In a great variety of laboratory experiments over large intervals in stress, strain, and frequency, rocks display pronounced nonlinear elastic behavior. Here we describe nonlinear response in rock from resonance experiments. Two important features of nonlinear resonant behavior are a shift in resonant frequency away from the linear resonant frequency as the amplitude of the disturbance is increased and the harmonics in the time signal that accompany this shift. We have conducted Young's mode resonance experiments using bars of a variety of rock types (limestone, sandstone, marble, chalk) and of varying diameters and lengths. Typically, samples with resonant frequencies of approximately 0.5–1.5 kHz display resonant frequency shifts of 10% or more, over strain intervals of 10−7 to 10−6 and under a variety of saturation conditions and ambient pressure conditions. Correspondingly rich harmonic spectra measured from the time signal progressively develop with increasing drive level. In our experiments to date, the resonant peak is observed to always shift downward (if indeed the peak shifts), indicating a net softening of the modulus with drive level. This observation is in agreement with our pulse mode and static test observations, and those of other researchers. Resonant peak shift is not always observed, even at large drive levels; however, harmonics are always observed even in the absence of peak shift when detected strain levels exceed 10−7 or so. This is an unexpected result. Important implications for the classical perturbation model approach to resonance results from this work. Observations imply that stress‐strain hysteresis and discrete memory may play an important role in dynamic measurements and should be included in modeling. This work also illustrates that measurement of linear modulus and Q must be undertaken with great caution when using resonance.

S U M M A R YWe derive the explicit analytical expressions of the phase velocities of the three bulk waves (gP, qS, and qS,) propagating in an arbitrary direction in a homogeneous strongly anisotropic medium of arbitrary symmetry (triclinic). The mathematical formulation is concise and symmetrical with respect to all the wave types. A simple geometrical interpretation of the formulae is proposed. The conciseness of the formalism hides the great complexity of the solutions when they are explicitly expressed in terms of the elastic constants of the media and of the direction cosines of the unit propagation vector, which makes further analytical developments daunting. We use perturbation techniques to derive simplified approximate expressions of these solutions. In a continuation of the work of Thomsen ( 1986) in weakly transversely isotropic media, we extend the work of Sayers (1994) in arbitrary weakly anisotropic media, but restricted to qP waves, by deriving the complete directional dependence of the velocities of the three bulk waves (qP, $3, and qS,). Furthermore, contrary to the previous references, we do not need to assume that the anisotropy is weak since our model is based on the perturbation of a reference model which can exhibit strong S-wave birefringence. We do not use spherical harmonic decomposition, as Sayers ( 1994) did, but only elementary rotations of the components of the elastic tensor of the considered medium. The forms of the proposed solutions naturally introduce generalizations of Thomsen's classical parameters E, 6 and y , but they are relevant for arbitrary symmetry. Numerical comparisons between the exact and the approximate velocities using perturbation theory in transversely isotropic and triclinic media validate the proposed solutions, and show that the first-order approximations are reasonable even in the presence of a moderate amount of anisotropy.

The elastic nonlinear behavior of rocks as evidenced by deviations from Hooke's law in stress‐strain measurements, and attributable to the presence of mechanical defects (compliant features such as cracks, microfractures, grain joints), is a well‐established observation. The purpose of this paper is to make the connection between the elastic nonlinearity and stress‐induced effects on waves, in this case uniaxial‐stress‐induced transverse isotropy. The linear and nonlinear elastic coefficients constitute the most condensed manner in which to characterize the elastic behavior of the rock. We present both the second‐ and the third‐order nonlinear elastic constants obtained from experimental data on rock samples assumed homogeneous and isotropic when unstressed. As is normally the case, the third‐order (nonlinear) constants are found to be much larger than the second‐order (linear) elastic constants. Contrary to results from intact homogeneous solids (materials without mechanical defects), rocks exhibit weak to strong nonlinearity and always in the same manner (i.e., an increase of the moduli with pressure). As a consequence the stress‐induced P wave anisotropy and S wave birefringence can be large. The stress‐induced P wave anisotropy appears to be much larger than the S wave birefringence. The fast direction is parallel to the stress direction, and the anisotropy goes as sin2 θ, θ being the angle between the propagation direction and the stress direction. Experiments on rocks indicate that at low applied stresses, the proportionality of the stress and the induced S birefringence and P anisotropy predicted by theory is well corroborated.

We developed new experimental and theoretical tools for the measurement and the characterization of arbitrary elasticity tensors and permeability tensors in rocks. They include an experimental technique for the 3-D visualization of hydraulic invasion fronts in rock samples by monitoring the injection of salt solutions by X-ray tomography, and a technique for inverting the complete set of the six coefficients of the permeability tensor from invasion front images. In addition, a technique for measuring the complete set of the 21 elastic coefficients, a technique allowing the identification and the orientation in the 3-D space of the symmetry elements (planes, axes), and a technique for approximating the considered elastic tensor by a tensor of simpler symmetry with the quantification of the error induced by such an approximation have been developed.We apply these tools to various types of reservoir rocks and observed quite contrasted behaviors. In some rocks, the elastic anisotropy and the hydraulic anisotropy are closely correlated, for instance in terms of the symmetry directions. This is the case when elastic anisotropy and hydraulic anisotropy share the same cause (e.g., layering, fractures). In contrast, in some other rocks, hydraulic properties and elastic properties are clearly uncorrelated. These results highlight the challenge we have to face in order to estimate the rock permeability and to monitor the fluid flow from seismic measurements in the field.

[1] The complete permeability tensor of 18 porous rock cores was determined by means of X-ray tomography monitoring during the displacement of a salty tracer. To study the effect of the pore space geometry on the anisotropy of permeability, we compared the three-dimensional shape of the invasion front with the X-ray porosity maps obtained before injection. The samples (clean and shale-bearing sandstones, limestones, and volcanic rocks) belong to a broad variety of granulometry and pore space geometry. Their porosity ranges from 12 to 57%, and their permeability ranges from 1.5 Â 10 À14 to 4 Â 10 À12 m 2 . For sandstones the permeability anisotropy is well correlated with the presence of bedding. For volcanic rocks it is clearly related to the orientation of vesicles or cracks. However, for limestones, no evident link between the geometry of the porous network and the permeability anisotropy appears, probably because of the influence of the nonconnected porosity that does not contribute to the hydraulic transport. This systematic work evidences the ability and the limitations of the tracer method to characterize the anisotropy of permeability in the laboratory in a simple and rapid way.Citation: Clavaud, J.-B., A. Maineult, M. Zamora, P. Rasolofosaon, and C. Schlitter (2008), Permeability anisotropy and its relations with porous medium structure,

Abstract. We describe Young's mode resonant bar results obtained under effective pressure at two saturation states: dry and water saturated. We monitor primary manifestations of nonlinear response in these experiments: the harmonic content, the source extinction intensity, and fundamental resonant frequency shift. In addition, we describe the hysteretic behavior of the static pressure response, the linear modulus, and O. Because we currently lack a complete theoretical description of nonlinear behavior under resonance at pressure, we provide relative measures of nonlinear response rather

Un principe gŽnŽral esquissŽ par P. Curie (1894) concernant lÕinfluence de la symŽtrie sur les phŽnom•nes physiques dit, en langage actuel, que le groupe de symŽtrie des causes est un sousgroupe du groupe de symŽtrie des effets. Par exemple, en ce qui concerne lÕanisotropie sismique induite par les contraintes, la symŽtrie la plus complexe prŽsentŽe par un milieu initialement isotrope, sous contrainte triaxiale, est orthorhombique ou orthotrope, caractŽrisŽe par trois plans de symŽtrie orthogonaux deux ˆ deux (Nur, 1971).Ë dÕautres Žgards, Schwartz et al. (1994) ont montrŽ que deux mod•les de roches tr•s diffŽrents, un mod•le fissurŽ et un mod•le granulaire faiblement consolidŽ, conduisent toujours ˆ une anisotropie elliptique quand ils sont soumis ˆ une contrainte uniaxiale. La question posŽe est la suivante : est-ce que ce rŽsultat est vrai pour tous les mod•les de roches ? et, plus gŽnŽralement, est-ce que les roches initialement isotropes, quand elles sont soumises ˆ une contrainte triaxiale, forment un sous-ensemble bien dŽfini des milieux orthorhombiques ?Sous lÕhypoth•se dÕhyperŽlasticitŽ isotrope non linŽaire du troisi•me ordre (cÕest-ˆ-dire absence dÕhystŽrŽsis, et existence dÕune fonction dÕŽnergie Žlastique dŽveloppŽe au troisi•me ordre dans les composantes de la dŽformation), il est dŽmontrŽ que lÕanisotropie de lÕonde qP induite par les contraintes est toujours ellipso•dale, pour tout degrŽ dÕanisotropie. Par exemple, les sources ponctuelles engendrent des fronts dÕonde qP de forme ellipso•dale. Ce rŽsultat est gŽnŽral et est absolument indŽpendant du mod•le de roche, cÕest-ˆ-dire indŽpendant des causes de la non-linŽaritŽ, pour autant que les hypoth•ses de dŽpart soient vŽrifiŽes. Ceci constitue le principal rŽsultat de cet article.Thurston (1965) a remarquŽ, vis-ˆ-vis des propriŽtŽs Žlastiques, quÕun milieu Žlastique initialement isotrope, quand il est soumis d es contraintes non isotropes, nÕest jamais tout ˆ fait Žquivalent û n cristal anisotrope non soumis ˆ des contraintes. Par exemple, les composantes du tenseur dÕŽlasticitŽ du milieu sous contrainte ne prŽsentent pas la symŽtrie habituelle vis-ˆ-vis de la permutation des indices. Ceci interdit lÕemploi de la notation de Voigt sur les indices contractŽs. Toutefois, si lÕamplitude des composantes du dŽviateur de contrainte est petite par rapport aux modules dÕonde, ce qui est toujours vŽrifiŽ sur le terrain en exploration sismique, lÕŽquivalence parfaite est rŽtablie. Sous cette condition, les 9 rigiditŽs Žlastiques CÕ IJ (en notation contractŽe) dÕun solide initialement isotrope, soumis ˆ une contrainte triaxiale, sont toujours liŽes par les 3 conditions ci-apr•s dÕellipticitŽ dans les plans de coordonnŽes associŽs aux directions propres de la prŽcontrainte statique. STRESS-INDUCED SEISMIC ANISOTROPY REVISITEDA general principle outlined by P. Curie (1894) regarding the influence of symmetry in physical phenomena states, in modern language, that the symmetry group of the causes is a sub-group of the symmetry group of the effects. For instanc...

[1] Since the exhaustive work by Adams and Coker at the Carnegie Institute in the early 1900s and the work of F. Birch's group at Harvard University conducted in the 1940s-1950s, it has been well documented that the quasi-static stress-strain behavior of rock is nonlinear and hysteretic. Over the past 20 years, there has been an increasing body of evidence suggesting that rocks are highly elastically nonlinear and hysteretic in their dynamic stress-strain response as well, even at extremely small strain amplitudes that are typical of laboratory measurements. In this work we present a compendium of measurements of the nonlinear elastic parameter a extracted from longitudinal (Young's mode) and flexural-mode resonance experiments in eight different rock types under a variety of saturation and thermal conditions. The nonlinear modulus a represents a measure of the dynamic hysteresis in the wave pressure-strain behavior. We believe that hysteresis is the primary cause of nonclassical nonlinear dynamics in rock, just as it is responsible for elastic nonlinear behavior in quasi-static observations. In dynamics, a is proportional to the wave speed and modulus reduction as a function of wave strain amplitude due to the hysteresis, based on our current model. The rocks tested include pure quartz sandstone (Fontainebleau), two sandstones that contain clay and other secondary mineralization (Berea and Meule), marble (Asian White), chalk, and three limestones (St. Pantaleon, Estaillades, and Lavoux). The values of a range from $500 to >100,000, depending on the rock type, damage, and/or water saturation state. Damaged samples exhibit significantly larger a than intact samples (hysteresis increases with damage quantity), and water saturation has an enormous influence on a from 0 to 15-30% water saturation.