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This article presents a new methodology to estimate the effective permeability of random fractured media of any anisotropy
containing both microfractures and a large number of long fractures crosscutting the representative volume element. The fractures
are replaced by fictitious permeable materials for which the tangential permeability is deduced f...
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Citations
... Therefore, electrical surveys are widely employed for quantitatively detecting and characterizing fractures in fractured hydrocarbon reservoirs [12], [13], [14], [15], [16]. Accordingly, as the key to the successful explanation and interpretation of electrical survey data, the electrical properties of fractured reservoir rocks have been extensively studied both experimentally and theoretically [17], [18], [19], [20], [21], [22], [23], [24]. ...
... The electrical conductivity tensor of a rock containing fractures (K) can be regarded as the summation of the effect of the fractures in the TI background, in the general form of [17], [43], [44], [45] ...
... are the electrical conductivity tensors of the background TI rock, and the fracture-filling water, respectively, ϕ F is the porosity of the fractures, and A is the conductivity concentration tensor of the fractures, given as [17], [45], [46] ...
Fractures are widely existing in rocks and are one of the controlling factors for the formation and distribution of hydrocarbon reservoirs. Electrical surveys are effective means for the quantitative detection and characterization of fractures. However, although all rocks are experiencing pore pressure, the deformation of fractures with pore pressure and its effects on the anisotropic electrical properties of fractured rocks, which can help for the improved interpretation of electrical survey data, remain poorly understood. We bridge this knowledge gap in this work. We first implement dedicated laboratory measurements of the anisotropic electrical conductivity of artificial sandstone samples with and without aligned penny-shaped fractures as a function of pore pressure. We then invert from the experimental data for the fracture parameters that characterize the deformation of the fractures. We finally theoretically model the effects of pore pressure induced fracture deformation on the anisotropic electrical properties of fractured rocks. The results show that the inverted porosity and aspect ratio of the fractures increase exponentially with pore pressure and exhibit linear correlation with each other as an implicit function of varying pore pressure. We also demonstrate that the variations in the fracture porosity caused by the varying pore pressure are the first order parameter affecting the pore pressure dependent electrical properties of fractured rocks. The results not only reveal the deformation of aligned fractures with pore pressure and its effects on the anisotropic electrical properties of fractured rocks, but also provide new insights for the detection and prediction of overpressure in fractured rocks.
... The macroscopic scale considers the REV as an elementary particle with effective properties and the microscopic scale refers to the different phases composing the REV (solid grains, matrix phase and macropores here). An exhaustive theoretical development of effective medium calculation for permeability is presented in [25]. ...
In this work we focus on further understanding reactive transport in carbonate rocks, in particular limestones characterized by a bimodal pore size distribution. To this end, we performed injection experiments with CO_2-saturated water on a sample of Euville limestone and monitored the experiments with a medical CT scanner. Microscanner imaging was performed before and after alteration. Experiments showed that permeability increased by nearly two decades due to the alteration process. This increase could be attributed to the formation of a preferential dissolution path visualized on the CT images. Microscanner images show that preferential dissolution areas are characterized by the presence of numerous enlarged macropores. The preferential dissolution path created therefore retains a porous structure and does not correspond to a wormhole-type channel. To provide further knowledge of the small scale physics of reactive transport, we performed Lattice-Boltzmann simulations of flow in a numerically-generated model 2D porous medium having geometrical and topological features designed to approach Euville limestone. We showed that the fluid velocity increased in nearly percolating paths of macropores. Considering the experiments, this means that the CO_2-saturated water starts to enter high velocity zones earlier than low velocity zones, inducing an earlier onset of the alteration process and a more pronounced local dissolution. However, numerical results showed that alteration of non-connected macropores leads to an increase of permeability much smaller than the experimentally observed one. To explain this fact we used effective medium modelling that permits predicting the variation in permeability as a function of the fraction of macropores and consequently as function of alteration. It proved that as long as there is no alteration induced percolating path consisting of macropores, increase in permeability is relatively low as shown by the Lattice-Boltzmann simulations. An increase in permeability of several orders of magnitude is only observed when the macroporosity is close to the percolation threshold. This fact is in accordance with the experimentally observed results.
... where γ is the aspect ratio of radius and with shape function g(ξ) (see Barthélémy 2008;Giraud et al. 2019) those of 3D quadratic elements show that each mesh is well refined for the corresponding computations to obtain a precision of computation as accurate as possible. ...
In this work, we focus on the effect of non-ellipsoidal concave pore on thermal conduction properties of porous media with an infinite transversely isotropic matrix. This effect is described by the resistivity contribution tensors that will be computed via the Finite Elements (FE)-based numerical homogenization. The FE computations will be carried out with some adapted and bounded boundary conditions that are formulated as dependent of the Green function and its gradient for the three-dimensional Poisson’s equation in infinite anisotropic medium. It allows to incorporate the matrix anisotropy and the correction of the bias induced by the bounded character of the mesh domain. The boundary conditions are constructed and applied in such a way that they accelerate the convergence of numerical computations, and therefore preserve the accuracy of estimations. This is proved after several appropriate assessment and validation by comparing its predictions, in some particular cases, with analytical results and some available numerical ones. Finally, the effect of the pore concavity as well as that of the matrix anisotropy on the resistivity contribution tensor are quantitatively illustrated.
... Effective conductivity tensor of heterogeneous transversely isotropic materials has been extensively studied in the frame of multiscale homogenization method, by using single inclusion approach and considering inclusions of ellipsoidal shapes. See among many others (Giraud et al. 2007(Giraud et al. , 2019 for application to electrical and thermal conductivity of transversely isotropic mudstone rocks, and in Barthélémy (2008) for the case of an arbitrarily oriented ellipsoidal inhomogeneity embedded in an orthotropic matrix (application to conductivity of cracked porous anisotropic materials are presented). ...
... The Hill polarization tensor and resistivity contribution tensor of a spheroidal inclusion aligned in a transversely isotropic host matrix is detailed below. See Giraud et al. (2019), Barthélémy (2008) for the complete solution of arbitrarily oriented ellipsoidal inhomogeneity embedded in an orthotropic matrix. One considers a transversely isotropic matrix of conductivity tensor λ 0 (n denotes unit vector on the symmetry axis, in this paper n = e 3 ) ...
... with shape function g(ξ) (see Barthélémy 2008;Giraud et al. 2019) ...
The aim of this paper is to extend recent elastic work to thermal problem. In the first part of the paper, approximate relations for the resistivity contribution tensor of pores of two reference shapes, supersphere and axisymmetrical superspheroid, are developed on the basis of 3D Finite Element Modelling, presented in the companion paper, and known exact solutions for the limiting cases of spherical pores. In the second part application to effective elastic coefficients of transversely isotropic materials such as clay rocks, in the frame of homogenization theory, is presented to illustrate the impact of concavity parameter on overall properties.
... Almost all analytical studies are based on the homogenization method. [17][18][19][20][21][22][23][24] In the framework of the homogenization approach, the fracture is represented by an ellipsoidal inclusion (in 3D) or an elliptical inclusion (in 2D) 25 in which the fluid flow obeys Darcy's law. The flux through a single inclusion under far-field pressure gradient is obtained from the well-known Eshelby solution. ...
... In the literature, the effective permeability of a fractured porous medium has been widely estimated by using the homogenization method. [17][18][19][20][21][22][23][24] In such a method, a fracture is usually assumed to be a flattened ellipsoidal inclusion in which the fluid flow is described by Darcy's law (called here by a Darcy flattened ellipsoidal inclusion). As a reminder, the fracture in this study is considered as a circular surface with a uniform aperture in which fluid flow obeys Poiseuille's law. ...
This paper considers the fluid flow through a porous medium containing intersecting fractures and presents three main analytical findings, namely: (1) mass exchange between fractures and surrounding matrix at the fracture intersection; (2) fluid potential solution (pressure field) within the whole domain under the form of a single singular integral equation; and (3) closed-form solutions of fluid flow in and around a crack disc under a far field pressure gradient. The crack is represented mathematically by a 2D smooth surface (i.e., zero thickness) within a 3D porous medium, while physically by a constant aperture. The fluid flow within the crack obeys Poisseuille's law, while Darcy's law is used to represent the fluid flow in the surrounding matrix. The general solution of pressure field for the general case of multiple intersecting cracks is firstly derived under a singular integral equation form. The mass exchange between the porous matrix and the crack, as well as the mass conservation at the intersection between cracks are the keys to obtaining this general solution. Then, the general solution is written for the case of a single crack. Rigorous derivation of the latter equation allows obtaining a closed-form solution of flow through a single crack. Introducing this solution of flow into the general equation gives the pressure field around the crack. The solution derived in this paper for a crack disk with Poisseuille's flow is slightly different from the well-known Eshelby's solution for the case of flattened inclusion in which the flow obeys Darcy's law.
... On the other hand, when phases are not a clear matrix-inclusion microstructure or the inclusions interact with each other, the results predicted by MT and KT models will underestimate the diffusivities [71,72]. Instead, the self-consistent (SC) scheme is an implicit method which considers the inclusions between different composites and is more suitable in this case [68,71,73]. The calculation can be done by: ...
The transport properties of concrete are closely related to its heterogeneous features. In this study, by considering the multi-scale microstructural characteristics, the chloride transport properties of concrete are comprehensively analysed. Based on the selected representative elementary volumes from bulk cement paste to concrete, a multi-scale predictive model was first proposed for diffusivity prediction of bulk cement paste, mortar and concrete. By taking the hydration process, the presence of sand/interfacial transition zones (ITZs) and the aggregate shape into account, the predicted diffusivities of each scale were validated against the experimental results. To further analyse the weights of different heterogeneous characteristics on chloride transport properties of cementitious materials, a statistical analysis method (principal component analysis) was then adopted based on the modelled results. The analysed results indicated that the capillary pore and C-S-H parts dominate the diffusivity estimation of bulk cement paste, and the consideration of ITZ and multi-species ions interaction can enhance the prediction accuracy for diffusivities of mortars and chloride penetration depths in concrete, respectively. The proposed multi-scale framework provides a novel perspective to study the chloride penetration in concrete by considering the effects of both the multi-scale microstructural characteristics and multi-species interactions, which can also enhance the understanding of deterioration mechanisms for reinforced concrete structures serving in a complex or marine environment.
... The presence of micro-fractures is an important control on permeability 66 , which, in turn, accommodates the presence and flow of pore fluids that affect the stress and, potentially, the resultant behaviour of materials 12 . As the damage created during the extended tests was generally microscopic (i.e., only samples which underwent rupture showed macroscopic damage), we did not anticipate large changes in absolute permeability values 67,68 ; hence, we focused our attention on the pressure dependence of the permeability in the original materials versus experimental products, as fracture geometry (which controls permeability; e.g., aperture, connectivity, tortuosity) is pressure dependent 69 . We quantify the permeability change rate, ⍺, as a metric for relative changes in the fracture characteristics of our samples, by fitting a linear regression through the permeability values as a function of confining pressure (Fig. 7a). ...
Cycles of stress build-up and release are inherent to tectonically active planets. Such stress oscillations impart strain and damage, prompting mechanically loaded rocks and materials to fail. Here, we investigate, under uniaxial conditions, damage accumulation and weakening caused by time-dependent creep (at 60, 65, and 70% of the rocks’ expected failure stress) and repeating stress oscillations (of ± 2.5, 5.0 or 7.5% of the creep load), simulating earthquakes at a shaking frequency of ~ 1.3 Hz in volcanic rocks. The results show that stress oscillations impart more damage than constant loads, occasionally prompting sample failure. The magnitudes of the creep stresses and stress oscillations correlate with the mechanical responses of our porphyritic andesites, implicating progressive microcracking as the cause of permanent inelastic strain. Microstructural investigation reveals longer fractures and higher fracture density in the post-experimental rock. We deconvolve the inelastic strain signal caused by creep deformation to quantify the amount of damage imparted by each individual oscillation event, showing that the magnitude of strain is generally largest with the first few oscillations; in instances where pre-existing damage and/or the oscillations’ amplitude favour the coalescence of micro-cracks towards system scale failure, the strain signal recorded shows a sharp increase as the number of oscillations increases, regardless of the creep condition. We conclude that repetitive stress oscillations during earthquakes can amplify the amount of damage in otherwise mechanically loaded materials, thus accentuating their weakening, a process that may affect natural or engineered structures. We specifically discuss volcanic scenarios without wholesale failure, where stress oscillations may generate damage, which could, for example, alter pore fluid pathways, modify stress distribution and affect future vulnerability to rupture and associated hazards.
... Electrical methods are among the most effective geophysical survey applications for the detection and characterization of cracks (Li et al., 2015;Tsoflias & Becker, 2008). To aid the quantitative interpretation of electrical survey data, great efforts have been made both experimentally and theoretically to understand the effects of cracks on the electrical properties of cracked rocks (Barthélémy, 2009;Giraud et al., 2019;Han, 2018;Saevik et al., 2013). However, most of these studies focus on the contributions of the crack distribution (e.g., inclined, rotated and randomly oriented cracks) as well as the shape and concentration of the cracks to the effective electrical properties of a cracked rock, and the fundamental understanding of whether cracks will improve the conductive ability of porous rocks (the quantification of their electrical conductivity if their porosity and saturant are the same) remains largely unaddressed. ...
... where D is the depolarization tensor that can be obtained by the eigenvalues and the corresponding unit eigenvectors of T (Barthélémy, 2009), which is further determined by the electrical conductivity of the background, as well as the shape and orientation of the cracks ...
Cracks are widely existing in rocks and are important in various geophysical applications. However, although electrical methods are conventionally employed for the detection and characterisation of cracks, the fundamental question regarding if cracks improve the conductive ability of porous rocks (a quantification of their electrical conductivity if their porosity and saturant are the same) remains largely unaddressed. We address this knowledge gap through theoretical models with confirmed validity. We show that the conductive ability of a rock containing non‐interacting penny‐shaped cracks with random orientation will be improved only in the case when the aspect ratio of the cracks is below a certain value, which is referred to as the critical crack aspect ratio. We also show that the critical crack aspect ratio is uniquely determined by the porosity and electrical conductivity (or cementation exponent) of the porous rock where the cracks reside. We further demonstrate that the critical crack aspect ratio is some representation of the pore structure, and using two times the critical crack aspect ratio as the pore geometry can give rise to reasonable agreement between modelled and measured P‐ and S‐wave velocities. The critical crack aspect ratio offers a consistent microstructure for the joint elastic‐electrical modelling for improved characterisation of cracks through integrated seismic and electrical surveys. This article is protected by copyright. All rights reserved
... The above models assume isotropic background, while natural rocks are usually layered and therefore can be regarded as transversely isotropic (TI). For such TI background, Barthélémy (2009) obtained a general implicit model for the overall conductivity when the spheroidal inclusions have the same symmetry axes with the TI background by solving the Hill tensors (Hill, 1963) from the Green function of the TI media. To account for the distribution of the inclusions, Giraud et al. (2019) presented a theoretical model for the electrical conductivity when the spheroidal inclusions are parallel or perpendicular to the axes of the TI background. ...
... For a TI rock containing a single fracture with arbitrary orientation, its effective electrical conductivity tensor can be written as (Duan et al., 2006;Giraud et al., 2007;Barthélémy, ...
... ⎠ is the conductivity tensor of the water that saturates the fracture, where k w is the conductivity of the water, and A is the conductivity concentration tensor of the fracture, in the form of (Shafiro & Kachanov, 2000;Barthélémy, 2009;Saevik et al., 2013;Yan et al., 2020) ...
Understanding the electrical properties of reservoir rocks has important applications in oil and gas exploration. Small-scale fractures that are widely existing in reservoir rocks are usually aligned along a certain preferential direction due to reasons such as tectonic movement, making the electrical conductivity of the rocks anisotropic. Since all rocks are experiencing geological pressure, it is theoretically and practically important to study the influence of pressure on the fractures and its control on the anisotropic conductivity of reservoir rocks. We first prepared artificial porous sandstone samples with and without penny-shaped fractures and experimentally measured the anisotropic conductivity of the samples under differential pressure (the difference between confining pressure and pore fluid pressure). Then, we inverted from the experimental data for the fracture porosity and fracture aspect ratio at each differential pressure. The variation of the fracture parameters caused by the differential pressure and their influence on the anisotropic electrical conductivity of the rocks were further analysed theoretically. The results show that as the differential pressure increases, the measured anisotropic conductivity of the artificial samples with and without fractures all decreases exponentially. We also show that both the inverted fracture porosity and fracture aspect ratio decrease exponentially with the increase of differential pressure, and the fracture porosity and fracture aspect ratio are linearly correlated as an implicit function of differential pressure. The modelling results show that the differential pressure induced decrease in the fracture porosity reduces the electrical conductivity in all directions, and the conductivity reduction in the rock parallel to the direction of the fractures is most significant. Comparing with the influence of the fracture porosity resulting from the applied differential pressure, the pressure caused variation in the fracture aspect ratio was shown to play a secondary role on affecting the pressure dependent anisotropic electrical properties.
This article is protected by copyright. All rights reserved
... This allows one to reduce the computational cost of the problem by adopting the lubrication approximation, essentially reducing the complexity of the problem by one dimension. This assumption is well established in soils [22][23][24] and has been applied successfully to textile materials [11,25,26]. However, in order to use lubrication theory for RTM simulations, a compatible geometrical space is needed. ...
... In Fig. 4.2 we show an example from the µ-CT scan of a textile, in which the inter-yarn channels are highlighted. Lubrication theory has been used as the micro-scale flow model for permeability upscaling in porous media [25,26]. The model and its definition of hydraulic aperture was also used by Wong et al. in [11] to compute the upscaled permeability of textiles using a "stream surface" method. ...
... 25: Square packing: models comparison, permeability (left) and error (right) ...
In this work, we study continuous fiber preforms in the context of Resin Transfer Moulding (RTM) processes. The aim of the thesis is two-fold: propose a new methodology to obtain mesoscale geometrical data from preforms and provide a new numerical model able to predict permeability or perform mesoscale filling simulations in a computationally efficient way.
In the first part, the focus is on the acquisition of geometrical data from preforms: we propose a novel methodology based on the analysis of the pressure field experienced by a dry preform under compaction. By using a commercial pressure-sensitive film, the pressure field exerted by a stack of layers against mould walls is captured and analyzed. Taking advantage of the periodic morphology of textiles, geometric patterns revealed by the pressure field are interpreted according to spectral Moiré analysis to recover the orientation and spatial distribution of each individual layer in the stack.
In the second part, the reconstructed digital architecture of the preform is used to carry out numerical flow simulations at the scale of the yarns, to characterize permeability of the stack or directly perform filling simulations. The stack geometry is replaced by a skeletonized representation of the same, on which a two-dimensional flow problem can be solved numerically, greatly reducing the computational cost when compared to a full 3D approach. This “medial skeleton” model is first formulated in its single-scale version (flow in channels) and then extended to dual-scale (flow in channels and yarns). The model potential is illustrated through several test cases.
This research establishes a pathway going from the non-destructive acquisition of data to the simulation of the dual-scale flow inside a multi-layer layup of textiles.
