Shale reservoirs are typically very tight, and crack are only a small part of the reservoir. The directional arrangement of cracks leads to the anisotropic characteristics of shale, and the type of fluid filled in cra...Shale reservoirs are typically very tight, and crack are only a small part of the reservoir. The directional arrangement of cracks leads to the anisotropic characteristics of shale, and the type of fluid filled in cracks affects the shale reservoir evaluation and late development. Many rock physics theories and methods typically use second-and fourthorder crack density tensors to characterize the elastic anisotropy induced by cracks as well as the normal-to-tangential crack compliance ratio to distinguish between dry and saturated cracks. This study def ines an anisotropic crack f luid indicator for vertical transversely isotropy(VTI) media with vertical symmetry axis which is the integration of the normal-to-tangential crack compliance ratio in three directions. A new dimensionless fourth-order tensor, including crack f luid type, azimuth distribution, and geometric shape, is constructed by substituting the normal and tangential compliance into the fourth-order crack density tensor, which can also be used to identify the type of crack fluid in the VTI media. Using the Callovo–Oxfordian shale experimental data, the variation of the elastic properties of dry and saturated shale samples with axial stress is analyzed. The results demonstrate that the anisotropic crack f luid indicator of water-bearing shale samples is less than that of the dry shale samples and that the dimensionless fourth-order tensor of water-bearing shale samples is nearly one order of magnitude greater than that of the dry shale samples. Therefore, the anisotropic crack f luid indicator and dimensionless fourth-order tensor can ref lect the crack f luid type in shale samples and can be used for shale reservoir prediction and f luid identif ication.展开更多
A-manifolds and/3-manifolds, introduced by Gray （1978）, are two significant classes of Einstein-like Riemannian manifolds. A Riemannian manifold is Ricci parallel if and only if it is simultaneously an A-manifold an...A-manifolds and/3-manifolds, introduced by Gray （1978）, are two significant classes of Einstein-like Riemannian manifolds. A Riemannian manifold is Ricci parallel if and only if it is simultaneously an A-manifold and a B-manifold. The present paper proves that both focal submanifolds of each isoparametric hypersurface in unit spheres with g = 4 distinct principal curvatures are A-manifolds. As for the focal submanifolds with g = 6, m = 1 or 2, only one is an A-manifold, and neither is a B-manifold.展开更多
基金supported by the National Natural Science Foundation of China(Nos.41874146,41674130)National Key S&T Special Project of China(No.2017ZX05049-002,2016ZX05027-004-001)+2 种基金the Fundamental Research Funds for the Central University(No.18CX02061A)the Innovative Fund Project of China National Petroleum Corporation(No.2016D-5007-0301)the Scientific Research&Technology Development Project of China National Petroleum Corporation(No.2017D-3504)
文摘Shale reservoirs are typically very tight, and crack are only a small part of the reservoir. The directional arrangement of cracks leads to the anisotropic characteristics of shale, and the type of fluid filled in cracks affects the shale reservoir evaluation and late development. Many rock physics theories and methods typically use second-and fourthorder crack density tensors to characterize the elastic anisotropy induced by cracks as well as the normal-to-tangential crack compliance ratio to distinguish between dry and saturated cracks. This study def ines an anisotropic crack f luid indicator for vertical transversely isotropy(VTI) media with vertical symmetry axis which is the integration of the normal-to-tangential crack compliance ratio in three directions. A new dimensionless fourth-order tensor, including crack f luid type, azimuth distribution, and geometric shape, is constructed by substituting the normal and tangential compliance into the fourth-order crack density tensor, which can also be used to identify the type of crack fluid in the VTI media. Using the Callovo–Oxfordian shale experimental data, the variation of the elastic properties of dry and saturated shale samples with axial stress is analyzed. The results demonstrate that the anisotropic crack f luid indicator of water-bearing shale samples is less than that of the dry shale samples and that the dimensionless fourth-order tensor of water-bearing shale samples is nearly one order of magnitude greater than that of the dry shale samples. Therefore, the anisotropic crack f luid indicator and dimensionless fourth-order tensor can ref lect the crack f luid type in shale samples and can be used for shale reservoir prediction and f luid identif ication.
基金supported by National Natural Science Foundation of China(Grant No.11301027)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20130003120008)+1 种基金the Beijing Natural Science Foundation(Grant No.1144013)the Fundamental Research Funds for the Central Universities(Grant No.2012CXQT09)
文摘A-manifolds and/3-manifolds, introduced by Gray （1978）, are two significant classes of Einstein-like Riemannian manifolds. A Riemannian manifold is Ricci parallel if and only if it is simultaneously an A-manifold and a B-manifold. The present paper proves that both focal submanifolds of each isoparametric hypersurface in unit spheres with g = 4 distinct principal curvatures are A-manifolds. As for the focal submanifolds with g = 6, m = 1 or 2, only one is an A-manifold, and neither is a B-manifold.
