On the theory of pressure and temperature nonlinear waves in compressible fluid-saturated porous rocks

Abstract

Thermo-poro-elastic equations describing fluid migration through fluid-saturated porous media at depth in the crust are analyzed theoretically following recent formulations of Rice and Cleary (1976), McTigue (1986) and Bonafede (1991). In this study these ideas are applied to a rather general model, namely to a deep hot and pressurized reservoir of fluid, which suddenly enters into contact with an overlaying large colder fluid-saturated layer. In a one-dimensional idealization this system can be described by two nonlinear differential heat-like equations on the matrix-fluid temperature and on the fluid overpressure over the hydrostatic value. The nonlinear couplings are due to Darcy thermal advection and to the mechanical work rate. Here we first sketch nonlinear solutions corresponding to Burgers' "solitary shock waves", which have recently been found valid for rocks with very low fluid diffusivity. Subsequently other nonlinear transient waves are discussed, such as "thermal" and "compensated" waves, which are found to exist for every value of the parameters present in the equations involved. One interesting aspect of these mechanisms is that the resulting time-scales are particularly small. Moreover, in order to figure out the system time-evolution and the role played by the fluid diffusivityjthermal diffusivity ratio, a mechanical similitude is proposed, which we treat both analytically and numerically. Although for realistic systems these solutions are somewhat idealized, they allow one to gain fundamental insight into fluid migration mechanisms in volcanic areas and in fault regions under strong frictional heating. As already discussed by McTigue, the theory is also of interest in studying areas of nuclear waste disposal. Furthermore such a theoretical study allows one to investigate the site at depth at which such nonlinear waves are generated.

The Effect of Fluid Thermal Expansivity on Thermo-mechanical Solitary Shock Waves in the Underground of Volcanic Domains

“…These fundamental equations are the continuity equation, Darcy's law, the fluid-rock energy equation, the ideal gas state equation and the equilibrium equation of elasticity along with the constitutive relation between stress, strain, fluid pore pressure and fluid-rock temperature. We then write the two one-dimensional nonlinear heat-like equations of the thermo-poroelasticity theory which entails the stress-diffusion equation, as stated by MCTIGUE (1986), and suitably handled form of the energy equation, which starts with the formulation derived from a combination of the expressions proposed by BEJAN (1984), MCTIGUE (1986), NATALE andMERLANI et al (1997). Conceptual geological section of a stratified system for a hydrothermal domain.…”

“…The other equation to be associated to the stress-diffusion equation (1) is the energy equation, representing here an innovative aspect of the theory concerned as the fluid thermal expansivity is considered. Under the assumption that fluid and overburden porous-permeable rock are in local thermal equilibrium and neglecting the thermo-elastic coupling terms representing sources of adiabatic deformation (MCTIGUE, 1986;MERLANI et al, 1997), the one-dimensional fluid-rock energy equation (A4) in the Appendix is reduced to:…”

“…Therefore he concluded that the cause of ground deformation phenomena might not be attributed to a pressure increase of a deep magma chamber embedded within an elastic continuous medium (MOGI, 1958;WALSH and DECKER, 1971;DIETERICH and DECKER, 1975;OKADA, 1992), but likely to be due to thermal expansion of hot and pressurised water moving up towards the top of a shallower aquifer and thermo-elastically affecting the overburden rock. Theoretical developments of this study were then proposed by NATALE and SALUSTI (1996), MERLANI et al (1996MERLANI et al ( , 1997 and NATALE et al (1998) who found different classes of solutions related to different water migration mechanisms, and also concerning fracturing processes of the overburden rock as the fluid pore pressure approaches the breakdown pressure of the rock at depth. In particular NATALE andNATALE et al (1998), analytically developing the one-dimensional nonlinear heat-like equations, found that thermo-mechanical solitary shock waves as a solution of Burgers' equation can arise and propagate through a water-saturated porous-permeable horizon if a Reynolds' type number R, given by the ratio of convective effects to diffusive ones and fundamentally ruled by the value of rock permeability, is large.…”

Non-Polynomial Spline Method for a Time-Dependent Heat-Like Lane- Emden Equation

“…These physical models describe some important equations as the Navier-Stokes equations can be converted into various heat-like equation in some special cases [10]. Furthermore, thermo-pore-elastic equations describing fluid migration through fluid-saturated porous media at depth in the crust [11] and this theory [12] can be expressed in heat-like equations. The analytic solutions to several forms of the above problem were presented by Shawagfeh [4], Wazwaz [13][14][15][16] using the Adomian decomposition method.…”

Fracturing processes due to temperature and pressure nonlinear waves propagating in fluid‐saturated porous rocks

“…Since the classic solutions by McTigue [1986] and Rice and Cleary [1976] as well as the solitary T-P waves discussed by Natale and Salusti [1996] and Garcia and Natale [1999] and the nonlinear waves of both Merlani et al [1997] and Natale et al [1998] are all invariant for scale transformations, we limit ourselves to analyzing here the set of solutions f = f(z/X/•), which are invariant for such transformations, as this set of functions entails most of the physically interesting cases: indeed, novel nonlinear solutions are discussed here. Following up recent models discussed by Merlani et al [1997], a mechanical analogy of such processes, such as the motion of a material point under the effect of a time-dependent external force, is also presented and investigated both analytically and numerically.…”