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Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2002 SEG Annual Meeting, October 6–11, 2002

Paper Number: SEG-2002-0388

... Summary We present a series of example of synthetic data with and without random noise to illustrate the effect of

**transverse****isotropy**on conventional electric (**ES**) logs. The first part shows the process of inversion for single tools and how the presence of the borehole changes the sensibility...
Abstract

Summary We present a series of example of synthetic data with and without random noise to illustrate the effect of transverse isotropy on conventional electric (ES) logs. The first part shows the process of inversion for single tools and how the presence of the borehole changes the sensibility of the tools to anisotropy. We then proceed to show results for the joint inversion of the short normal, long normal, and lateral tools. In this, the models include invasion. We finally incorporate random noise to the data and find the threshold of random noise to achieve an adequate enversion.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, October 11–16, 2020

Paper Number: SEG-2020-3427730

... (m ?, u ?, a?) = 0. Variable lifting approach for FWI To mitigate the nonuniqueness issue of optimizing Equation 9, u ? = (S >S ) 1S >d , (16) we need additional information to regularize the problem. The original lifted problem 5 has the constraint of rank(X) = 1. We that satis

**es**the...
Abstract

A standard characterization of unconventional reservoirs is as a horizontally transverse isotropic (HTI) medium and there are many ways to interrogate such a medium with seismic data. In this study, two methods are independently employed to quantify the degree of HTI for a multicomponent 3D seismic survey in the Anadarko Basin, Oklahoma. The first is amplitude variation with azimuth (AVAz) using P-P data and the second is converted P-S wave splitting (SWS). Intriguingly, these methods produce dissimilar, and in some cases, opposite solutions. The discrepancy between the solutions may be due to their relative sensitivities – AVAz being an interfacial measure and SWS being a volumetric measure of anisotropy – or it may be a difference of the degree to which the analysis method adheres to the HTI assumption. Presentation Date: Tuesday, October 13, 2020 Session Start Time: 8:30 AM Presentation Time: 11:00 AM Location: 351D Presentation Type: Oral

Proceedings Papers

Paper presented at the ISRM International Symposium - EUROCK 96, September 2–5, 1996

Paper Number: ISRM-EUROCK-1996-114

... proposed. RESUME: On examine une methode analytique de calcule des blindages circulaires des tunnels situes dans Ie massif

**transversal**isotrope, que modele Ie massif vierge anisotrope ou bien dans Ie massif af- falbli par un systerne des fissures hiperiodique. ZUSAMMENFASSUNG:**Es**wird eine analitische...
Abstract

ABSTRACT: Analytical method of designing multi-layer circular tunnel linings in the transversely isotropic medium simulating naturally anisotropic solid rock or the rock weakened by a double periodic chink set is proposed. RESUME: On examine une methode analytique de calcule des blindages circulaires des tunnels situes dans Ie massif transversal isotrope, que modèle Ie massif vierge anisotrope ou bien dans Ie massif affaibli par un système des fissures biperiodique. ZUSAMMENFASSUNG: Es wird eine analitische Methode del' Berechnung der Verkleidung von kreisförmige Tunnel im transversal-isotrop Medium, das natural anisotrop Gestein oder das Gestein mit doppel periodische System del' Kluft modelliert, vorgeschlagen. 1 MATHEMATICAL MODEL AND METHOD OF MULTI' LAYER TUNNEL LINING ANALYSIS The method proposed is based on mathematical modeling the interaction of the underground structure and the surrounding rock mass as elements of a united deformable system undergoing the actions of the rock's own weight, internal water head and external ground water pres- sure. For the analysis of the lining stressed state the elasticity theory corresponding plane contact problems are considered. The general de- sign scheme is given in Figure 1. The lining is simulated by the multi-layer ring consisting of n layers having the R;(i = 1, …, n) internal radii, isotropic materials of which possess the E i (i = 1,, n) deformation moduli and the Vi (i = 1,.., n) Poisson ratios, and supporting the opening of the R° radius in a transversely isotropic S° medium with an arbitrary directed plane of isotropy (in which the Ox axis inclined by the ß angle to the Ox' horizontal one is located), simulating the naturally anisotropic solid rock or the rock weakened by a double - periodic chink set. In the latter case the deformation characteristics of the equivalent transversely isotropic medium namely: the Eo,¹, Eo, 2 deformation moduli correspondingly in the plane of isotropy and in the direction of normal to that plane. The corresponding contact problems of the elasticity theory are solved with the application of the complex variables analytic functions theory. Then expressions (26) and (27) are substituted in the (16)-(19) boundary conditions. The coefficients at the same degrees of the s variable in the left and right parts of the equations obtained are equated to each other. It allows the recurrent correlations combining coefficients. Substituting expression obtained into the (20) boundary condition we come to an infinite system of linear algebraic equations relative to the unknown cv(j)(0) (j= 1,2) coefficients. On the above system being solved the stress state of the lining layers is determined. The computer program has been developed. 2 THE EXAMPLE OF THE DESIGN The example of designing tunnel lining upon the action of the rock's own weight is given below. The double-layer lining including the external concrete layer with R0 = 3.42 m, R1 = 3.02 m outer and inner radii correspondingly and the internal steel layer having R2 = 3.00 m inner radius supports the circular tunnel located in the transversely isotropic rock with the plane of isotropy inclined under the ß = 45° angle to the horizontal plane.

Proceedings Papers

Paper presented at the 1st ISRM Congress, September 25–October 1, 1966

Paper Number: ISRM-1CONGRESS-1966-062

... Faktoren werden einander gleich, wenn zwischen den Konstanten Am (m, n = 1, 2, 3) die nach- tragliche Bedingung (20) bestebt. Dann nirnmt die charakteristische Gleichung die Form an: Auch hier wird

**es**aus dem folgenden klar, da(3der letzte Sonderfall mit dem Fall der voIIkommenen**Isotropie**keineswegs...
Abstract

Zusammenfassung Das Grundsystem von drei partiellen Differentialgleichungen fuer die Verschiebungen u, v und w wird so umgeformt, daβ daraus fuer jede Verschiebung eine selbstandige Differentialgleichung entsteht. Diese selbstandigen Gleichungen fuehren zu einer charakteristiscben Gleichung (die gleiche fuer jede Verschiebung),woraus die Beziehung zwiscben den Koeffizienten a, b und c der linearen Form (ax+by+cz) erhalten wird, aus willkuerlichen (harmoniscben und biharmonischen) Funktionen von welcher die allgemeinen Lösungen aufgebaut werden. Als Sonderfall sind die einfacheren Formen behandelt, welche die charakteristische Gleichung bei transversal isotropen und vollkommen isotropen Medien annimmt. Gefunden sind die Faktoren, in welche die charakteristische Gleichung bei allgemeiner Orthotropie, transversaler Isotropie und vollkommener Isotropie zerlegt wird, aus welchen Faktoren die Transformationen fuer die direkte Integration der Ausgangs- -Differentialgleichungen folgen. Die direkte Integration wird an der vollkommenen Isotropie veranschaulicht, indem die vollstandigen Lösungen fuer die Verschiebungen erhalten sind, deren jede aus einer partikularen Lösung, die den durch Quadraturen dargestellten Einfluβ der Massen- krafte enthalt, und einer allgemeinen Lösung, die aus nicht mehr als 6 willkuerlichen Funktionen aufgebaut wird, besteht. Gegeben sind Anleitungen, wie Tabellen von Funktionen aufzustellen sind, welche die Differentialgleichungen befriedigen, und aus welchen passende Funktionen gewahlt werden können, die die Randbedingungen sowohl in Verschiebungen als auch in Spannungen erfuellen. Summary The initial system of three partial differential equations for the displacements u, v and ware transformed in a way to obtain three independent differential equations for every displacement. These independent equations lead to one characteristic equation (the same for every displacement) that gives the relations between the coefficients a, b and c in the linear form (ax+by+cz), by arbitrary functions (harmonic or biharmonic) of which the solutions are built. As particular cases the more simple forms are considered that the characteristic equation obtains in transverse isotropic and isotropic mediums. The factors are found to which the characteristic equation is discomposed in the cases of general orthotropy, transverse isotropy and isotropy, from which factors the transformations are derived for the direct integration of the initial differential equations. The direct integration is illustrated in the case of isotropy and the complete solutions for the displacements are obtained consisting of a particular solution in form of quadratures depending on the volume forces, and of a general solution consisting of not more than 6 arbitrary functions. Instructions are given how to compile tables of functions that satisfy the differential equations, and among which suitable functions should be chosen to satisfy the boundary conditions as well in displacements as in stresses. Resume Le système fondamental de trois equations aux derivees partielles par rapport aux deplacements u, v et w est transforme de façon qu'on obtient trois equations independantes pour chaque deplacement, qui conduisent à une equation caractöristique, la même pour les trois deplacements et de là resulte la relation entre les coefficients a, b et c de la forme lineaire (ax+by+cz). Les integrales generales sont des fonctions arbitraires, harmoniques ou biharmoniques, de cette forme lineaire, On a considere cornmes cas particuliers les formes simplifiees que prend l'equation caractèristique pour des milieux transversalement isotropes et entièrement isotropes. En cas d'orthotropie generale, d'isotropie transversale et d'isotropie generale on a trouve les facteurs de decomposition de l'equation caractöristique; de ces facteurs resultent les transformations necessaires à l'integration directe des equations differentielles fondamentales. L'integration directe est montree dans Ie cas d'isotropie generale; on a obtenu les solutions complètes pour les deplacements et chacune d'elles contient une solution particulière donnee en quad atures et dependant des forces de masse, et une solution generale qui ne depend que de 6 fonctions arbitraires. II est montre aussi comment on peut dresser des tables de fonctions qui satisfont aux equations differentielles; les fonctions choisies doivent egalement satisfaire aux conditions limites pour les deplacements aussi bien que pour les tensions. Der erste Schritt zum Ziel besteht in der Ermittlung der allgemeinem Teile der gesuchten Lösungen. Dazu ist es notwendig, daβ selbstandige Differentialgleichungen fuer jede der drei Verschiebungen abgeleitet werden. Die drei selbstandigen partiellen Dlfferentialgleichungen (7) unterscheiden sich voneinander nur durch die Störungsfunktionen rechterseits, da der Differentialoperator D ueberall der gleiche ist. Infolge seiner Entstehung enthalt der Differentialoperator D auβer den Differentiationen nach den raumlichen Veranderlichen x, y und Z naturgemaβ auch Differentiationen nach der Zeit. In dieser Arbeit sollen aber keine dynamischen, sondern nur statische Probleme behandelt werden. In Übereinstimmung mit der allgemeinen Theorie der linearen Differentialgleichungen sind auch hier die allgemeinen Lösungen der Differentialgleichungen (7) aus deren homogenisierten Formen zu ermitteln. Die partikularen Lösungen, welche von den Störungsfunktionen abhangig sind, werden dann den Einfluβ der spezifischen Massenkraft ausdruecken. Die Koeffizienten der drei quadratischen Formen sind aus den technischen Konstanten Amm zu berechnen, wenn diese zahlenmaβig gegeben sind. Wie das aus dem folgenden zu ersehen ist, sind diese beiden Sonderfalle mit dem Fall der transversalen Isotropie (Monotropie) bezw. mit, dem Fall der vollkommenen Isotropie keineswegs identisch.

Proceedings Papers

Paper presented at the SPWLA 52nd Annual Logging Symposium, May 14–18, 2011

Paper Number: SPWLA-2011-E

... 1, with or without 12-in. mud invasion. Finally, the black curve refers to the log response at the depth of 9,800 ft, which is marked with a red arrow in Figure 5. 5 10 15 20 25 30 35 40 2 4 6 8 10 12 14 16 18 Depth of Investigation (in) Ap pa re nt R

**es**ist ivi ty (o hm *m ) Multi-Laterolog...
Abstract

ABSTRACT: This paper summarizes 3-D modeling and analysis results of a study on the unusual responses from multiple resistivity tools in the Bossier Formation of the Haynesville Shale Play in Louisiana. The resistivity tools included a Multi-Laterolog, an Array Induction Log, and a Multi-Component Induction Log. The Multi-Laterolog and Array Induction Log responses indicate that the apparent resistivity increases with increasing depth of investigation (DOI). The observed curve separation from different sub-arrays could not be explained by mud invasion or a single drilling-induced fracture. To explain this effect, a model with a series of parallel vertical fractures or vertical, thin sand beds filled with gas was proposed and investigated. Good agreement between modeling results and logs was achieved. Further studies also indicated that the fracture model was reduced to a homogeneous model with horizontal transverse isotropy (HTI) when the fracture width decreased while keeping a fixed volume ratio with respect to the conductive formation. The fracture model alone, however, cannot explain the increase of apparent conductivity with increasing frequency (the "anti-skin" effect) in some induction instrument responses. Extra-large dielectric permittivity, in combination with a strong azimuthal effect, must be included to explain the abnormal antiskin effect as well as the unusual azimuthal effect in the frequency spectra of the Multi-Component Induction logs. This extremely large anisotropy in permittivity can be caused by the vertical fractures and their interaction with certain formation components, which eventually causes an unusually high induced polarization effect along a particular direction. I. INTRODUCTION The Haynesville Shale is mainly composed of consolidated, organic-rich clay particles deposited in East Texas and northwest Louisiana during the upper Jurassic age approximately 170 million years ago (Pujana et al., 2009, 2010). Its unique combination of ultra-low permeability and high porosity makes it a major resource for shale gas.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 52nd U.S. Rock Mechanics/Geomechanics Symposium, June 17–20, 2018

Paper Number: ARMA-2018-268

..., and the regime determined by yield stress of Herschel-Bulkley fluid are calculated. The aspect ratio for the toughness-do inated regime is in agreement with the known solution for the dry crack, while for other regi

**es**the aspect ratio is different. 1. INTRODUCTION Unconventional reservoir...
Abstract

ABSTRACT: This paper revisits the problem of an elliptical planar fracture in a transversely isotropic (TI) material. Transverse isotropy can be used to represent the behavior of finely layered shale formations subject to hydraulic fracturing (HF). In contrast to isotropic homogeneous formations, in which penny-shaped vertical fractures are produced, the elastic anisotropy changes the aspect ratio of fractures. The latter aspect ratio of a uniformly pressurized (or ‘dry’) elliptical crack can be estimated for given TI elastic parameters. We extend this result to hydraulically driven fractures by using the near-tip asymptotic solutions of the crack for different regimes of propagation. The crack aspect ratios for the toughness-dominated regime, the viscosity-dominated regime, and the regime determined by yield stress of Hers chel-Bulkley fluid are calculated. The aspect ratio for the toughness-dominated regime is in agreement with the known solution for the ‘dry’ crack, while for other regimes the aspect ratio is different.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Middle East Oil Show, March 17–20, 2001

Paper Number: SPE-68193-MS

... s modulus from the stress-strain data (static) along the triaxial compression stress path, as given below: s = 1 - Kb,s / Kma (10) Kb,s =

**Es**/ 3(1-2 s) (11) s = - d 3 /d 1 (12) and**Es**= d 1 /d 1 (13) Where d 1 and d 3 represent stresses in the axial and radial direction, and, d 1 and d 3 represent...
Abstract

Abstract Biot's constant is an important poroelastic parameter that has a majorinfluence on various petroleum-related rock mechanical applications. It isneeded to compute the in-situ stresses accurately (s'=s-a* p r , or effective stress=total stress - Biot's constant * porepressure). The in-situ stresses play a key role in the design of hydraulicfracturing, sanding tendency prediction, and in optimizing borehole trajectoryto combat instability problems in drilling. In regards to in-situ stress andBiot's constant, most rocks are not isotropic, as is commonly assumed in manyof the above applications. This leads to inaccurate determination of keyparameters resulting in hole failure or loss of productivity, which translateinto losses amounting to millions of dollars. This underscores the importanceof determining the Biot's constant experimentally. In most cases where Biot'sconstant is used, it is either assumed or derived empirically. Another aspectof this assumption is that the directional variation of Biot's constant is notconsidered at all, which results in further approximation. To address both these problems, we have developed a laboratory testingprocedure to evaluate the dynamic (ultrasonic) transverse anisotropy in Biot'sconstant on Cordoba Cream limestone samples obtained from a quarry in Texas.Cylindrical samples (NX-size) were cored from cubical blocks and were subjectedto triaxial and hydrostatic compression stress paths in a triaxial cell.Ultrasonic compressional and shear waves were transmitted in both axial andradial directions and travel times were recorded simultaneously with load anddeformation recordings. Biot's constant was evaluated as a function of axialand confining stresses in both axial (a v ) and lateral(a h ) directions, thus allowing for anisotropy determination.Dynamic values of Biot's constant obtained in the axial direction are comparedwith those obtained from static rock mechanical measurements. Results show that the limestone material behaves in a transverselyanisotropic manner, with dynamic a v values being lower thanthe corresponding a h at almost all confining and deviatoric(shear) stresses. With increasing deviatoric stress, a v wasfound to decrease slightly or remain constant, while a h increased considerably. A comparison between static and dynamic values showsthat a v obtained statically is more than a v obtained dynamically, for all confining pressures and deviatoric stresses. Theinformation presented in this paper shows that the Biot's constant isnot a constant; rather, it is dependent on stress magnitude as well asprincipal stress directions. We present a case study to show the optimizationof borehole trajectory in terms of mud weight as a function of boreholeinclination. It is found that the computed critical mud weight is considerablysensitive to Biot's constant and its directional variation, when the boreholeinclination is greater than 40°. Similar case studies can be shown on hydraulicfracture design and sanding tendency prediction also, which will be discussedin subsequent papers.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1999 SEG Annual Meeting, October 31–November 5, 1999

Paper Number: SEG-1999-0792

... obtained from SH. 5000 10000 15000 0 0.25 0.5 0.75 1 1.25 1.5 Offset ft SV 5000 10000 15000 Offset ft SH 5000 10000 15000 0 0.5 1 1.5 2 2.5 3 40 0 40 0 50 100 ms O cc ur re nc

**es**SP: Pass 1 40 0 40 ms SP: Pass 2 40 0 40 0 200 400 ms O cc ur re nc**es**RP: Pass 1 Figure 4: Histograms of shotpoint (SP) and...
Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1999 SEG Annual Meeting, October 31–November 5, 1999

Paper Number: SEG-1999-0728

... rectangular recording patch, Si; Ri is

**es**- sentially SV at the far offsets, while Sx; Rx is essentially SH at the far offsets. P-wave energy is apparent on Si; Ri preceding the shear-wave first arrival. Azimuthal rotation to SV-SH coordinates produces the gathers seen in Figure 3. The effect of the azimuthal...
Proceedings Papers

Paper presented at the The Third International Offshore and Polar Engineering Conference, June 6–11, 1993

Paper Number: ISOPE-I-93-184

... major part of the available test data rela~

**es**to in-situ cantilever ice beams, as shown in figure 1. A large parnon of th~se tests have been carried out at relatively high loading rates. Thus, If the loading rate is high, it may be justified to assume that ice behaves approximately as a linearly elastic...
Abstract

ABSTRACT: In this paper, the finite element method has been used as the numerical modelling tool. An assumption of an elastic material and transverse inhomogeneity was included. The stress distributions over the thickness of the ice cover were produced with a finite element program. These stress distributions were compared with calculations based on the assumption of a homogeneous material. In the creeping mode the calculations were based on an assumption of an homogeneous material. The inhomogeneity was introduced both as a variation of Young's modulus and the shear modulus Glen's law for creep was chosen to account for creep deformations: mainly because it is a simple and well documented equation. Results from these calculations were compared with experimental data from four-point beam tests. INTRODUCTION The mechanical behaviour of ice is very complex and there are many important material properties that has to be considered when doing numerical modeling. For example, both inhomogeneity and isotropy 10 an Ice cover have to be taken in to account when a calculation of the deformation and stress state is performed. !his paper focuses attention of the transverse in homogeneity that IS almost always present 10 an Ice cover. In homogeneity can be caused by temperature variations, variations in brine volume, and the crystallographic structure of the ice. The effect of transverse inhomogeneity on deformation and stress distributions is studied, both in the elastic and the creeping mode. It is of practical interest to know how the stresses distributes over the thickness of an ice cover since an incorrect assumption can spoil the conclusions that can be drawn. Most authors still make assumptions of an elastic homogeneous material when calculating stress results from field tests. The first part of the paper presents the effect of inhomogeneity on stresses and deformations in the elastic range.

Proceedings Papers

Paper presented at the SPWLA 55th Annual Logging Symposium, May 18–22, 2014

Paper Number: SPWLA-2014-OOOO

... and ck strength test dulus and Poisso RESS FOR IS S rd or traditiona stress for an iso ). ertically polariz (9) tensor for a V measurement a ratio. sor for a transvers

**ese**components horizontal Youn ing device and n s ratio. OTROPIC AN l equation used tropic linear ela SPWLA 55 3 ed TI nd ely are g s...
Abstract

Abstract It is well-known that organic shale reservoirs have very low permeability. Any stimulated fracture system is influenced by extensive horizontal laminations that are pervasive in shale reservoirs. The laminations will strongly influence the hydraulic fracture height because of the difference in measured and predicted rock mechanical properties normal and parallel to the bedding planes. To accurately predict fracturing height and even fracture width from logs in this environment, these mechanical property differences must be considered. This is accomplished by predicting from logs the Young's modulus and Poisson's ratio parallel to the bedding planes (referred to as the horizontal Young's Modulus ( E horz ) And Poisson's ratio ( V horz ), respectively) and also the Young's modulus and Poisson's ratio normal to the bedding planes (referred to as the vertical Young's modulus ( E vert ) and Poisson's ratio ( V vert ), respectively). These predictions are typically derived by density/sonic measurements used in conjunction with the ANNIE equations (Schoenberg et al. 1996). Review of the publications using ANNIE model (Higgins et al. 2008; Waters et al. 2011) reveals that ANNIE always predicts v vert = v horz . This is shown to be the general case for ANNIE.

Proceedings Papers

Paper presented at the ISRM Regional Symposium - 7th Asian Rock Mechanics Symposium, October 15–19, 2012

Paper Number: ISRM-ARMS7-2012-031

... th MPa, which , the laborat and this valu et al. (2012

**es**) vars et al., act orientati lap and "slid tact model a wly assigne rength and c be solved a nd 2D spec frees loaded mulate the b this paper, h um anisotro sion in joint , the assigne are explain ence, the str sed. at the failur bbass and non...
Abstract

ABSTRACT DEM is now recognized as a powerful tool for simulating behaviors in rock mechanics. In particular, bonded particle model has been successfully applied in emulating the elastic modulus, Poisson's ratio and strength parameters of isotropic rock by controlling the microparameters in DEM model. In this study, the smooth joint contact model was introduced to represent the bedding planes in order to eventually model the transversely isotropic rock. Consideration of anisotropy is important for shale gas production because shale is shown to have a significant anisotropy in terms of elastic constants and compressive and tensile strengths. The chosen anisotropy model is transversely isotropic model which is believed to model the mechanical behavior of shale to a reasonable extent. The properties of Boryeong shale (Cho. et al., 2012) were used as a reference for transversely isotropic rock. Transversely isotropic rock model using DEM behaved in good agreement with the mechanical behaviors of Boryeong shale from the laboratory. The results can be evidence that development of anisotropic numerical model is promising through DEM. This anisotropic modelling is expected to pave the way for wide variety of engineering application ranging from traditional rock mechanics application to the emerging applications such as shale gas production.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2011 SEG Annual Meeting, September 18–23, 2011

Paper Number: SEG-2011-2216

... previous section, the error in results 3311S 3311S 10-3 10-2 10-1 100 10-3 10-2 10-1 100 -0.5 0 0.5 1 Aspect ratio

**Es**he lb y te ns or S1111 S3333 S1133 S3311 S1313 S1212 S1122 Exact Approx. Mura Iso 0 0.2 0.4 0.6 0.8 1 Aspect ratio**Es**he lb y te ns or S1111 S3333 S1133 S3311 S1313 S1212 S1122 Exact Approx...
Abstract

ABSTRACT A common feature of various averaging methods for calculating effective elastic properties of rocks is the need for a solution when an inclusion is embedded in a homogeneous background of the rocks. Inclusion-based rock-physics models generally use Eshelby''s theory, which links the strain component of the inclusion to the eigenstrain (Eshelby, 1957). In this study, we derive an approximate expression for the Eshelby tensor using Thomsen''s parameters for transversely isotropic media. Such an approximation allows us to more efficiently evaluate the effective rock properties for anisotropic media.

Proceedings Papers

Paper presented at the 9th ISRM Congress, August 25–28, 1999

Paper Number: ISRM-9CONGRESS-1999-174

... Fliessfunktion und das Bruchkriterium durch Verwendung eines transformierten Spannungstensors enthalten: Σ ij = A ijkl σ kl . Die Eignung des Modells wird demonstriert, indem

**es**bei einem geschichteten Sedimentgestein, Tournemire Tonschiefer, angewendet wird. 1 INTRODUCTION In many rocks, due to the...
Abstract

ABSTRACT: An invariant formulation of an elastic/viscoplastic model for an initially transversely isotropic rock is presented. To describe the nonlinearity of the elastic response, the elastic moduli are considered to be stress dependent. Furthermore, it is shown that in order to have a conservative elastic response, the elastic moduli may depend on stress only through the mean stress and a mixed stress-structural tensor invariant tr M σ, where M = S 1 S 1 , S 1 being the axis of symmetry. The influence of the anisotropy on the irreversible response and strength is described by means of a fourth order symmetric anisotropic tensor A ijkl , which is involved in the expressions of the flow rule, yield function, and failure criterion in the form of a transformed stress tensor: Σ ij = A ijkl σ kl The model adequacy is demonstrated by applying it to a sedimentary rock, Toumemire shale. The comparison between the model predictions and data is within the natural scatter of the data. RÉSUMÉ: On presente un modèle elasto-viscoplatique d'une roche initiallement isotrope transverse. Le modele rend compte de la nonlinearite de la reponse elastique. On montre que Ie choix des lois de variation des modules elastiques avec l'etat de contrainte ne peut pas être arbitraire: afin que la reponse soit conservative les paramètres doivent s'exprimer que par des fonctions de la contrainte moyenne et de l'invariant tr M σ, ou M = S 1 S 1 , S 1 etant l'axe de symmetrie. L'anisotropie du domaine plastique et de la rupture est decrite à l'aide d'un tenseur d'order quatre A, invariant par rapport à toute transformation orthogonale appartenant au groupe de symmetrie. A intervient dans les expressions de la surface de charge, de la regie d'ecoulement et du critère du rupture sous Le forme d'un tenseur de contrainte transforme Σ ijj = A ijkl σ kl Le modèle a ete: applique à une roche sedimentaire, l'argilite de Toumemire. La comparison entre les predictions du modèle et les donnees experirnentales est satisfaisante. ZUSAMMENFASSUNG: Eine invariante elasto-viskoplastische nicht asoziierte Gleichung fuer ein anfangs transversal isotropes Gestein wird vorgestellt. Gezeigt wird, wie die elastischen Moduli von Spannungszustand abhangen können. Die Anisotrote wird durch einen Tensor vierter Ordnung A ijkl beschrieben. Er ist in den Ausdrucken fur die Fliessregel, die Fliessfunktion und das Bruchkriterium durch Verwendung eines transformierten Spannungstensors enthalten: Σ ij = A ijkl σ kl . Die Eignung des Modells wird demonstriert, indem es bei einem geschichteten Sedimentgestein, Tournemire Tonschiefer, angewendet wird. 1 INTRODUCTION In many rocks, due to the existence of well-defined fabric elements such as bedding, layering, foliation or lamination planes, or due to the existence of linear structures, anisotropy can be important. The symmetries most frequently encountered are: transverse isotropy and orthotropy. By adopting both theoretical and experimental approaches, many authors have investigated the effect of the presence within the rock of pronounced anisotropic feature on strength. The experimental studies have been carried out mainly on cylindrical specimens subjected to axisymmetrical state of compressive stresses. It has been found that at zero or low degree of confinement the compressive strength varies significantly with the orientation angle β (β is the angle between the strata's planes and the direction of maximum compression). Most rocks undergo a Significant reduction in strength when the anisotropic plane is aligned at about 30° to the axis of major principal stress (for most rocks β ε (30°, 45°)) while the maximum compressive strength occurs at an orientation angle β of 0° or 90° depending upon the rock type. As the amplitude of the confined pressure is raised the rock responds in a more ductile manner.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 4–7, 2009

Paper Number: SPE-124236-MS

... Elastic undrained -80 -40 0 40 80 0 0.2 0.4 0.6 0.8 1 Normalized Radial Distance Ef fe ct iv e R ad ia l S tr

**es**s (M Pa ) Poroelastic Elastic drained Elastic undrained a) b) 0 30 60 90 120 150 180 0 0.2 0.4 0.6 0.8 1 Normalized Radial Distance Ef fe ct iv e Ta ng en tia l S tr**es**s (M Pa ) Poroelastic...
Abstract

Abstract Even with the emerging gas and oil shale plays, little laboratory data exists for the simulation of openhole completion and production accounting for the inherent anisotropy of shale. Contrary to the success of the thick wall cylinder test in simulating openhole production stability and/or solids production in sandstone and carbonate formations, the application and modeling of this test in shale, while subjecting the sample to in-situ openhole conditions, presents considerably greater complexity. Shale in general possesses anisotropic mechanical material properties that will alter the stress distribution around an open hole. Moreover, time-dependent pore pressure and permeability effects will modify the effective stresses and can cause delayed formation failure. In this study, the response of anisotropic hollow cylindrical shale samples under laboratory time-dependent loading, simulating near-wellbore production stresses, is investigated under the realm of coupled fluid and shale matrix interaction to address these challenging issues. Experimental results on thick wall cylinder testing of an anisotropic and low permeability shale were analyzed using the developed physical and mathematical model. The results showed that significant pore pressure could be generated and sustained in even small shale specimens (3.81 cm outer diameter and 1.27 cm inner diameter), depending on shale anisotropic properties, permeability, and loading rate, and cannot be ignored when the collapse strength from thick wall cylinder tests is interpreted for field conditions. Furthermore, commonly employed elasticity analyses for the thick wall cylinder test fail to account for the coupled fluid and shale matrix interaction under time-dependent loading condition and produce oversimplified and inaccurate results for not only the effective stress distribution but also failure analysis. The new modeling, therefore, will be essential for designing, simulating, and interpreting laboratory thick wall cylinder tests on shales as well as the subsequent modeling of sustainable gas and oil production from these formations. Introduction The thick wall cylinder test (Antheunis et al., 1976) has become a routine test in the petroleum industry to characterize openhole stability and/or sanding/solids production. In this test, a rock sample in the form of a hollow cylinder core is externally stressed until failure. The testing configuration has the distinct advantage of producing non-uniform stress and strain distributions around the central hole, thus mimicking field conditions of near wellbore or near perforation production stress and strain regime. There are a number of different approaches to analyzing the thick wall hollow cylinder test. This test has been analyzed analytically using classical elasticity (Ewy et al., 2001) or bifurcation analysis (van den Hoek et al., 2000). Alternatively, numerical studies using either finite difference method (Nouri et al., 2006) or finite element method (Papamichos et al., 2001) have been conducted. All existing analyses assume isotropic rock properties. Shaly formations, however, regularly exhibit transverse isotropy which could significantly influence the coupled responses, in particular pore pressure and effective stress distribution, thereby affecting the stability of the tested specimen in the lab as well as the wellbore or perforation in the field. The analytical solution for the behavior of anisotropic poroelastic hollow cylinders subjected to time-dependent axial displacement/force loading, fluid pressure on the inner boundary, and confining pressure or fluid flux on the outer boundary, in addition to thermal gradients, has been derived previously by Kanj and Abousleiman (2005). Although these solutions potentially cover many combinations of time-dependent loading boundaries, they have only been numerically illustrated for the case of an idealized Heaviside unit step loading of axial strain or average axial stress. Poroelastic anisotropic behavior of hollow cylinders subjected to various pressure/displacement loading scenarios has also been investigated by Kanj and Abousleiman (2007).

Proceedings Papers

Paper presented at the ISRM International Symposium - EUROCK 2005, May 18–20, 2005

Paper Number: ISRM-EUROCK-2005-014

... defined by: -PeP=L. PI = l;pP -+ l;eP = 1 PI ' P P where 7/ is the intrinsic mass density of the ,Bth fluid component C;/ = pfJ I¢» and PI is the total fluid mass density (per unit volume of the fluid). The chemical potential s RT1 eSJ.J n MS can be linearized by approximating In

**eS**such that: fJ -fJ-fJa...
Abstract

ABSTRACT This paper presents a linear cherno-poroelastic model for analyzing borehole stability in transversely isotropic shales. The theory couples mechanical, hydraulic and chemical processes in fluid-saturated porous media. In particular, the model couples stresses and pore pressure to hydraulic conduction, chemical osmosis, and solute diffusion. Field equations are obtained by requiring the rock constitutive and transport models to satisfy the momentum, fluid mass, and solute mass balance equations. The field equations are solved analytically for the problem of vertical and inclined boreholes in shales to yield the distributions of the solute mass fraction, pore pressure, and the stresses around the borehole. The analytical solution is used to demonstrate the impact of the degree of anisotropy, chemical osmosis, solute diffusion and reflection, swelling, and chemical properties of shales on the pore pressure and the stress fields around a borehole drilled in anisotropic shale. INTRODUCTION Oil and gas wells represent the fundamental infrastructure in hydrocarbon exploitation. Drilling the well bore is the first and, usually, the most expensive step in oil and gas production activities. Therefore, designing a stable and safe well has become a critical issue in the industry, especially due to the recent and ever-increasing complexity of geological settings requiring continuous improvement in drilling technology. Well bore stability, mainly in shales, is one of the most challenging problems encountered during drilling. These problems often depart [Tom the classical mechanical failure mechanisms exhibiting time-dependent mud support changes and loss of strength according to physico-chemical processes. Presently, the design of improved water-base muds for shale stability is of primary concern for the drilling industry. As a results, a key factor in selecting the appropriate drilling fluid is a better understanding of the swelling phenomena and transport processes in such formations; being associated with the chemical composition and characteristics of the material. Prior to drilling, a shale formation is in a state of mechanical, hydraulic, thermal, and chemical equilibrium. Drilling disturbs this natural equilibrium, resulting in the modification of local stresses accompanied by deformation of the borehole, as well as the enforcement of hydraulic, temperature, and chemical potential gradients (Hale et al., 1993; Mody & Hale, 1993; van Oort et al., 1996; and Dick et al., 1996). This paper presents an extension of poroelasticity for fully coupling mechanical, hydraulic, and chemical processes in transversely isotropic porous media saturated by a compressible liquid consisting of a solvent and one or more solute. WELLBORE STABILITY MODELS Many different theories, (Low & Anderson, 1958; Chenevert et al., 1970, 1993; Fritz & Marine, 1983; Mody & Hale, 1993; Pashley & Israclachvili, 1984) have been presented to explain the swelling phenomenon of shales (capillary suction, osmosis pressure, and hydraulic pore pressure imbalance). However, until now, the experimental data have not totally and effectively been explained or even understood. Moreover, the extension of the classic poroelastic theory of Biot (1941) for coupling isothermal processes (mechanical, hydraulic, and chemical), that causing shale deterioration and borehole instability while drilling, have been studied by a number of researchers.

Proceedings Papers

Paper presented at the 9th ISRM Congress, August 25–28, 1999

Paper Number: ISRM-9CONGRESS-1999-190

... wird die Wellenausbreitung manchmal stark behindert durch die Interferenz mehrfach gestreuter Wellen (sogenanntes "stop-pass" Verhalten).

**Es**konnte gezeigt werden, dass Wellen, welche sich parallel zu den Kluften ausbreiten, mehrere gebundelte Eigenfunktionen aufweisen mit Geschwindigkeiten und...
Abstract

ABSTRACT: Frequency-dependent acoustic wave propagation in fractured rock is examined for a set of equally-spaced, parallel fractures. Exact solutions for the velocity of the waves are obtained as a function of wave frequency and fracture stiffness and spacing. Waves propagating obliquely to the fractures show strongly anisotropic velocities for low fracture stiffnesses and high frequencies. For high frequencies, propagation of the waves is sometimes prohibited due to the interference of multiply scattered waves (stop-pass behavior). For the waves propagating parallel to the fractures, multiple guided modes are shown to exist with velocities and particle motions that are dependent on fracture stiffness and frequency. RÉSUMÉ: Ici, on examine la propagation des ondes acoustiques dans une roche fissuree avec une serie de fissures parallèles et egalement espacer, dependant de la frequence d'onde. Des solutions exactes pour la vitesse des ondes sont decritent comme fonction de frequence d'onde, la rigidite des fissures, et la distance entre fissures. Les ondes qui se propagent de direction oblique au fissures demontrent des vitesses fortement anisotrope en cas de fissures avec peu de rigidite et des frequences elevees. En cas des frequences elevees, la propagation des ondes est parfois arrêtee à cause de l'interference entre ondes multiple, dispersees (comportement arrête-aller). En cas d'ondes qui se propagent de direction parallèle au fissures, on demontre qu'il existe des modes multiples guides avec des vitesses and movement de particles dependant de la frequence et la rigidite des fissures. ZUSAMMENFASSUNG: In dieser Abhandlung untersuchen wir die frequenzabangige Ausbreitung akustischer Wellen in Festgesteinen mit parallel verlaufenden, aequidistanten Klueften, Exakte Lösungen wurden gefunden fuer die Geschwindigkeit der Wellen als Funktion der Frequenz, der Steifigkeit der Kluft sowie des Kluftabstandes. Wellen, die unter einem flachen Winkel auf die Kluefte treffen zeigen stark anisotrope Geschwindigkeiten bei geringen Steifigkeiten und hohen Frequenzen. Bei hohen Frequenzen wird die Wellenausbreitung manchmal stark behindert durch die Interferenz mehrfach gestreuter Wellen (sogenanntes "stop-pass" Verhalten). Es konnte gezeigt werden, dass Wellen, welche sich parallel zu den Klueften ausbreiten, mehrere gebuendelte Eigenfunktionen aufweisen mit Geschwindigkeiten und Teilchenbewegungen, welche von der Kluftsteifigkeit und Frequenz abhangen. 1. INTRODUCTION In recent years, fractures in reservoir rock have been recognized as important conduits for hydrocarbon production (Nelson, 1987). Fractures that serve as conduits for fluid and gas transport must be partially open. This imperfect contact can result in large normal and tangential compliances when filled with gas and large tangential compliances when filled With fluid. Compliant fractures are seismically detectable by the changes they impart to wave velocities, amplitudes, and Spectral content. By examining these characteristics, reservoir properties that arise from fractures can be investigated. One of the common geologic structures encountered in reservoir rock is that of closely-spaced parallel fractures (Laubach, 1991; Lorenz & Finley, 1991). Such structures are possibly formed by large regional compressive stresses (Lorenz et al., 1991). Gas, fluid flow and seismic wave propagation in rock with a single set of parallel fractures can exhibit strong transverse isotropy. When the seismic wavelength is much longer than the fracture spacing, the medium can be replaced by an equivalent transversely Isotropic medium (Schoenberg and Douma, 1988; Schoenberg and Muir, 1989; Hood and Schoenberg, 1989). Such an approximation is possible because the stress distribution in a representative volume containing fractures is nearly uniform because of the small spatial variations in stress.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE/ISRM Rock Mechanics Conference, October 20–23, 2002

Paper Number: SPE-78162-MS

... t = 24 hrs t = 6 hrs t = 3 hrs t = 1 hr t = 0.1 hr = 0.4 0 = 4 MPa D = 1.0E-08 m2/sec r / R P or e pr

**es**su re , M P a Figure 3. Pore pressure distribution around the borehole. -5 0 5 10 15 1 2 3 4 5 t = 120 hrs t = 72 hrs t = 24 hrs t = 6 hrs t = 3 hrs t = 1 hr t = 0.1 hr = 0.4 0 = 4.0 MPa...
Abstract

Abstract It is generally accepted that shale deterioration and borehole instability are significantly influenced by excessive pore pressure, ion exchange between drilling fluids and shale, and the anisotropy of the in-situ stress state as well as the formation. This paper describes a model for estimating the influence of elastic and chemo-mechanical anisotropy on the distribution of stress and pore pressure around a well in transversely isotropic shale. The model is based on a practical theory that couples ion diffusion, chemical osmosis, and hydraulic flow to stresses and pore pressure. The field equations of the model are derived within the framework of a continuum chemoporoelastic theory that linearly relates total stresses and variation of fluid content to the strains, pore pressure, and solute mass fraction through anisotropic material coefficients. The field equations are solved analytically for the problem of a wellbore in transversely isotropic shale to yield the solute mass fraction, pore pressure, and the stress distributions around the borehole. The solution has been applied to a typical field situation and the results indicate that the osmotic pressure tends to stabilize the borehole. However, osmotic pressure dissipates with time due to ion diffusion. Furthermore, it is found that the anisotropy in chemo-mechanical material coefficients strongly affects the effective stresses around the borehole and enhances the potential for borehole failure. Introduction When drilling for oil and gas, a significant portion of the drilled sections contain shales that are the major source (90%) of wellbore instability problems. These problems often depart from the classical mechanical failure mechanism exhibiting time-dependent mud support requirements and loss of strength associated with physico-chemical processes. Prior to drilling, a shale formation may be considered in a state of mechanical, hydraulic, chemical, and thermal equilibrium. Drilling disturbs this virgin state and imposes hydraulic, chemical potential, and thermal gradients. The ensuing influx of mud filtrate into the formation weakens the rock and leads to excessive stress, pore pressure, and ionic exchange between drilling fluids and shale. As a result, rock deforms and may faile leading to borehole collapse.

Proceedings Papers

Paper presented at the 9th ISRM Congress, August 25–28, 1999

Paper Number: ISRM-9CONGRESS-1999-301

... temporaryand permanentliningsnormal to the tunnelaxis (leastfavourablerock zone) aftercompleteexploita- tionof the,

**es**~matedconvergence ' ; '. " ~ ,; u 1 I In the following we leave out a discussion of the computational investigations and mention only that' the characteristic line method provides useful...
Abstract

ABSTRACT: The planning of long deep-lying base tunnels through the Alps highlights the problem of tunnelling in squeezing rock conditions. After a brief description of the constructional experience gained to date and a retrospective look at the first scientific attempts to explain the various forms of rock pressure, the main aspects of the project for the 57 m long Gotthard Base Tunnel are presented. Following this a description is given of the results of the planning work for the construction lot Sedrun, for which long stretches of squeezing rock conditions have been predicted. The concept is based on full face excavation with systematic support of the face and control of the convergence using a steel lining with yielding joints after Toussaint-Heintzmann as well as systematic rock bolting. RÉSUMÉ: La planification de longs tunnels de base sous les Alpes fait ressortir Ie problème de la construction de tunnels dans des conditions de roche poussante. Les aspects principaux du projet pour le Tunnel de Base du Gotthard, de 57 km de long, sont abordes après une brève description de l'experience acquise à ce jour dans la construction et un regard en arrière sur les premiers essais scientifiques pour expliquer les differents types de pression des roches. Ensuite, on fournit une description des resultats de planification du lot de Sedrun pour lequel on a predit de long tronçon de roche poussante. Le concept d'excavation est base sur une attaque à pleine section avec un soutènement systematique uniforme du front et un contrôle de la convergence estimee par un revêtement en acier selon la methode de Toussaint-Heintzmann ainsi que l'utilisation de grilles d'ancrages radiaux. ZUSAMMENFASSUNG: Die Planung langer, tiefliegender Basistunnel durch die Alpen verleiht dem Problem des Tunnelbaus in druckhaftem Gebirge hohe Aktualitat. Nach einer kurzen Schilderung der Bauerfahrung und einem Rueckblick auf die ersten wissenschaftlichen Ansatze zur Erklarung der mannigfaltigen Äusserungen des echten Gebirgsdrucks wird das Projekt des 57 km langen Gotthard Basistunnels in groben Zuegen dargestellt. Anschliessend werden die Ergebnisse der Projektierungsarbeit fuer das Los Sedrun erlautert, wo Iangere Strecken druckhaften Gebirges prognostiziert sind. Das Konzept beruht auf dem Vollausbruch mit systematischer Sicherung der Ortsbrust und der Kontrolle einer eingeplanten Konvergenz mit Stahlausbau nach Toussaint-Heintzmann sowie radialer Systemankerung. 1. INTRODUCTION The term "squeezing rock" originates from the pioneering days of tunnelling in the Alps. The various descriptions of rock pressure were already in the early days classified into three groups, namely loosening rock pressure, swelling pressure and squeezing pressure. Thus the observed rock behaviour was often described using terms like spalling, swelling and squeezing. As long ago as the last century one already understood that these three types of rock pressure were caused by fundamentally different physical mechanisms (Kovari 1975)., They may also act in a superimposed way; and thus it is for example conceivable that in a rock of low strength containing clay minerals the failure processes are accompanied by the swelling phenomenon. Squeezing rock is characterised by the tendency to reduce the cross-section of the opening (Figure 1). The reduction in size of the opening in course of time is called "convergence". The actual creep potential of the rock under the given stresses is a basic requirement for the occurrence of squeezing rock. Since the lining resists the convergence, the pressure acts as a reaction, so that rock pressure and rock deformation are directly related to one another. With respect to the lining the, rock pressure is regarded as a loading, and with respect to the rock it acts as a lining resistance; thereby, two distinct aspects (action and reaction) of the same phenomenon are expressed. If the rock pressure exceeds the bearing capacity of the lining it will be damaged or even destroyed.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1998 SEG Annual Meeting, September 13–18, 1998

Paper Number: SEG-1998-1483

..., perform depth pro- cessing in TTI media. Using the exact NMO equation, we examined the propagation of errors in the moveout velocities into the estimated values of the anisotropic parameters and

**es**- tablished the necesary conditions for a stable inversion procedure. Determination of the individual values...Advertisement