Three efficient numerical models to analyse the step problem in shallow water

In this paper, the problem of acoustic wave propagation in a waveguide of infinite extent is modelled, taking into account constant depth in each section of the sea. Efficient numerical strategies in the frequency domain are addressed to investigate two-dimensional acoustic wave propagation in a shallow water configuration, considering a step in the rigid bottom and a flat free surface. The time domain responses are obtained by means of an inverse Fast Fourier Transform (FFT) of results computed in the frequency domain. The numerical approaches used here are based on the Boundary Element Method (BEM) and the Method of Fundamental Solutions (MFS). In the numerical models only the inclined or vertical interface between the sub-regions of different depth are discretized, as Green׳s functions that take into account the presence of free and rigid surfaces are used. These Green׳s functions are obtained either by eigenfunction expansion or by Ewald׳s method. A detailed discussion on the performance of these formulations is carried out, with the aim of finding an efficient numerical formulation to solve the step problem in shallow water

A time-stepping technique to solve wave propagation problems using the boundary element method

SIGLELD:D49963/84 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

O metodo dos elementos de contorno aplicado ao problema da onda escalar: Derivadas espaciais e temporal

As equaçóes correspondentes As derivadas espaciais e temporal da representaçáo integral de Volterra do problema de propagaçáo da onda escalar sáo escritas empregando-se o conceito de parte finita da integral. Admitindo variaçóes linear e constante para o potencial e sua derivada na direçáo normal, respectivamente, a integrasao no tempo é efetuada analiticamente. Apresenta-se, no final do trabalho, um exemplo para testar a eficiencia da formulaçáo, comparando-se as respostas numérica e analítica

Numerical Simulation of MZF Design with Non-planar Hydraulic Fracturing from Multi-lateral Horizontal Wells

In recent years, developments in the oil and gas industry have evolved significantly in advancing the mechanical systems technology to perform hydraulic fracturing. However, further developments will require an in-depth understanding of the impacts of fracture spacing, stress anisotropy, and reservoir characterization. In order to develop a comprehensive and robust completion design for hydraulic fracturing from multi-lateral wellbores with closely spaced fractures, it is important to consider stress shadowing effects. In this work the Cohesive Segments Method is combined with the Phantom Node Method, a combination termed CPNM. This is capable of not only simulating non-planar hydraulic fracture propagation with an unpredictable path, but also simulating the emergence of multiple cohesive cracks within a porous medium. This paper focuses on the “Modified Zipper-Frac” (MZF) design, which has been introduced to design the clusters from multi-lateral wells with the aim of increasing the fracture complexity. Validation of the numerical technique has been performed by comparing the solution for an individual hydraulic fracture with a Khristianovic-Geertsma-de Klerk (KGD) solution. In addition, a study of the development of double fractures has been conducted in the presence of stress shadowing to verify the simulation results. Taking the stress shadowing effects into account, a large number of numerical simulations are conducted using CPNM to investigate the stress anisotropy as well as the in-plane shear stress in the area between the two wells. The main contribution of this work is the detailed investigation of the effects of stress shadowing as a function of the fracture spacing on the horizontal stress contrast, direction of maximum local stress, leak-off flow rate, in-plane shear stress, and pore pressure of the formation