A symmetry analysis of Richard s equation describing flow in porous media. Ron Wiltshire

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1 BULLETIN OF THE GREEK MATHEMATICAL SOCIETY Volume 51, ) A symmetry analysis of Richard s equation describing flow in porous media Ron Wiltshire Abstract A summary of the classical Lie method is presented as it applies to Richard s equation for water flow in an unsaturated uniform soil. In addition the more general potential symmetries for Richard s equations presented as a system are also given. These results are extended to give a new non-classical symmetry analysis based upon the method of Bluman and Cole. An example of a nonclassical symmetry reduction of Richard s equation is presented. Furthermore a new class of potential symmetries is derived. Keywords: Richard s equation, symmetry analysis, non-classical, potential. 000 Mathematics subject classification: 37L0 1. Introduction The symmetry analysis presented here is motivated by problems associated with water flow in unsaturated soils as described by for example Philip [1]. It is normal to describe such flow by means of Richard s equation [] which is considered here in the one dimensional form: x, t, u,u t,u x,u xx ) u t Du x ) x K u u x = 0 1) and where the soil will be taken to be homogeneous so that both the diffusivity D and hydraulic conductivity K are functions of u alone. In addition a suffix indicates a partial derivative. The use of symmetry or similarity methods to describe flow in unsaturated soil is not new and has found practical application in the theory of infiltration. For example, in the case of horizontal absorption Philip [3] describes travelling wave solutions based upon an ansatz using the Boltzmann similarity variable ω: u = u ω) ω = xt 1 ) although the resulting mathematical forms for the diffusivity are not well adapted for fitting empirical data. 93

2 94 Ron Wiltshire In a second example, with application to horizontal flow, with Neuman boundary conditions the infiltration process is described using the ansatz: u x, t) = ψ ω) t 1 n+ ω = xt n+1 n+ D u) = cu n 3) Similarity solutions of Richard s equation have been described by Sposito [4], Edwards [5] and El-labany et al [6] who have conducted a comprehensive classical Lie analysis of this equation and Sophocleous [7] has presented a partial classical analysis of the equation in its potential form. In addition, Gandarias [8] presents potential symmetries for a form of heterogeneous porous media with power law diffusivity and hydraulic conductivity. There is however no detailed analysis of the important nonclassical approach of Bluman and Cole [9] applied to equation 1) although Gandarias et al [10] has presented a non-classical analysis of the equation in a restricted form. It is the aim here to summarize the main classical symmetries of Richard s equation, excluding obvious cases, for example translation symmetries and furthermore to introduce new symmetries by undertaking both a non-classical and a potential symmetry analysis.. Classical results for Richard s equation In the classical Lie group method one-parameter infinitesimal point transformations,with group parameter ε are applied to the dependent and independent variables x, t, u) In this case the transformation are x = x + εη 1 x, t, u) + O ξ ) t = t + εη x, t, u) + O ξ ) 4) ū = u + εφ x, t, u) + O ξ ) and the Lie method requires invariance of the solution set Σ {u x, t), = 0}. This results in a system of overdetermined, linear equations for the infinitesimals η 1, η, φ. The corresponding Lie algebra of symmetries is the set of vector fields X = η 1 x, t, u) x + η x, t, u) + φ x, t, u) t u The condition for invariance of 1) is the equation: where the second prolongation operator X ) E 5) X ) E ) =0 = 0 6) is written in the form X ) E = X + φ[t] + φ [x] + φ [xx] 7) u t u x u xx and where φ [t], φ [x] and φ [xx] are defined through the transformations of the partial derivatives of u. In particular: ū x = u x + +εφ [x] x, t, u) + O ξ ) ū t = u t + +εφ [t] x, t, u) + O ξ ) ū x x = u xx + +εφ [xx] x, t, u) + O ξ ) 8)

3 A symmetry analysis of Richard s equation describing flow in porous media 95 Once the infinitesimals are determined the symmetry variables may be found from condition for invariance of surface u=u x, t): Ω=φ η 1 u x η u t = 0 9) Throughout the following it has been found convenient to set D = H u 10) and also, the MACSYMA program symmgrp.max due to Champagne et. al. [11] has been used to calculate the determining equations. In the case Richard s equation 1) the nine well known, for example, Sposito [4], Edwards [5], over-determined linear determining equations are : η u = 0 η x = 0 11) η 1u H u u η 1u u H u = 0 η u H u u + η u u H u = 0 1) η x H u u + η u x H u + η 1u = 0 φ x K u + φ xx H u φ t = 0 13) η x H u K u φh u u +η xx H u η t H u + η 1x H u = 0 14) η 1u H u K u + φh u H u u u φh u u + φ u H u H u u + φ u u H u η 1ux H u = 0 15) φh u K uu φh uu K u + η 1x H u K u +φ x H u H uu + φ ux H u η 1xx H u + η 1t H u = 0 16) As may be seen from Table 1 the classical symmetries are given for power and exponential functions of H and K in which the infinesimal η 1 and η are linear functions of x and t and where φ is linear in u. Note that each of these symmetries has been used by Edwards [5] to reduce Richard s equation to an ordinary differential equation. Functions H and K Symmetries λ K=ku µ λ K=k ln u λ K=ku ln u u) H=ce λu K=ke µu η 1=λ µ)x, φ=u η =λ µ+1)t η 1=λx,φ=u η =λ+1)t η 1 =λ 1)x kt η =λ 1)t, φ=u η 1 =λ µ)x η =λ µ)t, φ=1 H=ce λu, K=ku η 1 =λx kt, η =λt, φ=1, K=ku K=ku η 1 = x, η = t, φ=u η 1= kxt, η = kt φ=x+kut, K=ku η 1= kt, η =0, φ=1 Table 1. Classical symmetries of Richard s equation based on the comprehensive analysis of Sposito [4], Edwards [5], and excluding obvious cases such as translation symmetries)

4 96 Ron Wiltshire 3. Non-classical point symmetry The non-classical method is a generalisation of the classical Lie group approach due to Bluman & Cole [9] that incorporates the invariant surface condition 9) into the condition 6) for form invariance of Richard s equation 1). It follows that nonclassical symmetries of Richard s equation may be found by solving the non-linear set of determining equations: X ) E ) =0, Ω=0 = 0 17) To apply 17) two cases must be considered as follows Case A η = 1 η 1 η x, t, u) It is straight forward to show that there are four non-linear determining equations as follows: H u η u H u u η u u H u ) = 0 18) φ x H u K u + φ H u u + φ xx H u φ t H u η x φh u = 0 19) η u H u K u + φh u H u u u φhu u + φ u H u H u u + φ u u Hu η ux Hu + ηη u H u = 0 0) φh u K u u φh u u K u + η x H u K u + φ x H u H u u ηφh u u 1) + φ ux Hu η xx Hu η u φh u + η t H u + ηη x H u = 0 Solutions of these equations will be generated below. 3.. Case B η = 0 η 1 η 1 In this case the four determining equations are: φh u u = 0 ) φ x K u + φ xx H u φ t = 0 3) φh u H u u u φh u u + φ u H u H u u + φ u u H u = 0 4) φh u K u u φh u u K u + φ x H u H u u + φ ux H u = 0 5) There is only one solution of these equations as follows, namely the infinite symmetry where g = g x, t) and satisfies η = 1 H = cu K = ku φ = u x = c 0u + g 6) cg xx + kg x g t = 0 7)

5 A symmetry analysis of Richard s equation describing flow in porous media Examples of symmetries for Case A Using 18) and 0) the following explicit forms for η x, t, u) and φ x, t, u) may be obtained: η x, t, u) = f x, t) H u) + g x, t) 8) φ x, t, u) = Hu 1 { fx H fz u) + fgx u) +f W u) + HS x, t) + R x, t) } 9) where f, g, R, S depend on x, t and W, X and Z depend only on u. Examples of explicit non-classical symmetries may now be found by considering two sub-cases f = 0 and f 0 separately Sub-case f = 0 When f = 0 equations 8) and 9) become: η = g φ = H 1 u {HS x, t) + R x, t)} 30) and the determining equations 19) and 1) are now H H u u K u HH u K u u + g H u u ) R + HS) + Hu 3 g xx S x ) Hu g x K u + g t + g g x ) = 0 31) and Hu 3 R xx + HS xx ) + H u u R + HS) + HuK u R x + H S x ) g x Hu R + H S) Hu R t + H S t ) = 0 3) The following three solutions of these equations have found. In the first case: f = 0 H = c u K = k u 33) where c and k are constants and where the infinitesimals are η = c e kx c φ = c ku e kx c 34) c To generate a similarity solution of Richard s equation these infinitesimals may be substituted into the surface invariant condition 9) and the method of characteristics employed to determine the following ansatz for u x, t): u x, t) = ψ ω) e kx c 35)

6 98 Ron Wiltshire The similarity variable ω x, t) is given by: e kω c = e kx c kc t c 36) Substitution of these relationships into Richard s equation 1) gives the ordinary differential equation: 4cψ ωω ψ 8cψ ω ψ 3 kψ ω ψ c e kω c ψω = 0 37) Clearly the equations 36) and 37) together define ψ ω) and ω x, t) to give the final form of the solution 35). For the second symmetry the following solutions of 31) and 3) have also been found: f = 0 H = arbitrary K = arbitrary 38) with infinitesimals: η = η t) φ = 0 39) and in a third case solutions of 31) and 3) may be obtained when: This gives rise to the infinitesimals: η = x kt + c t + c 0 f = 0 H = cu K = ku 40) φ = c 4 + c 1 u + c 6 e c3cc3t kt x) t + c 0 41) The similarity solutions corresponding to the second and third symmetry follows as for the first symmetry. However the details will not be presented here. 4.. The sub-case f 0 For this sub-case substitution of 8) and 9) into the determining equation 0) gives rise to the following condition for f and g : a 0 ga fa 4 = 0 4) where a i are constants for which the functions Z, X and W satisfy: Z u K = a 0 u + a 1 H u W u + H u X u H u + u = a u + a 3 43) Hdu = a 4 u + a 5 44)

7 A symmetry analysis of Richard s equation describing flow in porous media 99 The remaining two determining equations 19) and 1) have the lengthy form: f x 4H u Z u H H u K u u + H K u H u u H H uk u ) + f Z H u K u u K u H u u ) + f H H u u Z H uz H u K u u W + K u H u u W ) + f g H u u Z H u K u u X + K u H u u X) + f x g 4HuX u + H H u u H Hu) + f gx H u X u HHu ) + f g H H u u X + Hu X + H u u W ) + f g H u u X + ff x H 3 H u u 8H uw u ) + f 3 W HH u u + H u ) H 3 u S x + K u H u u H u K u u ) R + HS) + f HH u u + H u) R + HS) + g H u u R + HS) g x H uk u 3 f xx H H 3 u + g xx H 3 u f t H H u g t H u gg x H u = 0 45) 4 f H u u Z + f f x 4 H H u u Z + 4 H H uz + 4 H uk u W ) + f xx H H u K u H 3 u Z ) + f g x 4H u Z + H u K u X ) 8 f g H u u X Z 8 f 3 H u u W Z 4 f H u u Z R + HS) f x Hu K u Z + f t Hu Z + 4 f g H u u X + f f x g 4 H H u u X 4 H Hu X ) + f x g x 4 H 3 u X H Hu ) + 8 f 3 g H u u W X + 4 f gh u u X R + HS) + f x ghuk u X + f g xx HuX 3 + f xx g Hu 3 X f g t Hu X 4 f g g x Hu X f t g HuX + 4 f 4 H u u W + f f x 4 H H u u W 4 H Hu W ) + fx 4 H 3 u W + H 4 H u u H 3 Hu ) + 4 f H u u W R + HS) + 4 f f xx H 3 uw 4 f g x H uw 4 f f t H u W H u R t + HS t ) + H 3 u R xx + HS xx ) + H u K u R x + HS x ) + H u u R + HS) + f x H H u u H H u) R + HS) g x H u R + HS) + f xxx H H 3 u f xt H H u = 0 46) The following three symmetry solutions have been found have been found from these equations. In the first sub-case: f = c 0 K u = c 0 H 47) where c 0 is constant and the corresponding infinitesimals are: η = c 0 H φ = 0 48)

8 100 Ron Wiltshire Secondly the sub-case f = k c H = cu K = ku 49) gives rise to the infinitesimals: η = k u + h) φ = k u u + h) c 50) where h = h x, t) such that h t = ch xx + khh x 51) and finally the sub-case: c 1 λ f = c 0 + c 1 x + c t H = ce λu K = ke λu 5) yields: η = cc 1λe λu c 0 + c 1 x + c t φ = kc 1λe λu c λ c 0 + c 1 x + c t) The corresponding similarity solutions for these sub-cases giving the solutions of Richard s equations will be considered elsewhere. 53) Functions H and K H= u c K= u k H, K arbitrary H=Hu) K u= c 0H K=ku H=ce λu K=ke λu Symmetries φ = ηu x +u t η=c e kx c φ= c ku c e kx c η=c 0H φ=0 η=ηt) φ=0 η=ku+h) φ= k u u+h) c h t =ch xx +khh x cc η= 1 λe λu c 0 +c 1 x+c t φ= kc 1 λeλu c λc 0 +c 1 x+c t) Table : Examples of non-classical symmetries of Richard s equation 5. Potential symmetries Richard s equation may also be written as the potential system 1, ) = 0 where: 1 = v x u = 0 = v t H u u x K = 0 54)

9 A symmetry analysis of Richard s equation describing flow in porous media 101 and this form gives rise to many new important symmetries. In this case the classical Lie analysis is based upon the infinitesimal transformations of the four variables: x = x + εη 1 x, t, u, v) + O ξ ) t = t + εη x, t, u, v) + O ξ ) ū = u + εφ 1 x, t, u, v) + O ξ ) v = v + εφ x, t, u, v) + O ξ ) 55) Note that since u = v x these define contact transformations for v. The associated generator: X = η 1 x + η t + φ 1 u + φ 56) v and the condition for form invariance of 54) is found by applying the first prolongation so that now X 1) E ) 1 =0, =0 = 0 57) It may be shown that the seven linear determining equations are: η u = 0 η v H u η 1u = 0 58) uη v + η x = 0 φ u uη 1u = 0 59) η 1u K + φ 1 H u u φ v H u + φ 1u H u + η t H u η 1x H u = 0 60) φ u K uη 1u K uφ v H u φ x H u + φh u + u η 1v H u + uη 1x H u = 0 61) φ 1 H u K u η 1u K φ 1 H u u K φ 1 H u K + uη 1v H u K + η 1x H u K + uφ 1v H u + φ 1x H u φ t H u + uη 1t H u = 0 6) There are two main cases. to be considerd. In the first η 1v second, considered in detail by Sophocleous [7]) η 1v = 0. 0 whilst in the 5.1. The case η 1v 0 The determining equations have the following new symmetry solution: H = c u c 0 + uc 1 ) 1 c 0 1 K = ku u c 0 + uc 1 ) 1 c 0 1 when c 0 0, 1 63) H = ce 1 c 1 u K = kue 1 c 1 u when c 0 = 0 64) ) u H = c ln K = ku when c 0 = 1 65) 1 + c 1 u

10 10 Ron Wiltshire with infinitesimals η 1 = c 0 x + c 1 v η = c 3 + t φ 1 = u c 0 + c 1 u) φ = 0 66) A detailed discussion of the corresponding similarity solutions of Richard s equation is the subject of on going research The case η 1v = 0 In this case it follows that η = η t) and the corresponding results have been analysed by Sophocleous [7]. Table 3 is based upon his comprehensive work in which all the known symmetries for this case are presented excluding the obvious translational symmetries. Functions H and K K=ku µ +k0u K=ke λu +k0u K=k ln u+k0u K=ku ln u+k0u Symmetries η 1 =1 µ)x k 0 t), φ 1 =u η =1 µ)t, φ = µ)v η 1 =µk0t x), η = µt φ 1 =1, φ = µv+x+k 0 t η 1 = kt, η =t φ 1 =u, φ =v+kt η 1 =x k0t, η =0 φ 1 =u, φ =v η 1 = kt, η =0 K=ku φ 1 =1, φ =x K=ku H=ce λu K=ku H=ce λu K=ke µu +k0u λ K=ku µ +k0u λ λ K=ku K=ku lnu)+k0u λ K=k lnu)+k0u η 1 = kxt, η = kt φ 1 =x+kut, φ =ct+ t η 1 =λx kt, η =λt φ 1 =1, φ =x+λv η 1 =λ µ)x k0µt, η =λ µ)t φ 1 =1, φ =x+k0t+λ µ)v η 1 =λ µ)x+k 0 µ 1)t, φ 1 =u η =λ µ+1)t, φ =λ µ+1)v η 1 = λ 1) x+kt), η =0 φ 1 =u, φ = λ+1) v η 1 =λ 1)x kt, φ 1 =u η =λ 1)t, φ =λv η 1 =λx k0t, φ 1 =u η =λ+1)t, φ =λ+1)v+kt Table 3 Potential symmetries of Richard s equation based upon the comprehenvive analysis of the case η = η t) by Sophocleous [7])

11 A symmetry analysis of Richard s equation describing flow in porous media 103 References [1] Philip, JR., Quasianalytic & analytic approaches to unsaturated flow. In Flow & Transport in the natural environment: Advances & applications. WL Steffen & OT Denmead Eds). Springer-Verlag [] Richard s, LA. Capillary conduction of liquids through porous mediums Physics [3] Philip JR., The theory of infiltration: 1. Soil Science [4] Sposito, G., Lie group invariance of Richard s equation. Fluids in heirarchical porous medium. ed J. Cushman New York: Academic) [5] Edwards, MP, Classical symmetry reductions of nonlinear diffusion-convection equations. Phys Lett.A [6] El-labany, Sk, Elhanbaly, AM & Sabry R., Group classification and symmetry reduction of variable coefficient non-linear diffusion-convection equation. J. Phys. A: Math. Gen [7] Sophocleous. C., Potential symmetries of nonlinear diffusion convection equations. J. Phys. A. Math. Gen , [8] Potential symmetries of a porous medium equation. J. Phys. A: Math. Gen [9] Bluman, GW & Cole, JD, The general similarity solution of the heat equation. J. Math. Mech. 1969, 18, [10] Gandarias, Ml, Romero, JL & Diaz, JM. Nonclassical symmetry reductions of a porous medium equation with covection. J. Phys A: Math. Gen., [11] Champagne, B, Hereman, W & Winternitz, P., The computer calculation of Lie point symmetries of large systems of differential equations. Comput Phys. Commun. 1991, 66, Ron Wiltshire Division of Mathematics & Statistics School of Technology University of Glamorgan Pontypridd CF37 1DL Wales - UK rjwiltsh@glam.ac.uk