Two-dimensional propagators and spatio-temporal correlations for flow in porous media: A comparative study

The two-dimensional displacement joint probability densityPΔ(X,Z) and the two-time probability density W₂(Z₁,Δ₁;Z₂,A₂) for water flowing through several porous systems have been measured by means of pulsed field-gradient nuclear magnetic resonance (PFG-NMR). The simultaneous particle displacementsX and Z perpendicular and parallel to the pressure gradient, respectively, at a given encoding time Δ are obtained from an experiment employing orthogonal magnetic field gradients. Time-correlated propagators which relate the displacement spectra at two consecutive times Δ_l, and Δ₂ with each other were derived by applying rephasing gradients in two steps. Flow through a random packing of glass beads and through natural sandstone is compared to flow through arrays of either oriented or unoriented fibers with different solid volume fractions. The dependence of the dispersion tensor D^* as a function of time is discussed and related to a characteristic length ξ_tT transverse to the flow direction. Within a certain range of Z values, displacements inX and Z are related by a power law 2(Z)> ∝Z _γ. The spreading exponent γ is found to increase with growing orientational order in the porous system and is largest for fiber bundles being twisted with respect to the mean pressure gradient axis. The evolution of the correlation coefficient px²,z with time gives a measure for the typical correlation length of the system parallel to the flow axis, ξ_‖. Analyzing the shape ofW ₂(Z₁,Δ₁Z₂,Δ₂) allows one to investigate the loss of coherence in flow by an alternative approach. The decay of the two-time correlation coefficient,pZ ₁,Z₂, is sensitive to the change of the z-component of the particle velocity and probes a different lengthscale thanpx ₂z^.