Perturbation solution for pulsatile flow of a non-Newtonian Williamson fluid in a rock fracture

In this work we present the analytical study of the motion of a pseudoplastic fluid injected in a fracture; the injection of a pseudoplastic fluid is often used in enhanced oil recovery operations. The forcing action considered is owing to a constant pressure gradient and a small sinusoidal variation superposed to it. Two parallel plates are used to schematise the geometry of the fracture. However, a real fracture in rock masses has rough surfaces and, unlike parallel plates, the walls can be in contact at some locations if subjected to normal stress. Nevertheless, the scheme of plane motion gives reliable results if the normal stress is less than ~10 Mpa and if the aperture variation along a flow path is not large (Tsang [1], Neuzil [2]). The complexity generated by the non linear behaviour of the fluid and by the unsteady flow suggest to adopt this simplified geometry to allow analytical treatment of the problem. In this paper the effects of roughness are neglected, although they are very important in fracture flow, as emphasized in the reference work of Zimmerman and Bodvarsson [3].