3-dimensional physically consistent diffusion in anisotropic media with memory
Caputo, Michele
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 9 (1998), p. 131-143 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

Some data on the flow of fluids exhibit properties which may not be interpreted with the classic theory of propagation of pressure and of fluids [21] based on the classic D’Arcy’s law which states that the flux is proportional to the pressure gradient. In order to obtain a better representation of the flow and of the pressure of fluids the law of D’Arcy is here modified introducing a memory formalisms operating on the flow as well as on the pressure gradient which implies a filtering of the pressure gradient without singularities; the properties of the filtering are also described. We shall also modify the second constitutive equation of diffusion, which relates the density variations of the fluid to its pressure variations, by introducing the rheology of the fluid also represented by derivatives of fractional order operating on the pressure as well as on the density. Moreover the medium will be considered anisotropic. We shall obtain the diffusion equation with these conditions in an anisotropic medium and find the Green function for a point source.

Alcuni dati sperimentali sul flusso di fluidi in mezzi porosi mostrano proprietà che non possono essere spiegate in base alla legge di D’Arcy che stabilisce proporzionalità fra flusso e gradiente di pressione. Per spiegare questa fenomenologia in questa Nota si modifica la legge di D’Arcy introducendo un formalismo di memoria, che opera sia sul flusso che sul gradiente di pressione, che genera un filtraggio senza singolarità fisicamente inaccettabili. Nella Nota si modifica anche l’equazione che lega la variazione di pressione con quella del fluido mediante l’introduzione di un secondo formalismo di memoria. Entrambi i formalismi di memoria sono rappresentati da derivate di ordine frazionario. Il mezzo è assunto anisotropico. Si trovano la trasformata di Laplace della funzione di Green in generale e la sua antitrasformata in casi particolari.

Publié le : 1998-06-01
@article{RLIN_1998_9_9_2_131_0,
     author = {Michele Caputo},
     title = {3-dimensional physically consistent diffusion in anisotropic media with memory},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {9},
     year = {1998},
     pages = {131-143},
     zbl = {0948.76075},
     mrnumber = {1677270},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1998_9_9_2_131_0}
}

Caputo, Michele. 3-dimensional physically consistent diffusion in anisotropic media with memory. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 9 (1998) pp. 131-143. http://gdmltest.u-ga.fr/item/RLIN_1998_9_9_2_131_0/

[1] Barry, D. A. - Sposito, G., Analytical solution of a convection dispersion model with time dependent transport coefficient. J. Geophys. Res., 25,12, 1989, 2407-2416.

[2] Lee Bell, M. - Amos Nur, , Strength Changes Due to Reservoir-Induced Pore Pressure and Stresses and Application to Lake Oroville. J. Geophys. Res., 83, 89, 1978, 4469-4483.

[3] Bella, F. - Biagi, P. F. - Caputo, M. - Cozzi, E. - Della Monica, G. - Ermini, A. - Plastino, W. - Sgrigna, V., Aquifer-induced seismicity in the central Apennines (Italy ). Pure and Applied Geophysics, 1998, in press.

[4] Biot, M. A., General theory of three dimensional consolidation. J. Appl. Phys., 12, 1941, 155-164. | JFM 67.0837.01

[5] Biot, M. A., General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech., 78, 1956, 91-96. | MR 75784 | Zbl 0074.19101

[6] Biot, M. A., Thermoelasticity and irreversible thermodynamics. J. Appl. Phys., 27, 1956, 240-253. | MR 77441 | Zbl 0071.41204

[7] Biot, M. A., Non linear and semilinear rheology of porous solids. J. Geophys. Res., 78, 1973, 4924-4937.

[8] Biot, M. A. - Willis, D. G., The elastic coefficients of the theory of consolidation. J. Appl. Mech., 24, 1957, 594-601. | MR 92472

[9] Boley, B. A. - Tolins, I. S., Transient coupled thermoelastic boundary value problem in the half space. J. Appl. Mech., 29(4), 1962, 637-646. | MR 147052 | Zbl 0114.16003

[10] Booker, J. R., Time dependent strain following faulting of a porous medium. J. Geophys. Res., 79, 1974, 2037-2044.

[11] Caputo, M., Elasticità e dissipazione. Zanichelli, Bologna1969.

[12] Caputo, M., Vibrations of an infinite plate with frequency independent Q. J. Acoust. Soc. Am., 60, 3, 1976, 634-639.

[13] Caputo, M., Relaxation and free modes of a selfgravitating planet. Geophys J. R. Astr. Soc., 77, 1984, 789-808.

[14] Caputo, M., Velocity of propagation of precursors of strong earthquakes and reduction of the alarm area. Rend. Fis. Acc. Lincei, s. 9, v. 3, 1992, 5-10.

[15] Caputo, M., The Green function of diffusion of fluids in porous media with memory. Rend. Fis. Acc. Lincei, s. 9, v. 7, 1996, 243-250. | Zbl 0879.76098

[16] Caputo, M., Mean fractional-order-derivatives differential equations and filters. Annali Univ. Ferrara, sez. VII, Scienze Matematiche, 41, 1995, 73-83. | MR 1444906 | Zbl 0882.34007

[17] Caputo, M., Modern rheology and electric induction: multivalued index of refraction, splitting of eigenvalues and fatigue. Annali di Geofisica, 39, 5, 1996, 941-966.

[18] Caputo, M., Diffusion of fluids in porous media with memory. Geothermics, in press.

[19] Hu, X. - Cushman, H., Non equilibrium statistical mechanical derivation of a nonlocal Darcy’s law for insaturated/saturated flow. Stochastic Hydrology and Hydraulics, 8, 1994, 109-116. | Zbl 0809.76087

[20] Körnig, H. - Müller, G., Rheological models and interpretation of postglacial uplift. Geophys. J. R., Astr. Soc., 98, 1989, 243-253.

[21] Mainardi, F., Fractional diffusive waves in viscoelastic solids. Appl. Mech. Rev., 46, 549, 1993, 93-97.

[22] Mcnamee, J. - Gibson, R. E., Displacement functions and linear transforms applied to diffusion through porous elastic media. Quart. J. Mech. Appl. Math., 13, 1960, 99-111. | MR 122169 | Zbl 0097.42103

[23] Nowacki, W., Green function for thermoelastic medium, 2. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys., 12, 1964, 465-472. | Zbl 0138.21003

[24] Rice, J. R. - Cleary, M. P., Some basic stress-diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev. Geophys. Space Phys., 14, 1976, 227-241.

[25] Roeloffs, E. A., Fault Stability Changes Induced Beneath a Reservoir with Cyclic Variations in Water Level. J. Geophys. Res., 93, B3, 1988, 2107-2124.

[26] Schneider, W. R. - Wyss, W., Fractional diffusion and wave equations. J. Math. Phys., 30, 1989, 134-144. | MR 974464 | Zbl 0692.45004

[27] Terzaghi, K., Die Berechnung der Durchassigkeitsziffer des Tones aus dem Verlauf der Hydrodynamischen Spannungsercheinungen. Sitzungsber. Akad. Wiss. Wien Math.-Naturwiss. Kl., Abt. 2A, 132, 1923, 105.

[28] Terzaghi, K., The shearing resistance of saturated soils. Proc. Int. Conf. Soil Mech. Found. Engin. Ist., 1, 1936, 54-55.

[29] Wyss, W., Fractional diffusion equation. J. Math. Phys., 27, 1986, 2782-2785. | MR 861345 | Zbl 0632.35031