One‐dimensional analytical solution for mesoscopic flow induced damping in a double‐porosity dual‐permeability material

Heterogeneities, such as fractures and cracks, are ubiquitous in porous rocks. Mesoscopic heterogeneities, that is, heterogeneities on length scales much larger than typical pore size but much smaller than the wavelength, are increasingly believed to be responsible for significant wave energy loss in the seismic frequency band. When a compressional wave stresses a material containing mesoscopic heterogeneities, the more compliant parts of the material (e.g., fractures and cracks) respond with a greater fluid pressure than the stiffer portions (e.g., matrix pores). The induced fluid flow, resulting from the pressure gradients developed on such scale, is called mesoscopic flow. In the present study, the double‐porosity dual‐permeability model is adopted to incorporate mesoscopic heterogeneities into rock models to account for the attenuation of wave energy. Based on the model, the damping effect due to mesoscopic flow in a one‐dimensional porous structure is investigated. Analytical solutions for several boundary‐value problems are obtained in the frequency domain. The dynamic responses of infinite and finite porous layer are examined. Numerical calculations show that the damping effect of mesoscopic flow is significant on the pore pressure response and the resulting effective stress. For the displacement, the effect is seen only at the very low frequency range or near the resonance frequencies. Copyright © 2017 John Wiley & Sons, Ltd.     

The method of fundamental solution for 3‐D wave scattering in a fluid‐saturated poroelastic infinite domain

By using a complete set of poroelastodynamic spherical wave potentials (SWPs) representing a fast compressional wave P_I, a slow compressional wave P_II, and a shear wave S with 3 vectorial potentials (not all are independent), a solution scheme based on the method of fundamental solution (MFS) is devised to solve 3‐D wave scattering and dynamic stress concentration problems due to inhomogeneous inclusions and cavities embedded in an infinite poroelastic domain. The method is verified by comparing the result with the elastic analytical solution, which is a degenerated case, as well as with poroelastic solution obtained using other numerical methods. The accuracy and stability of the SWP‐MFS are also demonstrated. The displacement, hoop stress, and fluid pore pressure around spherical cavity and poroelastic inclusion with permeable and impermeable boundary are investigated for incident plane P_I and SV waves. The scattering characteristics are examined for a range of material properties, such as porosity and shear modulus contrast, over a range of frequency. Compared with other boundary‐based numerical strategy, such as the boundary element method and the indirect boundary integral equation method, the current SWP‐MFS is a meshless method that does not need elements to approximate the geometry and is free from the treatment of singularities. The SWP‐MFS is a highly accurate and efficient solution methodology for wave scattering problems of arbitrary geometry, particularly when a part of the domain extends to infinity.     

Analytical 3D transient elastodynamic fundamental solution of unsaturated soils

Unsaturated soils are considered as porous continua, composed of porous skeleton with its pores filled by water and air. The governing partial differential equations (PDE) are derived based on the mechanics for isothermal and infinitesimal evolution of unsaturated porous media in terms of skeleton displacement vector, liquid, and gas scalar pressures. Meanwhile, isotropic linear elastic behavior and liquid retention curve are presented in terms of net stress and capillary pressure as constitutive relations. Later, an explicit 3D Laplace transform domain fundamental solution is obtained for governing PDE and then closed‐form analytical transient 3D fundamental solution is presented by means of analytical inverse Laplace transform technique. Finally, a numerical example is presented to validate the assumptions used to derive the analytical solution by comparing them with the numerically inverted ones. The transient fundamental solutions represent important features of the elastic wave propagation theory in the unsaturated soils. Copyright © 2010 John Wiley & Sons, Ltd.