Homotopy analysis of 1D unsteady, nonlinear groundwater flow through porous media

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1 Hootopy analysis of D unsteady, nonlinear groundwater flow troug porous edia Autor Song, Hao, Tao, Longbin Publised 7 Journal Title Journal of Coastal Researc Copyrigt Stateent 7 CERF. Te attaced file is reproduced ere in accordance wit te copyrigt policy of te publiser. Please refer to te journal's website for access to te definitive, publised version. Downloaded fro ttp://dl.andle.net/7/84 Link to publised version ttp:// Griffit Researc Online ttps://researc-repository.griffit.edu.au

2 Journal of Coastal Researc SI 5 pg - pg ICS7 (Proceedings) Australia ISBN Hootopy analysis of D unsteady, nonlinear groundwater flow troug porous edia H. Song and L. Tao Griffit Scool of Engineering Griffit University PMB5 GCMC QLD, 976, Australia.song@griffit.edu.au l.tao@griffit.edu.au ABSTRACT SONG, H. and TAO, L., 7. Hootopy analysis of D unsteady, nonlinear groundwater flow troug porous edia. Journal of Coastal Researc, SI 5 (Proceedings of te 9t International Coastal Syposiu), pg pg. Gold Coast, Australia, ISBN In tis paper, te D unsteady, nonlinear groundwater flow troug porous edia, corresponding to flood in an aquifer between two reservoirs, is studied by ass conservation equation and Forceier equation instead of 's law. Te coupling nonlinear equations are solved by ootopy analysis etod (HAM), an analytic, totally explicit ateatic etod. Te etod uses a apping tecnique to transfer te original nonlinear differential equations to a nuber of linear differential equations, wic does not depend on any sall paraeters and is convenient to control te convergence region. Coparisons between te present HAM solution and te nuerical results deonstrate te validity of te HAM solution. It is furter revealed te strong nonlinear effects in te HAM solution at te transitional stage. ADDITIONAL INDEX WORDS: Hootopy analysis etod, Forceier equation, porous edia INTRODUCTION Groundwater flow troug porous edia is traditionally described by s law and it is norally valid for low Reynolds pore-scale nubers. For oderate and ig velocity flow, owever, Forceier equation sould be applied due to nonlinear effects. STARK (97) nuerically solved te Navier- Stokes lainar flow equations and tested te relations of s law, Forceier equation and oters; INNOCENTINI et al. (999) copared s law and Forceier equation and recoended igly te latter in order to take into account pereability; NIELD () discussed te inertial effects on viscous dissipation for te case of, Forceier and Brinkan odels. Forceier equation could be derived in different approaces (e.g. AHMED and SUNADA, 969; HASSANIZADEH and GRAY, 987; WHITAKER, 996) and as been proved in teoretical and experiental way (MACDONALD et al., 979; THAUVIN and MOHANTY, 998). Extensive studies on te paraeters in Forceier equation ave been carried out. COULAUD et al. (988) introduced a nonlinear ter into s equation and solved it nuerically. In is approac, te ydrodynaic constants in te Forceier equation were expressed by te expression of porosity and geoetrical data. WANG and LIU (4) investigated te scaling relations for te fluid pereability and te inertial paraeter in te Forceier equation, by solving te Navier-Stokes equation for flow in a two-diensional percolation porous edia. Altoug te application of te Forceier flow is very useful and practical, very liited attepts on te analytical approac ave been reported in te literature. MOUTSOPOULOS and TSIHRINTZIS (5) solved te Forceier flow troug porous edia in D for by perturbation etod, dividing te proble into two stages and solving te by two sets of equations. For nuerical etod, GREENLY and JOY (996) used onediensional finite eleent etod and Forceier equation to investigate te groundwater flow troug a valley fill. EWING et al. (999) used finite difference, Galerkin finite eleent and ixed finite eleent tecniques to investigate Forceier flow in a ydrocarbon reservoir. KIM and PARK (999) and PARK (5) used ixed finite eleent etod to analyse te flow of a singlepase fluid in a porous ediu governed by Forceier equation. Recently, a new ateatical tecnique, naely ootopy analysis etod (HAM) as been applied to nonlinear fluid dynaics probles (LIAO, 995, 4). Te approac does not depend on sall or large paraeters and is easy to adjust te convergence region and rate of approxiation series. In tis paper, ootopy analysis etod is applied to solve te D unsteady, nonlinear groundwater flow troug porous edia, corresponding to a flood in a long aquifer between two reservoirs, or to a flow in a laboratory colun restrained by two external tanks. Te coupling equations are transfored by siilarity law and a global solution, wic is analytical, totally explicit is obtained. Te piezoetric ead fro te present HAM solution for nonlinear flow agrees well wit nuerical results and te previous perturbation solutions. THEORETICAL CONSIDERATION Consider a one-diensional (D) flow in a confined porous ediu sown in Figure. Before t=, te piezoetric ead is a constant I. After t=, te piezoetric ead at left end raises Δ, wile te piezoetric ead at te far rigt end reains uncanged. Journal of Coastal Researc, Special Issue 5, 7

3 Hootopy analysis of te D non-steady, nonlinear groundwater flow troug porous edia Figure. Te sketc of te proble. Te oveent of te flow satisfies te equation of ass conservation as following: S r + ( Bq) R, t () were S is te storage coefficient; is te piezoetric ead; t is te tie; is te D Nabla operator; B is te tickness of te aquifer; q r is te velocity; and R is te external sink-source ter, wic is assigned zero in tis particular proble. Assuing tat te properties of te aquifer are oogeneous, te Forceier or Forceier-Dupuit equation is: = aq + bq q () were a and b are coefficients. Equations () and () can be expressed in D for as q S + B =, t () = aq + bq. (4) Te initial condition is: = I at t =. (5) Te boundary conditions are: = I +Δ for x= at t >, (6) = I for x= at any t. (7) Introducing % = I, Δ (8) Equations () and () are transfored as: S % q Δ + =, B t x (9) % Δ = aq+ bq. () Fro Equation (), te velocity can be expressed as % a+ a 4bΔ q =. b () If 4bΔ > a, te inertial ter is doinant, oterwise te (viscous) ter is doinant. Substituting Equation () into Equation (9), we ave: % % % / = C +C, () t subject to te initial and boundary conditions: % = for t =, () = for x= (4) = for x= (5) were C ( ) = 4 S B bδ, S C ( ) = B a. Using te siilarity transforations: xt % / (, ) = t f, ξ = x/ t, (6) Equation () becoes f + ξ f / 6 = Cξ f + C, (7) f ξ f wit boundary conditions: f () = C, (8) f ( ) =, (9) were C = / t /, wic is a constant for a given tie t. Alternatively, Equation (7) can be expressed as, f + ξ f ξ 6 C [ Cf. () = / ] f ξ f SOLUTION METHOD Define τ = + λξ, () were λ is a constant paraeter to be deterined later, Equation () can be transfored to Equation () (see Appendix), and te boundary conditions becoe f () = C, () f ( ) =. (4) Fro te above boundary conditions, f(τ) can be expressed by a set of base functions { τ }. (5) Establis te zerot-order deforation equation: ( L [ F( τ; f] = ph N [ F( τ; ], (6) subject to te boundary conditions: F( ; = C, F( ; =, (7) were p [, ] is an ebedding paraeter and L is a linear auxiliary operator; F(τ; is a real function of τ and p. Te auxiliary nonzero paraeter ħ is used to adjust te convergence rate and region. By introducing HAM, we set up a apping f () τ F( τ;. It is seen fro Equation (6) tat te solution F(τ; continuously varies fro te initial estiate f (τ) to te exact solution f(τ) as te ebedding paraeter p increases fro to. Te linear auxiliary operator L as a various coices, wic will affect te convergence of te solution. We select ( ) [ ( )] τ; p L F τ; p = τ + F( τ;, τ (8) wit te property L [ C4/ τ ] =. (9) Te nonlinear operator N is defined in Equation () (see Appendix). Expand F(τ; in Taylor series wit respect to p, we ave F( τ; = f + f p, () were = F( τ ;. () f( τ ) =! p p= Journal of Coastal Researc, Special Issue 5, 7

4 H. Song and L. Tao If te auxiliary paraeter ħ is properly cosen tat te series are convergent at p=, ten Define f( τ ) = f + f ( τ ). () = r f = { f() τ, f() τ, f() τ, L, f ()} n n τ. (4) Differentiating Equation (6) ties wit respect to p, ten setting p=, and finally dividing te by!, te t-order deforation equation can be obtained: r L [ f( τ ) χf ] = H R( f, τ), (5) subject to te boundary conditions: f() =, f( ) = for, (6) were, χ, (7) =, > and [ F( ] R ( f τ ;, τ ) = ( )! p= Substituting N and F(τ; in Equations () and () respectively into Equation (8), we can ave te detailed for of R. Suppose f( τ ) = C/ τ, (9) ten te wole proble can be solved by iteration: τ r C4 f = χf + H( α) R( f, τ) dα τ +, (4) τ were H(τ) can be cosen as H(τ)= τ. Fro te first few orders of te solution, it can be concluded tat f (τ) can be expressed as 9+ n f( τ ) = β, nτ, (4) n= were β,n is a coefficient. Substituting Equation (4) into te ig-order deforation equation (5) and equating te sae power of τ, te recurrence forulae of β,n can be obtained, wic is very long and oitted ere. Fro Equation (6), β, can be deterined uniquely. β 9+ = β. (4),, n n= If te series () is convergent, it ust be an exact solution of Equation (), since te following equation stands wen te series () is convergent: li f ( τ ) =. (4) Using Equations (6), (5) and (7), we ave r n H() τ R ( f, τ) = li L [ f () τ χ f ()] τ n = = n (44) = L li [ f() τ χf ()] τ = L li fn() τ =, n n = wic gives R ( ), (45) f r, τ = = for any τ. Under te definition (8), it is easy to prove tat Equation () olds. Fro Equations (6) and (9), te boundary conditions (8) and (9) also old. Terefore, if ħ and λ are properly cosen to ensure te series convergence, f(ξ) is an exact solution of te siilarity questions. RESULTS AND DISCUSSION In tis section, te pysical proble of groundwater flow troug porous edia is solved by ootopy analysis etod described above and te HAM solutions are copared to te perturbation solution of MOUTSOPOULOS and TSIHRINTZIS (5) in Figure. Te principle data of te first tree figures are a=.5s/, b=5s /, S =S/B=. - and Δ= wile te data of te last two figures are a=.64s/, b=.8s /, S =. - and Δ=. Te tie instances corresponding to te results sown in te five figures are (a) 4s, (b) 99s, (c) 54s, (d) 97s and (e) 445s respectively. As can be seen in te figures, te first tree figures are nonlinear doinated and te rest two are quasi-. Te nuerical results fro are calculated by pdepe function wit Δx= and Δt=.s for te distance and tie intervals, assuing x= is sufficiently large. For ootopy analysis etod, we coose ħ = - and λ=/5 respectively. It can be seen tat te results fro ootopy analysis etod agree well wit te nuerical etod and MOUTSOPOULOS and TSIHRINTZIS (5) in nonlinear doinated cases, but less accurate in quasi-linear cases. If te solution contains a linear function, te results sould be ore reasonable for quasi-linear flow. In te present study, nonlinear base functions were eployed, so tat te penoena of unsteady, nonlinear flow troug porous edia could be described accurately. Witout te coponent of a linear function, te present HAM solution is ore suitable for te initial unsteady, nonlinear stage, wic is igly concerned by scientists and engineers. For quasi-linear flow, s law could be applied directly. Tus te proble could be siplified to a linear proble and easy to be solved. Table is an exaple of te convergence of te series at t=4s and x=. A rapid convergence rate of te series is evident. CONCLUSIONS In tis paper, a new analytic etod, naely ootopy analysis etod (HAM) as been applied to give an analytic, totally explicit and unifor valid solution to te proble of unsteady, nonlinear groundwater flow troug porous edia. Te HAM analytic solution agrees well wit nuerical results and previous perturbation etod for nonlinear flow. Te present approac sows a great potential to oter unsteady nonlinear probles. ACKNOWLEDGEMENTS Tis paper is based on work funded by te Australian Researc Council under Grant No. ARC-DP4596. Te support fro te above Grant is greatly appreciated. Table : Te convergence of te series at t=4s and x=. Order % Journal of Coastal Researc, Special Issue 5, 7

5 Hootopy analysis of te D non-steady, nonlinear groundwater flow troug porous edia.9.8 Moutsopoulos & Tsirintzis Hootopy.9.8 Moutsopoulos & Tsirintzis Hootopy (a) t =4 s.6 (d) t =97s x() x().9.8 Moutsopoulos & Tsirintzis Hootopy.9.8 Moutsopoulos & Tsirintzis Hootopy (b) t =99 s.6 (e) t =445s x() x().9.8 Moutsopoulos & Tsirintzis Hootopy Figure. Coparison of diensionless piezoetric ead in MOUTSOPOULOS and TSIHRINTZIS (5)..7.6 (c) t =54s x() Journal of Coastal Researc, Special Issue 5, 7

6 H. Song and L. Tao LITERATURE CITED AHMED, N. and SUNADA, D.K., 969. Nonlinear flow in porous edia. Journal of Hydraulic Division ASCE, 95, COULAUD, O., MOREL, P. and CALTAGIRONE, J.P., 988. Nuerical odelling of nonlinear effects in lainar flow troug a porous ediu. Journal of Fluid Mecanics, 9, EWING, R.E.; LAZAROV, R.D.; LYONS, S.L.; PAPAVASSILIOU, D.V.; PASCIAK, J. and QIN, G., 999. Nuerical well odel for non- flow troug isotropic porous edia. Journal Coputational Geosciences, (-4), GREENLY, B.T. and JOY, D.M., 996. One-diensional finiteeleent odel for ig flow velocities in porous edia. Journal of Geotecnical Engineering, (), HASSANIZADEH, S.M. and GRAY, W.G., 987. Hig velocity flow in porous edia. Transport in Porous Media, (6), 5-5. INNOCENTINI, M.D.M.; PARDO, A.R.F.; SALVINI, V.R. and PANDOLFELLI, V.C., 999. How accurate is 's law for refractories. Aerican Ceraic Society Bulletin, 78(), KIM M.-Y. and PARK E.-J., 999. Fully discrete ixed finite eleent approxiations for non- flows in porous edia. Coputers and Mateatics wit Applications, 8(), - 9. LIAO, S.J., 995. An approxiate solution tecnique not depending on sall paraeters: a special exaple. International Journal of Non-Linear Mecanics, (), 7-8. LIAO, S.J., 4. Beyond Perturbation: Introduction to te Hootopy Analysis Metod. Florida: Capan & Hall / CRC. MACDONALD, I.F.; EL-SAYED, M.S.; MOW, K. and DULLIEN, F.A.L., 979. Flow troug porous edia - te Ergun equation revisited. Industrial and Engineering Ceistry Fundaentals, 8(), MOUTSOPOULOS, K.N. and TSIHRINTZIS, V.A., 5. Approxiate analytical solutions of te Forceier equation. Journal of Hydrology, 9, 9. NIELD, D.A.,. Resolution of a paradox involving viscous dissipation and nonlinear drag in a porous ediu. Transport in Porous Media, 4(), PARK, E.J., 5. Mixed finite eleent etods for generalized Forceier flow in porous edia. Nuerical Metods for Partial Differential Equations, (), -8. STARK, K.P., 97. A nuerical study of te nonlinear lainar regie of flow in an idealised porous ediu. In: Fundaentals of Transport Penoena in Porous Media, New York: Elsevier Publising Copany, 86-. THAUVIN, F. and MOHANTY, K.K., 998. Network odelling of non- flow troug porous edia. Transport in Porous Media, (), 9-7. WANG, X.-H. and LIU, Z.-F., 4. Te Forceier equation in two-diensional percolation porous edia. Pysica A: Statistical and Teoretical Pysics, 7(-4), WHITAKER, S., 996. Te Forceier equation: A teoretical developent. Transport in Porous Media, 5(), 7-6. APPENDIX C f 8 C ( τ ) f f + [4 C ( τ ) C λ ] f f 4 + [8 C ( τ ) λ C ( τ ) ] f f + [56λ 8 C ( τ ) λ + 6 C ( τ ) ] f 8 C ( τ ) λ f f f C( τ ) λ f f f [ ( τ ) λ 5 C ( τ ) λ ] f f 8 C ( τ ) λ f f + 5 C ( τ ) λ f f f + [644( τ ) λ 5 C ( τ ) λ ] f f + 89( τ ) λ f f + 496( τ ) λ f 4 C ( τ ) λf f + C( τ ) λf f 96 C ( τ ) λf f + 8 C ( τ ) λf f 64 C ( τ ) λf = F( τ; N [ F( τ; ] = CF( τ; 8 C( τ ) F( τ; F( τ; F( τ; + [4 C ( τ ) C λ ] F( τ; [ ] + [8 C ( τ ) λ C ( τ ) ] F( τ; [ ] 4 4 F( τ; 4 F( τ; F( τ; + [56λ 8 C( τ ) λ + 6 C( τ ) ][ ] 8 C( τ ) λ F( τ; F( τ; F( τ; 4 F( τ; F( τ; + 5 C( τ ) λ F( τ; [ ] + [48( τ ) λ 5 C ( τ ) λ ][ ] F( τ; F( τ; F( τ; 8 C ( ) F( ; [ ] + 5 C ( ) F( ; [ ] τ λ τ τ λ τ + [644( τ ) 4 4 F( τ; F( τ; 4 F( τ; F( τ; λ 5 C ( τ ) λ ][ ] [ ] + 89( τ ) λ [ ] F( τ; F( τ; F( τ; + 496( τ ) λ [ ] 4 ( τ ) λ ( τ; ) [ ] + ( τ ) λ ( τ; ) [ ] C 4 F p C F p F( τ; 96 C ( τ ) λf( τ; [ ] 4 4 F( τ; 5 5 F( τ; + 8 C 6 ( τ ) λf( τ; [ ] 64 C( τ ) λ[ ] () () Journal of Coastal Researc, Special Issue 5, 7