Stochastic effects on single phase fluid flow in porous media P. Mansfield*, M. Bencsik Magnetic Resonance Centre, School of Physics and Astronomy, University of Nottingham, Nottingham, UK Abstract The flow encoded PEPI technique has been used to measure the fluid velocity distribution and fluid flow of water passing through a phantom comprising randomly distributed 10 mm glass beads. The object of these experiments is to determine the degree of causality between one steady-state flow condition and another. That is to say, knowing the mean fluid velocity and velocity distribution, can one predict what happens at a higher mean fluid velocity? In a second related experiment flow is established at a given mean fluid velocity. The velocity distribution is measured. The flow is then turned off and later re-established. In both kinds of experiment we conclude that the errors in predicting the flow velocity distribution and the errors in re-establishing a given velocity distribution lie well outside the intrinsic thermal noise associated with velocity measurement. It follows, therefore, that the causal approach to prediction of flow velocity distributions in porous media using the Navier-Stokes approach is invalid. © 2001 Elsevier Science Inc. All rights reserved. Keywords: Porous media; Flow; PEPI; Stochastic irreproducibility 1. Introduction The study of fluid transport in porous rocks is of consid- erable interest to the oil industry where relevant models are still being sought to understand the oil recovery process. The starting point for such calculations is Darcy’s Law which involves a knowledge of rock permeability as well as fluid viscosity. Advances in NMR imaging and in localised flow and velocity measurements now allow a detailed study of the flow processes, either directly in bore cores or in phantom systems comprising glass beads. In earlier papers [1,2] the characteristics of the fluid velocity distribution within a porous matrix measured using the -EPI (PEPI) technique [3] were evaluated using a stochastic theory of flow in porous rocks. This theory led to a Gaussian velocity distribution and in particular a linear relationship between velocity variance and the mean fluid velocity through the rock, the Mansfield-Issa equation. Sub- sequent work in glass bead phantoms has further substan- tiated this work [4]. In our original approach the porous material was mod- elled on the basis of an assembly of Poiseuille flow channel pairs [5] passing through the porous medium, each pair having a single connecting Bernoulli flow channel. Remark- ably this simple model is able to explain the broad details of flow. If extra flow channels are introduced between the pairs of Poiseuille flow channels, the theory predicts a slight change of shape of the velocity distribution which is ob- served in the data [6,7]. One of the important assertions of the stochastic theory to which we were led by experimental results [1], is that the details of a particular velocity distribution within a cross- section of the porous matrix can change if the flow is turned off and then re-established. This aspect of irreproducibility of the details of the velocity distribution has again been observed in glass bead phantoms. Further details of these new experiments will be presented. 2. Velocity and flow maps Fig. 1 shows a water velocity map from a 10 mm slice through a glass bead phantom comprising 10 mm glass spheres contained in a cylindrical volume of diameter 75 mm and length 150 mm. The data were obtained at 0.5 T using the PEPI sequence. The pixel resolution in Fig. 1 is 2  2 mm 2 . In order to better approximate to the theoretical model of velocity distribution in which porosity is assumed to be homogeneous over the grain size, it is necessary to coarsen * Corresponding author. Tel.: 01159 514740; fax: 01159 515166. E-mail address: bencsik@magres.nottingham.ac.uk (P. Mansfield). Magnetic Resonance Imaging 19 (2001) 333–337 0730-725X/01/$ – see front matter © 2001 Elsevier Science Inc. All rights reserved. PII: S0730-725X(01)00245-4