回线源瞬变电磁法有限体积三维任意各向异性正演及分析
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摘要
目前,瞬变电磁法(TEM)数据基本都是基于各向同性模型进行反演解释,这对于存在明显电性各向异性的勘探区域会产生较大的反演解释误差.为分析电各向异性对回线源瞬变电磁信号的影响方式与程度,本文通过求解离散化的全张量电导率时间域Helmholtz方程,实现了基于有限体积法的TEM任意各向异性的三维正演算法.该算法采用基于交错网格的拟态有限体积法(MFV)对时域Maxwell方程组进行空间域离散,并利用后退欧拉算法(Backward Euler Method)进行时间域离散.为提高时域电磁场的求解精度与效率,该算法将时间分段等步长算法与方程直接求解法相结合.通过对一维各向异性模型以及三维复杂各向同性模型进行测试,验证了本算法对于回线源瞬变电磁响应计算的正确性及有效性.最后,通过对几类典型电各向异性介质中大回线源瞬变电磁信号响应的分析,总结了不同电各向异性类型对TEM电磁信号的影响模式,结果表明,主轴各向异性情况下TEM信号主要受水平方向电导率的影响,倾斜各向异性对TEM信号的影响程度远大于水平各向异性,而通过水平各向异性信号能较清晰判断出各向异性主轴方向.
Electrical anisotropy of strata has been long recognized by field and laboratory observations.However,nearly all of interpretations of transient electromagnetic(TEM)data is based on the assumption of electric isotropy of media,which can cause misleading data interpretation in regions with strong electrical anisotropy.To clarify the influence of electricalanisotropy on loop-source TEM responses,we present a three dimensional(3 D)robust finitevolume(FV)algorithm for simulating TEM responses in an arbitrarily anisotropic medium through solving Helmholtz equations in the time domain.The time domain Maxwell equations are discretized using the mimetic finite-volume method(MFV)on a conventional staggered grid in the space domain and discretized in the time domain using the backward Euler method.To reduce time steps required by computation,the modeling time is first divided into several intervals,each of which has a constant time step size.Then,the resulting system matrix in each interval is factored and solved by a direct solver,which allows the factored matrix to be used to calculate the TEM responses for subsequent time steps in the same time interval.The accuracy of our algorithm is validated against quasi-analytic solutions of a 1-D layered anisotropic model.Finally,by numerical experiments for 3 Dmodels with different types of electrical anisotropy,we analyze the influences of electrical anisotropy on TEM responses.The results demonstrate that TEM responses are mainly affected by the horizontal conductivity.The effect of dipping anisotropy on TEM responses is much greater than horizontal anisotropy.Besides,the horizontal principleaxis direction of electrical anisotropy could be inferred from the TEM signal.
Electrical anisotropy of strata has been long recognized by field and laboratory observations.However,nearly all of interpretations of transient electromagnetic(TEM)data is based on the assumption of electric isotropy of media,which can cause misleading data interpretation in regions with strong electrical anisotropy.To clarify the influence of electricalanisotropy on loop-source TEM responses,we present a three dimensional(3 D)robust finitevolume(FV)algorithm for simulating TEM responses in an arbitrarily anisotropic medium through solving Helmholtz equations in the time domain.The time domain Maxwell equations are discretized using the mimetic finite-volume method(MFV)on a conventional staggered grid in the space domain and discretized in the time domain using the backward Euler method.To reduce time steps required by computation,the modeling time is first divided into several intervals,each of which has a constant time step size.Then,the resulting system matrix in each interval is factored and solved by a direct solver,which allows the factored matrix to be used to calculate the TEM responses for subsequent time steps in the same time interval.The accuracy of our algorithm is validated against quasi-analytic solutions of a 1-D layered anisotropic model.Finally,by numerical experiments for 3 Dmodels with different types of electrical anisotropy,we analyze the influences of electrical anisotropy on TEM responses.The results demonstrate that TEM responses are mainly affected by the horizontal conductivity.The effect of dipping anisotropy on TEM responses is much greater than horizontal anisotropy.Besides,the horizontal principleaxis direction of electrical anisotropy could be inferred from the TEM signal.
引文
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Druskin V,Knizhnerman L.1994.Spectral approach to solving three-dimensional Maxwell′s diffusion equations in the time and frequency domains.Radio Science,29(4):937-953.
Fu H H,Wang Y Q,Um E S,et al.2015.A parallel finite-element time-domain method for transient electromagnetic simulation.Geophysics,80(4):E213-E224.
Hünziker J,Thorbecke J,Slob E.2015.The electromagnetic response in a layered vertical transverse isotropic medium:A new look at an old problem.Geophysics,80(1):F1-F18.
Haber E,Ascher U M,Oldenburg D W.2004.Inversion of 3Delectromagnetic data in frequency and time domain using an inexact all-at-once approach.Geophysics,69(5):1216-1228.
Haber E,Oldenburg D W,Shekhtman R.2007.Inversion of time domain three-dimensional electromagnetic data.Geophysical Journal International,171(2):550-564.
Haber E,Ruthotto L.2014.A multiscale finite volume method for Maxwell′s equations at low frequencies.Geophysical Journal International,199(2):1268-1277.
Han B,Li Y G,Li G.2018.3Dforward modeling of magnetotelluric fields in general anisotropic media and its numerical implementation in Julia.Geophysics,83(4):F29-F40.
Haroon A,Adrian J,Bergers R,et al.2015.Joint inversion of long-offset and central-loop transient electromagnetic data:Application to a mud volcano exploration in Perekishkul,Azerbaijan.Geophysical Prospecting,63(2):478-494.
Hyman J M,Shashkov M.1999.Mimetic discretizations for Maxwell′s equations.Journal of Computational Physics,151(2):881-909.
Key K.2012.Is the fast Hankel transform faster than quadrature?.Geophysics,77(3):F21-F30.
Li H,Xue G Q,Zhong H S,et al.2016.Joint inversion of CMPgathers of multi-channel transient electromagnetic data.Chinese Journal of Geophysics(in Chinese),59(12):4439-4447,doi:10.6038/cjg20161206.
Li J H,Zhu Z Q,Liu S C,et al.2011.3Dnumerical simulation for the transient electromagnetic field excited by the central loop based on the vector finite-element method.Journal of Geophysics and Engineering,8(4):560-567.
Li J H,Hu X Y,Zeng S H,et al.2013.Three-dimensional forward calculation for loop source transient electromagnetic method based on electric field Helmholtz equation.Chinese Journal of Geophysics(in Chinese),56(12):4256-4267,doi:10.6038/cjg20131228.
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Li J H,Farquharson C G,Hu X Y,et al.2016b.A vector finite element solver of three-dimensional modelling for a long grounded wire source based on total electric field.Chinese Journal of Geophysics(in Chinese),59(4):1521-1534,doi:10.6038/cjg20160432.
Li J H,Farquharson C G,Hu X Y.2016a.3Dvector finite-element electromagnetic forward modeling for large loop sources using a total-field algorithm and unstructured tetrahedral grids.Geophysics,82(1):E1-E16.
Li J H,Farquharson C G,Hu X Y.2016b.Three effective inverse Laplace transform algorithms for computing time-domain electromagnetic responses.Geophysics,81(2):E113-E128.
Li J H,Lu X S,Farquharson C G,et al.2018.A finite-element time-domain forward solver for electromagnetic methods with complex-shaped loop sources.Geophysics,83(3):E117-E132.
Li Y G,Spitzer K.2005.Finite element resistivity modelling for three-dimensional structures with arbitrary anisotropy.Physics of the Earth and Planetary Interiors,150(1-3):15-27.
Li Z H,Huang Q H.2014.Application of the complex frequency shifted perfectly matched layer absorbing boundary conditions in transient electromagnetic method modeling.Chinese Journal of Geophysics(in Chinese),57(4):1292-1299,doi:10.6038/cjg20140426.
Liu Y H,Yin C C,Cai J,et al.2018.Review on research of electrical anisotropy in electromagnetic prospecting.Chinese Journal of Geophysics(in Chinese),61(8):3468-3487,doi:doi:10.6038/cjg2018L0004.
Newman G A,Hohmann G W,Anderson W L.1986.Transient electromagnetic response of a three-dimensional body in a layered earth.Geophysics,51(8):1608-1627.
Oldenburg D W,Haber E,Shekhtman R.2013.Three dimensional inversion of multisource time domain electromagnetic data.Geophysics,78(1):E47-E57.
Pek J,Santos F A M.2006.Magnetotelluric inversion for anisotropic conductivities in layered media.Physics of the Earth and Planetary Interiors,158(2-4):139-158.
Peng R H,Hu X Y,Han B,et al.2016.3Dfrequency-domain CSEM forward modeling based on the mimetic finite-volume method.Chinese Journal of Geophysics(in Chinese),59(10):3927-3939,doi:10.6038/cjg20161036.
Qiu Z P,Li Z H,Li D Z,et al.2013.Non-orthogonal-Grid-based three dimensional modeling of transient electromagnetic field with topography.Chinese Journal of Geophysics(in Chinese),56(12):4245-4255,doi:10.6038/cjg20131227.
Sugeng F.1998.Modeling the 3D TDEM response using the 3Dfull-domain finite-element method based on the hexahedral edge-element technique.Exploration Geophysics,29(4):615-619.
Sun H F,Li X,Li S C,et al.2013.Three-dimensional FDTDmodeling of TEM excited by a loop source considering ramp time.Chinese Journal of Geophysics(in Chinese),56(3):1049-1064,doi:10.6038/cjg20130333.
Swidinsky A,Edwards R N.2009.The transient electromagnetic response of a resistive sheet:straightforward but not trivial.Geophysical Journal International,179(3):1488-1498.
Um E S,Harris J M,Alumbaugh D L.2010.3Dtime-domain simulation of electromagnetic diffusion phenomena:A finiteelement electric-field approach.Geophysics,75(4):F115-F126.
Um E S,Commer M,Newman G A,et al.2015.Finite element modelling of transient electromagnetic fields near steel-cased wells.Geophysical Journal International,202(2):901-913.
Wang T,Hohmann G W.1993.A finite-difference,time-domain solution for three-dimensional electromagnetic modeling.Geophysics,58(6):797-809.
Weiss C J,Newman G A.2002.Electromagnetic induction in a fully3-D anisotropic earth.Geophysics,67(4):1104-1114.
Xu Y C,Lin J,Li S Y,et al.2012.Calculation of full-waveform airborne electromagnetic response with three-dimension finitedifference solution in time-domain.Chinese Journal of Geophysics(in Chinese),55(6):2105-2114,doi:10.6038/j.issn.0001-5733.2012.06.032.
Xue G Q,Li X,Di Q Y.2008.Research progress in TEM forward modeling and inversion calculation.Progress in Geophysics(in Chinese),23(4):1165-1172.
Xue G Q,Chen W Y,Zhou N N,et al.2013.Short-offset TEMtechnique with a grounded wire source for deep sounding.Chinese Journal of Geophysics(in Chinese),56(1):255-261,doi:10.6038/cjg20130126.
Xue G Q,Wang H Y,Yan S,et al.2014.Time-domain Green function solution for transient electromagnetic field.Chinese Journal of Geophysics(in Chinese),57(2):671-678,doi:10.6038/cjg20140230.
Yee K S.1966.Numerical solution of initial boundary value problems involving maxwell′s equations in isotropic media.IEEE Transactions on Antennas and Propagation,14(3):302-307.
Yin C C,Qi Y F,Liu Y H.2016.3Dtime-domain airborne EMmodeling for an arbitrarily anisotropic earth.Journal of Applied Geophysics,131:163-178.
Yogeshwar P,Tezkan B.2017.Two-dimensional basement modeling of central loop transient electromagnetic data from the central Azraq basin area,Jordan.Journal of Applied Geophysics,136:198-210.
Yu L,Evans R L,Edwards R N.1997.Transient electromagnetic responses in seafloor with triaxial anisotropy.Geophysical Journal International,129(2):292-304.
Zhang Y,Wang H N,Tao H G,et al.2012.Finite volume algorithm to simulate 3D responses of multi-component induction tools in inhomogeneous anisotropic formation based on coupled scalarvector potentials.Chinese Journal of Geophysics(in Chinese),55(6):2141-2152,doi:10.6038/j.issn.0001-5733.2012.06.036.
Zhou J M,Liu W T,Li X,et al.2018.Research on the 3Dmimetic finite volume method for loop-source TEM response in biaxial anisotropic formation.Chinese Journal of Geophysics(in Chinese),61(1):368-378,doi:10.6038/cjg2018K0598.
李海,薛国强,钟华森等.2016.多道瞬变电磁法共中心点道集数据联合反演.地球物理学报,59(12):4439-4447,doi:10.6038/cjg20161206.
李建慧,胡祥云,曾思红等.2013.基于电场Helmholtz方程的回线源瞬变电磁法三维正演.地球物理学报,56(12):4256-4267,doi:10.6038/cjg20131228.
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李建慧,Farquharson C G,胡祥云等.2016b.基于电场总场矢量有限元法的接地长导线源三维正演.地球物理学报,59(4):1521-1534,doi:10.6038/cjg20160432.
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薛国强,王贺元,闫述等.2014.瞬变电磁场时域格林函数解.地球物理学报,57(2):671-678,doi:10.6038/cjg20140230.
张烨,汪宏年,陶宏根等.2012.基于耦合标势与矢势的有限体积法模拟非均匀各向异性地层中多分量感应测井三维响应.地球物理学报,55(6):2141-2152,doi:10.6038/j.issn.0001-5733.2012.06.036.
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Amestoy P R,Guermouche A,L′Excellent J Y,et al.2006.Hybrid scheduling for the parallel solution of linear systems.Parallel Computing,32(2):136-156.
Commer M,Newman G.2004.A parallel finite-difference approach for 3Dtransient electromagnetic modeling with galvanic sources.Geophysics,69(5):1192-1202.
Dennis Z R,Cull J P.2012.Transient electromagnetic surveys for the measurement of near-surface electrical anisotropy.Journal of Applied Geophysics,76:64-73.
Druskin V,Knizhnerman L.1994.Spectral approach to solving three-dimensional Maxwell′s diffusion equations in the time and frequency domains.Radio Science,29(4):937-953.
Fu H H,Wang Y Q,Um E S,et al.2015.A parallel finite-element time-domain method for transient electromagnetic simulation.Geophysics,80(4):E213-E224.
Hünziker J,Thorbecke J,Slob E.2015.The electromagnetic response in a layered vertical transverse isotropic medium:A new look at an old problem.Geophysics,80(1):F1-F18.
Haber E,Ascher U M,Oldenburg D W.2004.Inversion of 3Delectromagnetic data in frequency and time domain using an inexact all-at-once approach.Geophysics,69(5):1216-1228.
Haber E,Oldenburg D W,Shekhtman R.2007.Inversion of time domain three-dimensional electromagnetic data.Geophysical Journal International,171(2):550-564.
Haber E,Ruthotto L.2014.A multiscale finite volume method for Maxwell′s equations at low frequencies.Geophysical Journal International,199(2):1268-1277.
Han B,Li Y G,Li G.2018.3Dforward modeling of magnetotelluric fields in general anisotropic media and its numerical implementation in Julia.Geophysics,83(4):F29-F40.
Haroon A,Adrian J,Bergers R,et al.2015.Joint inversion of long-offset and central-loop transient electromagnetic data:Application to a mud volcano exploration in Perekishkul,Azerbaijan.Geophysical Prospecting,63(2):478-494.
Hyman J M,Shashkov M.1999.Mimetic discretizations for Maxwell′s equations.Journal of Computational Physics,151(2):881-909.
Key K.2012.Is the fast Hankel transform faster than quadrature?.Geophysics,77(3):F21-F30.
Li H,Xue G Q,Zhong H S,et al.2016.Joint inversion of CMPgathers of multi-channel transient electromagnetic data.Chinese Journal of Geophysics(in Chinese),59(12):4439-4447,doi:10.6038/cjg20161206.
Li J H,Zhu Z Q,Liu S C,et al.2011.3Dnumerical simulation for the transient electromagnetic field excited by the central loop based on the vector finite-element method.Journal of Geophysics and Engineering,8(4):560-567.
Li J H,Hu X Y,Zeng S H,et al.2013.Three-dimensional forward calculation for loop source transient electromagnetic method based on electric field Helmholtz equation.Chinese Journal of Geophysics(in Chinese),56(12):4256-4267,doi:10.6038/cjg20131228.
Li J H,Cao X F,Ling C P,et al.2016a.Geoelectric models and their corresponding successful cases for transient electromagnetic prospecting.Progress in Geophysics(in Chinese),31(1):232-250,doi:10.6038/pg20160127.
Li J H,Farquharson C G,Hu X Y,et al.2016b.A vector finite element solver of three-dimensional modelling for a long grounded wire source based on total electric field.Chinese Journal of Geophysics(in Chinese),59(4):1521-1534,doi:10.6038/cjg20160432.
Li J H,Farquharson C G,Hu X Y.2016a.3Dvector finite-element electromagnetic forward modeling for large loop sources using a total-field algorithm and unstructured tetrahedral grids.Geophysics,82(1):E1-E16.
Li J H,Farquharson C G,Hu X Y.2016b.Three effective inverse Laplace transform algorithms for computing time-domain electromagnetic responses.Geophysics,81(2):E113-E128.
Li J H,Lu X S,Farquharson C G,et al.2018.A finite-element time-domain forward solver for electromagnetic methods with complex-shaped loop sources.Geophysics,83(3):E117-E132.
Li Y G,Spitzer K.2005.Finite element resistivity modelling for three-dimensional structures with arbitrary anisotropy.Physics of the Earth and Planetary Interiors,150(1-3):15-27.
Li Z H,Huang Q H.2014.Application of the complex frequency shifted perfectly matched layer absorbing boundary conditions in transient electromagnetic method modeling.Chinese Journal of Geophysics(in Chinese),57(4):1292-1299,doi:10.6038/cjg20140426.
Liu Y H,Yin C C,Cai J,et al.2018.Review on research of electrical anisotropy in electromagnetic prospecting.Chinese Journal of Geophysics(in Chinese),61(8):3468-3487,doi:doi:10.6038/cjg2018L0004.
Newman G A,Hohmann G W,Anderson W L.1986.Transient electromagnetic response of a three-dimensional body in a layered earth.Geophysics,51(8):1608-1627.
Oldenburg D W,Haber E,Shekhtman R.2013.Three dimensional inversion of multisource time domain electromagnetic data.Geophysics,78(1):E47-E57.
Pek J,Santos F A M.2006.Magnetotelluric inversion for anisotropic conductivities in layered media.Physics of the Earth and Planetary Interiors,158(2-4):139-158.
Peng R H,Hu X Y,Han B,et al.2016.3Dfrequency-domain CSEM forward modeling based on the mimetic finite-volume method.Chinese Journal of Geophysics(in Chinese),59(10):3927-3939,doi:10.6038/cjg20161036.
Qiu Z P,Li Z H,Li D Z,et al.2013.Non-orthogonal-Grid-based three dimensional modeling of transient electromagnetic field with topography.Chinese Journal of Geophysics(in Chinese),56(12):4245-4255,doi:10.6038/cjg20131227.
Sugeng F.1998.Modeling the 3D TDEM response using the 3Dfull-domain finite-element method based on the hexahedral edge-element technique.Exploration Geophysics,29(4):615-619.
Sun H F,Li X,Li S C,et al.2013.Three-dimensional FDTDmodeling of TEM excited by a loop source considering ramp time.Chinese Journal of Geophysics(in Chinese),56(3):1049-1064,doi:10.6038/cjg20130333.
Swidinsky A,Edwards R N.2009.The transient electromagnetic response of a resistive sheet:straightforward but not trivial.Geophysical Journal International,179(3):1488-1498.
Um E S,Harris J M,Alumbaugh D L.2010.3Dtime-domain simulation of electromagnetic diffusion phenomena:A finiteelement electric-field approach.Geophysics,75(4):F115-F126.
Um E S,Commer M,Newman G A,et al.2015.Finite element modelling of transient electromagnetic fields near steel-cased wells.Geophysical Journal International,202(2):901-913.
Wang T,Hohmann G W.1993.A finite-difference,time-domain solution for three-dimensional electromagnetic modeling.Geophysics,58(6):797-809.
Weiss C J,Newman G A.2002.Electromagnetic induction in a fully3-D anisotropic earth.Geophysics,67(4):1104-1114.
Xu Y C,Lin J,Li S Y,et al.2012.Calculation of full-waveform airborne electromagnetic response with three-dimension finitedifference solution in time-domain.Chinese Journal of Geophysics(in Chinese),55(6):2105-2114,doi:10.6038/j.issn.0001-5733.2012.06.032.
Xue G Q,Li X,Di Q Y.2008.Research progress in TEM forward modeling and inversion calculation.Progress in Geophysics(in Chinese),23(4):1165-1172.
Xue G Q,Chen W Y,Zhou N N,et al.2013.Short-offset TEMtechnique with a grounded wire source for deep sounding.Chinese Journal of Geophysics(in Chinese),56(1):255-261,doi:10.6038/cjg20130126.
Xue G Q,Wang H Y,Yan S,et al.2014.Time-domain Green function solution for transient electromagnetic field.Chinese Journal of Geophysics(in Chinese),57(2):671-678,doi:10.6038/cjg20140230.
Yee K S.1966.Numerical solution of initial boundary value problems involving maxwell′s equations in isotropic media.IEEE Transactions on Antennas and Propagation,14(3):302-307.
Yin C C,Qi Y F,Liu Y H.2016.3Dtime-domain airborne EMmodeling for an arbitrarily anisotropic earth.Journal of Applied Geophysics,131:163-178.
Yogeshwar P,Tezkan B.2017.Two-dimensional basement modeling of central loop transient electromagnetic data from the central Azraq basin area,Jordan.Journal of Applied Geophysics,136:198-210.
Yu L,Evans R L,Edwards R N.1997.Transient electromagnetic responses in seafloor with triaxial anisotropy.Geophysical Journal International,129(2):292-304.
Zhang Y,Wang H N,Tao H G,et al.2012.Finite volume algorithm to simulate 3D responses of multi-component induction tools in inhomogeneous anisotropic formation based on coupled scalarvector potentials.Chinese Journal of Geophysics(in Chinese),55(6):2141-2152,doi:10.6038/j.issn.0001-5733.2012.06.036.
Zhou J M,Liu W T,Li X,et al.2018.Research on the 3Dmimetic finite volume method for loop-source TEM response in biaxial anisotropic formation.Chinese Journal of Geophysics(in Chinese),61(1):368-378,doi:10.6038/cjg2018K0598.
李海,薛国强,钟华森等.2016.多道瞬变电磁法共中心点道集数据联合反演.地球物理学报,59(12):4439-4447,doi:10.6038/cjg20161206.
李建慧,胡祥云,曾思红等.2013.基于电场Helmholtz方程的回线源瞬变电磁法三维正演.地球物理学报,56(12):4256-4267,doi:10.6038/cjg20131228.
李建慧,曹晓峰,凌成鹏等.2016a.瞬变电磁法勘探的地电模型及其成功案例分析.地球物理学进展,31(1):232-250,doi:10.6038/pg20160127.
李建慧,Farquharson C G,胡祥云等.2016b.基于电场总场矢量有限元法的接地长导线源三维正演.地球物理学报,59(4):1521-1534,doi:10.6038/cjg20160432.
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