International Conference on Energy and Environmental Protection (ICEEP 2016)

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1 Inrnaional Confrnc on Enrgy and Environmnal Procion (ICEEP 6) Discussion abou diffrnial quaion of diffusion y in h Submarin Inducd Polarizaion Elcrical Proscing Wang Yuanshng,a Yan LinBo,b, LU Guiying,,c School of Mchanics and Elcrical Informaion, China Univrsiy of Go-Scinc, Wuhan Hubi, 474 School of Susainabl Enginring and h Buil Environmn, Arizona Sa Univrsiy, Tm, AZ , USA a wangyshr@sohu.com,b946465@qq.com, C75478@qq.com Kywords: Maxwll quaion; On-aramr ransformaion grou; mhod of variabl saraion; Hrmi quaion; Absrac: Hydrohrmal sulfid or in submarin minral rsourcs is a ho oic rcn ocan xloraion. From h fundamnal quaions of lcromagnic fild, his ar firs us forward h modl of με μ σ A a ; Thn, Analyical soluion of h simlifid nonlinar diffusion - ha conducion modl u u q w u is givn, which rovs ha hr is no On-aramr ransformaion grou maks h ψ (q w)ψ ψ() form rmain h sam. Finally, closd form analyical soluion of h linar diffusion - ha conducion modl u u qu is givn, and h roagaion ruls of h lcromagnic fild in h sa and h non frromagnic hydrohrmal sulfid or ar simly xlaind. Th horical sudy has imoran guiding significanc o rojc imlmnaion of frquncy domain inducd lcric diol drag sysm. Inroducion Org Hydrohrmal sulfid dosis in marin minral rsourcs ar abundan, and h hydrohrmal sulfid dosis can b dvlod in h background of h mid ocan ridg, h volcanic and h back arc basin. Rsarch found ha h sabd hydrohrmal sulfid or is mainly disribud in h low laiud Ocan nar h ridg axial Vally and crar, whr h dh gnral in hundrds of mrs o 7 mrs, h mraur of abou o 4 and a rssur of abou MPa-4MPa.In h ocan ridg sing, h dh of h hydrohrmal aciviy ar mor concnrad in abou mrs; In h back arc basin conic nvironmn, dh of h hydrohrmal aciviy focus a 8 mrs; h dh of island arc volcanic conic nvironmn of h hydrohrmal aciviy ar mor concnrad in lss han mrs undr h sa. Thrfor, h mrs in h ocan of oly mallic sulfid or is a ho so in h rcn ocan xloraion. Nauilus Canada and Nun Minrals and ohr comanis as on of h ionrs of hydrohrmal minral xloraion, mining is undrway of safloor hydrohrmal sulfid dosi xloraion, rscivly in in h corrsonding mining wr s mining []. Th ransin lcromagnic mhod and inducd olarizaion mhod ar sudid in h domsic. Transin lcromagnic mhod of horical rsarch shows ha drag and dro dcion, h rob is no mor han 5 mrs from h high of h or body []. Prliminary xrimnal rsuls show ha h inducd olarizaion of mal sulfids can b ffcivly found by IP mhod. In ordr o rduc h suly currn, h frquncy domain inducd olarizaion mhod is usd o dc h [,4]. Low frquncy lcromagnic wavs in sawar by diffusion ha conducion quaion [5,6]. Sawar lcromagnic aramrs including rmabiliy, conduciviy, dilcric consan, hy and sawar saliniy, mraur, dnsiy. Th sa war is a non frromagnic subsanc wih a mraur of 7 Safloor hydrohrmal sulfid Minral 6. Th auhors - Publishd by Alanis Prss 9

2 y, mal conn, mraur, and void dnsiy. In his ar, h influnc of im varying conduciviy on h marin lcric fild masurmn was sudid by using h singl aramr ransformaion grou (OPG) [7] mhod. Basic quaion Inducd olarizaion mhod is a mhod basd on h diffrnc of hysical and lcrochmical ffcs of rock and or in h fild of arificial lcric fild. Th Maxwll quaion of lcromagnic fild in im domain is as follows[8,9]: d h j () b ˆ () b () d (4) Th h, ê,b,d, j, ρin h formula ar magnic fild srngh, lcric fild srngh, magnic inducion innsiy, onial shif, currn dnsiy, charg dnsiy and is im cofficin. Th hysical rory quaion is: ˆ j σ (5) d ε ˆ (6) b μ h (7) Th σ, ε, μ in h form ar conduciviy, dilcric consan, rmabiliy. Bcaus h ha ransfr a roblm has h form of h soluion of T T, so s u: A, A, U, U,, A, a U / KT U / KT a A A A (8) ar consans, and K is h Klvin consan. d ˆ h h ˆ (9) h ˆ ˆ h ˆ ˆ ˆ ˆ h h h () () () () 5 Formula () () ar diffusion ha conducion diffrnial quaion. Whn Hz is μεω ˆ μσω, h conducion currn lays an imoran rol in ˆ, which is h diffusion quaion. Sady sa ˆ his is h Poisson quaion. Sinc h A a (A and a ar consans) is a nonlinar im-varying facor, h diffrnial quaion of diffusion ha conducion can b sudid by h mhod of Li grou [7].

3 Th OPG soluions of nonlinar quaions Tha Equaion (8) is subsiud ino quaion () can b obaind: ˆ A a ˆ ˆ (4) Du o symmry, h olarizaion fild only Z comonn, alicaion of xrssion, nglcing h highr-ordr rms, hr ar: n n n! sris A Aa Aa ˆ ˆ ˆ (5) zz By using h Smyon Quinn Lov onial, h onial quaion of h olarizaion fild is rwrin as follows: Th u ux, q wu u u (6), q A Aa, w Aa ar consans. using h mhod of saraion of variabls and s u u( x, ) ( ( ), i can b concludd ha: ( ) ( ( q w) ( (7) ( q w) C ( ( ) (8), C is a consan. C ( can b obaind by C(, and hn hav : g cos( g sin(, C (9) ( Thos, g ( q w) g ar consans. According o h formula, i can g ha : ( ) ( q w) ( ) () I is a nonlinar homognous quaion, using h following OPG: (, ) ( ) (, ) ( ) () I can h sam form of : q w ) ( ) ( - ( q w) - ( ) / F(,, ) () F F F F F F F () Pu h F w F F w q,, ino h formula, hr is: w q w q ( q w) w - ( ) - q w w q (4)

4 In ordr o g h quaion s: Th formula (7 ) can b obaind: : (5) : (6) : ( w q)/ (7 ) (7) Thn : / can rfr ha: (8) ( q w) w w q : (9) w q : () S g g4 g5 g6 w q w () w q () Thr is no OPG o mak h quaion ( q w) ( ) form unchangd. Similar o h y xy y quaion soluion Hrmi, o h quaion ( q w) ( ), sing q w ( ) i can rfr ha: Th quaion ( q w) ( ) q w q w q w ( ) () can b convrd o: d q w q w d (4) d d Thrfor: ( ) lnh n( ) n ln ( ) H n n! n n H n( ) n ( ) q w d n cos d (5) H n () is h Hrmi of h olynomial. Thus, h soluion of h in siu onial quaion (6) can b wrin as:

5 g cos( g sin( l H ( u u( x, ) ( ( ) n n ) (6) n Analyical soluion of linar diffusion y quaion In h inducd olarizaion mhod, h charg discharg frquncy of h arificial lcric fild is low (.g..6hz, 9Hz), h discharg im is long nough and h full discharg, hn h quaion (4) can b simlifid o h quaion (7). ˆ ˆ ˆ (7) In h sam way, using h Smyon Quinn Lov onial, h onial quaion of h olarizaion fild is as follows: ˆ ˆ ˆ (8) zz u u qu (9), q Among hm.using h mhod of saraion of variabls and sing u u( x, ) ( ( ),i can g: ( ) ( q( (4) q C ( ( ) (4) C is a consan. According o C ( g C(x ),and hr is: g cos( g sin(, C (4) ( Thos g, g ar Arbirary consans. and According o q g: ( ) q ( ) (4) This is a linar homognous diffrnial quaion wih consan cofficins, which can b solvd by Lalac ransform. s s () () qs () (44) s () () q () (45 ) ( s) s g qs g ar ral characrisic roo 4 s q L g 5 g6 is mulil characrisic roo (46) s qs R cosim g7 g8 sinim,is conjuga comlx roos g,g 4,g 5,g 6,g 7,g ar consans and 8 λ, q 4. R, Im ar h ral and imaginary ars of h comlx. Thus, h soluion of h in siu onial quaion (9) can b wrin as: I can g ha: u u( x, ) ( ( ). g cos( g sin( g g g cos( g sin( g5 g6 R g cos( g sin( g cosim g sinim s () () q () g cos( g sin( L s qs (47) 7 4 8

6 , 4 (48) 4 Sa war as a non frromagnic marial, whn h mraur 7, Is conduciviy is σ (Ω m), is rmabiliy is -7 μ μ 4π H/m, h sa war's dilcric consan ε and saliniy and mraur, by Dby xrssion, simlify ak ε 8ε F/m, hn 6, 6..Thrfor, in h non mining ara, h onial variaion of h inducd lcric fild is: g x g x g g cos( ) sin( ) 4 u (49) Th variaion of conduciviy of safloor hydrohrmal sulfid or is σ.8.(ω m),th 7 rmabiliy of sulfid minrals conaining Cu, Zn and Mn is μ μ 4π H/m, and h dilcric consan of sulhid or is F / m,so hr is : 6 Th onial variaion of h inducd lcric fild λ, μσ 4μμ μσ..6 is in h non frromagnic on: u x, [ g cosx g sinx] [ g g ] (5) 4 6 Conclusions In his ar, basd on h basic quaions of lcromagnic fild, h lcric fild modl ˆ a ˆ ˆ A is roosd. hn an analyical soluion is rsnd for h nonlinar diffusion ha conducion modl u u q wu.and i is rovd ha hr is no OPG in h form of ( q w) ( ).Finally, h closd form analyical soluion of h linar diffusion ha conducion modl u u qu is givn. Th ransin roagaion of h lcric onial in h non frromagnic hydrohrmal sulfid or is brifly xlaind. On h magnic fild and magnic onial modl, I can also b usd for similar rocssing. Th rsarch on h hory of h roscing of h marin hydrohrmal sulfid or has imoran guiding significanc for h nginring ralizaion of h frquncy domain inducd diol drag sysm. According o h acual siuaion, his ar simlifis h rocss, hrfor, h abov conclusions nd o b vrifid by racic. In addiion, h analyical soluion of ( q w) ( ) (quaion (5) )mus b dly sudid. Rfrncs [] Jing Chunli. Rgional gological background and or conrolling facors of oly mallic sulfid minralizaion in h sa floor [D]. Qingdao: h firs Insiu of h Sa Ocanic Adminisraion,.6. [] Zhou Shng, D drag y ransin lcromagnic rsons ruls [J]. Cnral Souh Univrsiy Journal of (NATURAL SCIENCE EDITION):. Of 4 ():86-89: [] Xiao Hong Yang. H Jishan Frquncy domain IP fini lmn numrical simulaion [J]. Progrss in Gohysics: 8, (4):65-6. [4] Zhou Shng, D drag y ransin lcromagnic rsons ruls [J]. Cnral Souh Univrsiy Journal of (NATURAL SCIENCE EDITION):. Of 4 ():6-4. [5] Huang Jung, Ruan Bai Yao XJ.Undrwar DC rsisiviy mhod simulaion [J]. Gohysical xloraion of comuing chnology, 4, 6 () 4

7 [6] Chn Yun, Wu Wu. Marin lcromagnism and is alicaion [J]. Marin Scinc: 99, :9-6. [7] Pan Zuliang. Mahmaical mhods and alicaions of nonlinar roblms [M]. Hangzhou: Zhjiang Univrsiy rss, [8] Zhao Jingxiang, Wang Yanjun. Exloraion gohysical lcromagnic mhod firs volum [M]. Bijing: Gological Publishing Hous, [9] Zhang Shngy, Pan Yuling. Alicaion of h rincil of Gohysics [M]. Wuhan: China Univrsiy of Goscincs rss,