A Short Note on Self-Similar Solution to Unconfined Flow in an Aquifer with Accretion

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1 Open Journal o Flui Dynamics, 5, 5, 5-57 Publishe Online March 5 in SciRes. A Short Note on Sel-Similar Solution to Unconine Flow in an Aquier with Accretion Arieh Pistiner Unit or Hyrocarbon Pollution Prevention, Ministry o the Environmental Protection, Haia, Israel ariehpistiner@gmail.com Receive 6 February 5; accepte 7 February 5; publishe March 5 Copyright 5 by author an Scientiic Research Publishing Inc. This work is license uner the Creative Commons Attribution International License (CC BY). Abstract In this stuy we reer to a non-steay state, one-imensional (on the x-axis), unconine an saturate low in an aquier, escribe by the Boussinesq equation, combine with accretion. In accorance with the above, the moving bounary o the saturate area (towar x + ) serves as a horiontal water lux source to the unsaturate area. As time avances, the horiontally saturate one, lying on the x-axis, becomes wier. A sel-similar solution is erive that, ater some mathematical manipulation, it is escribe in terms o Hypergeometric unctions. The long-time behaviors o the solution escribe the situation at which the water lux, that penetrates horiontally to the non-saturate one, is equal to the water lux entering into the saturate one. Keywors Boussinesq Equation, Sel-Similar Solution, Hypergeometric Function. Introuction In this stuy, the equation escribing unsteay low in a semi-ininite phreatic aquier with accretion []-[] is analye. In the above equation, (, ) h h h = N xt t x + x (, ), () h xt is the hyraulic hea in the aquier; x an t are the normalie position an time coorinates, respectively (i.e., x ), an N( xt, ) is a time an position epenent unction, representing the rain intensity istribution impose on the aquier that is given by How to cite this paper: Pistiner, A. (5) A Short Note on Sel-Similar Solution to Unconine Flow in an Aquier with Accretion. Open Journal o Flui Dynamics, 5,

2 ( x N xt, ) = A, A> t A is the rain intensity. We consier a situation in which the water hea istribution in a boy o water, lying in the porous meium, at time t <, is unknown. Initially, at time t =, the water level on the inlet ace o the aquier suenly rops, accoring to the ollowing power law (, ) a h t = t, (a) is a scaling parameter o the porous meium, an a is a negative constant to be etermine hereater. This bounary conition woul correspon to an inluent stream that supplies water to the aquier. In aition to this, rainwater begins to penetrate into the aquier accoring to () an as rainwater to the saturate water boy. As a response to that, water lux at the inlet ace is create an possesses the ollowing orm h h = qt x x = a (), (b) q is a imensionless inlet lux parameter an a is a negative constant to be etermine hereater. X t is given by The ownstream bounary conitions or the saturate water boy on the moving bounary h( xt, ) at x ( t ) X ( t) an the ownstream water lux on the moving bounary is given by = (c) h h = qt x xt ( ) = X ( t ) a, () q is the imensionless lux parameter o the moving bounary, the area in the omain X ( ) t < x< +, is suppose to be a non-saturate one. In general, the problem must be solve or speciie initial conitions impose upon h( x,). However, as will be shown below, the long-time proile o h( xt, ) is inepenent o the precise orm o the initial conition h( x,), which governs the hyraulic hea at early stages only. However, the long-time proile will be investigate in the next section by the similarity metho.. Sel-Similar Moel We will now reer to the circumstances in which the hyraulic hea in the aquier (, ) asymptotic, an is escribe by a single inepenent sel-similar variable ξ [4]: (, ) ( ξ ) h xt achieves a certain a h xt = t, (4a) b x = ξt, (4b) ( ξ ) is a similarity positive unction, a an b are parameters to be etermine later. Substituting (), (4a), (4b) in () an ater certain mathematical manipulation we obtain In this stuy we reer to the particular case Introucing (5a) into (5b), we obtain 4 ξ ξ ξ + b ξ + A A = ( a + b) ξ, (5) ξ ξ a = b. (5a) a+ b =. (5b) 5

3 Substituting (6b) in (5) an integration we obtain λ is an integration constant.. Metho o Solution 5 a = ; b = ; a = (6a) (6b) (6c) A ξ ξ + ξ + = λ, (7) ξ 4 4 The similarity unction ( ξ ) may be eine via a new inepenent unction φ( ξ ) as ollows Introucing (8) into (7) combine to yiel Deine a new epenent variable η Introucing () into (9) we obtain an φ = ξ. (8) φ A φ φξ ξ λξ ξ =. (9) η 6 = ξ. () φ = 6A, () η 5 φ φ λη η We now eine two new unctions, θ an v( θ ) respectively λ = λ 5 (a) φ = ηθ, () = + v θ λη θ θ. () Dierentiating v( θ ) with respect to θ, using () an () an selecting a value or the rain intensity, i.e., A =, we obtain an Abel-type equation o the secon kin [5] We now eine a new unction τ ( θ ) as ollows [5] v v = v θ + θ θ θ + θ θ 7 5. (4) 4 v( θ) θτ( θ) θ θ 5 4 = +. (5) The substitution o (5) in (4) leas to a Riccati equation with respect to θ = θ( τ) θ 4. (6) 5 τ 5 τ + = θ + θτ + We now eine the unction u ( τ ) an apply the Riccati transormation [5], as ollows 5

4 θ 9 u. (7) 4 45 u τ = τ Substituting (7) in (6), we obtain the ollowing linear ODE u 7 + u u =, (8) ( τ ) τ ( τ ) τ τ We now eine ( τ ) as ollows which is vali in the omain 4 τ = τ. (8a) 5 = τ, (9) The substitution o (9) in (8) then yiels the hypergeometric equation which possesses the general solution In the above <. (9a) u 5 u ( ) + + u =, () 6 48 u = C CF + F. () F F =, ; ; 4, (a) 5 5 F = F, ; ; 4, (b) are expresse via hypergeometric unctions [6], an C an C are constants to be etermine below. Using the u properties o the hypergeometric series, we obtain rom () an (a), (b) the expression or the hypergeometric unctions F ( ) an F4 Substituting (9) into (7) using (8a) we obtain u C = C F + F + F4 6 6, () are given by 4 F = F, ; ; 4, (a) F4 = F, ; ; 4. (b) θ = u 6 ( ) 5 u. () The introuction o () an () into () we obtain the inal solution or θ 54

5 an θ C F + F + F CF F = 6 ( ) Substituting () in () we obtain / = 6 +. (4) v θ λ ξ θ θ. (5) The introuction o (8) an () into () gives the ollowing expression or 6 = ξθ. (6) Using the expression or v( θ ) in Equations (5) an (5) an combine with (6), the unctions ( ) are given by ξ 6 = 4 θ θ 9 = 6 τ can be easily obtaine rom (9) θτ λ + 5 λ τ θ 5θ θ τ The inlet ace position, i.e. ξ =, is obtaine rom (7a) as ollows 4 ξ an, (7a), (7b) =. (8) ξ = as θ. (9) It can be observe rom () that the requirement appearing in (9) can be achieve only i u =. In ac- = corance with the above, we obtaine the value or C by equating () to ero at =, i.e. ( ) an in accorance with (9a), the constant C exists in the ollowing range F C = F, ().95 < C <. () Substituting (9) in (7b) yiels the bounary conition parameter eine in (a) = λ. () 6 From the above, it can be observe that λ must be negative λ <. () The bounary conition (c), impose on the moving ront, is etermine by equating (7b) to ero by introucing θ = = (see ()). Hence, the ownstream parameter ξ (i.e. ( ξ ) = ) is obtaine ater introucing = into (7a). Using the property o the hypergeometric unctions (i.e., F 4 = ) we obtain the ownstream parameter 55

6 ξ λ = C 5 ( 4 ) ξ 4, (4) X t = t. (5) In accorance with the above (i.e., λ < ), the enominator o Equation (4) must obey the ollowing inequality which automatically shows that < C 5, (6) < C <, (6a) an it is in accorance with the range or the parameter C in (). The behavior o as approaches ero can be obtaine rom (7b) an is given by ξ K as, (7) λ 5 5 K is a positive constant which is equal to C 4 C. The lux parameter q or the saturate one, which appear in (b), can be obtaine by using (8)-() as ollow q ( λ ) 74 = =. (8) ξ The water lux parameter on the moving bounary, that serve as water source or the unsaturate one ξ (i.e., see (7)), can be obtaine rom (7) q 5 λ = = ξ ξ 8 ξ. (9) We will now assume that at the long-time limit, the water lux exchange between the inlet ace an the moving bounary (i.e., the water lux to the saturate one an the water lux to the unsaturate one) reach some equilibrium. As a result, an aitional conition can be ormulate as ollow q = q. (4) The introuction o (5) into (9), using (8) an (4) we obtain the ollowing equilibrium equation = C 5 C 5, (4) which is inepenent on the value o λ. Solving (4) implicitly an using () an (4), we obtain the value or 4. Short Discussion.. (4) Figure illustrates the evolution o the water hea in the aquier or three time intervals. 56

7 Figure. Hyraulic hea proiles or time intervals ( =., λ =, q = q = 5.858, ξ =.986, C =.58 ). It can be observe that the ownstream branch o the water hea proiles is characterie by a steep transition to ero (almost ininite graient) as can be expecte rom (8) an (9) (i.e., the water lux on the bounary between the saturate one an the non-saturate one possess inite value, as can be observe rom (b) an ()). In general, the solution here evelope escribes the evolution o the saturate one, stem rom penetration o rainwater an an inluent stream rom the inlet ace. The evelope analytical solution can be most useul or veriying numerical solutions involving grounwater transport in an unconine aquier. Reerences [] Bear, J. (988) Dynamics o Fluis in Porous Meia. Dover, New York. [] Knowles, I. an Yan, A. (7) The Reconstruction o Grounwater Parameters rom Hea Data in an Unconine Aquier. Journal o Computational an Applie Mathematics, 8, 7-8. [] Rai, S.N. an Manglik A. () An Analytical Solution o Boussinesq Equation to Preict Water Table Fluctuations Due to Time Varying Recharge an Withrawal rom Multiple Basins, Wells an Leakage Sites Water Resources Management, 6, [4] Barenblatt, G.I. (979) Similarity, Sel-Similarity an Intermeiate Asymptotics. Consultants Bureau, New York. [5] Polyanin, A.D. an Zaitsev, V.F. (). Hanbook o Exact Solutions or Orinary Dierential Equations. n Eition, Chapman & Hall/CRC Press, Boca Raton. [6] Abramowit, M. an Stegun, I. (97) Hanbook o Mathematical Functions. Dover, New York. 57