Fractured Porous Media Research Articles

Impact of salinity on fluid/fluid and rock/fluid interactions in enhanced oil recovery by hybrid low salinity water and surfactant flooding from fractured porous media

Wettability alteration of the porous medium from oil-wet (OW) to neutral-wet or even water-wet (WW) is proposed as one of the main mechanisms behind the enhanced oil recovery (EOR) by the Low Salinity Water Flooding (LSWF). However, our recent studies for a specific crude-oil/brine/rock of the oil reservoir in the Middle East, shows that LSW (diluted sea water) is not that much effective, neither to change the wettability of the surface nor to significantly increase the oil recovery factor. This obstacle becomes more challenging when we deal with fractured formations, where the imbibition of injected fluid from fracture into matrix and its propagation within the matrix, is the main macroscopic mechanism of the EOR. Chemical EOR, in which the chemical additive is added to the injecting brine, is one of the hybrid techniques which can boost oil recovery in such cases. In this study, CTAB as a cationic surfactant was added to brine solutions with different ionic strengths (i.e., sea water (SW), 10 times diluted sea water (10dSW) and 2 times concentrated sea water (2cSW)). Contact angle, zeta potential of both fluid/fluid and fluid/rock surfaces, and IFT tests were conducted to shed insight on interactions in crude oil/brine and rock/brine interfaces and the role of ionic strength. Finally, dynamic micromodel tests were performed in oil-wet fractured porous media to observe the impact of the added surfactant along with different ionic strength on displacement mechanisms and oil recovery.Experimental results show that neither electrical double layer expansion nor multiple ion exchange mechanisms cannot justify the observed detachment of oil droplets from the oil-wet substrates in the presence of the CTAB. Surfactant are absorbed on the oil/brine interface by the embedment of their hydrophobic tails in oil. Both surfactant molecules and the non-polar oil components (that are attracted to hydrophobic tail of surfactant molecules), are attracted to the solution via the hydrophilic parts of the surfactants. The effectiveness of this mechanism is dependent on both the concentration of the surfactant and concentration of the ions in brine.In the case that the aforementioned mechanism is adequately dominant and effective, the small droplets of oil will be detached from continuous oil bulk, resulting in oil to be broken to smaller droplets which enhances the detachment of the oil bulk from oil-wet substrates. This fluid/fluid interaction mechanism also promotes the imbibition of the aqueous phase into the matrix and also its propagation within the matrix.

Accurate modeling and simulation of seepage in 3D heterogeneous fractured porous media with complex structures

The past decades have witnessed an increasing interest in numerical simulation for flow in fractured porous media. To date, most studies have focused on 2D or pseudo-3D computational models, where the impact of 3D complex structures on seepage has not been fully addressed. This work presents a method for modeling seepage in 3D heterogeneous porous media. The complex structures, typically the stochastic discrete fractures and inclusions, are able to be simulated. A mesh strategy is proposed to discretize the complex domain. In particular, a treatment on the intersected elements is developed to ensure a conforming mesh. Then, numerical discretization is provided, in which the flux interactions of fractures, inclusions and surrounding rock matrix are included. Numerical tests are performed to analyze the hydraulic characteristics of 3D fractured media. First, the developed framework is validated by comparing numerical solutions with the results of embedded discrete fracture model. Next, the effects of orientation, aperture and radius of fractures on fluid flow and equivalent permeability tensor are analyzed. The variations of pressure distribution are studied in heterogeneous and homogeneous media. Finally, the hydraulic properties of a medium with complex structures are investigated to show the difference of hydraulic feature between fractures and inclusions.

Spatial fractional permeability and fractional thermal conductivity models of fractal porous medium

In order to describe the seepage and heat transfer problems of non-Newtonian fluids in porous media, a spatial fractional permeability model and a fractional thermal conductivity model for a fractal porous medium are developed based on the fractional non-Newtonian constitutive equation and the fractional generalized Fourier law. It is an innovative attempt to link fractional operators to the microstructure of pore porous media. The predictive capability of the proposed permeability and thermal conductivity model is verified by comparing with experimental data and the conventional capillary model, and the effects of fractal dimension, fractional parameters, and microstructural parameters on permeability and thermal conductivity are discussed. The results are as follows: (a) These two new models have higher accuracy than the conventional capillary model and reveal the relationship between the nonlocal memory and microstructural properties of complex fluids. (b) The permeability and thermal conductivity increase with increase in the fractional parameter α and radius ratio β and decrease with the increase in the fractal dimension (Dτ and Df) and microstructural parameters (length ratio γ, branching angle θ, and branching level m) of the porous medium. (c) When the radius ratio is larger than a certain value, the growth rate of permeability (β > 0.46) and thermal conductivity (β > 0.3) increases significantly, while the branch angle has the smallest influence on permeability and thermal conductivity, which can be ignored.

Analysis of the Fractional-Order Local Poisson Equation in Fractal Porous Media

This paper investigates the fractional local Poisson equation using the homotopy perturbation transformation method. The Poisson equation discusses the potential area due to a provided charge with the possibility of area identified, and one can then determine the electrostatic or gravitational area in the fractal domain. Elliptic partial differential equations are frequently used in the modeling of electromagnetic mechanisms. The Poisson equation is investigated in this work in the context of a fractional local derivative. To deal with the fractional local Poisson equation, some illustrative problems are discussed. The solution shows the well-organized and straightforward nature of the homotopy perturbation transformation method to handle partial differential equations having fractional derivatives in the presence of a fractional local derivative. The solutions obtained by the defined methods reveal that the proposed system is simple to apply, and the computational cost is very reliable. The result of the fractional local Poisson equation yields attractive outcomes, and the Poisson equation with a fractional local derivative yields improved physical consequences.

A robust fully mixed finite element model for flow and transport in unsaturated fractured porous media

A fully mixed finite element (MFE) model is developed for nonlinear flow and transport in unsaturated fractured porous media with matrix-fracture and fracture-fracture fluid and mass exchanges. The model is based on the discrete fracture matrix (DFM) approach and assumes cross-flow equilibrium in the fractures. The MFE method is employed for the spatial discretization of both flow and transport on the 2D-matrix elements as well as on the 1D-fracture elements. An upwind scheme is employed to avoid unphysical oscillations in the case of advection dominant transport. The temporal discretization is performed using high-order time integration methods and efficient automatic time-stepping schemes via the MOL.Two test problems dealing with flow and mass transport in saturated and unsaturated fractured porous media are simulated to show the validity of the new model by comparison against (i) a 1D-2D Comsol finite element model and (ii) a 2D-2D Discontinuous Galerkin (DG) model where both fractures and matrix continua are discretized with small 2D mesh elements. The robustness and efficiency of the developed 1D-2D MFE model are then investigated for a challenging problem dealing with infiltration of contaminated water into an initially dry soil involving a fracture network.The new model yields stable results for advection-dominated and advection-dispersion transport configurations. Further, the results of the 1D-2D MFE model are in very good agreement with those of the 2D-2D DG model for both configurations. The simulation of infiltration of contaminated water into a dry fractured soil shows that the 1D-2D MFE model is within 15 times more efficient than the 2D-2D DG model, which confirms the high benefit of using robust and efficient DFM models for the simulation of flow and transport in fractured porous media.

Two-Phase Flow, Heat and Mass Transfer and Tracer Transport to the Atmosphere from Underground Nuclear Cavities Through Fractured Porous Media

Underground nuclear explosions generate non-isothermal two-phase flow, heat and mass transfer, and tracer transport through fractured porous media. The governing equations of these coupled phenomena on the Darcy scale are presented and the corresponding numerical code is briefly described. Two configurations are considered. In the first one, the damaged zone generated by the explosion is replaced by equivalent macroscopic boundary conditions at the bottom of the medium. Vapor condensation at distance from this zone plays a crucial role and slows down the tracer transport to the ground surface, while the presence of hot liquid water and of hot rubble particles in it accelerates it. In the second configuration, the damaged zone itself is treated as a porous medium with specific properties and is integrated in the simulation domain. Several regions can be distinguished with different properties, such as the chimney, the cavity, the solidified magma, the fractured porous medium, and the intact porous medium. Phase change, heat transfer and tracer transport can be detailed in these regions and at the boundaries between them. Among the most important conclusions, the role of phase change, of the thermal properties and therefore of the temperature, is pointed out.

Evolution of hydrate habit and formation properties evolution during hydrate phase transition in fractured-porous medium

Natural gas hydrate, as an efficient and clean energy resource, are naturally distributed in porous and fractured-porous medium. With the most recent development of advanced micro-scale imaging techniques, hydrate habits evolution, hydrate occurrences, and pore structure evolution as well as seepage characteristics during hydrate phase transition in porous hydrate-bearing sediments have been studied extensively at pore scale. However, there are few studies on gas hydrates in fractured-porous sediment. In this work, xenon hydrate phase transition experiment by excess-gas method is carried out in a fractured sandstone core with in-situ micro computed tomography (micro-CT) scanning to explore the evolution of hydrate habits and physical parameters of the host sediment. The results indicate that hydrate-bearing sediment is a dynamic equilibrium system as hydrate synthesis and decomposition occur simultaneously at each moment of hydrate phase transition induced by pressure change. The hydrate occurrences in fractured hydrate reservoirs include contiguous-sheet, clustered and isolated, which are slightly different from that of porous hydrate formation; and the contiguous-sheet hydrate is the occurrence that dominantly determines the seepage characteristics of fractured hydrate-bearing sediments. In addition, the logic diagram for hydrate growth paths in fractured-porous medium is presented for the first time. These findings are significant for detailed understanding of pore-scale hydrate distribution throughout phase transition process and provide theoretical basis for precise modeling of permeability in host sediments.

Boundary-Value Problems for Space-Time Fractional Differential Filtration Dynamics in Fractured-Porous Media

Boundary-Value Problems for Space-Time Fractional Differential Filtration Dynamics in Fractured-Porous Media

Modelling anisotropic adsorption-induced coal swelling and stress-dependent anisotropic permeability

To investigate the anisotropy of coal swelling, this study proposes an effective stress model for saturated, adsorptive fractured porous media by considering gas adsorption induced surface stress change on solid-fluid interface. The effective stress model can be used to capture the anisotropic swelling of coal combining anisotropic mechanical properties and to link with the anisotropic permeability. Direction dependent fracture compressibility is used to describe the evolution of anisotropic stress-dependent permeability behaviour. Particularly, the impact of gas adsorption on fracture compressibility is considered in the model. The proposed models were tested against experimental results and compared to relevant existing models available in literatures. The model predicts that the coal swelling in the direction perpendicular to the bedding plane, is greater than that in the parallel plane. Coal permeability in each direction can be affected by the stress changes in any directions. The permeability parallel to the bedding plane is more sensitive to change in stresses than in perpendicular to the bedding due to higher fracture compressibility. The cleat compressibility could increase with gas adsorption, especially for carbon dioxide. Permeability loss in the direction parallel to the bedding plane is more significant than that in the direction perpendicular to the bedding plane. The presented models provide a tool for quantifying gas adsorption-induced anisotropic coal swelling and permeability behaviours.

ANALYTICAL ELECTRICAL CONDUCTIVITY MODELS FOR SINGLE-PHASE AND MULTI-PHASE FRACTAL POROUS MEDIA

Archie’s law is commonly applied to predict electrical conductivity of rock by correlating the homogeneous, water-wet reservoir rock conductivity with its porosity and saturation. However, Archie’s law is an empirical model consisting of unknown cementation exponent and saturation exponent. In this study, we propose analytical electrical conductivity models for both single-phase and multi-phase porous media based on the fractal theory. The established electrical conductivity model for fractal porous media saturated with a single fluid phase is the function of several factors including pore fluid electrical conductivity, porosity, pore fractal dimension and tortuosity fractal dimension. The established electrical conductivity model for fractal porous media containing multiple fluid phases is the function of wetting fluid saturation, contact angle and pore shape apart from the factors for saturated fractal porous media condition. The calculated formation factor based on the proposed model correlates well with the sandstone experimental data.

An efficient adaptive implicit scheme with equivalent continuum approach for two-phase flow in fractured vuggy porous media

This work investigates numerical method and equivalent continuum approach (ECA) of fluid flow in fractured porous media. The commonly used discrete fracture model (DFM) without upscaling needs full discretization of all fractures. It enjoys the merit of capturing each fracture accurately but will get in trouble with mesh partition and low computational efficiency, especially when a complex geometry is involved. In this study, we develop an efficient implicit scheme with adaptive iteration, in which an improved ECA is devised and then integrated in this scheme. Numerical studies show that the proposed numerical scheme improves the convergence condition and computational efficiency. Then, a test is conducted to demonstrate the feasibility of using superposition principle of permeability tensor in upscaling. Based on these, different strategies are applied to simulate fluid flow in fracture networks with a complex geometry. It is demonstrated that the proposed ECA is able to reproduce the results computed by DFM. The accuracy depends on resolution of background grids. The presented method enjoys a low computational cost and desirable convergence performance compared with the standard DFM in which equivalent continuum is not considered.

A New Multicontinuum Model for Advection-Diffusion Process of Single-Phase Nonlinear Flow in a Multiscale Fractured Porous Media

Fractured porous media modeling and simulation has seen significant development in the past decade but still pose a great challenge and difficulty due to the multiscale nature of fractures, domain heterogeneity, and the nonlinear flow fields due to the high flow velocity and permeability resulting from the presence of fractures. Therefore, modeling fluid transport that is influenced by both advection and diffusion in fractured porous media studies becomes a generic problem, which this study seeks to address. In this paper, we present a study on non-Darcian fluid transport in multiscale naturally fractured reservoirs via an upscaling technique. An averaged macroscopic equation representing pressure distribution in a three-phase multiscale fractured porous medium was developed, consisting of the matrix and a 2-scale fractured network of length-scales ℓ m and ℓ M . The resulting macroscopic model has cross-advective and diffusive terms that account for induced fluxes between the interacting domains, as well as a mass transfer function that is dependent on both physical and geometric properties of the domain, with both advective and diffusive properties. This model also has effective diffusive and advective coefficients that account for reservoir properties such as viscosity, fluid density, and flow velocity. From the numerical simulation, a radial, a horizontal-linear flow behavior, and a transient and quasi-steady-state flow regime that is typical of naturally fractured porous media was observed. The findings of this study will provide researchers a reliable tool to study fractured porous media and can also help for better understanding of the dynamics of flow in fractured reservoirs.

Coupled hydro-mechanical modelling of saturated fractured porous media with unified embedded finite element discretisations

This paper presents the first unified finite element strategy for coupled hydro-mechanical (HM) analysis of fractured porous media where cracks arbitrarily intersect standard finite elements. This strategy is built by employing a discrete strong discontinuity approach and a coupling finite elements framework for the mechanical displacement and fluid pressure fields, respectively. The unified nature of the formulation means that both fracture network and bulk do not need to conform while relying only on standard finite element shape functions. The crack framework is directly embedded in the standard finite elements in both mechanical and fluid field approximations by applying two simple coupling statements. These establish the transmission of displacements due to crack openings accounted by shear and normal stiffness, as well as the internal compatibility of the longitudinal pressure field within the crack network and bulk. Uniquely, the coupling techniques presented here do not require additional degrees of freedom. In addition, the traction at discontinuities and its interaction with the fluid pressure within the discontinuity have a direct physical meaning, and are automatically accounted for in the coupled HM model. Since only standard shape functions are used, the implementation remains simple despite the complexity of the problem being simulated, and no special integration procedures within split domains are required. This is a novelty that contrasts with many existing formulations based on the partition of the unity method and non-matching meshes. Five numerical verification examples are used to assess the performance of the proposed method against existing reference solutions. A good agreement is found between the proposed method and reference solutions.

FLUID–STRUCTURE COUPLING SIMULATION OF EXPLOSIVE FLOW FIELD IN HETEROGENEOUS POROUS MEDIA BASED ON FRACTAL THEORY

In this paper, through finite difference and finite element coupling algorithms with structured grid, the fluid–structure coupling problems of the pressure of the fluid as well as stress and strain of the heterogeneous porous medium under the initial pressure of MPa magnitude are calculated, respectively. The porous medium is a cavity cylinder with fractal characteristics. The parameters of porosity, elastic modulus and Poisson’s ratio of porous media with different fractal dimensions ([Formula: see text], 2.4 and 2.5) are obtained by theoretical calculation. Through theoretical derivation, the amplitude of WM function is expressed by the fractal parameter porosity, which can predict the uneven characteristics caused by the nonuniformity of pore size distribution after explosion. The results show that with the increase of fractal dimensions, the porosity increases, and both the elastic modulus and the Poisson’s ratio decrease. The Poisson’s ratio has a linear relationship with porosity, which is well consistent with the modified formula in the literature. Under the same initial pressure, the stress and strain values of the material without fractal characteristics are much larger than those of the material with fractal feature which has an obvious fluctuation by extracting the average values. It indicates that the fractal porous media with rough surface can effectively reduce the pressure wave peak at the wall surface, and reduce the size of the stress and strain concentrations to prevent the vibration of the cylinder. As expected, if the initial pressure increases with an equal fractal dimension, the stress and strain values of the fluid–structure coupling will increase too. When the initial pressure is equal, the peak value of pressure in wall decreases gradually with the increase of fractal dimension, and the stress and strain values also decrease.

Numerical treatment of temporal-fractional porous medium model occurring in fractured media

This paper proposes a temporal-fractional porous medium model (T-FPMM) for describing the co-current and counter-current imbibition, which arises in a water-wet fractured porous media. The correlation between the co-current and counter-current imbibition for the fractures and porous matrix are examined to determine the saturation and recovery rate of the reservoir. For different fractional orders in both porous matrix and fractured porous media, the homotopy analysis technique and its stability analysis are used to explore the parametric behavior of the saturation and recovery rates. Finally, the effects of wettability and inclination on the recovery rate and saturation are studied for distinct fractional values.

Multilevel Monte Carlo estimators for elliptic PDEs with Lévy-type diffusion coefficient

General elliptic equations with spatially discontinuous diffusion coefficients may be used as a simplified model for subsurface flow in heterogeneous or fractured porous media. In such a model, data sparsity and measurement errors are often taken into account by a randomization of the diffusion coefficient of the elliptic equation which reveals the necessity of the construction of flexible, spatially discontinuous random fields. Subordinated Gaussian random fields are random functions on higher dimensional parameter domains with discontinuous sample paths and great distributional flexibility. In the present work, we consider a random elliptic partial differential equation (PDE) where the discontinuous subordinated Gaussian random fields occur in the diffusion coefficient. Problem specific multilevel Monte Carlo (MLMC) Finite Element methods are constructed to approximate the mean of the solution to the random elliptic PDE. We prove a-priori convergence of a standard MLMC estimator and a modified MLMC—control variate estimator and validate our results in various numerical examples.

Hybrid-dimensional modeling for fluid flow in heterogeneous porous media using dual fracture-pore model with flux interaction of fracture–cavity network

Developing numerical method of fractured porous media is of paramount importance in geoscience applications. Previous studies have revealed that the discrete fractures and cavities as well as the heterogeneity have considerable influences on hydraulic property of porous media. This work presents a numerical investigation on fluid flow in heterogeneous porous media with the consideration of flux connection of fracture–cavity network. A hybrid-dimensional modeling approach combined with the dual fracture-pore model is presented. Then, the numerical scheme is derived from Galerkin finite element method. Especially, the numerical treatment on flux interaction of multiple fractures is elaborated. Next, this model is verified by a benchmark study, and grid convergence test is performed to show the grid independence. Later, a fractured porous medium is simulated with different states of cavity. The effects of impermeable and conductive fractures on fluid flow are studied. In contrast to the homogeneous situation, we consider the effects of heterogeneity. Meanwhile, a comparison study is conducted to investigate the impacts of heterogeneity, boundary conditions and conductivity of the fracture–cavity network on fluid flow. Furthermore, pressure deviation induced by heterogeneity is analyzed with different conductivities of fractures and cavities. It appears that pressure distribution is highly related to fractures conductivity and the state of cavities, where the influence of heterogeneity on the high-conductivity fractures is relatively smaller than the low-conductivity.

Matrix–Fracture Flow Transfer in Fractured Porous Media: Experiments and Simulations

Matrix–Fracture Flow Transfer in Fractured Porous Media: Experiments and Simulations

Analytical and numerical studies on a moving boundary problem of non-Newtonian Bingham fluid flow in fractal porous media

Non-Darcy flow with a threshold in fractal porous media has been widely used in the development of unconventional petroleum resources such as heavy oil and tight oil. Mathematical modeling of such challenging “threshold flow” problems with strong nonlinearity has great significance in improving petroleum science and technology. Based on a fractal theory, a new non-Darcy kinematic equation with a fractal threshold pressure gradient (TPG) is mathematically deduced in order to describe the non-Darcy flow of a non-Newtonian Bingham fluid with a threshold in fractal porous media. Then mathematical modeling of planar radial non-Darcy flow in a fractal heavy oil reservoir is performed as a nonlinear moving boundary problem. In addition, a steady analytical solution method and a transient numerical solution method are developed. The analytical solution of an ordinary differential equation system for a simple steady model is derived directly, and the transient numerical solution of a partial differential equation system for an unsteady flow model is obtained based on the finite element method with good convergence. These two model solutions are validated by cross-comparisons. It is found from the calculation results that for the steady state, the extremely disturbed moving boundary and its corresponding pressure distribution are affected only by a TPG, production pressure, and a transport exponent; by contrast, for the unsteady state, the moving boundary and its corresponding pressure distribution are affected by many more factors including the fractal dimension. Furthermore, neglect of the fractal TPG and the induced moving boundary can lead to high overestimation of well productivity.

Finite Element Modeling of Diffusion in Fractured Porous Media by Using Hierarchical Material Properties

AbstractFractured porous media challenge modeling approaches due to high computational costs and excessive mesh refinement imposed by the extreme scale variability of fractures and the heterogeneity of the surrounding porous rock. To overcome such difficulties, we utilize the hierarchical finite element method (Hi‐FEM) that has been developed previously to simulate the electrical potential distribution in complex geologic environments. The method employs the hierarchical basis functions in classical finite element analysis to enable representation of material properties on each dimensional component of a given 3D unstructured finite element, thereby inherently allowing for interactions at the boundary between fracture and a host rock. In this study, we extend its application to transient fluid flow and heat conduction in the Laplace domain. Time‐domain flow solutions are obtained by numerical inverse Laplace transform. We evaluate the accuracy of the method using different flow models and demonstrate its robustness for large‐scale, rock mass models featuring complex fracture networks. Moreover, for the computation of nodal Darcian velocity fields in fractured porous media where the fractures are represented as 2D features, a new approach that employs the Yeh's Galerkin model for both volume and facet elements is proposed. Results show that Hi‐FEM can produce accurate flow solutions for fractured porous media without any need of coupling or transfer mechanism while still being computationally economical and numerically robust, even for large‐scale simulations.