Fractured Porous Media Research Articles

Geothermal modeling in complex geological systems with ComPASS

In deep geothermal reservoirs, faults and fractures play a major role, serving as regulators of fluid flow and heat transfer while also providing feed zones for production wells. To accurately model the operation of geothermal fields, it is necessary to explicitly consider objects of varying spatial scales, from the reservoir scale itself, to that of faults and fractures, down to the scale of the injection and production wells.Our main objective in developing the ComPASS geothermal flow simulator, was to take into account all of these geometric constraints in a flow and heat transfer numerical model using generic unstructured meshes. In its current state, the code provides a parallel implementation of a spatio-temporal discretization of the non-linear equations driving compositional multi-phase thermal flows in porous fractured media on unstructured meshes. It allows an explicit discretization of faults and fractures as 2D hybrid objects, embedded in a 3D matrix. Similarly, wells are modeled as one dimensional graphs discretized by edges of the 3D mesh which allows arbitrary multi-branch wells. The resulting approach is particularly flexible and robust in terms of modeling.Its practical interest is demonstrated by two case studies in high-energy geothermal contexts.

Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media

This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and a theoretical study of the stability and convergence of the method was carried out using the method of a priori estimates. The results were confirmed through a comparative analysis of the empirical and theoretical orders of convergence based on computational experiments. Furthermore, we analyzed the effect of variable-order functions of fractional derivatives on the process of fluid flow in a heterogeneous medium, presenting new practical results in the field of modeling the fluid flow in complex media. This work is an important contribution to the numerical modeling of filtration in porous media with variable orders of fractional derivatives and may be useful for specialists in the field of hydrogeology, the oil and gas industry, and other related fields.

Digital rock modeling of deformed multi-scale media in deep hydrocarbon reservoirs based on in-situ stress-loading CT imaging and U-Net deep learning

Deep and ultra-deep hydrocarbon reservoirs are regarded as an attractive target for onshore oil and gas exploration and development in China. The multi-scale fractures are widely distributed in this type of reservoir. The fluid-solid coupling effect is strong, which can greatly affect the fluid flow, resulting in a low hydrocarbon recovery with the average value less than 15%. To explore the underlying flow dynamics during efficient development of deep and ultra-deep hydrocarbon reservoirs, it is of great importance to characterize the multi-scale fracture-pore structure evolution of rock affected by strong geo-stress field variation. To tackle the above issues, an actual rock drilled from a typical ultra-deep reservoir in Tarim Basin are selected to conduct in-situ stress-loading computed tomography (CT) scanning experiments, and the CT gray images of rock microstructure under different dynamic loading conditions are obtained. A fully convolutional neural network (U-Net) deep learning segmentation algorithm is introduced to accurately distinguish the rock skeleton, pore space and fractures in deep rock. The three-dimensional (3-D) digital rock model of deformed multi-scale fracture-porous media under different dynamic loading conditions is ultimately established to investigate the evolution of fracture morphology and deep rock microstructure as the in-situ stress is gradually loaded. It indicates that, the U-Net deep learning semantic segmentation algorithm can accurately segment CT images of deep rocks, and the established 3D digital rock model of fractured porous media can accurately represent the pore-throat distribution, fracture morphology, and pore-scale topological connectivity. At the initial stage of stress loading, there are few micro-fractures inside the rock, and the fracture connectivity is poor. Due to effect of rock compaction, the average pore throat radius increases while pore throat ratio, coordination number and tortuosity greatly decrease. As the effective stress is gradually loaded, the micro-fractures begin to propagate, leading to stronger heterogeneity of micro-fractures’ distribution and better topological connectivity, and both the coordination number and tortuosity gradually increase. After the deep rock is fractured, micro-fractures run through a fracture network and all the above topological parameters increase greatly.

A Fully Coupled Discontinuous Deformation Analysis Model for Simulating Hydromechanical Processes in Fractured Porous Media

Numerical simulations play a key role in the optimization of fracturing operation designs for unconventional reservoirs. Because of the presence of numerous natural discontinuities and pores, the rock masses of reservoirs can be regarded as fractured porous media. In this paper, a fully coupled discontinuous deformation analysis model is newly developed to simulate the hydromechanical processes in fractured and porous media. The coupling of fracture seepage, pore seepage, and fracture network propagation is realized under the framework of DDA. The developed model is verified with several examples. Then, the developed DDA model is applied to simulate the hydraulic fracturing processes in fractured porous rock masses, and the effects of rock mass permeability on fracturing are investigated. Our findings suggest that high rock permeability may inhibit the stimulation of fracture networks, while increasing the viscosity of fracturing fluids can enhance the fracturing efficiency. This study provides a valuable numerical tool for simulating hydromechanical processes in fractured and porous media and can be used to analyze various geo-mechanical problems related to fluid interactions.

An Improved Probabilistic Collocation Method for Uncertainty Quantification of Oil–Water Transport through Fractured Porous Media: Effect of Uncertain Gravity Imbibition

An Improved Probabilistic Collocation Method for Uncertainty Quantification of Oil–Water Transport through Fractured Porous Media: Effect of Uncertain Gravity Imbibition

Stochastic Modeling of Two-Phase Transport in Fractured Porous Media Under Geological Uncertainty Using an Improved Probabilistic Collocation Method

Summary Simulation of multiphase transport through fractured porous media is highly affected by the uncertainty in fracture distribution and matrix block size that arises from inherent heterogeneity. To quantify the effect of such uncertainties on displacement performance in porous media, the probabilistic collocation method (PCM) has been applied as a feasible and accurate approach. However, propagation of uncertainty during the simulation of unsteady-state transport through porous media could not be computed by this method or even by the direct-sampling Monte Carlo (MC) approach. Therefore, with this research, we implement a novel numerical modeling workflow that improves PCM on sparse grids and combines it with the Smolyak algorithm for selection of collocation points sets, Karhunen-Loeve (KL) decomposition, and polynomial chaos expansion (PCE) to compute the uncertainty propagation in oil-gas flow through fractured porous media in which gravity drainage force is enabled. The effect of uncertainty in the vertical dimension of matrix blocks, which are frequently an uncertain and history-matching parameter, on simulation results of randomly synthetic 3D fractured media is explored. The developed numerical model is innovatively coupled with solving governing deterministic partial differential equations (PDEs) to compute uncertainty propagation from the first timestep to the last timestep of the simulation. The uncertainty interval and aggregation of uncertainty in ultimate recovery are quantified, and statistical moments for simulation outputs are presented at each timestep. The results reveal that the model properly quantifies uncertainty and extremely reduces central processing unit (or CPU) time in comparison with MC simulation.

An Enriched Petrov-Galerkin Method for Darcy Flow in Fractured Porous Media

An Enriched Petrov-Galerkin Method for Darcy Flow in Fractured Porous Media

Multiscale model reduction for the time fractional thermoporoelasticity problem in fractured and heterogeneous media

In this paper, we consider the time fractional thermoporoelasticity problem in fractured and heterogeneous media. The mathematical model with a time memory formalism is described by a coupled system of equations for pressure, temperature and displacements. We use an implicit finite difference approximation for temporal discretization. We present a fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) for two-dimensional model problems. Further, we use the Generalized Multiscale Finite Element Method (GMsFEM) for coarse grid approximation. The primary concept behind the proposed method is to streamline the complexity inherent in the thermoporoelasticity problem. Given that our model equation incorporates multiple fractional powers, leading to multiple unknowns with memory effects, we aim to address this intricacy by optimizing the problem’s dimensionality. As a result, the solution is sought on a coarse grid, a strategic choice that not only simplifies the computational cost but also contributes to significant time savings. We present numerical results for the two-dimensional model problems in heterogeneous fractured porous media. We derive relative errors between the reference fine grid solution and the multiscale solution for different numbers of multiscale basis functions. The results confirm that the proposed method is able to achieve good accuracy with a few degrees of freedoms on the coarse grid.

Modelling of mixed-mechanism stimulation for the enhancement of geothermal reservoirs.

Hydraulic stimulation is a critical process for increasing the permeability of fractured geothermal reservoirs. This technique relies on coupled hydromechanical processes induced through pressurized fluid injection into the rock formation. The injection of fluids causes poromechanical stress changes that can lead to fracture slip and shear dilation, as well as tensile fracture opening and propagation, so-called mixed-mechanism stimulation. The effective permeability of the rock is particularly enhanced when new fractures connect with pre-existing fractures. While hydraulic stimulation can significantly improve the productivity of fractured geothermal reservoirs, the process is also related to induced seismicity. Hence, understanding the coupled physics is central, for both reservoir engineering and seismic risk mitigation. This article presents a modelling approach for simulating the deformation, propagation and coalescence of fractures in porous media under the influence of anisotropic stress and fluid injection. It uses a coupled hydromechanical model for poroelastic, fractured media. Fractures are governed by contact mechanics and a fracture propagation model. For numerical solutions, we employ a two-level approach, combining a finite volume method for poroelasticity with a finite element method for fracture propagation. The study investigates the impact of injection rate, matrix permeability and stress anisotropy on stimulation outcomes.This article is part of the theme issue 'Induced seismicity in coupled subsurface systems'.

Phase-field modelling of dynamic hydraulic fracturing in porous media using a strain-based crack width formulation

This paper presents a comprehensive study on phase-field modelling in COMSOL MultiPhysics for simulating dynamic hydraulic fracturing in porous media based on Biot’s poro-elasticity theory. The focus is on addressing the challenges associated with crack width estimation in this context. A new strain-based crack width formulation is proposed, offering improved accuracy in predicting fracture permeability and ease of implementation in numerical approaches. The model’s capabilities are extended to consider dynamic crack propagation by incorporating the kinetic energy in the governing coupled hydro-mechanical-damage equations. The numerical implementations in COMSOL MultiPhysics are thoroughly explained, providing insights into the techniques used to solve the governing equations. Verification examples, including the benchmark KGD verification, are presented to demonstrate the model’s capabilities in simulating hydraulic fractures in porous media and validate its accuracy and reliability. A final numerical example focusing on the dynamics of crack propagation in a gravity dam is simulated, allowing for a comprehensive examination of the model’s performance. The proposed strain-based crack width formulation and consideration of dynamic crack propagation contribute to improved accuracy in predicting fracture permeability.

A Roundness Calculation Method and Physical Parameter Analysis of A Porous Medium Model

The rounding degree of particles is an important index of particle morphology. It represents the degree of rounding of the original edges and corners of debris particles. To a certain extent, it can reflect the transportation and deposition process of debris particles and the source properties. Therefore, it is of great significance to explore the rounding law of gravel and the influencing factors of rounding degree for the study of geological sedimentary environment, source analysis and mass transfer process. Therefore, this paper proposes a rounding degree calculation algorithm for fractal porous media model particles with rounding characteristics. Through numerical simulation and theoretical analysis, combined with the geometric parameters of particle morphology, the relationship between particle roundness and area and perimeter parameters is discussed in detail. The results show that roundness is an important factor in the geometric parameters of particles. The change of roundness can significantly affect the geometric shape of particles, and then affect the size and area of particles. It is of great significance for understanding and optimizing the flow characteristics of porous media models.

Physics-informed neural network simulation of two-phase flow in heterogeneous and fractured porous media

Physics-informed neural networks (PINNs) have received great attention as a promising paradigm for forward, inverse, and surrogate modeling of various physical processes with limited or no labeled data. However, PINNs are rarely used to predict two-phase flow in heterogeneous and fractured porous media, which is critical to lots of subsurface applications, due to the significant challenges in their training. In this work, we present an Enriched Physics-Informed Neural Network (E-PINN) to overcome these barriers and realize the simulation of such flow. Specifically, the Embedded Discrete Fracture Model (EDFM) is adopted to explicitly represent fractures, and then the finite volume method (FVM) instead of the Automatic Differentiation (AD) is used to evaluate spatial derivatives and construct the physics-informed loss function, so that the flux continuity between neighboring elements with different properties (e.g. matrix and fracture) can be defined rigorously. Besides, we develop a novel physics-informed neural network (NN) architecture adopting the adjacency-location anchoring, adaptive activation function, skip connection and gated updating to enrich the pressure information and enhance the learning ability of NN. Additionally, the initial and boundary conditions are constrained through a hard approach, which encodes them into network design, to improve the accuracy and efficiency of network training. In order to further reduce the difficulty of training, the Implicit-Pressure Explicit-Saturation (IMPES) scheme is used to calculate pressure and saturation, in which only the pressure needs to be solved by training NN. Finally, the superiority and applicability of E-PINN to complex practical problems is demonstrated through the simulations of immiscible displacement in 2D/3D heterogeneous and fractured reservoirs.

Multiscale analysis-based peridynamic simulation of fracture in porous media

Multiscale analysis-based peridynamic simulation of fracture in porous media

The Pathologically Evolving Aggregation-State of Cells in Cancerous Tissues as Interpreted by Fractal and Multi-Fractal Dispersion Theory in Saturated Porous Formations.

A recent author's fractal fluid-dynamic dispersion theory in porous media has focused on the derivation of the associated nonergodic (or effective) macrodispersion coefficients by a 3-D stochastic Lagrangian approach. As shown by the present study, the Fickian (i.e., the asymptotic constant) component of a properly normalized version of these coefficients exhibits a clearly detectable minimum in correspondence with the same fractal dimension (d ≅ 1.7) that seems to characterize the diffusion-limited aggregation state of cells in advanced stages of cancerous lesion progression. That circumstance suggests that such a critical fractal dimension, which is also reminiscent of the colloidal state of solutions (and may therefore identify the microscale architecture of both living and non-living two-phase systems in state transition conditions) may actually represent a sort of universal nature imprint. Additionally, it suggests that the closed-form analytical solution that was provided for the effective macrodispersion coefficients in fractal porous media may be a reliable candidate as a physically-based descriptor of blood perfusion dynamics in healthy as well as cancerous tissues. In order to evaluate the biological meaningfulness of this specific fluid-dynamic parameter, a preliminary validation is performed by comparison with the results of imaging-based clinical surveys. Moreover, a multifractal extension of the theory is proposed and discussed in view of a perspective interpretative diagnostic utilization.

Quantitative characterization of imbibition in fractured porous media based on fractal theory

In low-permeability reservoirs, such as shale and tight sandstone, imbibition is an important mechanism for enhancing oil recovery. After hydraulic fracturing treatment, these reservoirs create a network of fracture pathways for fluid flow. Therefore, understanding the imbibition mechanisms in fractured porous media and quantitatively characterizing oil–water distribution are crucial for the development of low-permeability reservoirs. In this study, a mathematical model of two-phase flow in porous media with branching fractures was established. The phase-field method was employed to track the oil–water interface, and quantitative characterization of imbibition was conducted based on fractal theory, and the effects of wetting phase injection rate, the number of disconnected fractures, fracture spacing, and fracture morphology on imbibition in branched fracture porous media were discussed. The research findings indicate that in branched fracture porous media, both co-current and countercurrent imbibition processes occur simultaneously, and there exists a diffusion interface layer with a certain thickness at the oil–water interface. The hydraulic pressure generated by the wetting phase injection rate provides the driving force for imbibition oil recovery, but it also affects the contact time between the wetting and non-wetting phases. The presence of disconnected fractures hinders the propagation of hydraulic pressure, reducing the effectiveness of imbibition. The imbibition displacement zone is limited and occurs only within a certain range near the fractures. As the number of branching fractures increases, the channels for the wetting phase to enter matrix pores are enhanced, resulting in higher efficiency of imbibition displacement of the oil phase. The results of this research can provide guidance for the design of fracturing programs and recovery prediction in low-permeability reservoirs.

Non-Integer Dimensional Analysis of Ultrasonic Wave Propagation in Fractal Porous Media

This paper explores the acoustics of porous media characterized by fractal, or self-similar, structures. Employing a fractal approach, we use differential operators in non-integer dimensional spaces to address the fundamental equations of acoustics in such media. The primary aim is to examine the transmission of ultrasonic waves within a fractal porous medium. Our findings reveal that the fractal dimension significantly influences wave transmission. In fractal porous materials, waves travel along more complex and intricate paths, resulting in increased tortuosity and attenuation. We introduce the concept of an effective path length leff , which is dependent on the fractal dimension Dx , to describe the actual trajectory of wave propagation. Additionally, we define an effective tortuosity αDx , directly proportional to leff2 , to quantify the additional tortuosity brought about by the fractal structure. The insights gained from this study are crucial, as they enhance our understanding of wave behavior in self-similar porous media, which are prevalent in various natural settings and have multiple practical applications, including sound insulation and the design of acoustic materials. Furthermore, understanding the impact of fractal dimensions on wave behavior is vital for developing more efficient acoustic solutions. This research also sets the stage for further theoretical and experimental work on applying fractal geometry to analyze wave propagation in porous structures.

Some Boundary-Value Problems Corresponding to the Model of Fractional Differential Filtration Dynamics in a Fractured Porous Media Under the Condition of Time Non-Locality

Some Boundary-Value Problems Corresponding to the Model of Fractional Differential Filtration Dynamics in a Fractured Porous Media Under the Condition of Time Non-Locality

Numerical Simulations of Viscous Fingering in Fractured Porous Media

AbstractThe effect of heterogeneity induced by highly permeable fracture networks on viscous miscible fingering in porous media is examined using high-resolution numerical simulations. We consider the planar injection of a less viscous fluid into a two-dimensional fractured porous medium that is saturated with a more viscous fluid. This problem contains two sets of fundamentally different preferential flow regimes; the first is caused by the viscous fingering, and the second is due to the permeability contrasts between the fractures and the rock matrix. We study the transition from the regime where the flow is dominated by the viscous instabilities, to the regime where the heterogeneity induced by the fractures define the flow paths. Our findings reveal that even minor permeability differences between the rock matrix and fractures significantly influence the behavior of viscous fingering. The interplay between the viscosity contrast and permeability contrast leads to the preferential channeling of the less viscous fluid through the fractures. Consequently, this channeling process stabilizes the displacement front within the rock matrix, ultimately suppressing the occurrence of viscous fingering, particularly for higher permeability contrasts. We explore three fracture geometries: two structured and one random configuration and identify a complex interaction between these geometries and the development of unstable flow. While we find that the most important factor determining the effect of the fracture network is the ratio of fluid volume flowing through the fractures and the rock matrix, the exact point for the cross-over regime is dependent on the geometry of the fracture network.

Experimental and Numerical Studies of Water–Sand Flow in Fractured Porous Media

Experimental and Numerical Studies of Water–Sand Flow in Fractured Porous Media

A critique on “Modeling the spontaneous imbibition of non-Newtonian fluids into the fractal porous media of tight reservoirs”

In February 2022, this journal published the paper “Modeling the spontaneous imbibition of non-Newtonian fluids into the fractal porous media of tight reservoirs” (Zhu et al., 2022), which develops and validates an analytical model for the capillary absorption of (non-)Newtonian liquids in sandstone, founded on a fractal-based bundle-of-tubes model. This critique – a post-publication review – expresses concerns on that model development and validation. It is shown that agreement between calculated and measured absorbed moisture masses and moisture front heights is only attained with two strongly distinct sets of material and fluid properties, a dichotomy that is however not disclosed in the paper. It is moreover revealed that the input material and fluid properties and the output absorbed moisture masses and moisture front heights often defy physical plausibility. Detailed scrutiny of the mathematical model complementarily exposes concrete defects in the calculation of the ‘sample’ cross-section. It is displayed that this cross-section calculation is exclusively governed by the material properties, implying that the measured cross-section can (almost) never be equated. It is furthermore established that the cross-section calculation assumes the equality of the areal and volumetric porosity, which is incorrect for a bundle-of-tubes with tortuous capillaries. Additionally it is demonstrated that the cross-section calculation merges two conflicting relations between the porosity and the fractal properties. All these concerns bring about doubts concerning the reliability of (Zhu et al., 2022)'s model development and validation, and the authors are hence invited to placate these concerns in their reply.