR permeability correlations shows that three conditions should be met for the validity of these correlations. The surface relaxivity should not vary significantly. The formation factor should depend only on porosity. And the characteristic pore body radius should be proportional to the characteristic throat radius. The correlations are improved by including surface relaxivity and formation factor.     

Modeling Fluid Flow in Fractured Porous Media with the Interfacial Conditions Between Porous Medium and Fracture

One of the most popular models that has been applied to predict the fluid velocity inside the fracture with impermeable walls is the cubic law. It highlights that the mean flux along the fracture is proportional to the cubic of fracture aperture. However, for a fractured porous medium, the normal and tangential interface conditions between the fracture and porous matrix can change the velocity profile inside the fracture. In this paper, a correction factor is introduced for flow equation along the fracture by imposing the continuity of normal and tangential components of velocity at the interface between the fracture and porous matrix. As a result, the mean velocity inside the fracture depends not only on the fracture aperture, but also on a set of non-dimensional numbers, including the matrix porosity, the ratio of intrinsic permeability of fracture to that of matrix, the wall Reynolds number, and the ratio of normal velocity on one wall to the other. Finally, the introduced correction factor is employed within the extended finite element method, which is widely used for numerical simulation of fluid flow within the fractured porous media. Several numerical results are presented for the fluid flow through a specimen containing single fracture, in order to investigate the deviation from the cubic law in different case studies.

     

A New Mathematical Model for Force Gravity Drainage in Fractured Porous Media

In force gas/oil gravity drainage process in fractured porous media, gas is flowing in both matrix and fractures leading to produce a finite gas pressure gradient. Consequently, viscous force plays an important role for displacing matrix oil toward fractures in addition to gravity force that is required to be modeled appropriately. A new analytical model for estimation of steady state oil saturation distribution with assumption of fixed gas pressure gradient throughout the matrix is presented. Moreover, based on some results of this analytical model a different numerical formulation is developed to predict the performance of oil production process. Comparison of the results obtained from this numerical model with the results of a conventional simulator demonstrates that the newly developed model can be applied with satisfactory accuracy. Numerical simulations show that the viscous displacement in fractured porous media can reduce the capillary threshold height, and thus it suggests the force gravity drainage as a favorable production mechanism when the matrix length is close to the threshold height.     

Modeling of Two-Phase Flow in Fractured Porous Media on Unstructured Non-Uniformly Coarsened Grids

In this article, we investigate two strategies for coarsening fractured geological models. The first approach, which generates grids that resolve the fractures, is referred to as explicit fracture-matrix separation (EFMS). The second approach is based on a non-uniform coarsening strategy introduced in Aarnes et al. (Adv Water Resour 30(11):2177–2193, 2007a). A series of two-phase flow simulations where the saturation is modeled on the respective coarse grids are performed. The accuracy of the resulting solutions is examined, and the robustness of the two strategies is assessed with respect to number of fractures, degree of coarsening, well locations, phase viscosities, and fracture permeability. The numerical results show that saturation solutions obtained on the non-uniform coarse grids are consistently more accurate than the corresponding saturation solutions obtained on the EFMS grids. The numerical results also reveal that it is much easier to tune the upscaling factor with the non-uniform coarsening approach.     

Upscaled Unstructured Computational Grids for Efficient Simulation of Flow in Fractured Porous Media

Discrete fracture modeling (DFM) is currently the most promising approach for modeling of naturally fractured reservoirs and simulation of multiphase fluid flow therein. In contrast with the classical double-porosity/double permeability models, in the DFM approach all the interactions and fluid flow in and between the fractures and within the matrix are modeled in a unified manner, using the same computational grid. There is no need for computing the shape factors, which are crucial to the accuracy of the double-porosity models. We have exploited this concept in order to develop a new method for the generation of unstructured computational grids. In the new approach the geological model (GM) of the reservoir is first generated, using square or cubic grid blocks. The GM is then upscaled using a method based on the multiresolution wavelet transformations that we recently developed. The upscaled grid contains a distribution of the square or cubic blocks of various sizes. A map of the blocks’ centers is then used with an optimized Delauney triangulation method and the advancing-front technique, in order to generate the final unstructured triangulated grid suitable for use in any general reservoir simulator with any number of fluid phases. The new method also includes an algorithm for generating fractures that, contrary to the previous methods, does not require modifying their paths due to the complexities that may arise in spatial distribution of the grid blocks. It also includes an effective partitioning of the simulation domain that results in large savings in the computation times. The speed-up in the computations with the new upscaled unstructured grid is about three orders of magnitude over that for the initial GM. Simulation of waterflooding indicates that the agreement between the results obtained with the GM and the upscaled unstructured grid is excellent. The method is equally applicable to the simulations of multiphase flow in unfractured, but highly heterogeneous, reservoirs.     

High-Resolution Temporo-Ensemble PIV to Resolve Pore-Scale Flow in 3D-Printed Fractured Porous Media

Transport in Porous Media - Fractures are conduits that can enable fast advective transfer of (fluid, solute, reactant, particle, etc.) mass and energy. Such fast transfer can significantly affect...     

Flow in Fractured Porous Media: A Review of Conceptual Models and Discretization Approaches

Transport in Porous Media - The last decade has seen a strong increase of research into flows in fractured porous media, mainly related to subsurface processes but also in materials science and...     

饱和不可压正交各向异性多孔弹性板的动力弯曲分析

基于多孔介质理论,在Kirchhoff直法线假定以及小变形和线性本构关系前提下,建立了饱和不可压正交各向异性多孔弹性板的线性动力分析模型.针对流体的面内扩散问题,在忽略面内惯性项的影响下,进一步简化了分析模型,给出了相应的基本控制方程以及初始和边界条件的一般描述.根据所建立的模型,采用Fourier级数展开法研究了四边简支透水正交各向异性矩形多孔弹性板在冲击载荷作用下的拟静态和动力弯曲响应,数值分析了不同参数下孔隙流体压力等效弯矩、固相有效应力等效弯矩以及挠度的变化规律和动力特征.研究表明在外载荷作用初始阶段,孔隙流体对板弯曲变形的影响不可忽视.     

Comparison of Three FE-FV Numerical Schemes for Single- and Two-Phase Flow Simulation of Fractured Porous Media

We benchmark a family of hybrid finite element–node-centered finite volume discretization methods (FEFV) for single- and two-phase flow/transport through porous media with discrete fracture representations. Special emphasis is placed on a new method we call DFEFVM in which the mesh is split along fracture–matrix interfaces so that discontinuities in concentration or saturation can evolve rather than being suppressed by nodal averaging of these variables. The main objective is to illustrate differences among three discretization schemes suitable for discrete fracture modeling: (a) FEFVM with volumetric finite elements for both fractures and porous rock matrix, (b) FEFVM with lower dimensional finite elements for fractures and volumetric finite elements for the matrix, and (c) DFEFVM with a mesh that is split along material discontinuities. Fracture discontinuities strongly influence single- and multi-phase fluid flow. Continuum methods, when used to model transport across such interfaces, smear out concentration/saturation. We show that the new DFEFVM addresses this problem producing significantly more accurate results. Sealed and open single fractures as well as a realistic fracture geometry are used to conduct tracer and water-flooding numerical experiments. The benchmarking results also reveal the limitations/mesh refinement requirements of FE node-centered FV hybrid methods. We show that the DFEFVM method produces more accurate results even for much coarser meshes.     

Efficiency of Capillary Imbibition Dominated Displacement of Nonwetting Phase by Wetting Phase in Fractured Porous Media

The critical and optimum injection rates as well as the critical fracture capillary number for an efficient displacement process are determined based on the experimental and numerical modeling of the displacement of nonwetting phase (oil) by wetting phase (water) in fractured porous media. The efficiency of the process is defined in terms of the nonwetting phase displaced from the system per amount of wetting phase injected and per time. Also, the effects of injection rate on capillary imbibition transfer dominated two-phase flow in fractured porous media are clarified by visualizing the experiments. The results reveal that as the injection rate is increased, fracture pattern begins to become an effective parameter on the matrix saturation distribution. As the rate is lowered, however, the system begins to behave like a homogeneous system showing a frontal displacement regardless the fracture configuration.     

Stochastic Analysis of Reactive Solute Transport in Heterogeneous,Fractured Porous Media: A Dual-Permeability Approach

A nonlocal, first-order, Eulerian stochastic theory is developed for reactive chemical transport in a heterogeneous, fractured porous medium. A dual-permeability model is adopted to describe the flow and transport in the medium, where the solute convection and dispersion in the matrix are considered. The chemical is under linear nonequilibrium sorption and first-order degradation. The hydraulic conductivities, sorption coefficients, degradation rates in both fracture and matrix regions, and interregional mass transfer coefficient are all assumed to be random variables. The resultant theory for mean concentrations in both regions is nonlocal in space and time. Under spatial Fourier and temporal Laplace transforms, the mean concentrations are explicitly expressed. The transformed results are then numerically inverted to the real space via Fast Fourier Transform method. The theory developed in this study generalizes the stochastic studies for a reactive chemical transport in a one-domain flow field (Hu et al., 1997a) and in a mobile/immobile flow field (Huang and Hu, 2001). In comparison with one-domain transport, the dual-permeability model predicts a larger second moment in the longitudinal direction, but smaller one in the transverse direction. In addition, various simplification assumptions have been made based on the general solution. The validity of these assumptions has been tested via the spatial moments of the mean concentration in both fracture and matrix regions.     

Leak-off During Placement of Cr(III)-Partially Hydrolyzed Polyacrylamide Gelling Solution in Fractured Porous Media

During waterflooding of a fractured formation, water may channel through the fracture or interconnected network of fractures, leaving a large portion of oil bearing rock unswept. One remedial practice is injection of a gelling solution into the fracture. Such placement of a gelling mixture (referred as gelant) is associated with leak-off from the face of the fracture into the adjoining matrix. As the gelant gets more crosslinked, the gelant encounters more resistance in flowing into the porous matrix. This article addresses the build-up of flow resistance as the Cr(III)-partially hydrolyzed polyacrylamide gelant, at various stages of crosslinking flows into the matrix. Flow experiments were conducted at constant injection pressure in unfractured Berea rocks that represent a matrix adjoining a fracture. Before entering the core, gelants underwent post-mixing delays, shorter than their gel time. On continued displacement, flow resistance developed that reduced the flow rate further. More delay, after mixing of gelant hastened, the build-up of resistance to flow and the resistance was contained nearer to the inlet face. Effect of flow over fracture face on the build-up of flow resistance in the matrix was also evaluated by conducting displacement of gelant in two fractured slabs. In one case, a part of the injected fluid came out of the fracture outlet with the rest leaking off into matrix. In the other case, all the fluid that entered into the fracture leaked off into the matrix. Build-up of flow resistances in the matrix for the two cases was compared. A simple conceptual model is presented that could explain the flow of gelant and build-up of resistance in porous rock at constant injection pressure.