Towards a Non-linear Theory for Induced Seismicity in Shales

Abstract We analyze the pore transmission of fluid pressure p and solute density ρ in porous rocks. To investigate shale deformation, fundamental for deep oil drilling and for gas extraction ("fracking"), a non-linear model of mechanic and chemo-poroelastic interactions among fluid, solute and the solid matrix is discussed. The solutions of the model are quick non-linear Burger's solitary waves, potentially destructive for deep operations. Following Civan [2], diffusive and shock waves are applied to fine particles filtration. Then the delaying effects of fine particles filtration is compared with fractional model time delay and the fractional order parameter can be realistically estimate

On the propagation of nonlinear transients of temperature and pore pressure in a thin porous boundary layer between two rocks.

The dynamics of transients of fluid-rock temperature, pore pressure, pollutants in porous rocks are of vivid interest for fundamental problems in hydrological, volcanic, hydrocarbon systems, deep oil drilling. This can concern rapid landslides or the fault weakening during coseismic slips and also a new field of research about stability of classical buildings. Here we analyze the transient evolution of temperature and pressure in a thin boundary layer between two adjacent homogeneous media for various types of rocks. In previous models, this boundary was often assumed to be a sharp mathematical plane. Here we consider a non-sharp, physical boundary between two adjacent rocks, where also local steady pore pressure and/or temperature fields are present. To obtain a more reliable model we also investigate the role of nonlinear effects as convection and fluid-rock “frictions”, often disregarded in early models: these nonlinear effects in some cases can give remarkable quick and sharp transients. All of this implies a novel model, whose solutions describe large, sharp and quick fronts. We also rapidly describe transients moving through a particularly irregular boundary layer

Additional Recursion Relations, Factorizations, and Diophantine Properties Associated with the Polynomials of the Askey Scheme

In this paper, we apply to (almost) all the "named" polynomials of the Askey scheme, as defined by their standard three-term recursion relations, the machinery developed in previous papers. For each of these polynomials we identify at least one additional recursion relation involving a shift in some of the parameters they feature, and for several of these polynomials characterized by special values of their parameters, factorizations are identified yielding some or all of their zeros—generally given by simple expressions in terms ofintegers(Diophantinerelations). The factorization findings generally are applicable for values of the Askey polynomials that extend beyond those for which the standard orthogonality relations hold. Most of these results are not (yet) reported in the standard compilations

Copernicus Marine Service Ocean State Report

This is the final version. Available from Taylor & Francis via the DOI in this record

Integrability, analyticity, isochrony, equilibria, small oscillations, and Diophantine relations: results from the stationary Burgers hierarchy

An isochronous system is introduced by modifying the Nth ODE of the stationary Burgers hierarchy, and then, by investigating its behaviour near its equilibria, neat Diophantine relations are identified, involving (well-known) polynomials of arbitrary degree having integer zeros, or equivalently matrices the determinants of which yield such polynomials. The basic idea to arrive at such relations is not new, but the specific application reported in this paper is new, and it is likely to open the way to several analogous new findings

Isochronous variant of the Ruijsenaars-Toda model: equilibrium configuration, behavior in their neighborhood, Diophantine relations

Gli articoli su questa rivista (e molte altre) vengono ora identificati con un numero e con la indicazione del numero di pagine

Memory Effects on Nonlinear Temperature and Pressure Wave Propagation in the Boundary between Two Fluid-Saturated Porous Rocks

The evolution of strong transients of temperature and pressure in two adjacent fluid-saturated porous rocks is described by a Burgers equation in an early model of Natale and Salusti (1996). We here consider the effect of a realistic intermediate region between the two media and infer how transient processes can also happen, such as chemical reactions, diffusion of fine particles, and filter cake formations. This suggests enlarging our analysis and taking into account not only punctual quantities but also “time averaged” quantities. These boundary effects are here analyzed by using a “memory formalism”; that is, we replace the ordinary punctual time-derivatives with Caputo fractional time-derivatives. We therefore obtain a nonlinear fractional model, whose explicit solution is shown, and finally discuss its geological importance

Tridiagonal matrices, orthogonal polynomials and Diophantine relations: I

It is well known that the eigenvalues of tridiagonal matrices can be identified with the zeros of polynomials satisfying three-term recursion relations and being therefore members of an orthogonal set. A class of such polynomials is identified some of which feature zeros given by simple formulae involving integer numbers. In the process certain neat formulae are also obtained, which perhaps deserve to be included in standard compilations, since they involve classical polynomials such as the Jacobi polynomials and other 'named' polynomials

Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials

Certain Diophantine conjectures are proven, and to do so certain remarkable classes of orthogonal polynomials are identified, yielding additional Diophantine findings