Multi-gradient pulse investigations of fluid transport in porous media

NMR pulsed field gradient (PFG) experiments employing the application of n gradient pulses k(1) ellipsis k(n) are discussed in a general way as an n-fold encoding of position at successive times. The experiments are then represented by a sampling of n-dimensional k-space, K(n). Various parameters of motion can be derived from the evolution of correlations within the n-dimensional position (r-)space, R(n), which is the Fourier conjugate space to K(n). A wide class of NMR experiments may be described by this formalism, where the dimension of the experiment is often reduced by imposing conditions to the free variables. This is demonstrated for the case of displacement measurements where the condition summation operatork(i) = 0 is met. The two simplest pulse sequences which allow one to correlate displacements at two different times with each other are presented. While the three-pulse version of SERPENT encodes displacements in two interleaved time intervals Delta(1) and Delta(2), the four-pulse VEXSY experiment includes a mixing time tau(m) in between both encoding intervals Delta. The behaviour of fluid transport subject to external pressure through a model porous system is demonstrated by means of numerical simulations of SERPENT and VEXSY experiments for water flowing through a packed bed of monosized spherical particles. Displacements parallel (Z) and perpendicular (X) to the main flow direction are determined and the 2-D joint probability densities and the conditional probabilities are discussed along with the correlation coefficients related to the displacements at different encoding times. It is shown that all possible correlations between Z and X(2) in VEXSY decay with time constants comparable to the average time needed for a fluid molecule to cover one bead diameter, while a negative correlation is observed between transverse (X) displacements which is explained by molecules flowing along streamlines which follow the circumference of the spherical particles. Correlations for displacements during the different times in SERPENT generally decay much slower and provide complementary information about the evolution of displacements with time.