Permeability of matrix-fracture systems under mechanical loading – constraints from laboratory experiments and 3-D numerical modelling

Abstract. The permeability of single fractures is commonly approximated by the cubic law assumption, which is however only valid under the condition of a single phase laminar flow between parallel plates. Departure from cubic law are related to many features like aperture fluctuations due to fracture surface roughness, relative shear displacement, the amount of flow exchange between the matrix and the fracture itself, etc. In order to quantify constitutive relationships among the aforementioned aspects, we have conducted a flow-through experiment with a porous rock sample (Flechtinger sandstone) containing a single macroscopic fracture. Based on this experiment, we obtained range of variations of intrinsic rock parameters, permeability and stress-strain relationships of the combined matrix-fracture system under hydrostatic loading. From the measured deformation of the matrix-fracture system, we derived the evolution in the mechanical aperture of the fracture. In order to quantify the processes behind the laboratory observations, we carried out coupled hydro-mechanical simulations of the matrix-fracture system. Navier–Stokes flow was solved in the 3-dimensional open rough fracture domain, and back-coupled to the Darcy flow and the poroelastic behaviour of the rock matrix. The results demonstrate that the elastic behaviour and the related permeability alteration of the fracture domain could be captured by the numerical simulation. Furthermore, the stress-strain values obtained in the vicinity of the fracture asperities suggest that inelastic deformation develops at low mechanical load. An attempt was made to quantify the inelastic deformation by using the failure envelope obtained by laboratory experiments (whether tensile, shear, compaction, or a combination of those). However, change in permeability observed in the experiments are significantly larger than that in the simulation showing the importance of plastic deformation during opening and closure of the fracture and its impact on the cubic law approximation.

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Fractured Reservoir Simulation

Society of Petroleum Engineers Journal ◽

10.2118/9305-pa ◽

1983 ◽

Vol 23 (01) ◽

pp. 42-54 ◽

Cited By ~ 123

Author(s):

L. Kent Thomas ◽

Thomas N. Dixon ◽

Ray G. Pierson

Keyword(s):

Fractured Reservoirs ◽

Fracture Flow ◽

Fracture System ◽

Fractured Reservoir ◽

Three Phase ◽

Matrix Fracture ◽

Naturally Fractured ◽

Matrix Blocks ◽

The Matrix ◽

Saturation History

Abstract This paper describes the development of a three-dimensional (3D), three-phase model for simulating the flow of water, oil, and gas in a naturally fractured reservoir. A dual porosity system is used to describe the fluids present in the fractures and matrix blocks. Primary flow present in the fractures and matrix blocks. Primary flow in the reservoir occurs within the fractures with local exchange of fluids between the fracture system and matrix blocks. The matrix/fracture transfer function is based on an extension of the equation developed by Warren and Root and accounts for capillary pressure, gravity, and viscous forces. Both the fracture flow equations and matrix/fracture flow are solved implicitly for pressure, water saturation, gas saturation, and saturation pressure. We present example problems to demonstrate the utility of the model. These include a comparison of our results with previous results: comparisons of individual block matrix/fracture transfers obtained using a detailed 3D grid with results using the fracture model's matrix/fracture transfer function; and 3D field-scale simulations of two- and three-phase flow. The three-phase example illustrates the effect of free gas saturation on oil recovery by waterflooding. Introduction Simulation of naturally fractured reservoirs is a challenging task from both a reservoir description and a numerical standpoint. Flow of fluids through the reservoir primarily is through the high-permeability, low-effective-porosity fractures surrounding individual matrix blocks. The matrix blocks contain the majority of the reservoir PV and act as source or sink terms to the fractures. The rate of recovery of oil and gas from a fractured reservoir is a function of several variables, included size and properties of matrix blocks and pressure and saturation history of the fracture system. Ultimate recovery is influenced by block size, wettability, and pressure and saturation history. Specific mechanisms pressure and saturation history. Specific mechanisms controlling matrix/fracture flow include water/oil imbibition, oil imbibition, gas/oil drainage, and fluid expansion. The study of naturally fractured reservoirs has been the subject of numerous papers over the last four decades. These include laboratory investigations of oil recovery from individual matrix blocks and simulation of single- and multiphase flow in fractured reservoirs. Warren and Root presented an analytical solution for single-phase, unsteady-state flow in a naturally fractured reservoir and introduced the concept of dual porosity. Their work assumed a continuous uniform porosity. Their work assumed a continuous uniform fracture system parallel to each of the principal axes of permeability. Superimposed on this system was a set of permeability. Superimposed on this system was a set of identical rectangular parallelopipeds representing the matrix blocks. Mattax and Kyte presented experimental results on water/oil imbibition in laboratory core samples and defined a dimensionless group that relates recovery to time. This work showed that recovery time is proportional to the square root of matrix permeability divided by porosity and is inversely proportional to the square of porosity and is inversely proportional to the square of the characteristic matrix length. Yamamoto et al. developed a compositional model of a single matrix block. Recovery mechanisms for various-size blocks surrounded by oil or gas were studied. SPEJ P. 42

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Black rubber and the non-linear elastic response of scale invariant solids

Journal of High Energy Physics ◽

10.1007/jhep09(2020)013 ◽

2020 ◽

Vol 2020 (9) ◽

Cited By ~ 1

Author(s):

Matteo Baggioli ◽

Víctor Cáncer Castillo ◽

Oriol Pujolàs

Keyword(s):

Strain Curve ◽

Effective Field ◽

Elastic Response ◽

Elastic Behaviour ◽

Scale Invariant ◽

Stress Strain ◽

Nonlinear Stress ◽

Hyper Elastic ◽

Simple Class ◽

Nonlinear Elastic

Abstract We discuss the nonlinear elastic response in scale invariant solids. Following previous work, we split the analysis into two basic options: according to whether scale invariance (SI) is a manifest or a spontaneously broken symmetry. In the latter case, one can employ effective field theory methods, whereas in the former we use holographic methods. We focus on a simple class of holographic models that exhibit elastic behaviour, and obtain their nonlinear stress-strain curves as well as an estimate of the elasticity bounds — the maximum possible deformation in the elastic (reversible) regime. The bounds differ substantially in the manifest or spontaneously broken SI cases, even when the same stress- strain curve is assumed in both cases. Additionally, the hyper-elastic subset of models (that allow for large deformations) is found to have stress-strain curves akin to natural rubber. The holographic instances in this category, which we dub black rubber, display richer stress- strain curves — with two different power-law regimes at different magnitudes of the strain.

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Predictive Computational Model for Damage Behavior of Metal-Matrix Composites Emphasizing the Effect of Particle Size and Volume Fraction

Materials ◽

10.3390/ma14092143 ◽

2021 ◽

Vol 14 (9) ◽

pp. 2143

Author(s):

Shaimaa I. Gad ◽

Mohamed A. Attia ◽

Mohamed A. Hassan ◽

Ahmed G. El-Shafei

Keyword(s):

Finite Element ◽

Metal Matrix Composites ◽

Metal Matrix ◽

Volume Fraction ◽

Particulate Composites ◽

Aluminum Matrix Composites ◽

Matrix Composites ◽

Zone Method ◽

Stress Strain ◽

The Matrix

In this paper, an integrated numerical model is proposed to investigate the effects of particulate size and volume fraction on the deformation, damage, and failure behaviors of particulate-reinforced metal matrix composites (PRMMCs). In the framework of a random microstructure-based finite element modelling, the plastic deformation and ductile cracking of the matrix are, respectively, modelled using Johnson–Cook constitutive relation and Johnson–Cook ductile fracture model. The matrix-particle interface decohesion is simulated by employing the surface-based-cohesive zone method, while the particulate fracture is manipulated by the elastic–brittle cracking model, in which the damage evolution criterion depends on the fracture energy cracking criterion. A 2D nonlinear finite element model was developed using ABAQUS/Explicit commercial program for modelling and analyzing damage mechanisms of silicon carbide reinforced aluminum matrix composites. The predicted results have shown a good agreement with the experimental data in the forms of true stress–strain curves and failure shape. Unlike the existing models, the influence of the volume fraction and size of SiC particles on the deformation, damage mechanism, failure consequences, and stress–strain curve of A359/SiC particulate composites is investigated accounting for the different possible modes of failure simultaneously.

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On the scattering of sound by two semi-infinite parallel staggered plates - I. Explicit matrix Wiener-Hopf factorization

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1988.0121 ◽

1988 ◽

Vol 420 (1858) ◽

pp. 131-156 ◽

Cited By ~ 18

Keyword(s):

Asymptotic Expression ◽

Series Representation ◽

Auxiliary Function ◽

Sound Waves ◽

Parallel Plates ◽

Hopf Equation ◽

Product Decomposition ◽

Matrix Kernel ◽

Wave Fields ◽

The Matrix

This paper discusses the two-dimensional scattering of sound waves by two semi-infinite rigid parallel plates. The plates are staggered, so that a line in the plane of the motion passing through both edges is not in general perpendicular to the plane of either plate. The problem is formulated as a matrix Wiener-Hopf functional equation, which exhibits the difficulty of a kernel containing exponentially growing elements. We show how this difficulty may be overcome by constructing an explicit product decomposition of the matrix kernel with both factors having algebraic behaviour at infinity. This factorization is written in terms of a single entire auxiliary function that has a simple infinite series representation. The Wiener-Hopf equation is solved for arbitrary incident wave fields and we derive an asymptotic expression for the field scattered to infinity; the latter includes the possibility of propagating modes in the region between the plates. In part II of this work we will evaluate our solution numerically and obtain some analytical estimates in a number of physically interesting limits.

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An Integrally Embedded Discrete Fracture Model for Flow Simulation in Anisotropic Formations

Energies ◽

10.3390/en13123070 ◽

2020 ◽

Vol 13 (12) ◽

pp. 3070

Author(s):

Renjie Shao ◽

Yuan Di ◽

Dawei Wu ◽

Yu-Shu Wu

Keyword(s):

Flow Simulation ◽

Fracture Model ◽

Fluid Exchange ◽

Two Phase ◽

Fine Grid ◽

Discrete Fracture Model ◽

Matrix Fracture ◽

Discrete Fracture ◽

The Matrix ◽

Anisotropic Formation

The embedded discrete fracture model (EDFM), among different flow simulation models, achieves a good balance between efficiency and accuracy. In the EDFM, micro-scale fractures that cannot be characterized individually need to be homogenized into the matrix, which may bring anisotropy into the matrix. However, the simplified matrix–fracture fluid exchange assumption makes it difficult for EDFM to address the anisotropic flow. In this paper, an integrally embedded discrete fracture model (iEDFM) suitable for anisotropic formations is proposed. Structured mesh is employed for the anisotropic matrix, and the fracture element, which consists of a group of connected fractures, is integrally embedded in the matrix grid. An analytic pressure distribution is derived for the point source in anisotropic formation expressed by permeability tensor, and applied to the matrix–fracture transmissibility calculation. Two case studies were conducted and compared with the analytic solution or fine grid result to demonstrate the advantage and applicability of iEDFM to address anisotropic formation. In addition, a two-phase flow example with a reported dataset was studied to analyze the effect of the matrix anisotropy on the simulation result, which also showed the feasibility of iEDFM to address anisotropic formation with complex fracture networks.

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Numerical Modelling of Stress/Strain Field Arising in Diamond-Impregnated Cobalt

Archives of Metallurgy and Materials ◽

10.2478/amm-2014-0073 ◽

2014 ◽

Vol 59 (2) ◽

pp. 443-446 ◽

Cited By ~ 4

Author(s):

J. Borowiecka-Jamrozek ◽

J. Lachowski

Keyword(s):

Strain Field ◽

Diamond Crystal ◽

Metallic Matrix ◽

Stress And Strain ◽

Stress Strain ◽

Strain Fields ◽

Matrix Potential ◽

The Matrix ◽

Saw Blade ◽

Diamond Retention

Abstract The paper presents results of computer simulations of the stress/strain field built up in a cobalt matrix diamond impregnated saw blade segment during its fabrication and after loading the protruding diamond with an external force. The main objective of this work was to create better understanding of the factors affecting retention of diamond particles in a metallic matrix of saw blade segments, which are produced by means of the powder metallurgy technology. The effective use of diamond impregnated tools strongly depends on mechanical and tribological properties of the matrix, which has to hold the diamond grits firmly. The diamond retention capability of the matrix is affected in a complex manner by chemical or mechanical interactions between the diamond crystal and the matrix during the segment manufacture. Due to the difference between the thermal expansion coefficients of the diamond and metallic matrix, a complex stress/strain field is generated in the matrix surrounding each diamond crystal. It is assumed that the matrix potential for diamond retention can be associated with the amount of the elastic and plastic deformation energy and the size of the deformation zone occurring in the matrix around diamonds. The stress and strain fields generated in the matrix were calculated using the Abaqus software. It was found that the stress and strain fields generated during segment fabrication change to a large extent as the diamond crystal emerges from the cobalt matrix to reach its working height of protrusion.

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Mechanisms Governing the Inelastic Deformation of Cortical Bone

MRS Proceedings ◽

10.1557/proc-873-l2.2/k2.2 ◽

2005 ◽

Vol 873 ◽

Author(s):

C. Mercer ◽

R. Wang ◽

A. G. Evans

Keyword(s):

Cortical Bone ◽

Inelastic Deformation ◽

Collagen Fibrils ◽

Strain Response ◽

Flexural Test ◽

Test Protocol ◽

Stress Strain ◽

Permanent Strain ◽

Nano Scale ◽

Transverse Strains

AbstractTo understand the inelastic response of bone, a two-part investigation has been conducted. In the first, a flexural test protocol has been designed and implemented that monitors the axial and transverse strains on both the tensile and compressive surfaces of cortical bone. The results are used to assess the relative contributions of dilatation and shear to the inelastic deformation. Unload/reload tests have characterized the hysteresis and provided insight about the mechanisms causing the strain. These tests reveal strain healing attributed to sacrificial bonds. The second part devises a model for the stress/strain response, based on a recent assessment of the nano-scale organization of the collagen fibrils and mineral platelets. The model rationalizes the inelastic deformation in tension, as well as the permanent strain and hysteresis.

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Transverse Young's modulus of carbon/glass hybrid fiber composites

Journal of Composite Materials ◽

10.1177/0021998319871689 ◽

2019 ◽

Vol 54 (7) ◽

pp. 947-960

Author(s):

Ganesh Venkatesan ◽

Maximilian J Ripepi ◽

Charles E Bakis

Keyword(s):

Finite Element ◽

Glass Fiber ◽

Fiber Composites ◽

Stress Model ◽

Hybrid Fiber ◽

Stress Strain ◽

Element Analysis ◽

Strain Behavior ◽

The Matrix ◽

Transverse Modulus

Hybrid fiber composites offer designers a means of tailoring the stress–strain behavior of lightweight materials used in high-performance structures. While the longitudinal stress–strain behavior of unidirectional hybrid fiber composites has been thoroughly evaluated experimentally and analytically, relatively little information is available on the transverse behavior. The objective of the current investigation is to present data on the transverse modulus of elasticity of unidirectional composites with five different ratios of carbon and glass fiber and to compare the data with predictive and fitted models. The transverse modulus increases monotonically with the proportion of glass fiber in the composite. Finite element analysis was used to evaluate different ways to model voids in the matrix and allowed the unknown transverse properties of the carbon fibers to be backed out using experimental data from the all-carbon composite. The finite element results show that the transverse modulus can be accurately modeled if voids are modeled explicitly in the matrix region and if modulus is calculated based on stress applied along the minimum interfiber distance path between adjacent fibers arranged in a rectangular array. The transverse modulus was under-predicted by the iso-stress model and was well predicted by a modified iso-stress model and a modified Halpin–Tsai model.

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Using Green Supplementary Materials to Achieve More Ductile ECC

Materials ◽

10.3390/ma12060858 ◽

2019 ◽

Vol 12 (6) ◽

pp. 858 ◽

Cited By ~ 8

Author(s):

Yichao Wang ◽

Zhigang Zhang ◽

Jiangtao Yu ◽

Jianzhuang Xiao ◽

Qingfeng Xu

Keyword(s):

Fracture Toughness ◽

Tensile Strain ◽

Tension Test ◽

Construction And Demolition Waste ◽

Demolition Waste ◽

Strain Capacity ◽

Fiber Bridging ◽

Matrix Fracture ◽

The Matrix ◽

Tensile Strain Capacity

To improve the greenness and deformability of engineered cementitious composites (ECC), recycled powder (RP) from construction and demolition waste with an average size of 45 μm and crumb rubber (CR) of two particle sizes (40CR and 80CR) were used as supplements in the mix. In the present study, fly ash and silica sand used in ECC were replaced by RP (50% and 100% by weight) and CR (13% and 30% by weight), respectively. The tension test and compression test demonstrated that RP and CR incorporation has a positive effect on the deformability of ECC, especially on the tensile strain capacity. The highest tensile strain capacity was up to 12%, which is almost 3 times that of the average ECC. The fiber bridging capacity obtained from a single crack tension test and the matrix fracture toughness obtained from 3-point bending were used to analyze the influence of RP and CR at the meso-scale. It is indicated that the replacement of sand by CR lowers the matrix fracture toughness without decreasing the fiber bridging capacity. Accordingly, an explanation was achieved for the exceeding deformability of ECC incorporated with RP and CR based on the pseudo-strain hardening (PSH) index.

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Modeling of Matrix/Fracture Transfer with Nonuniform-Block Distributions in Low-Permeability Fractured Reservoirs

SPE Journal ◽

10.2118/191811-pa ◽

2019 ◽

Vol 24 (06) ◽

pp. 2653-2670 ◽

Cited By ~ 1

Author(s):

Didier–Yu Ding

Keyword(s):

Shale Gas ◽

Shape Factor ◽

Oil Reservoirs ◽

Dual Porosity ◽

Tight Oil ◽

Fracture Networks ◽

Matrix Block ◽

Matrix Fracture ◽

Discrete Fracture ◽

The Matrix

Summary Unconventional shale–gas and tight oil reservoirs are commonly naturally fractured, and developing these kinds of reservoirs requires stimulation by means of hydraulic fracturing to create conductive fluid–flow paths through open–fracture networks for practical exploitation. The presence of the multiscale–fracture network, including hydraulic fractures, stimulated and nonstimulated natural fractures, and microfractures, increases the complexity of the reservoir simulation. The matrix–block sizes are not uniform and can vary in a very wide range, from several tens of centimeters to meters. In such a reservoir, the matrix provides most of the pore volume for storage but makes only a small contribution to the global flow; the fracture supplies the flow, but with negligible contributions to reservoir porosity. The hydrocarbon is mainly produced from matrix/fracture interaction. So, it is essential to accurately model the matrix/fracture transfers with a reservoir simulator. For the fluid–flow simulation in shale–gas and tight oil reservoirs, dual–porosity models are widely used. In a commonly used dual–porosity–reservoir simulator, fractures are homogenized from a discrete–fracture network, and a shape factor based on the homogenized–matrix–block size is applied to model the matrix/fracture transfer. Even for the embedded discrete–fracture model (EDFM), the matrix/fracture interaction is also commonly modeled using the dual–porosity concept with a constant shape factor (or matrix/fracture transmissibility). However, in real cases, the discrete–fracture networks are very complex and nonuniformly distributed. It is difficult to determine an equivalent shape factor to compute matrix/fracture transfer in a multiple–block system. So, a dual–porosity approach might not be accurate for the simulation of shale-gas and tight oil reservoirs because of the presence of complex multiscale–fracture networks. In this paper, we study the multiple–interacting–continua (MINC) method for flow modeling in fractured reservoirs. MINC is commonly considered as an improvement of the dual–porosity model. However, a standard MINC approach, using transmissibilities derived from the MINC proximity function, cannot always correctly handle the matrix/fracture transfers when the matrix–block sizes are not uniformly distributed. To overcome this insufficiency, some new approaches for the MINC subdivision and the transmissibility computations are presented in this paper. Several examples are presented to show that using the new approaches significantly improves the dual–porosity model and the standard MINC method for nonuniform–block–size distributions.

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