Computing the permeability and Forchheimer tensor of porous rocks via closure problems and digital images

Current computing capabilities, in combination with available theoretical frameworks, allow for the petrophysical evaluation of porous rocks from simple images. This procedure represents a less expensive alternative and it complements, generally expensive, laboratory measurements. A porosity, permeability and Forchheimer tensors estimation is reported here through a numerical solution of associated closure problems within digital images of porous rock thin sections. The solution of these steady-state boundary-value problems allows direct computation of all permeability elements and Forchheimer tensors. The digital images were obtained from a computer procedure that identifies the network of pores in thin sections. The results of the numerical estimation of permeability and porosity and its scopes were compared with those obtained via experimental measurements in four samples representing three distinct lithologies (travertine, sandstone and limestone), and were found to be in acceptable agreement. Once the permeability was computed, the Forchheimer tensor was calculated as function of the pore - scale Reynolds number. The Forchheimer coefficient varied with fluid velocity at an exponent close to 2 (range from 1.7 up to 4.2) depending on the lithology and local microstructure. The critical Reynolds number at which the Forchheimer coefficient became relevant was approximately 0.25. We found that the Forchheimer tensor exhibited anisotropy not only according to the local microstructure but also according to the flow path.