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A model equation derived by B. B. Kadomtsev & V. I. Petviashvili (1970) suggests that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. The theory is variational in nature. A simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties using a generalisation of the Lyapunov-Schmidt reduction procedure. A nontrivial critical point of the reduced functional is detected using the direct methods of the calculus of variations.

Studied here is an initial-and boundary-value problem for the Korteweg-de Vries equationposed on a bounded interval I = {x: a x b}. This problem features non-homogeneous boundary conditions applied at x = a and x = b and is known to be well-posed in the L 2 -based Sobolev space H s (I) for any s > − 3 4 . It is shown here that this initialboundary-value problem is in fact well-posed in H s (I) for any s > −1. Moreover, the solution map that associates the solution tothe auxiliary data is not only continuous, but also analytic between the relevant function classes. The improvement on the previous theory comes about because of a more exacting appreciation of the damping that is inherent in the imposition of the boundary conditions. Published by Elsevier Inc.

Capillary-gravity waves of finite amplitude at the interface of a two-fluid flow are considered. The fluids of constant densities, extended to infinity in each direction, are inviscid and incompressible. The flow is two-dimensional and irrotational in each fluid. Assume that there is a permanent wave at the interface moving with a constant speed and the wave decays to zero at infinity with a certain rate. It is shown that the wave must decay at infinity quadratically and the solution of the exact governing equations corresponding to such a wave must satisfy two integral identities for the net displacement and the energy of the wave, while the former implies non-existence of waves of pure depression or pure elevation in the flow.

In this paper, we study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential and the dimensionless parameters are the ratio between densities rho = rho(2)/rho(1) and lambda = gh/c(2). We study special values of the parameters such that lambda(1 - rho) is near 1(-), where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where U = 0 corresponds to a uniform state (velocity c in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with, in addition, a double eigenvalue in 0, a pair of simple imaginary eigenvalues +/-ilambda at a distance O(1) from 0, and for lambda(1 - rho) above 1, another pair of simple imaginary eigenvalues tending towards 0 as lambda(1 - rho) --> 1(+). When lambda(1 - rho)

Ordovician carbonate is important for petroleum exploration and production in the Tarim Basin. The matrix porosity of the carbonate reservoirs is usually less than 2% because of their age-old sedimentary history and depth (deeper than 5500 m). The storage spaces are secondary dissolution pores and fractures which are dominated by visible dissolution caves, holes, and fractures (Zhou et al., 2006). The reservoirs are therefore called fractured-cavernous reservoirs. Good quality reservoir usually appears as a strong reflection (called “bead-like reflection” or “BR”) on poststack seismic sections, and the corresponding reservoir is called a fracture-cave body (FCB) which is the main drilling target of current E&P in Tarim Basin. The FCBs are unevenly distributed in carbonate strata, and the storage spaces inside an FCB are also unevenly distributed. Therefore, the reservoirs have both macroscopic and microscopic heterogeneities, which bring great challenges to petroleum exploration.

C a r b o n a t e r e s e a r c h i n C h i n a 152 The Leading Edge February 2012 SPECIAL SECTION: C a r b o n a t e r e s e a r c h i n C h i n aThe theory and application of DEM-Gassmann rock physics model for complex carbonate reservoirs T he storage spaces of carbonate reservoirs in the Tarim Basin are dominated by secondary pore porosity such as dissolution caves, holes, and cracks (Song et al., 2001). Research has shown that pore shapes can significantly influence the velocities of elastic waves (Eshelby, 1957;Kuster and Toksöz, 1974;Xu and White, 1995;Yan et al., 2002;Sun et al., 2004). Empirical V P -V S relationships such as Castagna's (1985) mudrock line, Han's relations (1986), and Greenberg-Castagna's (1992) relationship that ignore the effect of pore geometry are not applicable to carbonates (Xu and White, 1995;Wang et al., 2009). Cheng and Toksöz (1979) imaged various pore space structures using SEM, and proposed a pore aspect ratio spectrum that can be used to explain velocity prediction. Han (2004) measured velocities of 52 different carbonate rocks, indicating that caves influence seismic velocities due to high rigidity while micropores or cracks significantly decrease velocities of rocks due to the low rigidity. Regarding carbonate reservoirs that are dominated by secondary pore porosity, Eberli et al. (2003) discussed the velocity-porosity relationship for microporosity rocks, moldic-porosity rocks, interparticle porosity rocks, and densely cemented rocks. Wang et al. (2009) analyzed the practicality of three published models (Wyllie time-average equation, Gassmann's equation, and Kuster-Toksöz model) on velocity prediction for carbonate reservoirs in the Tarim Basin, proving that Kuster-Toksöz model is the best one. Meanwhile, they also pointed out that the Kuster-Toksöz model is a very high-frequency model, and is limited to dilute concentrations of the pores. Based on effective medium theories of Norris et al. (1985), Berryman (1992) proposed a new differential effective medium (DEM) model. The matrix in this model begins as phase 1 and is updated at each step when a new increment of phase 2 (inclusions) is added; the process continues until the expected proportion of the inclusions is reached. In this process, the former incremental inclusion (phase 2) and the matrix phase are taken as a new matrix, which makes the inclusion added at each step "dilute." Unfortunately, DEM is also a high-frequency model. At high frequencies, there is not enough time for wave-induced pore pressure to equilibrate, so it is not applicable to consider the effect of pore fluids on elastic properties. Wu's (1992) arbitrary aspect ratio and Berryman's (1995) 3D special pores are introduced into Berryman's (1992) DEM model to calculate the elastic moduli of dry rock in this paper. Spheres, needles, and penny-shaped cracks are used to represent dissolution vugs (small-sized caves), holes (needle-shaped), and cracks that have developed in the Tarim Basin. These "special" pores are incrementally added into the m...

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