Mathematical Model of Magnetometric Resistivity. Sounding for a Conductive Host. with a Bulge Overburden

Transcription

1 Applied Mathematical Sciences, Vol. 7, 13, no. 7, Mathematical Model of Magnetometic Resistivity Sounding fo a Conductive Host with a Bulge Ovebuden Teeasak Chaladgan Depatment of Mathematics Faculty of Science Silpakon Univesity Nakhon Pathom 73, Thailand Suabsagun Yooyuanyong Depatment of Mathematics Faculty of Science Silpakon Univesity Nakhon Pathom 73, Thailand Coesponding autho: suabkul@su.ac.th Abstact In this pape, the solution of magnetic field esponse fom DC souce located on a two layeed unifom conductive host medium is fomulated. Fo the fist laye, the conductivity of ovebuden is a function of depth( z ) and denoted by bz ( l) 1( z) = e, z h, whee b is constant, l is positive which is used to locate the peak of the bulge, h is the thickness of ovebuden and is positive constant. The second laye, the conductivity of host medium, z > h, is constant and

2 336 Teeasak Chaladgan and Suabsagun Yooyuanyong is given by ( z) =. The Hankel tansfoms ae intoduced to solve the magnetic fields which ae expessed in the fom of integal expessions. In ode to detemine the magnetic fields, numeical solutions ae computed to show the behavio of the field while some paametes ae given appoximately. The gaphs of computation esults pefom the same elationship to the conductivity pofile of the gound. Mathematics Subject Classification: 86A5 Keywods: Hankel tansfoms, Magnetometic esistivity, Bulge 1. Intoduction The magnetometic esistivity method has ecently become an additional electical pospecting technique used fo finding undegound esouces. This technique is based on the measuement of low-level, low-fequency magnetic fields associated with non-inductive cuent flow fom the electodes in the gound. Chen and Oldenbug [4] deived the magnetic field diectly fom solving a bounday value poblems which was simila to the appoach used by Edwad [5] and then discussed in a homogeneous and a -layeed eath model. Yooyuanyong and Sipanya [1] deived and pefomed the solutions of the steady state magnetic field due to a DC cuent souce in thee types of heteogeneous eath models. These solutions ae citical to intepet the magnetometic esistivity (MMR) data. In this pape, a -layeed conductive eath model is consideed simila to Chen and Oldenbug [4], but it is diffeent in the conductivity pofile. Fo the fist bz ( l) laye, the conductivity of ovebuden is denoted by 1( z) = e, z h, whee b is constant, l is positive which is used to locate the peak of the bulge, h is the thickness of ovebuden and is positive constant. The second laye, the conductivity of host medium, z > h, is constant and is given by ( z) =. The objective of this pape is to show the behavio of the field while some paametes ae given appoximately.

3 Mathematical model of magnetometic esistivity sounding 337. Fomulation of the Poblem The geneal steady state Maxwell s equation in the fequency domain [4] can be used to detemine the magnetic field fo this poblem, namely uv v E = and uuv uv H = E, () whee uv E is the vecto electic field, uuv H is the vecto magnetic field, is the conductivity of the medium in Siemens pe mete ( S / m ) which is assumed to be a function of z only and is the del opeato. Eliminating uv E fom equations (1) and (), we obtain (1) 1 uuv v H =. (3) This can be expessed in cylindical coodinates (, φ, z) as H 1 H Hz ( Hφ ) e φ φ z z 1 1 H H z φ H + ( Hφ ) e z φ z φ 1 H Hz H H z φ + e = z z φ φ z φ (4) whee H, H φ and H z ae the components of uuv H in e, e φ and e z diections, espectively. Since the poblem is axisymmetic and uuv H has only the azimuthal component in cylindical coodinates, fo simplicity, we use H to epesent the azimuthal component H φ in the following deivations. Simplifying equation (4) yields 1 H 1 H ( H ) ( H ) ( H ). z z + + z + =

4 338 Teeasak Chaladgan and Suabsagun Yooyuanyong In ou study, we denote as a function of only depth z, and we now have H 1 H H 1 H 1 + H. z z + + = z (5) The Hankel tansfom [1] is intoduced and defined by and % ( λ, ) (, ) ( λ ) (6) H z = H z J1 d (, ) λ % ( λ, ) ( λ ) λ, (7) H z = H zj1 d whee J 1 is the Bessel function of the fist kind of ode one and λ is the Hankel vaiable. Taking the tansfomation on both sides of equation (5), we obtain H% 1 H % + λ H =. z z z % (8) Since the electode is in the ovebuden which the end of the electode is positioned at z = h, so that a magnetic field will be sepaated into two pats. The magnetic field is come fom the gound laye that can be descibed by the geneal solution of equation (8) and the magnetic field aising fom pobe souces H, which is only one element. It can be explained by the Ampee s law [7, 11], as H (, z) 1 = π, whee I is the cuent at the pobe on the gound suface. Fom equation (6), we have 1 1 H% (, ) λ z = J1( λ) d =. π πλ Theefoe, the magnetic field in each laye can be obtained by taking the invese

5 Mathematical model of magnetometic esistivity sounding 339 Hankel tansfom to the solution of equation (8), which satisfies the following bounday conditions[, 8]: 1. The vetical component of the cuent density must be zeo at the gound suface ( z =, ) z ( z) E ( z) 1 1,, z= = (9) z whee E 1 is the vetical component of the electic field in ovebuden.. The azimuthal component of the magnetic field needs to be continuous on each of the bounday planes in the eath, ( λ ) = ( λ ) lim H%, z lim H%, z, (1) 1 + z h z h whee H1 and H ae magnetic fields in the fist and second laye, espectively. 3. The adial component of the electic field, denoted by E, needs to be continuous on each of the bounday planes in the eath, lim % ( λ, ) = lim % E z E ( λ, z ), (11) 1 + z h z h whee E1 and E ae the adial component of electic fields in the fist and second laye, espectively. To detemine the adial and vetical components of the electic field elated to the azimuthal component of the magnetic field, we expand equation () and obtain 1 H 1 H z φ 1 H 1 Hz 1 1 H E = e + eφ ( Hφ) ez. φ z z + φ Since the poblem is axisymmetic and uuv H has only the azimuthal component in cylindical coodinates, fo simplicity as equation (4), we use H to epesent the azimuthal component H φ in the above deivations. This yield E 1 H =, z z E 1 = ( H). By using equation (6), we have

6 34 Teeasak Chaladgan and Suabsagun Yooyuanyong 1 1 E ( λ %, z) H(, z) J1 ( ) d H(, z). ( z) z λ = = % ( z) z λ 3. A Geometic -Layeed Eath Model In ou geometic model, a two-layeed eath model is consideed which the inteface between the layes is a plane paallel to the gound suface. A point souce of diect cuent I is located into the ovebuden which the end of the electode is positioned at z = h. Fo the fist laye, the conductivity of ovebuden bz ( l) is denoted by 1( z) = e, z h, whee b is constant, l is positive and it is used to locate the peak of the bulge, h is the thickness of ovebuden and is positive constant. The second laye, the conductivity of host medium, z > h, is constant and is given by ( z) =. Electode Gound suface z = e ( ) 1 ( z) = bz ( l) Z z = z = h Figue 1: A geometic -layeed conductive eath model. 4. Solution of Magnetic Field fo a -Layeed Eath Model An ovebuden has a vaiation of conductivity 1 ( z) with thickness h ove a conductive host medium having constant conductivity ( z). Hence, the equation fo the magnetic field in ovebuden and conductive host medium can be simplified by substituting 1 ( z) and ( z) into equation (8), thus, we obtain

7 Mathematical model of magnetometic esistivity sounding 341 and H% 1 H% 1 + bz ( l) λ H% 1 =, (1) z z H% z λ H% =. (13) The powe seies method and auxiliay equation ae used to find the magnetic field fomulation in an ovebuden, denoted by H % 1, and conductive host medium, denoted by H %, espectively. Theefoe, the solutions of the equation (1) and (13) ae witten by [6, 9] and ( λ ) I λ z blz 1 λ = H% (, z) 1 z A 1 A z, (14) πλ % ( z h) ( z h) H ( λ, z) = A e λ + A e λ, 3 4 espectively, whee A 1, A, A 3 and A 4 ae abitay constants, which can be detemined by using the bounday conditions. The condition at z, the magnetic field tends to zeo, that leads H % to % ( z h) = 3 H ( λ, z) A e λ. (15) Fo the fist bounday condition in equation (9), we obtain 1 1( z) ( H1(, z) ) =. 1( z) Since 1 ( z ) and ae not zeo, then o ( H z ) 1 (, ) =, H (, z) = H (, z) d = () d =. ( ) 1 1

8 34 Teeasak Chaladgan and Suabsagun Yooyuanyong Since >, thus H 1 (, z ) =, and H% 1( λ, z) = H1(, z) J1( λ) d. Hence, ( λ ) λ 1 1 I z blz 1+ z + A A z+ = () J ( λ) d =. πλ Since no electic cuent acoss at the ai-eath inteface, I =, then λ z blz A A z+ =. z = Thus, A 1 =. By the second bounday condition, we obtain Thus, lim ( 1 λ ) lim. πλ I blz ( z h) z A z A3e λ = z h z h + ( λ ) I blh A3 = 1 + h + A h +. (16) πλ Applying the thid bounday condition, we have o 1 1 lim H% ( λ, z) = lim H% ( λ, z), ( ) ( ) 1 + z h 1 z z z h z z Hence, 1 I ( blz 1 λ lim 1 ) ( ( z h z A ) + λ + z+ = lim A3e ). bz ( l) + z πλ z z h z h e 1 I h A = λ + A 1 + blh. ( ) bh ( l) λe π 3 (17)

9 Mathematical model of magnetometic esistivity sounding 343 Fom equation (16) and (17), we obtain Theefoe, l 1 λ ( + λ h ) + A h+ = + A ( + blh) I b h I h 1 1. πλ bh ( l) λe π A ( h ) I 1+ λ α1 + λh = h + blh + 1+ blh π λα1 λα1 ( ), (18) whee α bh ( l) 1 = e. 1 Iλh Since A = + A ( + blh) 3 λα1 π 1, then with the use of equation (18), we obtain A (( 1+ λ h ) α1 + λh)( 1+ blh) I = λh. h + bl h + ( 1+ blh ) 3 πλα1 λ α1 λ α1 (19) Hence, with the use of invese Hankel tansfoms, the magnetic field in ovebuden and conductive host medium ae shown, espectively, as ( h ) I Iλ 1+ λ α + λh blz 1 = ( + ) λ λ λ () π π hλα + blh λα + ( 1+ blh) H (, z) 1 z z J( d ), and (( 1+ λ h ) α1 + λh)( 1+ blh) I H( λ, z) = λh e J1( λ) dλ λ( z h) πα λhα + blλh α + ( 1+ blh (1) )

10 344 Teeasak Chaladgan and Suabsagun Yooyuanyong 5. Numeical Expeiments In ou numeical expeiments, the magnetic field due to a diect cuent souce on the gound suface of the model is calculated. Chave s algoithm [3] is used fo numeical calculating the invese Hankel tansfom of the magnetic field solutions. The cuent of 1-Ampee is injected to the gound by the pobe length of 1 and 3 metes pependicula to the gound suface, = S / m, b =.5. The esults of magnetic field esponse ae pefomed as the gaphs in Figue, 3, 4 and 5. The gaphs ae shown the behavio of the magnetic field against souce-eceive spacing () while the values of h, l and z ae adjusted. As we fix the value of h, the magnetic field cuves ae quite diffeent as z and l ae vaied between h l and h l. The magnetic field intensities dop vey fast when the value of z close to the value of h. At lage depth ( z ), the magnitude of magnetic fields tends to be small values as we expect. As the thickness of ovebuden is inceased, the shape of gaph is simila to the conductivity pofile of the gound. This is the advantage of magnetic field that can be pefomed some elationship to the conductivity pofile of the gound. Figue : The behavio of magnetic field against at diffeent depth z =.1,.,...,.7,.8 m. whee h= 1 m. and l =.5 m.

11 Mathematical model of magnetometic esistivity sounding 345 Figue 3: The behavio of magnetic field against at diffeent depth z =.,.4,...,1.8, m. whee h= 3 m. and l = 1.5 m. Figue 4: The behavio of magnetic field against at diffeent depth z =1.1,1.,...,1.7,1.8 m. whee h= 1 m. and l = 1.5 m.

12 346 Teeasak Chaladgan and Suabsagun Yooyuanyong Figue 5: The behavio of magnetic field against at diffeent depth z = 3.,3.4,...,4.8,5 m. whee h= 3 m. and l = 4.5 m. 6. Conclusions In this pape, the magnetic fields of an eath having the electical conductivity bz ( l) ( z) = e fo the depth z h, and ( z) = fo the depth 1 z > hae consideed. The integal expessions ae deived and computed the values of the magnetic field which is used to detemine the behavio of the magnetic field against souce-eceive spacing. The cuves of the magnetic field against souce-eceive spacing ae plotted and shown the advantage in the gound exploation. Refeences [1] I. Ali and S. Kalla, A genealized Hankel tansfom and its use fo solving cetain patial diffeential equation, J. Austal. Math. Soc. Se.B, 41, (1999),

13 Mathematical model of magnetometic esistivity sounding 347 [] B. Banejee, B.J. Sengupta and B.P. Pal, Appaent esistivity of a multilayeed eath with a laye having exponential conductivity, Geophysical Pospecting, 8 (198), [3] A.D. Chave, Numeical integation of elated Hankel tansfoms by quadatue and continued faction expansion, Geophysics, 48 (1983), [4] J. Chen and D.W. Oldenbug, Magnetic and electical fields of diect cuents in layeed eath, Expl. Geophys., 77 (4), [5] R.N. Edwods, A downhole MMR technique fo electical sounding beneath a conductive suface laye, Geophysics, 53 (1988), [6] A. Jeffey, Handbook of Mathematical Fomulas and Integals, Academic Pess, (1995). [7] W. Sipunya, Magnetic Field of Diect Cuent in Heteogeneous Media, Maste of Science Thesis. Silpakon Univesity, (5). [8] C.H. Stoye and J.R. Wait, Resistivity pobing of an exponential eath with a homogeneous ovebuden, Geoexploation, 15 (1977), [9] P. Yehuda and R. Jacob, An Intoduction to Patial Diffeential Equations, Cambige Univesity Pess, (5). [1] S. Yooyuanyang and W. Sipanya, Mathematical Modelling of Magnetometic Resistivity Sounding Eath Stuctues, Thai J. Math, 3() (5), [11] S. Yooyuanyang and W. Sipanya, Magnetic field of diect cuent in heteogeneous gound, Songklanakain J. Sci. Technol, 9() (7),

14 348 Teeasak Chaladgan and Suabsagun Yooyuanyong Received: Octobe, 1