scholarly journals Productivity Formulae of an Infinite-Conductivity Hydraulically Fractured Well Producing at Constant Wellbore Pressure Based on Numerical Solutions of a Weakly Singular Integral Equation of the First Kind

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Chaolang Hu ◽  
Jing Lu ◽  
Xiaoming He

In order to increase productivity, it is important to study the performance of a hydraulically fractured well producing at constant wellbore pressure. This paper constructs a new productivity formula, which is obtained by solving a weakly singular integral equation of the first kind, for an infinite-conductivity hydraulically fractured well producing at constant pressure. And the two key components of this paper are a weakly singular integral equation of the first kind and a steady-state productivity formula. A new midrectangle algorithm and a Galerkin method are presented in order to solve the weakly singular integral equation. The numerical results of these two methods are in accordance with each other. And then the solutions of the weakly singular integral equation are utilized for the productivity formula of hydraulic fractured wells producing at constant pressure, which provide fast analytical tools to evaluate production performance of infinite-conductivity fractured wells. The paper also shows equipotential threads, which are generated from the numerical results, with different fluid potential values. These threads can be approximately taken as a family of ellipses whose focuses are the two endpoints of the fracture, which is in accordance with the regular assumption in Kuchuk and Brigham, 1979.

2007 ◽  
Vol 51 (4) ◽  
pp. 1-7
Author(s):  
Yu. R. Agachev ◽  
R. K. Gubaidullina

1995 ◽  
Vol 62 (2) ◽  
pp. 312-319 ◽  
Author(s):  
Y. Mikata

Reflection and transmission of an SH-wave by a disordered periodic array of coplanar cracks is investigated, and subsequently its application to the dispersion and attenuation of an SH-wave in a disorderedly cracked medium is also treated. This is a stochastic boundary value problem. The formulation largely follows Mikata and Achenbach (1988b). The problem is formulated for an averaged scattered field, and the governing singular integral equation is derived for a conditionally averaged crack-opening displacement using a quasi-crystalline-like approximation. Unlike our previous study (Mikata and Achenbach, 1988b) where a point scatterer approximation was used for the regular part of the integral kernel, however, no further approximation is introduced. The singular integral equation is solved by an eigenfunction expansion involving Chebyschev polynomials. Numerical results are presented for the averaged reflection and transmission coefficients of zeroth order as a function of the wave number for normal incidence, a completely disordered crack spacing, and various values of the ratio of crack length and average crack spacing. Numerical results are also presented for the dispersion and attenuation of an SH-wave in a disorderedly cracked medium.


1996 ◽  
Vol 33 (02) ◽  
pp. 400-410 ◽  
Author(s):  
Gustaf Gripenberg ◽  
Ilkka Norros

Integration with respect to the fractional Brownian motionZwith Hurst parameteris discussed. The predictoris represented as an integral with respect toZ,solving a weakly singular integral equation for the prediction weight function.


1970 ◽  
Vol 42 (3) ◽  
pp. 447-464 ◽  
Author(s):  
Li-San Hwang ◽  
Ernest O. Tuck

A theory is developed for calculating oscillations of harbours of constant depth and arbitrary shape. This theory is based on the solution of a singular integral equation. Numerical results have been calculated for rectangular harbours so as to check the accuracy of the method. Examples for wave amplification factor and velocity field for both rectangular and actual complex-shaped harbours are given.


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