Effective-Medium Anisotropic models of Fractured Rocks of TI Symmetry: Analysis of Constraints and Limitations in Linear Slip model

Abstract

The simplest effective-medium model of fractured rocks known as the Linear Slip (LS) model of Schoenberg (1980) represents a single fracture set in an isotropic background rock. In the LS model, the stiffness C 13 is not independent as in the overall transversely isotropic (TI) model, but it is related to other stiffnesses by the equation, C 11 C 33-C 2 13 = 2C 66 (C33+C13). We have studied a physical sense of this constraint on the C 13 and found out that in terms of the TI elastic compliance tensor S it leads to the equality 12 = 13 , where 13 and 12 are the two different horizontal Poisson's ratios. In contrast to the Linear Slip model, in the overall TI model, one of these Poisson's ratios, 13 , is always greater than the other one, 12 , that is validated by numerous static and dynamic laboratory measurements of these Poisson's ratios in VTI-type rocks. Thus we have revealed a contradiction and inconsistency in the constraint on the C 13 for the LS model. Moreover, the restriction on C 13 for the LS model doesn't work for the overall TI medium in which there are physical constraints on the C 13 , namely, C 13_min < C 13 < C 13_max (e.g., Yan et al., 2013). We have revealed that mathematical expression for its lower bound, C 13_min , coincides with that for the constraint on the C 13 for the LS model. This means that the restriction on C 13 for the LS model, C 13 = C 13_min , does not satisfy the physical constraint for the overall TI model that is inequality C 13 > C 13_min. The LS model is not a universal model for real rocks. It may work successfully only in several special cases, or under certain conditions, for example, when the normal fracture weakness Δ N =0 (fluid-saturated cracks), or in the case of Δ N = Δ T (dry cracks). Also we have revealed that the LS model may suit better for sandstones and carbonates than for shales.

When do fractured media become seismically anisotropic? Some implications on quantifying fracture properties

“…The second observation is more concerning given that the LS model is used pervasively in the seismological community. Chichinina et al (2015) analyse the limitations of Schoenberg & Sayers (1995) LS model and find that their model is not generally adequate for real rocks. Chichinina et al (2015) note that the LS model is only valid for two conditions: (1) when Z N = 0 (i.e., case of fluid-filled fractures) or (2) when the scalar crack Z N /Z T = 1 is assumed (i.e., Bakulin et al, 2000).…”

“…Chichinina et al (2015) analyse the limitations of Schoenberg & Sayers (1995) LS model and find that their model is not generally adequate for real rocks. Chichinina et al (2015) note that the LS model is only valid for two conditions: (1) when Z N = 0 (i.e., case of fluid-filled fractures) or (2) when the scalar crack Z N /Z T = 1 is assumed (i.e., Bakulin et al, 2000). Hildyard (2001) observed that the LS model was only accurate for high-stiffness fractures and became increasingly inaccurate as the stiffness decreased, which is consistent with the first condition Z N = 0.…”

Critical analysis of theoretical basics of displacement discontinuity model for media with oriented fractures.

“…И не только оно, а и исходное условие-огра-ничение на s 13 в модели LS VTI, s 13 = s 12 , из которого оно получено при переходе от матрицы VTI S к матри-це VTI C : VTI VTI 1 ( ) − = C S . Об этом факте мы сообщи-ли в докладах на конференциях EAGE и SEG [Чичи-нина и др., 2015а; Chichinina et al, 2015].…”

“…По-казаны ограниченные возможности использования данной модели, которые связаны с произволом в вы-боре пятой константы с 13 , в результате которого полу-чаемая матрица жесткости теряет физический смысл и не отвечает принципу симметрии Кюри. Выполнен-ное нами исследование модели разрыва смещений представлено в сжатом виде в расширенных тезисах докладов на конференциях SEG и EAGE [Чичинина и др., 2015а; Chichinina et al, 2015].…”

Estimating applicability of the displacement-discontinuity model using experimental data on measurements of elastic-wave velocities

“…Причинами ее популярности являются умень шенное количество независимых элементов мат-рицы жесткости, описывающей среду с параллельными трещинами (четыре вместо пяти), и введение характе-ристик трещиноватости непосредственно в элементы матрицы жесткости. Однако как показал выполненный нами анализ этой модели [Чичинина и др., 2015, 2016Chichinina et al, 2015], она не имеет физического смыс ла и не соответствует принципу симметрии Кюри. Применять ее можно только в случае распространения упругих волн по нормали к плоскостям трещин, т. е. вдоль оси симметрии среды, и в плоскости трещин, плоскости изотропии.…”