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A NEW METHOD FOR THE DETECTION AND QUANTIFICATION OF DEEP-OCEAN METHANE HYDRATES USING SEISMICS Wojtowitz, Gabrielle 2008
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Title | A NEW METHOD FOR THE DETECTION AND QUANTIFICATION OF DEEP-OCEAN METHANE HYDRATES USING SEISMICS |
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Wojtowitz, Gabrielle Zervos, Antonis Clayton, Chris R.I. |
Date Created | 2008-09-17 |
Date Issued | 2008-07-01 |
Description | Methane gas hydrates have attracted significant international interest as a potential future energy resource, but also as a geotechnical hazard for offshore operations related to hydrocarbon recovery. In this context, the abilities to detect the presence of hydrate in marine sediments and to quantify the amount of hydrate contained therein, have become increasingly important over the years. Detection and quantification of hydrates are done on the basis of seismic surveys, which measure indirectly the bulk dynamic properties of large volumes of sediment in situ. Seismic data are then interpreted using an effective medium model, which employs theoretical assumptions to relate wave velocities to gas hydrate content of the sediment. Wave velocity can then be used to infer hydrate concentration levels. A host of such effective medium models exists in the literature. Many of these models have been calibrated on and tested on specific sites, and are not readily transferable to other settings. In addition, many models ignore the existence of heterogeneities of the host sediment, or the inhomogeneous distribution of hydrate within it. These, however, are factors that may have a significant impact on the seismic signature of the sediment-hydrate system, and thus on the predicted quantity of hydrate. This paper presents a review of existing effective medium models and identifies general areas for improvement. A new numerical modelling method is outlined that enhances existing effective medium approaches, by taking explicitly into account different hydrate morphologies within the host sediment. |
Extent | 177242 bytes |
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Gas Hydrates Effective Medium Modeling Numerical Modeling ICGH 2008 International Conference On Gas Hydrates 2008 |
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Conference Paper |
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File Format | application/pdf |
Language | Eng |
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6th International Conference on Gas Hydrates |
Date Available | 2008-09-17 |
DOI | 10.14288/1.0041117 |
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Citation | Wojtowitz, Gabrielle; Zervos, Antonis; Clayton, Chris R.I. 2008. A NEW METHOD FOR THE DETECTION AND QUANTIFICATION OF DEEP-OCEAN METHANE HYDRATES USING SEISMICS. Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008. |
Peer Review Status | Unreviewed |
Copyright Holder | Wojtowitz, Gabrielle |
URI | http://hdl.handle.net/2429/2162 |
Digital Resource Original Record | https://open.library.ubc.ca/collections/1022/items/1.0041117/source |
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A NEW METHOD FOR THE DETECTION AND QUANTIFICATION OF DEEP-OCEAN METHANE HYDRATES USING SEISMICS Gabrielle Wojtowitz School of Civil Engineering & the Environment University of Southampton Highfield, Southampton, Hampshire, SO17 1BJ UNITED KINGDOM Antonis Zervos School of Civil Engineering & the Environment University of Southampton Highfield, Southampton, Hampshire, SO17 1BJ UNITED KINGDOM Chris R. I. Clayton School of Civil Engineering & the Environment University of Southampton Highfield, Southampton, Hampshire, SO17 1BJ UNITED KINGDOM ABSTRACT Methane gas hydrates have attracted significant international interest as a potential future energy resource, but also as a geotechnical hazard for offshore operations related to hydrocarbon recovery. In this context, the abilities to detect the presence of hydrate in marine sediments and to quantify the amount of hydrate contained therein, have become increasingly important over the years. Detection and quantification of hydrates are done on the basis of seismic surveys, which measure indirectly the bulk dynamic properties of large volumes of sediment in situ. Seismic data are then interpreted using an effective medium model, which employs theoretical assumptions to relate wave velocities to gas hydrate content of the sediment. Wave velocity can then be used to infer hydrate concentration levels. A host of such effective medium models exists in the literature. Many of these models have been calibrated on and tested on specific sites, and are not readily transferable to other settings. In addition, many models ignore the existence of heterogeneities of the host sediment, or the inhomogeneous distribution of hydrate within it. These, however, are factors that may have a significant impact on the seismic signature of the sediment-hydrate system, and thus on the predicted quantity of hydrate. This paper presents a review of existing effective medium models and identifies general areas for improvement. A new numerical modelling method is outlined that enhances existing effective medium approaches, by taking explicitly into account different hydrate morphologies within the host sediment. Keywords: gas hydrates, effective medium modeling, numerical modeling Corresponding author: Phone: +44 2380 593827 E-mail: gabi.wojtowitz@soton.ac.uk Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008. INTRODUCTION Methane gas hydrates are ice-like compounds that can only exist under specific thermobaric conditions of low-temperature and high pressure restrictions. Due to these conditions, gas hydrates naturally occur in submarine sediments or in terrestrial sediments in permafrost regions. The occurrence of gas hydrates within sediments significantly alters the physical properties of the host sediments. Hydrates have stimulated significant international interest as a potential future energy resource due to their very high methane content, wide geographical distribution, especially in continental margins, and occurrence at shallow sediment depths within 2,000m of the surface of the earth. The potentially hazardous consequences of dissociating hydrates on the physical properties of the host sediment, resulting in seafloor instability, have highlighted the immediate concern of gas hydrates as a geotechnical hazard for offshore operations related to hydrocarbon recovery. In this context, the ability to quantify and detect the presence and concentration of hydrate in marine sediments and understand the effect it has on these host sediments has become increasingly important over the years. Hydrates have been recovered in a variety of sediments such as clastic marine sediments, siltstones, unconsolidated sands, conglomerates and poorly cemented sandstones. The growth of gas hydrate depends on the lithology of the sediment and subsequently four different gas hydrate morphologies are found within the sediment based upon an overview of several years of research. Hydrates can occur as nodules, be finely disseminated, as massive sheets of hydrate or as layers between sediment layers (veins). The location of hydrate in the pore space and its distribution within the sediment can affect the influence it has on the geotechnical properties of the sediment. The detection and quantification of gas hydrates and their effect on hydrate-bearing sediments has been inferred via exploratory seismic methods, which measure indirectly the bulk dynamic properties of large volumes of in-situ sediment. In order to use seismic techniques to quantify the gas hydrate concentration or to infer the physical properties of gas hydrate-bearing sediments, a predictive model that relates the amount of gas hydrate within a sedimentary sequence to the seismic velocities is required. At present, effective medium models have been used for the interpretation of the seismic data, by employing theoretical assumptions to relate wave velocities to the gas hydrate content of the sediment. Wave velocity can then be used to infer hydrate concentration levels within marine sediments. However, various limitations are associated with the theoretical assumptions used in the models. This paper presents a general review of the existing effective medium models, identifying typical areas for improvement. The aim of the research, which is work in progress, is to develop a new numerical modeling method that enhances existing effective medium approaches, by taking explicitly into account different hydrate morphologies within the host sediment. EFFECTIVE MEDIUM MODELING Overview An effective medium is defined as a theoretical medium that has the same overall bulk physical properties (seismic velocity and elastic moduli) as a physical medium which is composed of more than one constituent, each with its own different physical properties. An effective medium model relates the overall physical properties of a medium to the individual properties of the constituents making up the medium; however, the models are not expected to incorporate a detailed description of the microstructure. Such models include methods which are empirical and/or physical and reproduce many of the characteristics observed for sedimentary rocks. A variety of models have been developed to determine theoretically defined upper and lower bounds or attempting to come to an exact solution. Gas Hydrate Effective Medium Models Gas hydrates are often associated with higher seismic velocities than those observed for marine sediments. If the gas hydrate concentration and wave velocity relationship for hydrate-bearing sediment is known, the concentration of hydrate can be determined from anomalies in seismic data such as the latter mentioned higher velocities. This concentration/hydrate relationship has been predicted by a range of theoretical approaches, all of which are essentially effective medium models. However, it is unclear which of the published available methods to apply for the interpretation of seismic velocities for gas hydrate content [1]. A general review of the most widely used existing effective medium models for gas hydrate-bearing sediments was conducted by Chand et al. [1] and the results used to identify areas for improvement. The models differ in the assumptions applied with respect to, for example, the representation of the hydrate-bearing sediment on the particle scale, the exact location of hydrate within the medium and the underlying theories that are applied. The theories discussed are the empirical weighted equation [2, 3], a three-phase effective-medium theory [4, 5, 6], a three-phase Biot theory [7, 8] and two approaches using a combination of self- consistent approximation (SCA) and differential effective-medium (DEM) theory [9, 10]. Three-phase weighted equation The three-phase weighted equation uses a combination of Wyllie‟s three-phase time average equation [11] and Wood‟s weighted equation [12] to derive an empirical relationship between compressional wave velocity and the amount of hydrate filling the pore space. This combination accounts for the two extremes represented by the two equations. Wood‟s equation was developed for dilute suspensions where only bulk modulus exists and the grains are not in contact. Wyllie‟s three-phase time average equation represents the effective wave velocity as the sum of the transit times in each constituent phase and is more appropriate for consolidated sediments. Lee et al. [2] developed the three-phase weighted equation as shown: 21 1111 p n p n p V SW V SW V (1) The parameters required are the porosity Φ, concentration of hydrate in the pore space S, the compressional wave velocity from the Wood equation Vp1, the compressional wave velocity from Wyllie‟s time average equation Vp2, a weighting factor W and a constant n, representing the lithification rate with hydrate concentration. The parameters W and n are determined empirically by fitting equation (1) to seismic in situ data for non-hydrate-bearing sediments and they therefore have no physical meaning [1]. Due to the application of a weighting factor and a constant, it appeared to offer a more flexible, robust treatment of conditions in which hydrate is likely to occur. The equation is based on real observations and is simple to implement, however a substantial data set is required to constrain parameters. The equation can be fitted to any data set but only if all the sediment under investigation is of a similar type as variations in lithology lead to poor results. However, it is limited in its physical meaning, lacks theoretical formulation and is not valid in a geological setting different from those it was formulated from as it is difficult to establish a rational pattern of adapting „free‟ parameters to site-specific conditions [13]. The equation does not consider effective pressure, particle aspect ratio and size and is unable to handle anisotropy [1]. The change in shear wave velocity, produced when hydrate starts contributing to the sediment frame strength, is difficult to account for with this method as the shear wave velocity is empirically estimated from the compressional wave velocity [1]. Three-phase effective medium theory The three-phase effective medium theory, based on rock-physics contact models, considers the two mechanically extreme cases of hydrate morphology with two forms of models [4, 5, 6]. The two models are based on the assumption of a granular medium comprised of a random pack of identical spheres, the chosen idealized shape for representation of the grains. The first model is a dissemination/non-cementing model where hydrate is considered as either part of the pore fluid, located as an inclusion in the pore space or as part of the load-bearing solid phase matrix, treated as an extra mineral in the frame and located away from the grain contacts. Figure 1 shows a graphical representation of this model. Both alternative dissemination models predict small increases of seismic velocities at low concentrations of gas hydrate within sediments; however results show that the load-bearing assumption produced more accurate estimates of the in-situ hydrate concentrations. The second method is the cementation model which models hydrate as cement either at the grain contacts or enveloping the grains, resulting in strongly increasing the stiffness of the granular medium at low hydrate saturation values. The two forms for hydrate cementation are shown in Figure 1. The theories predict a dramatic increase of seismic velocities even with a small amount of gas hydrate because intergranular gas hydrate cementation significantly increases the elastic moduli of the dry rock frame. The cementation model predicts velocities that are significantly higher that those generally observed in nature [4]. Therefore, the model is only applicable for small amounts of cement at intergranular contacts. Recent studies are favoring dissemination models for gas hydrate occurrence instead of cementation at grain boundaries, especially at low saturation [14]. The applied approximation of a random pack of identical spheres for both models is unrealistic in representing the heterogeneous nature of hydrate-bearing sediments. These contact models can not model an anisotropic response. Isotropic assumptions are acceptable for sands; however they can not always be applied for clay rich sediments [15]. The three-phase effective medium theory is limited in its application as it is sensitive to the input parameters, some of which are dependent on the clay content and porosity of the sediment. The theory is only valid when the applied contact model is relevant to site-specific conditions and the velocity data is accurate. The application of Gassmann‟s equation within the model formulation limits its application as the equation is a low frequency approximation and not suitable for laboratory measurements. Figure 1 Dissemination and Cementation models of gas hydrate-bearing sediments for three-phase effective medium theory (redrawn from [16]) Three-phase Biot theory The three-phase Biot theory assumes, for the effective medium calculation, a medium composed of two frames, the solid mineral frame and hydrate frame, plus a fluid phase. This assumption reflects the existence of two solid phases and one fluid phase in the medium compared to a two phase (solid-fluid) medium. In essence, Biot theory relates the seismic velocity of a dry rock to that of a fully saturated rock at all frequencies. The three- phase Biot theory approach was developed by Carcione & Tinivella [7] and Gei & Carcione [8]. Calibration of the model is required using in-situ or laboratory data to estimate the various empirical coefficients that are involved [1]. The bulk and shear moduli of the solid and hydrate frame need to be determined before the model can be applied. Combined SCA/DEM theory Following approaches developed by Hornby [17] and Sheng [18], Jakobsen et al. [9] extended the combined self-consistent approximation/differential effective medium (SCA/DEM) method for hydrate bearing sediments. Jakobsen et al. [9] developed a method relating the seismic properties of clay-rich hydrate-bearing sediments to their mineralogy, porosity, hydrate saturation, microstructure and clay particle anisotropy. Of the latter effective medium models, this is the only approach that explicitly incorporates the effects of anisotropy due to clay fabric by applying a combination of the self-consistent approximation, differential effective medium theory and a method of smoothing crystalline aggregates [1]. The modeled material is considered to be made up of blocks of fully aligned composite, arranged at different orientations, so due to the alignment of clay particles, the building blocks are considered to be transversely isotropic. Firstly, the effective medium is calculated with a self-consistent method using a single inclusion within a matrix having the elastic properties of the yet to be determined effective medium at a critical porosity, reflecting the point of bi-connectivity. Secondly, by application of the differential effective medium theory, the elastic properties at other porosities can be determined by successive calculations of adding or removing infinitesimal volumes of the host material and replacing it with equal volumes of another component. The load- bearing and non-load bearing hydrate structures are represented by either replacing hydrate with the pore fluid using the differential effective medium, or by starting with a material composed of hydrate and sediment grains into which pore fluid is added as shown in Figure 2 [15]. Once the effective stiffness has been calculated, a „method of smoothing‟ is used to determine the effects of interactions between surrounding grains, in order to minimize the error due to effects at edges which occurs when the orientation of the components is different [1]. GR GR GH GH GH GH 1. Cement at contacts 4. Pore filling 3. Load-bearing matrix/grain 2. Grain enveloping Figure 2 Three-phase medium where hydrate is load-bearing or non-load bearing [15]. The model is purely physical and is not affected by the limitations associated with the application of empirical constants. The connectivity of phases is preserved, producing sediment that is bi-connected at all porosities [1]. Due to the consideration of using building blocks arranged at different orientations, the approach is limited in its lack of accountability for weaker bonding and the greater compliance likely to exist at the individual preferential particle alignment domain edges and does not consider pressure effects other than through the resulting changes in porosity [1]. However, the success of the application of this theory with clay-hydrate as a starting model suggests that hydrates can form a connected phase affecting the rock framework rather than disconnected inclusions. The combination of the self-consistent approximation and differential effective medium model is suitable for materials with low shear moduli such as uncemented sediments. Building on the combined SCA/DEM theory and applying an alternative approach to the modeling of the hydrate location, Chand et al. [19] developed the effective medium inversion algorithm. The algorithm is also based on the self- consistent approximation and differential effective medium theory but Biot and squirt flow mechanisms of fluid flow are applied. A possible problem with the load-bearing hydrate structure modeled with the method developed by Jakobsen et al. [9] is that at low hydrate saturations hydrate becomes completely cementing; implying that the sediment is not completely cemented for a very small range of hydrate saturations which is not entirely valid. Chand et al. [19] attempted to solve this problem by using a different approach to the latter models and modeling a portion of the hydrate saturation as load-bearing cement and the remaining portion as pore-filling inclusions. Application of this approach simulates partial cementation by hydrate at low saturations and incomplete cementation at 100 per cent saturation. The approach is limited by its lack of consideration of pressure effects other than through the resulting changes in porosity [19]. The authors of this algorithm identified that their model does not consider the case where hydrate fills cracks and fissures, as observed in nature. Results of comparison The various effective medium models yield results that differ by orders of magnitude, yet it is unknown which method is the most accurate. This uncertainty is due to the lack of representative samples for calibration with seismic results and comparison with modeling outputs. The equations used for the effective medium modeling techniques cannot be expected to incorporate a detailed description of the microstructure limiting their simulation of real in situ behavior. The different approaches applied to model the gas hydrate location within the pore space and sediment all consider the gas hydrate as homogeneously distributed in the sediments which is not realistic. The presented models consider hydrate located only at a grain scale and do not consider morphological heterogeneities at a larger scale such as fractures and nodules. The models are calibrated for site-specific conditions and no robust model applicable to all sites with different conditions exists. NUMERICAL MODELING APPROACH To address the above limitations we develop a new modeling method that takes explicitly into account different hydrate morphologies within the host sediments. We take a different path to that usually followed in constructing effective media, and Non-load bearing hydrate Load bearing hydrate Hydrate Pore fluid Grains employ the ideas behind first-order computational homogenization methods [20-22] First-order computational homogenization is used in material science to model the elastic response of heterogeneous materials, especially with the method of finite elements. The method essentially consists of determining the constitutive response at each macro-scale point of a heterogeneous material, through the solution of a separate, appropriate boundary value problem formulated at the micro-scale. The deformation gradient at the macro-scale point during the current iteration is used to “drive” the boundary conditions of the micro-volume, the microstructure of which is modeled explicitly. The resulting stress increment field is averaged over the micro-volume, and it is reported as the stress increment at the corresponding macro-scale point for that iteration. An advantage of the above approach is that the constitutive response of the heterogeneous material need not be defined explicitly; it is derived every time through detailed modelling of its microstructural response. Nevertheless this can be computationally demanding, as a separate micro-scale boundary value problem must be solved for every Gauss point, at every iteration of the macro-model. In this work we will retain the idea of deriving the equivalent properties of a heterogeneous material using detailed finite element analyses at the microstructural level. However, based on the numerical results, we will produce an “empirical” analytical description of the constitutive response which, although not a mathematically rigorous equivalent, will be sufficiently accurate for practical purposes, e.g. for use in higher-level FE analyses. The analytical description will aim to give the constitutive response in terms of physical parameters that can be quantified, like hydrate content, vein thickness, orientation and spacing. One of the issues to resolve, when carrying out finite element analyses on the microstructural level, is that of the boundary conditions applicable to the micro-volume. Different boundary conditions will yield different results: uniform stress or strain result to lower (Reuss) and upper (Voigt) bounds to the constitutive response, while combinations will give results inbetween. Recent work suggests that the use of periodic boundary conditions at the micro-volume is the most appropriate choice [23], and this is an idea we are currently exploring. This work has also certain similarities with homogenisation approaches for jointed rock found in the rock mechanics literature, and in particular with the Non-Representative Volume Element (NRVE) approach [24]. In NRVE, rock joints are modelled as planar inclusions of softer material in a cube of solid rock. An analytical expression can be derived by the solution of appropriate “unit” boundary value problems, and the effect of different joints is assumed to be additive. It is noted that the volume considered is not representative for the whole rock mass, hence the name of the method; a new derivation of properties is needed for every location where joint number, orientation or spacing is different, or for finite element meshes of different sizes. There is some similarity between NRVE and our proposed approach, as we too model hydrate veins within sediment as planar inclusions of stiffer, in our case, material, and aim at an analytical description of the constitutive behaviour. However, we consider volumes that are representative, and hence aim to use the same material description for the whole model if possible. Furthermore, it is not clear how NRVE would work if parallel veins were included in the same volume; it is not reasonable to assume the effect to be simply additive. Finally, NRVE has been shown to work when the volume fraction of veins/joints is small, of the order of few percent. This is consistent with the theory having been developed to deal with rock joints, rather than genuine planar inclusions. On the other hand, hydrate saturation can be much higher than a few percent; our work will take that into account by employing appropriate microstructural models, representing realistic hydrate contents. Once an appropriate analytical description of the constitutive response has been constructed, the second stage of this research will consist of full- scale FE analyses of wave propagation through a half-space, to determine how it responds to the transmission of plane waves. The analyses will focus in particular on determining the differences in the response of a hydrate-bearing layer, modelled using material properties derived as above, from the response of a non-hydrate bearing layer, modelled as an isotropic half-space. REFERENCES [1] Chand S, Minshull TA, Gei D, Carcione JM. Elastic velocity models for gas hydrate-bearing sediments – a comparison. Geophysical Journal Institute 2004;159:573-590. [2] Lee MW, Hutchinson DR, Collet, TS, Dillon WP. Seismic velocities for hydrate-bearing sediments using weighted equation. Journal of Geophysical Research 1996;101(B9):20347- 20358. [3] Lee MW, Collett TS. Elastic properties of gas hydrate-bearing sediments. Geophysics 2001;66(3):763-771. [4] Ecker C, Dvorkin J, Nur A. Sediments with gas hydrates; Internal structure from seismic AVO. Geophysics 1998;63(5):1659-1669. [5] Helgerud MB, Dvorkin J, Nur A, Sakai A, Collett T. Elastic wave velocity in marine sediments with gas hydrates: Effective medium modeling. Geophysical Research Letters 1999;26(13):2021-2024. [6] Ecker C, Dvorkin J, Nur A. Estimating the amount of gas hydrate and free gas from marine seismic data. Geophysics 2000;65(2):565-573. [7] Carcione JM, Tinivella U. Bottom-simulating reflectors: seismic velocities and AVO effects. Geophysics 2000;65(1):54-67. [8] Gei D, Carcione JM.. Acoustic properties of sediments saturated with gas hydrate, free gas and water. Geophys. Prospect.. 2003;51:141-157. [9] Jakobsen M, Hudson JA, Minshull TA, Singh SC. Elastic properties of hydrate-bearing sediments using effective medium theory. Journal of Geophysical Research 2000;105(B1):561-577. [10] Chand S, Minshull TA, Priest JA, Best AI, Clayton CRI, Waite WF. An effective medium inversion algorithm for gas hydrate quantification and its application to laboratory and borehole measurements of gas hydrate-bearing sediments. Geophysical Journal International 2006;166:543- 552. [11] Wyllie MRJ, Gregory AR, Gardner LW. Elastic wave velocities in heterogeneous and porous media. Geophysics 1956;23:459-493. [12] Wood AB. A text book of sound. New York: Macmillan, 1941. [13] Helgerud MB, Dvorkin J, Nur A. Rock physics characterization for gas hydrate reservoirs. Elastic properties. Annals New York Academy of Sciences 116-125. [14] Lee MW. Biot-Gassmann theory for velocities of gas hydrate-bearing sediments. Geophysics 2002;67(6):1711-1719. [15] Ellis M. Joint seismic/electrical measurements of gas hydrates in continental margin sediments. PhD Thesis. University of Southampton, United Kingdom, 2008 [16] Xu H, Dai J, Snyder F, Dutta N. Seismic detection and quantification of gas hydrates using rock physics and inversion. In: Taylor CE, Kwan JT, editors. Advances in the study of gas hydrates, 2004., Kluwer Academic/Plenum Publishers, New York p. 117-139. [17] Hornby BE, Schwartz LM, Hudson JA. Anisotropic effective-medium modeling of the elastic properties of shales. Geophysics 1994;59:1570-1581. [18] Sheng P. Effective-medium theory of sedimentary rocks. Phys. Rev. B. 1990;41:4507- 4512. [19] Chand S, Minshull TA, Priest JA, Best AI, Clayton CRI and Waite WF. An effective medium inversion algorithm for gas hydrate quantification and its application to laboratory and borehole measurements of gas hydrate-bearing sediments. Geophysical Journal International 2006;166:543- 552. [20] Miehe, C. and Koch, A. Computational micro-to-macro transition of discretized microstructures undergoing small strain. Arch. Appl. Mech., 2002;72:300–317. [21] Miehe, C., Schotte, J., and Schröder, J. Computationalmicro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput. Mater. Sci., 1999;16:372– 382. [22] Kouznetsova, V., Brekelmans, W. A. M., and Baaijens, F. P. T. An approach to micro-macro modeling of heterogeneous materials. Comput. Mech., 2001:27:37–48. [23]Kaczmarczyk L, Pearce CJ. Bicanic N. Scale transition and enforcement of RVE boundary conditions in second-order computational homogenization, International Journal for Numerical Methods in Engineering 2007;74(3): 506–522. [24] Pariseau WG. An equivalent plasticity theory for jointed rock masses. International Journal of Rock Mechanics and Mining Sciences 1999;36:907-918.
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