This article delves Chern's conjecture for hypersurfaces with constant second fundamental form squared length S in the spherical space.At present,determining whether the third gap point of S is 2n remains unsolved...This article delves Chern's conjecture for hypersurfaces with constant second fundamental form squared length S in the spherical space.At present,determining whether the third gap point of S is 2n remains unsolved yet.First,we investigate the height functions and their properties of the position vector and normal vector in natural coordinate vectors,and then prove the existence of a Simons-type integral formula on the hypersurface that simultaneously includes the first,second,and third gap point terms of S.These results can provide new avenues of thought and methods for solving Chern's conjecture.展开更多
Petrophysicists and reservoir engineers utilise the capillary pressure and saturation-height function for calculating the water saturation of any reservoir,at a given height above the free water level.The results have...Petrophysicists and reservoir engineers utilise the capillary pressure and saturation-height function for calculating the water saturation of any reservoir,at a given height above the free water level.The results have a big impact on reserve estimation.The majority of capillary pressure studies are carried out in labs with core data.Cores,on the other hand,are usually altered from their original state before being delivered to laboratories.Moreover,the accuracy of discrete sets of core data in describing entire reservoir parameters,is still up for debate.Prediction of the capillary pressure curve in reservoir condition is an important subject that is challenging to perform.The use of nuclear magnetic resonance(NMR)logs for oil and gas exploration has grown in popularity over the last few decades.However,most of the time the utilization of the data is limited for evaluating porosity-permeability,distributions and computation of irreducible water saturation.After the advent of fluid substitution methods,NMR T_(2)distributions may now be used to synthesize core equivalent capillary pressure curves.Using fluid substitution workflow,our study introduces a better approach for obtaining capillary pressure curves from the NMR T_(2)distribution.The available core data has been used to calculate calibration parameters for better saturation height modelling.The workflow introduces a novel approach in resistivity independent saturation computation.In the exploratory wells of our study area,computed water saturation derived from the saturation height function exhibits encouraging agreement with resistivity based water saturation from multi-mineral model.The NMR based saturation height modelling approach has been included in study area for the first time so far.展开更多
We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the ...We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus, the upper bounds of height, and the degree of the field of definition.展开更多
文摘This article delves Chern's conjecture for hypersurfaces with constant second fundamental form squared length S in the spherical space.At present,determining whether the third gap point of S is 2n remains unsolved yet.First,we investigate the height functions and their properties of the position vector and normal vector in natural coordinate vectors,and then prove the existence of a Simons-type integral formula on the hypersurface that simultaneously includes the first,second,and third gap point terms of S.These results can provide new avenues of thought and methods for solving Chern's conjecture.
基金The authors gratefully appreciate the support of Oil and Natural Gas Corporation,for providing data and permission to carry out the work under the CoEOGE project:RD/0120-PSUCE19-001.
文摘Petrophysicists and reservoir engineers utilise the capillary pressure and saturation-height function for calculating the water saturation of any reservoir,at a given height above the free water level.The results have a big impact on reserve estimation.The majority of capillary pressure studies are carried out in labs with core data.Cores,on the other hand,are usually altered from their original state before being delivered to laboratories.Moreover,the accuracy of discrete sets of core data in describing entire reservoir parameters,is still up for debate.Prediction of the capillary pressure curve in reservoir condition is an important subject that is challenging to perform.The use of nuclear magnetic resonance(NMR)logs for oil and gas exploration has grown in popularity over the last few decades.However,most of the time the utilization of the data is limited for evaluating porosity-permeability,distributions and computation of irreducible water saturation.After the advent of fluid substitution methods,NMR T_(2)distributions may now be used to synthesize core equivalent capillary pressure curves.Using fluid substitution workflow,our study introduces a better approach for obtaining capillary pressure curves from the NMR T_(2)distribution.The available core data has been used to calculate calibration parameters for better saturation height modelling.The workflow introduces a novel approach in resistivity independent saturation computation.In the exploratory wells of our study area,computed water saturation derived from the saturation height function exhibits encouraging agreement with resistivity based water saturation from multi-mineral model.The NMR based saturation height modelling approach has been included in study area for the first time so far.
文摘We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus, the upper bounds of height, and the degree of the field of definition.
