A variationally separable splitting for the generalized-α method for parabolic equations

We present a variationally separable splitting technique for the generalized-α method for solving parabolic partial differential equations. We develop a technique for a tensor-product mesh which results in a solver with a linear cost with respect to the total number of degrees of freedom in the system for multidimensional problems. We consider finite elements and isogeometric analysis for the spatial discretization. The overall method maintains user-controlled high-frequency dissipation while minimizing unwanted low-frequency dissipation. The method has second-order accuracy in time and optimal rates (h^p+1 in L² norm of u and h^p in L² norm of ∇u) in space. We present the spectral analysis on the amplification matrix to establish that the method is unconditionally stable. Various numerical examples illustrate the performance of the overall methodology and show the optimal approximation accuracy.     

A novel approach to determine the Biot’s coefficient using X-ray computed tomography

Linear poroelastic effective stress law is often used to relate in situ total stresses to pore fluid pressure in underground formations. The indirect estimations of effective stress coefficient (Biot coefficient) based on porosity relationships are often not accurate, and therefore the Biot coefficient measurements on retrieved samples using hydro-mechanical laboratory testing is considered. In this study, we propose a new indirect technique to infer the Biot coefficient from micro X-ray computed tomography (XRCT) images of porous granular samples whenever images are available using a simple principle of continuum mechanics. The performance of the method was assessed through a series of experiments conducted on sandstone and unconsolidated sand. The proposed method enables calculating the anisotropy of the Biot coefficient of the sample in three dimensions. We also conducted the conventional hydro-mechanical experiment to validate the obtained Biot coefficient results from the XRCT technique. The proposed technique shows promising results on estimation of the Biot coefficient. The analysis of the images confirms that the method performs well when the connectivity between grains (skeleton structure) can be extracted from XRCT images with enough resolution. The method is also able to (a) estimate the Biot coefficient of both consolidated and unconsolidated sand structure and (b) map the anisotropy of Biot coefficient in three dimensions.