When a poro-elastic beam is suddenly bent, pressure is imposed on the liquid in the pores, so it begins to flow. This leads to a decrease in the amount of force needed to sustain the deflection of the beam. The time that it takes for the pressure in the pores to be released is related to the permeability of the beam, so a measurement of the force on the beam can be used to measure permeability. This technique has been used extensively in the Scherer lab for a variety of materials, including gels, porous glass, cement, and stone. The group is presently using the beambending method to measure the permeability of shale, as described below. The analysis of the pressure relaxation is based on an approximate analytical method that has yielded good agreement with experiments. Recently, they have made detailed comparisons of this solution to “exact” finite-element simulations to ascertain its range of validity.
Predicting the stresses and strains during bending of a beam is an apparently simple problem for which no exact solution exists. The situation is more complicated when the beam is a porous solid with fluid in the pores (as in the case of a saturated rock). The Scherer-Prévost Group has developed analytical solutions for fluid-saturated beams under pure bending, subject to certain plausible (and widely used) assumptions about the stresses at the boundaries. A solution was also obtained for the strains (i.e., the shape change) corresponding to the stresses in the beam, but those stresses are only approximately correct. The analytical solution is exactly right for the initial and final states, but comparison to a finite-element simulation shows that it is deficient during the transient phase, while liquid is flowing through the pores. An alternative approach to this problem is to recognize that the other stresses in a slender beam are small compared to the uniaxial (stretching) stress, so they can be neglected. This leads to a simple result for the stress and strain in a bent beam as liquid flows within it. Comparison of finite-element simulations to the stresses calculated in this way shows that the results are quite good. This gives increases their confidence in the results of permeability measurements by the beam-bending method.
Dr. Jie Zhang and undergraduate Karambir Khangoora have been measuring the permeability of shale samples sent to Princeton by Rusty Riese (BP). Using the beam-bending method, they have succeeded in measuring permeabilities in the nanodarcy range. The values correlate with independent measurements that they have made of the pore size distribution and the organic content of the shale.