APPLICATIONS OF A NOVEL INTEGRAL TRANSFORM TO THE CONVECTION-DISPERSION EQUATIONS

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1 S33 APPLICATIONS OF A NOVEL INTEGRAL TRANSFORM TO THE CONVECTION-DISPERSION EQUATIONS by Xin LIANG, Gunnn LIU b*, nd Shnjie SU Stte Key Lbortory for Geomechnics nd Deep Underground Engineering, Chin University of Mining nd Technology, Xuzhou, Chin b School of Mechnics nd Civil Engineering, Chin University of Mining nd Technology, Xuzhou, Chin Originl scientific pper In this pper, we etend the novel integrl trnsform of some functions by the dulity reltionship between it nd Lplce trnsform. Additionlly, pplying the novel integrl trnsform, we solve -D convection-dispersion eqution describing the dispersion process of chemicl dditives in porous rocks during the hydrulic frcturing. The results indicte tht the novel integrl trnsform cn provide new ide to obtin more ect solutions of different convection-dispersion problems. Key words: integrl trnsform, convection-dispersion eqution, hydrulic frcturing Introduction The convection-dispersion equtions (CDE) re widely pplied to describe the trnsport of suspended prticles in nture porous medi, which is touched upon mny engineering brnches of the environment, the rchitecture, s well s the energy [-3]. In prticulr, for energy eploittion (such s shle gs eploittion), with the mssive ppliction of hydrulic frcturing in recent yers [4-6], the problem of the groundwter pollution hs brought bout the widespred ttentions due to the fct tht the hydrulic frcturing fluid contining numerous chemicl dditives is dispersed into the quifer. Therefore, it is of criticl importnce for understnding the dispersion process of the chemicls in quifer [7, 8]. Usully, the nlytic solutions of the CDE cn be intuitively used to provide quntittive nlysis of the dispersion of prticles. Presently, mny reserchers hve solved the CDE with specil initil nd boundry conditions by the integrl trnsforms (IT), such s Lplce trnsform (LT), nd Fourier trnsform (FT) [9]. However, with the development of the IT, some novel IT cn be pplied to provide new ides for obtining the nlytic solutions of the CDE. Recently, some novel IT were proposed s the etensions of clssicl FT nd LT to solve the het trnsfer equtions nd diffusion equtions [0-5]. Prticulrly, in [], the dulity reltionship between novel integrl trnsform (NIT) nd LT ws studied in detil nd the NIT ws proved to be effective to solve some PDE like LT. This indictes tht the NIT cn be pplied to obtin the ect solutions of the CDE. * Corresponding uthor, e-mil: gunnnliu@cumt.edu.cn

2 S34 In view of the nlysis, we pln to etend the NIT of some functions by the dulity chrcteristic between the NIT nd LT. Menwhile, the -D CDE is nlyzed by the NIT nd the corresponding solution is discussed grphiclly. The NIT of some bsic functions According to the ide [0], the definition of the NIT is described: e Ω ( ) = NI φ( ) = φ( ) d, > 0 () 0 where φ ( ), > 0, is rel function, e / is the kernel function, nd N I is the opertor of NIT. ( ) ( ) ( ) F = L φ = φ e d, > 0 () 0 where φ ( ), > 0, is rel function, e is the kernel function, Ω ( ) = F ( ) nd L is the opertor of the LT. Obviously, from previous definitions, the NIT is similr to NIT LT the LT in form. The min difference between them is reflected F ( ) = Ω( ) in the selection of the kernel function [6]. It is indicted tht Figure. The dulity of the NIT nd LT re correlted, s well s mutully different. In the NIT nd LT order to verify the pplicbility of dulity reltionship s shown in fig., we derive the NIT of five bsic functions. In detil, the NIT of five functions re proved s follows. For φ ( ) = nd > 0, we hve [7]: F = = ( ) L[ ] According to the dulity reltionship in fig., we hve the following eqution: For φ ( )= nd > 0, we hve [8]: The NIT of φ ( ) cn be written: For φ ( ) = e nd > 0, we hve [7]: Ω = = = ( ) F( ) L[ ] F ( ) = L [ ] = Ω = = = ( ) F( ) L [ ] 3 F ( ) = L e = (3) (4) (5) (6) (7) The NIT of φ ( ) is derived by the dulity reltionship:

3 S35 Ω ( ) = ( ) = e = F L For φ ( ) = sin( ) nd > 0, we hve [7]: The NIT of φ ( ) becomes: ( ) = sin ( ) = F L + ( ) Ω ( ) = F ( ) = L sin ( ) = For φ ( ) = cos( ) nd > 0, we hve [7]: The NIT of φ ( ) is: ( + ) Ω ( ) = F ( ) = L cos ( ) = ( + ) Ω ( ) = F ( ) = L cos ( ) = ( + ) The results of eqs. (4), (6), (8), (0), nd () re sme s those given in [0]. It is illustrted tht the dulity reltionship between the NIT nd LT is pplicble to obtin the NIT of more functions. Thus, we etend the NIT of more functions in the ppendi nd dopt them to solve the CDE in net section. (8) (9) (0) () () Solving the CDE by the NIT As displyed in fig., the quifer is ssumed to be semi-infinite porous medium nd contins no chemicl dditives eisting in frcturing fluid in initil stte. During hydrulic frcturing, the frcturing fluid flows into the quifer with the constnt velocity, u. Menwhile, the concentrtion of chemicl dditives in quifer boundry is treted s constnt, C 0. Thus, the dispersion process of chemicl dditives in the quifer cn be described: Overburden rock Aquifer Shle gs reservoirs Shle gs well Frcturing fluid Figure. The schemtic digrm of hydrulic frcturing k u = ct (,) ct (,) ct (,) t (3) where ct (,) is the concentrtion of the chemicl dditives in the quifer, k the dispersion coefficient, nd u the verge trvel velocity of the frcturing fluid. The initil-vlue condition (IVC) nd the boundry-vlue conditions (BVC) re:

4 S36 c (,0) = 0 (IVC) c(0,) t = C0, c(,) t = 0(BVC) respectively, where C 0 is rel constnt. According to the properties of the NIT in [0], the NIT of eq. (3) with respect to t is obtined s: C (, ) C (, ) c (,0) k u = C (, ) (4) Correspondingly, the NIT of BVC becomes: C0 C(0, ) =, C(, ) = 0 (5) Substitution of the IVC into eq. (4) results in: C (, ) C (, ) k u C (, ) = 0 (6) From the eq. (6), we cn obtin its eigenvlue eqution [8]: kλ uλ =0 (7) Then, the eigenvlues of eq. (7) re given: u+ u + 4k u u + 4k λ =, λ = (8) k k Thus, the solution of eq. (6) is presented: ( ) u+ u + 4k u u + 4k k k C, = me + m e (9) Substituting the BVC eq. (5) into eq. (9), we hve: C0 m + m, 0 = m = (0) Tking eq. (0) into the eq. (9), we obtin: 4 u u + k u u + 4k k 0 C 0 k k e e e C C = = () According to the result in tb. (Appendi), we hve the NIT of specil function: b b NI erfc bt (e e ) e + = t + b / / where, b re the constnts, nd erfc[/( / t) ( bt) ] is the error function [7]. Finlly, we solve the inverse NIT of eq. () with respect to t nd hve: () C C + ut ut e erfc erfc kt kt u 0 k = + (3)

5 S37 The convection-dispersion solutions of eq. (3) for vried prmeters re illustrted in figs. 3() nd 3(b). The results show tht the dispersion rnges of chemicl dditives will be wider nd wider s the time increses nd the concentrtions of chemicl dditives in quifer lso will increse. For emple, fig. 3() demonstrtes tht the dispersion rnges rech pproimtely 5 meters in 60 seconds. In ddition, 70% chemicl dditives re discovered t 30 meters in 60 minutes [red curve () in fig. 3(b)], while pproimtely 0 in 60 seconds [red curve () in fig. 3()]. Reltive concetrtion of chemicl dditives in frcturing fluids () k = 0.8 u = 0.03 C 0 = 0.5 t = 0 ~ 60 Reltive concetrtion of chemicl dditives in frcturing fluids () k = 0.8 u = 0.03 C 0 = 0.5 t = 0 ~ 360 () (b) Figure 3. The convection-dispersion curves with different prmeters: () seconds time intervl for ech curve, (b) minutes time intervl for ech curve; red curve () (for color imge see journl web site) Conclusion In this study, the NIT ws pplied to obtin the ect solution of -D CDE for n efficient description of the dispersion process of chemicl dditives in porous rocks during the hydrulic frcturing. With the help of the dulity reltionship between the NIT nd LT, the etended NIT tble presented in the Appendi ws convenient to obtin the solutions of some PDE. The results were given to show tht the dispersion rnge of chemicl dditives in the quifer is wide (5 meters in 60 seconds) nd tht the dispersion scle is lrge (70% within 60 minutes). It is demonstrted tht the chemicl dditives ffect the groundwter qulity with the time increses due to the strong dispersion process. Acknowledgment This work is finncilly supported by the Fundmentl Reserch Funds for the Centrl Universities (No. 07BSCXA) nd the Postgrdute Reserch & Prctice Innovtion Progrm of Jingsu Province (No. KYCX7_53). Nomenclture c concentrtion of the chemicl dditives, [gl ] k dispersion coefficient, [cm s ] t time, [s] u flow velocity, [cms ] diffusion distnces, [m]

6 S38 References [] Ahfir, N. D., et l., Trnsport nd Deposition of Suspended Prticles in Sturted Porous Medi: Hydrodynmic Effect, Hydrogeology Journl, 5 (007), 4, pp [] Benncer, L., et l., Suspended Prticles Trnsport nd Deposition in Sturted Grnulr Porous Medium: Prticle Size Effects, Trnsport in Porous Medi, 00 (03), 3, pp [3] Ahfir, N. D., et l., Influence of Internl Structure nd Medium Length on Trnsport nd Deposition of Suspended Prticles: A Lbortory Study, Trnsport in Porous Medi, 76 (009),, pp [4] Wng, L., et l., Element Mobiliztion from Bkken Shles s Function of Wter Chemistry, Chemosphere, 49 (06), Apr., pp [5] Li, Y., et l., The Sttus Quo Review nd Suggested Policies for Shle Gs Development in Chin, Renewble & Sustinble Energy Reviews, 59 (06), June, pp [6] Hou, P., et l., Eperimentl Investigtion on the Filure nd Acoustic Emission Chrcteristics of Shle, Sndstone nd Col under Gs Frcturing, Journl of Nturl Gs Science nd Engineering, 35 (06), Sept., pp. -3 [7] Torres, L., et l., A Review on Risk Assessment Techniques for Hydrulic Frcturing Wter nd Produced Wter Mngement Implemented in Onshore Unconventionl Oil nd Gs Production, Science of the Totl Environment, 539 (06), Jn., pp [8] Kng, Y., et l., Comprehensive Evlution of Formtion Dmge Induced by Working Fluid Loss in Frctured Tight Gs Reservoir, Journl of Nturl Gs Science & Engineering, 8 (04), 8, pp [9] Wng, H., et l., Prticle Trnsport in Porous Medium: Determintion of Hydrodispersive Chrcteristics nd Deposition Rtes, Comptes Rendus de l Acdémie des Sciences-Series IIA-Erth nd Plnetry Science, 33 (000),, pp [0] Yng, X. J., A New Integrl Trnsform Method for Solving Stedy Het Trnsfer Problem, Therml Science, 0 (06), Suppl. 3, pp. S639-S64 [] Yng, X. J., A New Integrl Trnsform with n Appliction in Het Trnsfer Problem, Therml Science, 0 (06), Suppl. 3, pp. S677-S68 [] Ling, X, et l., Applictions of Novel Integrl Trnsform to Prtil Differentil Equtions, Journl of Nonliner Science nd Applictions, 0 (07),, pp [3] Yng, X. J., A New Integrl Trnsform Opertor for Solving the Het-Diffusion Problem, Applied Mthemtics Letters, 64 (06), Feb., pp [4] Yng, X. J., et l., A New Technology for Solving Diffusion nd Het Equtions, Therml Science, (07), A, pp [5] Beerends, R. J., et l., Fourier nd Lplce Trnsforms, Cmbridge University Press, Oford, UK, 003 [6] Feynmn, R. P., et l., Quntum Mechnics nd Pth Integrls, Dover Publictions, Mineol, N. Y., USA, 00 [7] Debnth, L., Bhtt, D., Integrl Trnsforms nd Their Applictions, CRC Press, Boc Rton, Fl., USA, 05 [8] Polynin. A. D., Zitsev, V. F., Ect Solutions for Ordinry Differentil Equtions, CRC Press, Boc Rton, Fl., USA, 995

7 S39 Appendi Tble. The NIT of some functions sinh( ) ( ) 3 n e φ ( ) F( ) = L φ( ) ( ) cosh Γ ( n + ) ( ) ( n+ + ) 4 sin( ) e b ( ) b + b b + 5 cos( ) e b ( ) 6 sin( ) ( + ) 7 cos( ) 8 sinh( ) ( + ) ( ) 9 cosh( ) + ( ) 0 ( + ) e π ( ) sin ( ) π cos ( ) π 3 sinh ( ) π 4 cosh ( ) π 5 erf 5 e 3 e 5 e 3 e ( ) e

8 S40 Tble. (continution) 6 e erf ( ) 7 erfc 8 e erf ( ) φ ( ) F( ) = L φ( ) + ( ) 3 e π ( ) 9 0 b b b + b e erfc e erfc + b b b b + e erfc + e erfc + e ( b) e + b Pper submitted: Mrch 8, 07 Pper revised: April 8, 07 Pper ccepted: My 3, Society of Therml Engineers of Serbi Published by the Vinč Institute of Nucler Sciences, Belgrde, Serbi. This is n open ccess rticle distributed under the CC BY-NC-ND 4.0 terms nd conditions