ON THE LAUWERIER FORMULATION OF THE TEMPERATURE FIELD PROBLEM IN OIL STRATA

Transcription

1 ON THE LAUWERIER FORMULATION OF THE TEMPERATURE FIELD PROBLEM IN OIL STRATA LYUBOMIR BOYADJIEV, OGNIAN KAMENOV, AND SHYAM KALLA Received 3 January 5 Dedicated to Professor Dr. Ivan Dimovski, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria, on the occasion of his 7th birthday The paper is concerned with the fractional extension of the Lauwerier formulation of the problem related to the temperature field description in a porous medium sandstone) saturated with oil strata). The boundary value problem for the fractional heat equation of order <α and thelaplacetransform.thesolutionisobtainedinanintegralform,wheretheintegrand is expressed in terms of a convolution of two special functions of Wright type. is solved by means of the Caputo differintegration operator D α). Introduction The problem of describing the temperature field u = ux, y,z,t) of the strata arises when a hot water or steam is injected into the strata in order to facilitate the oil extraction process. Two cases of fluid injection are mainly considered: linear and radial injections. In the linear case, a hot fluid is forced into the strata in the positive and negative x-directions with constant velocity through an infinitely long vertical gallery. In the radial case, a hot fluid is forced through an infinitely thin well, which is considered as a linear source of incompressible fluid with positive volume rate. Theheatequationforaporousmediumisderivedin[] under several generally accepted assumptions on the model. Beside the exact formulation of the temperature field problem in oil strata, the following three approximate formulations are also treated: i) the lumped formulation, where the thermal conductivity of the strata is infinitely large in the vertical direction; ii) the incomplete lumped formulation, where the horizontal heat transfer in the cap and base rocks surrounding the strata is neglected; iii) the formulation of Lauwerier, where the horizontal heat transfer in the strata is also neglected. Our interest in this paper is particularly addressed to the fractional generalization of the Lauwerier formulation since analogous problems concerning the lumped formulations as well as the incomplete lumped formulation have been recently studied in []. The Lauwerier formulation relates to the temperature field of a single layer stratum in the case the velocity of the heat transfer between the fluid and the skeleton is finite. In this Copyright 5 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 5: 5) DOI:.55/IJMMS.5.577

2 578 Fractional Lauwerier problem in oil strata case, one has to consider separately the temperatures ux,t) and Θx,z,t) of the fluid and the cap rock, respectively. The heat equation for the regions containing the fluid and the skeleton, respectively, is derived [, Chapter 8, Section 8.7] under the main assumption that instead of having two regions containing the fluid and the skeleton separately, there is just a single region which is taken to be a porous medium. For a sufficiently large filtration velocity, one can neglect the heat transferred to the cap rock and stratum in the x-direction in comparison with the heat transfer in the z-direction. In this case, the Lauwerier formulation for the linear fluid injection reads as z = : u t Θ t = Θ z, <x,z,t<, u = γ αu Θ), x z = : Θ t = µ Θ + ku Θ), z <x,t<, <x,t<,.) x = :u =,.a) u,θ asx + z,.b) t = :u = Θ =..c) The physical meaning of the constants involved can be briefly described as follows: i) the constant γ> depends on the volume rate of the hot fluid forced into the strata, the porosity specified as the ratio of the pores volume to the whole volume, and the coefficient of thermal conductivity of the cap rock; ii) the constant α> depends on the porosity, the coefficients of thermal conductivity of the cap rock, and the volumic heat capacity of the fluid; iii) the constant µ>dependsonthecoefficient of thermal conductivity of the cap rock, the volumic heat capacity of the skeleton, and the coefficient of thermal conductivity of the cap rock; iv) the constant k> depends on the porosity and the volumic heat capacity of the fluid and the skeleton. By using the Laplace transform f p) = L [ f t) ] = e pt f t)dt, Rep) >,.3) the solution of the problem.) and.) is given by the following formulas [, Chapter 8, formulas 8.7.4) and )]: ux,t) t x/γ = e αx/γ e kτ µt x/γ τ) πt x/γ τ) 3/ e τµ/ t x/γ τ) + k erfc τµ t x/γ τ ) kαx I γ τ dτ,.4)

3 Lyubomir Boyadjiev et al. 579 ) t x/γ Θx,z,t) = ke αx/γ e kτ erfc z + µτ kαx I dτ, t x/γ τ γτ.5) where I z) is the modified Bessel function of the first kind with zero index [6]and erfcx) = e t dt.6) π is the complementary error function [5]. The main goal of this paper is to formulate in a reasonable way and to solve the fractional generalization of the problem.)and.). Further on in the text we refer to this problem as fractional Lauwerier formulation of the temperature field problem in oil strata.. Fractional heat equation The concept of a noninteger differentiation and integration of a function is almost as old as calculus itself. Many famous mathematicians including Leibniz, Euler, Lagrange, Laplace, Fourier, and Abel made some contribution to it. Nowadays there exist specialized treaties where mathematical aspects and applications of the fractional calculus are extensively discussed [3, 4, 7, 9]. Fractional calculus is a significant topic in mathematical analysis as a result of its increasing range of applications. One of the main advantages of the fractional calculus is that the fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. For our purpose, we adopt in this paper the Caputo fractional derivative D β Γm β) f t) = t x f m) τ) t τ) β+ m dτ, m <β<m, d m f t) dt m, β = m,.) where m is a positive integer. This definition was introduced by Caputo in the late sixties of the twentieth century and adopted by Caputo and Mainardi in the framework of the theory of linear viscoelasticity. The method we apply makes the following rule of the Laplace transform extremely important: L [ D β f t) ] = p β f m p) f k )p β k, m <β m..) k= The modelling of diffusion in a specific type of porous medium is one of the most significant applications of fractional-order derivatives [, 8]. The fractional diffusion equation deduced by Nigmatullin [5, 6]in the form β u t β = a u z, <β,.3)

4 58 Fractional Lauwerier problem in oil strata deserves special attention. This equation is also know as the fractional diffusion-wave equation [, ]. The reason is that when the order of the fractional derivative is β =, the equation becomes the classical diffusion equation, and if β =, it becomes the classical wave equation. For < β<, we have the so-called ultraslow diffusion. Values < β< correspond to the so-called intermediate processes [8]. In this paper, we consider the fractional diffusion equation.3) when the fractional time derivative is determined by the Caputo fractional derivative.). Thus we extend the problem.) and.) to the fractional Lauwerier formulation of the temperature field problem in oil strata that reads as D β Θ = Θ z, <x,z,t<, <β,.4) subject to the boundary conditions and the conditions z = :D β u = γ u αu Θ), x <x,t<, z = :D β Θ = µ Θ.5) z + ku Θ), <x,t<, 3. Auxiliary results x =, z = :u =,.6a) u,θ asx + z,.6b) t = :u = Θ =..6c) With the idea to make this paper relatively self-contained, we summarize in this section some important facts that enable us to solve the fractional problem stated above. i) First we mention that a key role in obtaining the solutions.4) and.5) isgiven to the followingtheorem [3, pages 35 36]. Theorem 3. Efros theorem). Assume there exist analytic functions Gp) and qp) and the relations Fp) = L [ f t) ], e τp) Gp) = L [ gt,τ) ]. 3.) Then Gp)F qp) ) [ ] = L f τ)gt,τ)dτ. 3.) Theorem 3. is simply a generalization of the convolutional theorem for the Laplace transform. Indeed, if we take qp) = p,then L [ gt,τ) ] = e pτ Gp), 3.3)

5 Lyubomir Boyadjiev et al. 58 and by the p-shift theorem, gt,τ) = gt τ). Thus formula 3.9)becomes [ ] [ t ] Gp)Fp) = L f τ)gt,τ)dτ = L f t)gt τ)dτ, 3.4) since for original functions we have gt τ) = ast<τ. ii) The fundamental solution of the basic Cauchy problem for the time fractional diffusion equation.3) can be expressed in terms of an auxiliary function defined as [] Mz;β) = dσ πi σ β, <β<, 3.5) H a e σ zσβ where H a denotes the Hankel path of integration that begins σ = ib b > ), encircles the branch cut that lies along the negative real axis, and ends up at σ = + ib b > ). It is also proved that the following relation takes place: Mz;β) = W z; β; β), 3.6) where Wz;λ,µ) = z n n!γλn + µ) = n= πi dσ e σ+zσ λ H a d µ, λ>, µ>, 3.7) is an entire function of z referred to as the Wright function cf. [7, Chapter 8]). In the particular case β = /, it holds that M z; ) = π m= ) m ) m z m m)! = e z/4. 3.8) π Further, we introduce the following auxiliary function: Nξ;β;λ) = dσ e σ ξσβ πi H a σ λ, λ>, <β, 3.9) and prove the following statement. Lemma 3.. If <β / and <t,τ,z<, then the following relations hold: i) ii) e z+µτ)pβ = L [ g t,τ,z;β) ], µ>; 3.) p λ e x/γ+τ)pβ = L [ g t,τ,β;λ) ], λ>, x>, γ>, 3.)

6 58 Fractional Lauwerier problem in oil strata where ) z + µτ)β z + µτ g t,τ,z,β) = t β+ M t β ;β, 3.) ) x/γ + τ g t,τ;β;λ) = t λ N ;β;λ. 3.3) t β Proof. The validity of i) is a direct consequenceof [, formulas 3.4) and 3.5)]. To prove part ii), we consider the Laplace transform L [ g p,τ;β;λ) ] = p λ e x/γ+τ)pβ, λ>. 3.4) According to the inversion formula for the Laplace transform, g t,τ;t;β;λ) = [ ] e pt πi H a p λ e x/γ+τ)pβ dp. 3.5) Setting σ = pt and ξ = x/γ + τ)/t β,weobtain Hence, if < β<, In particular, g t,τ;β;λ) = tλ dσ e σ ξσβ πi H a σ λ. 3.6) g t,τ;β;λ) = t λ W ξ; β;λ) = t λ Nξ;β;λ). 3.7) g t,τ; ) ;λ = tλ σ ξσ dσ e πi H a σ λ = [ ] e pt πi H a p λ e x/γ+τ)p dp. 3.8) Referring to [4, formulas 3) and 7)], we conclude that g t,τ; ) t ;λ s λ = Γλ) δ t x )ds, γ τ s 3.9) where Γλ) is the Euler gamma function and δt) is the Dirac delta function. It is also worth mentioning that if λ =, g t;τ; ), = H t x ) γ τ, 3.) where Ht a) is the Heaviside function.

7 4. Fractional Lauwerier formulation Lyubomir Boyadjiev et al. 583 We apply the Laplace transform method to prove the following main results. Theorem 4.. If <β / and λ>, the solutions of the fractional Lauwerier formulation of the temperature field problem in oil strata.4),.5),and.6) are given by the integrals ux,t) = e αx/γ+τk)[ ϕt,τ;β) g t,τ,;β) ] ) αkx I γ τ dτ, 4.) Θx,z,t) = k e αx/γ+τk)[ g t,τ;z;β) g t,τ,β;) ] ) αkx I γ τ dτ, 4.) where g t,τ,z;β) and g t,τ;β;) are defined by 3.)and3.3), respectively, and Proof. We denote ϕt,τ;β) = g t,τ;β; β)+µg t,τ;β; β)+kg t,τ;β;). 4.3) ūx, p) = L [ ux,t) ], Θx,z, p) = L [ Θx,z,t) ], 4.4) where L is the Laplace transformation operator. Applying the Laplace transform to.4),.5), and.6), we obtain, in accordance with.) and.6c), p β Θx,z, p) = Θx,z, p) z, <x, z<, 4.5) z = :p β ūx, p) ūx, p) = γ α [ ūx, p) Θx,z, p) ], <x<, x z = :p β ūx,z, p) = µ Θx,z, p) + k [ ūx, p) Θx,z, p) ] 4.6), <x<, x x = :ū, p) = p, 4.7a) The solution of 4.5) which remains bounded as z is ū,θ asx + z. 4.7b) Θx,z, p) = cx, p)e zpβ, 4.8) where cx, p) is still unknown function. Substituting 4.8) into4.6), we have p β ūx, p) ūx, p) = γ α [ ūx, p) cx, p) ], x 4.9) p β cx, p) = µp β cx, p)+k [ ūx, p) cx, p) ]. 4.) Solving 4.)forcx, p), we get cx, p) = kūx, p) p β + µp β + k. 4.)

8 584 Fractional Lauwerier problem in oil strata The substitution of this representation of cx, p) into4.9) leads to the equation ūx, p) x = γ The solution of this equation that conforms to 4.7a)is Then obviously cx, p) = and from 4.8) it follows that Θx,z, p) = k p ) p β αk + α p β + µp β ūx, p). 4.) + k ūx, p) = [ αkx p exp γ p β + µp β + k p β + α ) x γ ]. 4.3) k p β + µp β + k [ ] αkx p exp γ p β + µp β + k pβ + α)x, 4.4) γ To apply Theorem 3.,werepresent [ e zpβ αkx p β + µp β + k exp γ p β + µp β + k p β + α ) x γ ]. 4.5) ūx, p) = F [ qp;β) ] Gx, p;β), 4.6) where F [ qp;β) ] = [ ] αkx qp;β) exp γ, 4.7) qp;β) Gx, p;β) = pβ + µp β + k p [ exp x p β + α )]. 4.8) γ Thewell-knownformula[4, page 47,.4.8)] enables us to get Fp) = L [ f t) ] )] αkx = L [I γ t. 4.9) Further, we consider the function e τqp;β) Gx, p;β) = pβ + µp β + k p [ ) )] x αx exp γ + τ p β τµp β γ + τk. 4.) From Lemma 3. and the convolution theorem, we obtain e x/γ+τ)pβ p β e τµpβ = L [ g t,τ;β; β) g t,τ,;β) ], e x/γ+τ)pβ p β e τµpβ = L [ g t,τ;β; β) g t,τ,;β) ], e p e x/γ+τ)pβ τµpβ = L [ g t,τ;β;) g t,τ,;β) ]. 4.)

9 Taking into account 4.)as well as the formulas4.), we can write where Lyubomir Boyadjiev et al. 585 e τqp;β) Gx, p;β) = L [ gt,τ) ], 4.) gt,τ) = e αx/γ+τk)[ ϕt,τ;β) g t,τ,;β) ]. 4.3) The application of Theorem3. leads directly to the solution 4.). Following the same idea, we represent Θx,z, p) as the product where F[qp,β)] is defined by 4.7)and Obviously Θx,z, p) = F [ qp;β) ] G x, p;β), 4.4) G x, p;β) = k p e zpβ e x/γ)pβ e αx/γ. 4.5) e τqp;β) G x, p;β) = ke αx/γ+kτ) e z+µτ)pβ p e x/γ+τ)pβ. 4.6) Then as a direct consequence of Lemma 3. and the convolution theorem, we obtain e τqp;β) G x, p;β) = ke αx/γ+kτ)[ g t,τ,z,β) g t,τ;β;) ]. 4.7) Then 4.4), 4.7), and Theorem 3. lead to the solution 4.) and this completes the proofofthetheorem. Corollary 4.. For the particular case β = /, the solution 4.)reducesto.4). Proof. Because of 3.8), we evidently can write g t,τ,; ) = τµ t) 3/ e τµ/. 4.8) πt According to 3.3)and[9, Section 7.3, formula 7)], g t,τ; ) ; = x/γ + τ t N ; ) t ; = e pt[ e x/γ+τ)p] dp = δ t x ) πi H a γ τ H t x ) γ τ. 4.9) Likewise from 3.3), [9, Section 7.3, formula 3)], and the t-shifting theorem for the Laplace transform it follows that g t,τ; ; ) = x/γ + τ t / N ; t ; ) = [ ] Ht x/γ τ) 4.3) e x/γ+τ)p dp = πi p/. π t x/γ τ) H a e pt

10 586 Fractional Lauwerier problem in oil strata Taking into account 3.), we obtain that ϕ t,τ; ) = δ t x ) y τ + µ + k H π t x/γ τ) t x γ τ ). 4.3) Formulas 4.8) and4.3) yield that in the case β = / the convolution into the integrand of 4.) takes the form ϕ t,τ; ) g t,τ,;β) = τµht x/γ τ) t x/γ τ π s) e τµ/ δ t x ) s3/ γ τ s ds + τµ Ht x/γ τ) t t s) e τµ/ ds π x/γ+τ t s) 3/ s x/γ τ) + kτµ π H t x γ τ ) t x/γ+τ t s) e τµ/ ds. t s) 3/ 4.3) Thethirdintegralin4.3) can be computed as τµ t t x/γ τ t s) 3/ e tµ/ t s) ds = e τµ/ t x/γ τ s ) d τµ x/γ+τ t x/γ τ s = e w dw = π erfc τµ τµ/ t x/γ τ). t x/γ τ 4.33) By using [9, Section 3.47, formula 3)], one can see that t x/γ+τ t s) s 3/ x ) / t x/γ τ γ τ e tµ/ t s) ds = x t 3/ x ) / γ τ x e τµ/ x) dx = π e τµ/ t x/γ τ). τµ t x/γ τ) 4.34) For the first integral in 4.3), it is clear that t x/γ τ s) eτµ/ δ t x ) s3/ γ τ s ds = t x ) 3/ γ τ e τµ/ t x/γ τ). 4.35)

11 Lyubomir Boyadjiev et al. 587 The substitution of these integrals by their expressions into 4.3) results in ϕ t,τ; ) g t;τ,;β) µ [ t x/γ) τ ] = πt x/γ τ) 3/ e τµ/ t x/γ τ) + k erfc τµ t x/γ τ H t x ); γ τ 4.36) that proves the statement. Corollary 4.3. For the particular case β = /, the solution 4.)reducesto.5). Proof. From 3.8) and3.), it follows that g t,τ,z; ) = z + µτ t 3/ e z+µτ)/ t). 4.37) π Then according to 3.),the convolutioninto the integrand of 4.) takes the form g t,τ,z; ) g t,τ; ) ; t = z + µτ e z+µτ)/ πt s) 3/ = H t x ) γ τ π = H t x ) γ τ π t x/γ+τ t x/γ τ t s) H s x ) γ τ ds z + µτ t s) e z+µτ)/ ds t s) 3/ z + µτ t x/γ τ ξ) 3/ e z+µτ)/ t x/γ τ ξ) dξ = H t x ) t x/γ τ γ τ π e z+µτ)/ t x/γ τ ξ) d z + µτ t x/γ τ ξ = H t x ) γ τ π e w dw z+µτ)/ t x/γ τ = H t x ) γ τ erfc z + µτ, t x/γ τ and proves the statement. 4.38) Acknowledgment The first author was partially supported by NSF-Bulgarian Ministry of Education and Science under Grant no. MM 35/3.

12 588 Fractional Lauwerier problem in oil strata References [] M. Ya. Antimirov, A. A. Kolyshkin, and R. Vaillancourt, AppliedIntegralTransforms, CRM Monograph Series, vol., American Mathematical Society, Rhode Island, 993. [] L.BoyadjievandR.Scherer,Fractional extensions of the temperature field problem in oil strata, Kuwait J. Sci. Engrg. 3 4), no., 5 3. [3] V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus, Pergamon Press, Oxford, 965. [4], Formulaire pour le calcul opérationnel, Masson & Cie, Paris, 967. [5] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. Vol. I, McGraw-Hill, New York, 953. [6], Higher Transcendental Functions. Vol. II, McGraw-Hill, New York, 953. [7], Higher Transcendental Functions. Vol. III, McGraw-Hill, New York, 955. [8] R. Gorenflo and R. Rutman, On ultraslow and on intermediate processes, Transform Methods and Special Functions P. Rusev, I. Dimovski, and V. Kiryakova, eds.), SCT Publishers, Singapore, 995, pp [9] I. S. GradshteynandI. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 98. [] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Waves and Stability in Continuous Media Bologna, 993) S. Rionero and T. Ruggeri, eds.), Ser. Adv. Math. Appl. Sci., vol. 3, World Scientific, New Jersey, 994, pp [], Fractionalrelaxation-oscillationandfractionaldiffusion-wave phenomena, Chaos Solitons Fractals 7 996), no. 9, [] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. FreemanandCompany, California, 98. [3] A. C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions, Research Notes in Mathematics, vol. 3, Pitman, Massachusetts, 979. [4] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, 993. [5] R. R. Nigmatullin, To the theoretical explanation of the universal response, Phys. Status Solidi B)3 984), [6], The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status Solidi B) ), [7] K. B. Oldham and J. Spanier, The Fractional Calculus, Mathematics in Science and Engineering, vol., Academic Press, New York, 974. [8] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol. 98, Academic Press, California, 999. [9] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science, Yverdon, 993. Lyubomir Boyadjiev: Institut für Praktische Mathematik, Universität Karlsruhe TH), 768 Karlsruhe, Germany address: lyubomir boyadjiev@math-uni-karlsruhe.de Ognian Kamenov: Department of Applied Mathematics and Informatics, Technical University of Sofia, P.O. Box 384, Sofia, Bulgaria address: okam@abv.bg Shyam Kalla: Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat 36, Kuwait address: kalla@mcs.sci.kuniv.edu.kw

13 Mathematical Problems in Engineering Special Issue on Time-Dependent Billiards Call for Papers This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome. To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors Edson Denis Leonel, Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 4A, 55 Bela Vista, Rio Claro, SP, Brazil ; edleonel@rc.unesp.br Alexander Loskutov, Physics Faculty, Moscow State University, Vorob evy Gory, Moscow 999, Russia; loskutov@chaos.phys.msu.ru Manuscript Due December, 8 First Round of Reviews March, 9 Publication Date June, 9 Hindawi Publishing Corporation