Measurement of nonlinear elastic response in rock by the resonant bar method

In this work we are studying the behavior of the fundamental (Young's) mode resonant peak as a function of drive amplitude in rock samples. Our goal from these studies is to obtain nonlinear moduli for many rock types, and to study the nonlinear moduli as a function of water saturation and other changes in physical properties. Measurements were made on seven different room dry rock samples. For one sample measurements were taken at 16 saturation levels between 1 and 98%. All samples display a softening'' nonlinearity, that is, the resonant frequency shifts downward with increasing drive amplitude. In extreme cases, the resonant frequency changes by as much as 25% over a strain interval of 10[sup [minus]7] to [approximately]4 [times] 10[sup [minus]5]. Measurements indicate that the nonlinear response is extremely sensitive to saturation. Estimates of a combined cubic and quartic nonlinear parameter [Gamma] range from approximately [minus]300 to [minus]10[sup 9] for the rock samples

Reservoir stress path and induced seismic anisotropy: Results from linking coupled fluid-flow/geomechanical simulation with seismic modelling

We present a workflow linking coupled fluid-flow and geomechanical simulation with seismic modelling to predict seismic anisotropy induced by nonhydrostatic stress changes. We generate seismic models from coupled simulations to examine the relationship between reservoir geometry, stress path and seismic anisotropy. The results indicate that geometry influences the evolution of stress, which leads to stress-induced seismic anisotropy. Although stress anisotropy is high for the small reservoir, the effect of stress arching and the ability of the side-burden to support the excess load limit the overall change in effective stress and hence seismic anisotropy. For the extensive reservoir, stress anisotropy and induced seismic anisotropy are high. The extensive and elongate reservoirs experience significant compaction, where the inefficiency of the developed stress arching in the side-burden cannot support the excess load. The elongate reservoir displays significant stress asymmetry, with seismic anisotropy developing predominantly along the long-edge of the reservoir. We show that the link between stress path parameters and seismic anisotropy is complex, where the anisotropic symmetry is controlled not only by model geometry but also the nonlinear rock physics model used. Nevertheless, a workflow has been developed to model seismic anisotropy induced by non-hydrostatic stress changes, allowing field observations of anisotropy to be linked with geomechanical models

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Stress-Induced Seismic Anisotropy Revisited

This summary contains formulas (***) which can not be displayed on the screenA 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 instance, regarding stress-induced seismic anisotropy, the most complex symmetry exhibited by an initially isotropic medium when tri-axially stressed is orthorhombic, or orthotropic, symmetry characterized by three symmetry planes mutually perpendicular (Nur, 1971). In other respects, Schwartz et al. (1994) demonstrated that two very different rock models, namely a cracked model and a weakly consolidated granular model, always lead to elliptical anisotropy when uniaxially stressed. The addressed questions are : Is this result true for any rock model? and more generally : Do initially isotropic rock form a well-defined sub-set of orthorhombic media when triaxially stressed?Under the hypothesis of 3rd order nonlinear isotropic hyperelasticity (i. e. , no hysteresis and existence of an elastic energy function developed to the 3rd order in the strain components) it is demonstrated that the qP-wave stress-induced anisotropy is always ellipsoidal, for any strength of anisotropy. For instance point sources generate ellipsoidal qP-wave fronts. This result is general and absolutely independent of the rock model, that is to say independent of the causes of nonlinearity, as far as the initial assumptions are verified. This constitutes the main result of this paper. Thurston (1965) pointed out that an initially isotropic elastic medium, when non-isotropically pre-stressed, is never strictly equivalent to an unstressed anisotropic crystal. For instance the components of the stressed elastic tensor lack the familiar symmetry with respect to indices permutation. This would prohibit Voigt's notation of contracted indices. However if the magnitude of the components of the stress deviator is small compared to the wave moduli, which is always verified in practical situations of seismic exploration, the perfect equivalence is re-established. Under this condition, the 9 elastic stiffnesses C'ij (in contracted notation) of an initially isotropic solid, when triaxially stressed, are always linked by 3 ellipticity conditions in the coordinate planes associated with the eigen directions of the static pre-stress, namely :(***)Thus only 6 of the 9 elastic stiffnesses of the orthorhombic stressed solid are independent (Nikitin and Chesnokov, 1981), and are simple functions of the eigen stresses, and of the 2 linear (2nd order) and the 3 nonlinear (3rd order) elastic constants of the unstressed isotropic solid. Furthermore, given the state of pre-stress, the strength of the stress-induced P- or S-wave anisotropy and S-wave birefringence (but not the magnitude of the wave moduli themselves) are determined by only 2 intrinsic parameters of the medium, one for the P-wave and one for the S-waves. Isotropic elastic media, when triaxially stressed, constitute a special sub-set of orthorhombic media, here called ellipsoidal media , verifying the above conditions. Ellipsoidal anisotropy is the natural generalization of elliptical anisotropy. Ellipsoidal anisotropy is to orthorhombic symmetry what elliptical anisotropy is to transversely isotropic (TI) symmetry. Elliptical anisotropy is a special case of ellipsoidal anisotropy restricted to TI media. In other words, ellipsoidal anisotropy degenerates in elliptical anisotropy in TI media. In ellipsoidal media the qP-wave slowness surface is always an ellipsoid. The S-wave slowness surfaces are not ellipsoidal, except in the degenerate elliptical case, and have to be considered as a single double-valued self-intersecting sheet (Helbig, 1994). The intersections of these latter surfaces with the coordinate planes are either ellipses, for the S-vave polarized out of the coordinate planes, or circles, for the qS-wave polarized in the coordinate planes. The nearly exhaustive collection of experimental data on seismic anisotropy in rocks (considered as transverse isotropic) by Thomsen (1986) show that elliptical anisotropy is more an exception than a rule. Since stress-induced anisotropy is essentially elliptical when restricted to transversely isotropic media, as a consequence this work clearly shows that stress can be practically excluded as a unique direct cause of elastic anisotropy in rocks

Velocity Analysis Using Nonhyperbolic Move-Out in Anisotropic Media of Arbitrary Symmetry: Synthetic and Field Data Studies

A robust method for estimating the interval parameters (i. e. the normal move-out velocity Vnmo and the anisotropy parameter h) of horizontally layered transversely isotropic media from reflected P-waves data has been recently proposed by Alkhalifah (1997) based on move-out equation from Tsvankin and Thomsen (1994). The method, tested on synthetic and field data, is based first on semblance analysis on nonhyberbolic (i. e. long spread) move-out for the estimation of the effective parameters, and then on a layer stripping process. Sayers and Ebrom (1997) recently proposed another nonhyperbolic traveltime equation and a corresponding interval velocity analysis which can be used for azimuthally anisotropic layered media. The method was tested on synthetic and physical model data in homogeneous anisotropic media of various symmetry. Here we propose a generalization of the method proposed by Alkhalifah, which can deal with arbitrary, but moderately (i. e. anisotropy strength of roughly 20%), anisotropic layered media. The parametrization is a natural extension of the parametrization used by the previous author and based on generalized Thomsen's parameters (Thomsen, 1986) proposed by Mensch and Rasolofosaon (1997). The method is first applied to synthetic data on a six layer model of contrasted anisotropy (type and magnitude). The robustness of the method is demonstrated. All the interval parameters (here Vnmo and the horizontal velocity Vh) are estimated with reasonable errors (typically < 2%, to be compared with the considered anisotropies of about 15 to 20%) at all azimuths. The method is also tested on field data from the North Sea including three 2D seismic lines intersecting at a well location

Vérification expérimentale de la formule de Gassmann dans les calcaires poreux

Dans cet article, nous présentons une vérification expérimentale, originale et convaincante, de la formule de Gassmann. Cette dernière permet de calculer l'effet du fluide saturant l'espace poreux d'une roche, sur les vitesses sismiques. La méthode utilisée pour cette vérification consiste en la mesure précise de la variation de vitesse de propagation des ondes ultrasonores P et S, lors de la substitution de liquides de module d'incompressibilité variable, dans six roches calcaires et dans une lave perméable. On peut ainsi montrer l'existence d'une relation expérimentale linéaire entre le module d'incompressibilité de la roche (Ksat) et le module d'incompressibilité du liquide saturant (Kfl). Nous montrons que cette relation linéaire constitue une simplification de la formule de Gassmann tout à fait acceptable, quantitativement, pour les roches poreuses. L'ordonnée à l'origine de cette relation linéaire est égale au module d'incompressibilité de la roche sèche (Kdry) et sa pente est directement liée au coefficient b de Biot. On dispose donc d'un moyen de contrôle quantitatif par l'estimation du module d'incompressibilité du minéral formant la roche (Kgrain). Dans le cas de la calcite, constituant les échantillons de calcaire, la valeur de ce paramètre est bien connue. Sur un échantillon de calcaire, nous avons aussi pratiqué des substitutions de fluide en système diphasique (eau plus liquide non miscible à l'eau). Dans ce cas, la formule est encore vérifiée lorsque l'on prend pour valeur de la compressibilité du fluide (1/Kfl), la moyenne des compressibilités des fluides présents, pondérées par leur fraction volumique

Velocity Analysis Using Nonhyperbolic Move-Out in Anisotropic Media of Arbitrary Symmetry: Synthetic and Field Data Studies

International audienceA robust method for estimating the interval parameters (i. e. the normal move-out velocity Vnmo and the anisotropy parameter h) of horizontally layered transversely isotropic media from reflected P-waves data has been recently proposed by Alkhalifah (1997) based on move-out equation from Tsvankin and Thomsen (1994). The method, tested on synthetic and field data, is based first on semblance analysis on nonhyberbolic (i. e. long spread) move-out for the estimation of the effective parameters, and then on a layer stripping process. Sayers and Ebrom (1997) recently proposed another nonhyperbolic traveltime equation and a corresponding interval velocity analysis which can be used for azimuthally anisotropic layered media. The method was tested on synthetic and physical model data in homogeneous anisotropic media of various symmetry. Here we propose a generalization of the method proposed by Alkhalifah, which can deal with arbitrary, but moderately (i. e. anisotropy strength of roughly 20%), anisotropic layered media. The parametrization is a natural extension of the parametrization used by the previous author and based on generalized Thomsen's parameters (Thomsen, 1986) proposed by Mensch and Rasolofosaon (1997). The method is first applied to synthetic data on a six layer model of contrasted anisotropy (type and magnitude). The robustness of the method is demonstrated. All the interval parameters (here Vnmo and the horizontal velocity Vh) are estimated with reasonable errors (typically < 2%, to be compared with the considered anisotropies of about 15 to 20%) at all azimuths. The method is also tested on field data from the North Sea including three 2D seismic lines intersecting at a well location

Petroacoustic Characterization of Reservoir Rocks for Seismic Monitoring Studies. Laboratory Measurement of Hertz and Gassmann Parameters

The production of oil and gas reservoirs, or the storage of gas in geological formations, always has direct repercussions on the fluid content and pore pressure, and hence on the seismic properties of reservoir rocks. This article describes the laboratory methods used to measure the effect of variations in differential pressure and saturating fluid on elastic wave propagation velocities in reservoir rocks. The pressure effect is easy to measure in the laboratory, via the Hertz coefficient, exponent of the power function linking the velocity to the differential pressure. It is difficult to estimate the representativity of core samples that have undergone the sudden stress relaxation caused by coring. A statistical comparison of the measurement results on surface samples and core samples confirms the reality of this damage. The values measured in the laboratory are often values from above. They are very useful for setting the upper bounds of the anticipated effect of differential pressure. This effect is often negligible in many limestone reservoirs. It may be high or overpressurized in shallow sandstone reservoirs (underground storage facilities). The effect of the saturating fluid is quantified by the Gassmann formula, the value of which is usually confirmed by experiment. The use of this formula requires the knowledge of certain elastic properties of the rock. These moduli can be determined at the laboratory. We propose an original method that is also simple in principle, based on the experimental measurement of the quasi linear relation predicted by the Biot-Gassmann theory, between the bulk modulus Ksat of the saturated rock and the bulk modulus Kfl of the saturating fluid. In sandstones, during substitution experiments, the liquids used must not disturb the clay minerals (and weathered feldspars). Apart from the case of perfectly clean sandstones, it is therefore highly preferable to preserve an irreducible saturation of brine (Swi) and hence to work with two-phase saturation (brine/hydrocarbons). In limestones, which usually contain no clay, fluid substitution experiments are facilitated by the possibility of single-phase flushing by liquids with highly varied bulk modulus. The advantage procured by this experimental expedient is unfortunately diminished by the difficulty of signal processing caused by the "path dispersion" mechanism corresponding to the scattering on heterogeneities of nonnegligible size compared with wavelength. These heterogeneities (obviously associated with the complex diagenesis of limestones) are omnipresent but not always detectable by a conventional petrographic study. The use of phase velocities in processing transmitted signals is the safest means to help solve this difficulty. In the case of rocks of simple mineralogical composition (limestone, clean sandstone), the knowledge of the bulk modulus of the solid matrix Kgrain offers an excellent means to check the results, thereby substantially facilitating interpretation