Normal compression wave scattering by a permeable crack in a fluid-saturated poroelastic solid

A mathematical formulation is presented for the dynamic stress intensity factor (mode I) of a finite permeable crack subjected to a time-harmonic propagating longitudinal wave in an infinite poroelastic solid. In particular, the effect of the wave-induced fluid flow due to the presence of a liquid-saturated crack on the dynamic stress intensity factor is analyzed. Fourier sine and cosine integral transforms in conjunction with Helmholtz potential theory are used to formulate the mixed boundary-value problem as dual integral equations in the frequency domain. The dual integral equations are reduced to a Fredholm integral equation of the second kind. It is found that the stress intensity factor monotonically decreases with increasing frequency, decreasing the fastest when the crack width and the slow wave wavelength are of the same order. The characteristic frequency at which the stress intensity factor decays the fastest shifts to higher frequency values when the crack width decreases.

The Parameter Averaging Technique in Finite-Difference Modeling of Elastic Waves in Combined Structures with Solid,Fluid and Porous Subregions

To finite-difference model elastic wave propagation in a combined structure with solid,fluid and porous subregions,a set of modified Biot’s equations are used,which can be reduced to the governing equations in solids,fluids as well as fluidsaturated porous media.Based on the modified Biot’s equations,the field quantities are finite-difference discretized into unified forms in the whole structure,including those on any interface between the solid,fluid and porous subregions.For the discrete equations on interfaces,however,the harmonic mean of shear modulus and the arithmetic mean of the other parameters on both sides of the interfaces are used.These parameter averaging equations are validated by deriving from the continuity conditions on the interfaces.As an example of using the parameter averaging technique,a 2-D finite-difference scheme with a velocity-stress staggered grid in cylindrical coordinates is implemented to simulate the acoustic logs in porous formations.The finitedifference simulations of the acoustic logging in a homogeneous formation agree well with those obtained by the analytical method.The acoustic logs with mud cakes clinging to the borehole well are simulated for investigating the effect of mud cake on the acoustic logs.The acoustic logs with a varying radius borehole embedded in a horizontally stratified formation are also simulated by using the proposed finite-difference scheme.

Modified multiplying-factor integration method for solving exponential function dual integrals in crack problems

Crack problems are often reduced to dual integral equations,which can be solved by expanding the displacement integral equation as a series in the form of Chebyshev-like or Jacobi polynomials.Schmidt’s multiplying-factor integration method has been one of the most favorable techniques for determining the expansion coefficients by constructing a well-posed system of linear algebraic equations.However,Schmidt’s method is less efficient for numerical computation because the matrix elements of the linear equations are evaluated from dual integrals.In this study,we propose a modified method to construct linear equations to efficiently determine the expansion coefficients.The modified technique is developed upon the application of certain multiplying factors to the traction integral equation and then integrating the resulting equation over“source”regions.Such manipulations simplify the matrix elements as single integrals.By carrying out numerical examples,we demonstrate that the technique is not only accurate but also very efficient.In particular,the method only needs approximately 1/5 of the computation time of Schmidt’s method.Therefore,this method can be used to replace Schmidt’s method and is expected to be very useful in solving crack problems.