Stochastic Homogenisation of Fluid Flows in Heterogeneous Porous Media
We present here a general methodology for using homogenisation technique, based on a stochastically stationary description of heterogeneities in porous media, to provide an overall pressure drop/flow rate relation valid at scales larger than those of the correlation lengths for heterogeneity. A dual formulation arises depending on whether flow rate or pressure gradient are treated as the independent variable. The homogenisation technique combines perturbation of the local variables with stochastic linearisation of the fluctuation equations. Closure problems are solved by means of spectral Stieltjes-Fourier decomposition under the statistical stationarity assumption. We then require the dual formulation to be consistent in form. The methodology is illustrated on generalized Newtonian single phase flow.
KeywordsHeterogeneous porous media homogenisation stochastic Stieltjes-Fourier spectral representation filtration law Darcy non-Newtonian fluids
Unable to display preview. Download preview PDF.
- Bear, J (1987) Dynamics of fluids in porous media, Dover ed.Google Scholar
- Vossoughi, S (1999) Advances in the flow and rheology of non-Newtonian fluids, Part B., D.A. Siginer et al. ed., Elsevier, Rheology series 8, 1225–1230.Google Scholar
- Fadili, A (2001) Ecoulements diphasiques en réservoir pétroliers hétérogènes: homogénéisation stochastique, Ph.D, INP Toulouse, no 1762, pp. 301.Google Scholar
- Priestley, M B (1981) Spectral analysis and time series, 1: univariate series, Academic Press.Google Scholar
- Fadili, A Tardy, P M J and Pearson, J R A (2002) 3D macro-scale analytical filtration law for power-law fluids in heterogeneous porous media, Proc. Biot Conf. Poromechanics II, eds. Auriault et al., 431–436, Grenoble, France, August 2002.Google Scholar
- Fadili, A Tardy, P M J and Pearson, J R A (2002) Stochastic homogenisation of non-Newtonian flows in heterogeneous porous media: 1. Theory, in preparation.Google Scholar
- Tardy, P M J Fadili, A and Pearson, J R A (2002) Stochastic homogenisation of non-Newtonian flows in heterogeneous porous media: 2. Numerical validation, in preparation.Google Scholar