The Sichuan Basin is one of the vital basins in China,boasting abundant hydrocarbon reservoirs.To clarify the intensity of the tectonic stress field of different tectonic episodes since the Mesozoic and to identify th...The Sichuan Basin is one of the vital basins in China,boasting abundant hydrocarbon reservoirs.To clarify the intensity of the tectonic stress field of different tectonic episodes since the Mesozoic and to identify the regional dynamic background of different tectonic movements in the Sichuan Basin and its adjacent areas,the characteristics of the acoustic emission in rocks in different strata of these areas were researched in this paper.Meanwhile,the tectonic stress magnitude in these areas since the Mesozoic was restored.The laws state that the tectonic stress varied with depth was revealed,followed by the discussion of the influence of structural stress intensity on structural patterns in different tectonic episodes.These were conducted based on the paleostress measurement by acoustic emission method and the inversion principle of the stress fields in ancient periods and the present,as well as previous research achievements.The results of this paper demonstrate that the third episode of Yanshanian Movement(Yanshanian III)had the maximum activity intensity and tremendously influenced the structural pattern in the study area.The maximum horizontal principal stress of Yanshanian III varied with depth as follows:0.0168 x+37.001(MPa),R^2=0.8891.The regional structural fractures were mainly formed in Yanshanian III in Xujiahe Formation,west Sichuan Basin,of which the maximum paleoprincipal stress ranging from 85.1 MPa to 120.1 MPa.In addition,the law stating the present maximum horizontal principal stress varies with depth was determined to be 0.0159 x+10.221(MPa),R^2=0.7868 in Wuling Mountain area.Meanwhile,it was determined to be 0.0221 x+9.4733(MPa),R^2=0.9121 in the western part of Xuefeng Mountain area and 0.0174 x+10.247(MPa),R^2=0.8064 in the whole study area.These research results will not only provide data for the simulation of stress field,the evaluation of deformation degree,and the prediction of structural fractures,but also offer absolute geological scientific bases for the elevation of favorable shale gas preservation.展开更多
We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation in...We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate.And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant,we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well.Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function.The linearly dependent relation between two eigenfunctions is also studied.展开更多
We first convert the angular Teukolsky equation under the special condition ofτ≠0,s≠0,m=0 into a confluent Heun differential equation(CHDE)by taking different function transformation and variable substitution.And t...We first convert the angular Teukolsky equation under the special condition ofτ≠0,s≠0,m=0 into a confluent Heun differential equation(CHDE)by taking different function transformation and variable substitution.And then according to the characteristics of both CHDE and its analytical solution expressed by a confluent Heun function(CHF),we find two linearly dependent solutions corresponding to the same eigenstate,from which we obtain a precise energy spectrum equation by constructing a Wronskian determinant.After that,we are able to localize the positions of the eigenvalues on the real axis or on the complex plane whenτis a real number,a pure imaginary number,and a complex number,respectively and we notice that the relation between the quantum number l and the spin weight quantum number s satisfies the relation l=∣s∣+n,n=0,1,2….The exact eigenvalues and the corresponding normalized eigenfunctions given by the CHF are obtained with the aid of Maple.The features of the angular probability distribution(APD)and the linearly dependent characteristics of two eigenfunctions corresponding to the same eigenstate are discussed.We find that for a real numberτ,the eigenvalue is a real number and the eigenfunction is a real function,and the eigenfunction system is an orthogonal complete system,and the APD is asymmetric in the northern and southern hemispheres.For a pure imaginary numberτ,the eigenvalue is still a real number and the eigenfunction is a complex function,but the APD is symmetric in the northern and southern hemispheres.Whenτis a complex number,the eigenvalue is a complex number,the eigenfunction is still a complex function,and the APD in the northern and southern hemispheres is also asymmetric.Finally,an approximate expression of complex eigenvalues is obtained when n is greater than∣s∣.展开更多
基金The study associated with this paper was supported by projects of China Geological Survey(DD20190085,DD20160183,1212011120976).
文摘The Sichuan Basin is one of the vital basins in China,boasting abundant hydrocarbon reservoirs.To clarify the intensity of the tectonic stress field of different tectonic episodes since the Mesozoic and to identify the regional dynamic background of different tectonic movements in the Sichuan Basin and its adjacent areas,the characteristics of the acoustic emission in rocks in different strata of these areas were researched in this paper.Meanwhile,the tectonic stress magnitude in these areas since the Mesozoic was restored.The laws state that the tectonic stress varied with depth was revealed,followed by the discussion of the influence of structural stress intensity on structural patterns in different tectonic episodes.These were conducted based on the paleostress measurement by acoustic emission method and the inversion principle of the stress fields in ancient periods and the present,as well as previous research achievements.The results of this paper demonstrate that the third episode of Yanshanian Movement(Yanshanian III)had the maximum activity intensity and tremendously influenced the structural pattern in the study area.The maximum horizontal principal stress of Yanshanian III varied with depth as follows:0.0168 x+37.001(MPa),R^2=0.8891.The regional structural fractures were mainly formed in Yanshanian III in Xujiahe Formation,west Sichuan Basin,of which the maximum paleoprincipal stress ranging from 85.1 MPa to 120.1 MPa.In addition,the law stating the present maximum horizontal principal stress varies with depth was determined to be 0.0159 x+10.221(MPa),R^2=0.7868 in Wuling Mountain area.Meanwhile,it was determined to be 0.0221 x+9.4733(MPa),R^2=0.9121 in the western part of Xuefeng Mountain area and 0.0174 x+10.247(MPa),R^2=0.8064 in the whole study area.These research results will not only provide data for the simulation of stress field,the evaluation of deformation degree,and the prediction of structural fractures,but also offer absolute geological scientific bases for the elevation of favorable shale gas preservation.
基金Project supported by the National Natural Science Foundation of China(Grant No.11975196)partially by SIP,Instituto Politecnico Nacional(IPN),Mexico(Grant No.20210414)。
文摘We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate.And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant,we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well.Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function.The linearly dependent relation between two eigenfunctions is also studied.
基金supported by the National Natural Science Foundation of China(Grant No.11975196)partially by 20220355-SIP,IPN。
文摘We first convert the angular Teukolsky equation under the special condition ofτ≠0,s≠0,m=0 into a confluent Heun differential equation(CHDE)by taking different function transformation and variable substitution.And then according to the characteristics of both CHDE and its analytical solution expressed by a confluent Heun function(CHF),we find two linearly dependent solutions corresponding to the same eigenstate,from which we obtain a precise energy spectrum equation by constructing a Wronskian determinant.After that,we are able to localize the positions of the eigenvalues on the real axis or on the complex plane whenτis a real number,a pure imaginary number,and a complex number,respectively and we notice that the relation between the quantum number l and the spin weight quantum number s satisfies the relation l=∣s∣+n,n=0,1,2….The exact eigenvalues and the corresponding normalized eigenfunctions given by the CHF are obtained with the aid of Maple.The features of the angular probability distribution(APD)and the linearly dependent characteristics of two eigenfunctions corresponding to the same eigenstate are discussed.We find that for a real numberτ,the eigenvalue is a real number and the eigenfunction is a real function,and the eigenfunction system is an orthogonal complete system,and the APD is asymmetric in the northern and southern hemispheres.For a pure imaginary numberτ,the eigenvalue is still a real number and the eigenfunction is a complex function,but the APD is symmetric in the northern and southern hemispheres.Whenτis a complex number,the eigenvalue is a complex number,the eigenfunction is still a complex function,and the APD in the northern and southern hemispheres is also asymmetric.Finally,an approximate expression of complex eigenvalues is obtained when n is greater than∣s∣.