A simple derivation of the effective stress coefficient for seismic velocities in porous rocks

Understanding the effective stress coefficient for seismic velocity is important for geophysical applications such as overpressure prediction from seismic data as well as for hydrocarbon production and monitoring using time-lapse seismic measurements. This quantity is still not completely understood. Laboratory measurements show that the seismic velocities as a function of effective stress yield effective stress coefficients less than one and usually vary between 0.5 and 1. At the same time, theoretical analysis shows that for an idealized monomineral rock, the effective stress coefficient for elastic moduli ͑and therefore also for seismic velocities͒ will always equal one. We explore whether this deviation of the effective stress coefficient from unity can be caused by the spatial microheterogeneity of the rock. The results show that only a small amount ͑less than 1%͒ of a very soft component is sufficient to cause this effect. Such soft material may be present in grain contact areas of many rocks and may explain the variation observed experimentally.

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“…This agrees with Gassmann's limit ͑Berryman and Milton, 1991;Berryman, 1992;Gurevich, 2004͒ when only a single mineral is present.…”

Section: Analytical Solutionssupporting

confidence: 88%

Influence of microheterogeneity on effective stress law for elastic properties of rocks

et al. 2008

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“…For the bulk volumetric strain n (Π) is equal to Biot's poroelastic coefficient (e.g., Nur and Byerlee, 1971;Robin, 1973), which is the case in Equation (4). In contrast, for the porosity and for the drained bulk modulus n (Π) is equal to 1 in idealized model (e.g., Zimmermann, 1991;Gurevich, 2004). In other words, for these physical properties the effective pressure is equal to the differential pressure.…”

Section: Importance Of the Differential Pressurementioning

confidence: 99%

“…Both confining pressure P c and pore pressure P p (not to be confused with the pore-pressure increment induced by the seismic wave) vary in the subsurface and modify any physical property of rocks. In the framework of an idealized rock-physics model, Gurevich (2004) demonstrated that seismic velocities essentially depends on the differential pressure P diff = P c -P p (the difference between the confining pressure P c and the pore-pressure P p ) and not in an independent way on P c and on P p . This will be shown experimentally in the next sections.…”

Section: Abstract -Experimental Verification Of the Petroelastic Modementioning

confidence: 99%

Experimental Verification of the Petroelastic Model in the Laboratory – Fluid Substitution and Pressure Effects

Rasolofosaon

Zinszner

2012

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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Ce modèle doit prendre en compte à la fois les effets de substitution de fluides et les effets de variation de pression sur les paramètres sismiques mesurés (vitesses, impédances). Cet article décrit une vérification expérimentale au laboratoire de ce modèle. Concernant les effets de substitution de fluides, le modèle de Biot-Gassmann est le plus utilisé. Ce modèl considère que le module de cisaillement est indépendant de la nature du fluide saturant, quand celui-ci est non visqueux, et relie les variations des modules d'incompressibilité de la roche due à la substitution de fluides aux paramètres du milieu poreux et des fluides saturants. La validation expérimentale, portant donc sur les deux aspects, montre sur un échantillonnage varié de grès et de calcaires poreux que le module de cisaillement de la roche est indépendant du fluide saturant peu visqueux. Cette indépendance est vérifiée, même dans le cas de fluides visqueux (viscosité inférieure à 10 4 cP), si la pression différentielle est élevée (fermeture des microfissures) ; l'écart entre le module d'incompressibilité mesuré et calculé est toujours faible ; le module d'incompressibilité des cristaux formant une roche monominérale (calcaire, grès propre) déduit de la formule de Gassmann est proche des valeurs données dans la littérature pour ces minéraux ; lors de mesures sous une même pression différentielle, mais avec des pressions de pore différentes, les écarts observés sur les modules d'incompressibilité sont comparables à ceux prévu par la formule de BiotGassmann en prenant en compte la variation du module d'incompressibilité du fluide saturant sous l'effet de la pression de pore (non linéarité). Ces trois dernières observations convergent vers la validation de la formule de Biot-Gassmann pour le module d'incompressibilité. Concernant les effets de pression, le paramètre pertinent est la pression différentielle P diff = P c -P p , c'est-à-dire la différence entre la pression de confinement P c et la pression de pore P p . Plus précisément, nous démontrons que les vitesses des ondes P et S dépendent uniquement de la pression différentielle P diff , et non individuellement des pressions P c et P p . Cette pression, en contribuant à fermer les micro-défauts mécaniques (contacts entre grains, microfissures), a des conséquences très variables sur les vitesses et les atténuations, suivant l'abondance relative de ces micro-défauts. Les roches calcaires, quelle que soit la pression, sont souvent très peu sensibles à cet effet, à cause de la facilité avec laquelle le carbonate cimente les micro-défauts. Les grès consolidés sont souvent sensibles à la pression différentielle et les roches inconsolidées (sables) très sensibles. La relation Vitesse vs P diff est généralement du type loi puissance et l'exposant de cette relation est un excellent moyen de quantifier la sensibilité d'une roche à la pression.

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“…It can be shown that the effective pressure P eff for pore space deformation is just the differential pressure if the grain matrix material is homogeneous and elastic (e.g., ZIMMERMAN et al, 1986;SHAPIRO, 2003;GUREVICH, 2004), and/or the bulk porosity is low (SHAPIRO and KASELOW, 2003;KASELOW and SHAPIRO, 2004). As the in situ porosity of the metamorphic KTB rocks is below 1% (e.g., KERN et al, 1991) at least the last condition seems to be satisfied.…”

Section: Methodsmentioning

confidence: 99%

Stress Sensitivity of Seismic and Electric Rock Properties of the Upper Continental Crust at the KTB

Kaselow

Becker

Shapiro

2006

Pure appl. geophys.

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We test the hypothesis that the general trend of P-wave and S-wave sonic log velocities and resistivity with depth in the pilot hole of the KTB site Germany, can be explained by the progressive closure of the compliant porosity with increasingly effective pressure. We introduce a quantity h c characterizing the stress sensitivity of the mentioned properties. An analysis of the downhole measurements showed that estimates of the quantitiy h c for seismic velocities and electrical formation factor of the in situ formation coincide. Moreover, this quantity is 3.5 to 4.5 times larger than the averaged stress sensitivity obtained from core samples. We conclude that the hypothesis mentioned above is consistent with both data sets. Moreover, since h c corresponds approximately to the inverse of the effective crack aspect ratio, larger in situ estimates of h c might reflect the influence of fractures and faults on the stress sensitivity of the crystalline formation in contrast to the stress sensitivity of the nearly intact core samples. Finally, because the stress sensitivity is directly related to the elastic nonlinearity we conclude that the elastic nonlinearity (i.e., deviation from linear stress-strain relationship i.e., Hooke's law) of the KTB rocks is significantly larger in situ than in the laboratory.

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A simple derivation of the effective stress coefficient for seismic velocities in porous rocks

Cited by 84 publications

References 29 publications

Influence of microheterogeneity on effective stress law for elastic properties of rocks

Influence of microheterogeneity on effective stress law for elastic properties of rocks

Experimental Verification of the Petroelastic Model in the Laboratory – Fluid Substitution and Pressure Effects

Stress Sensitivity of Seismic and Electric Rock Properties of the Upper Continental Crust at the KTB

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