Accurate phase-shift velocimetry in rock at 7T

Considering the analytic potential of spatially resolved velocimetry in rocks and low porosity media, it is surprising that only few studies are found in the literature. This is probably due to the fact propagator imaging comes at an enormous time price necessary to acquire multiple q-space steps for each voxel. Phase-shift velocimetry, that appears as a time-efficient alternative for obtaining local velocity measurements seems to be sensitive to parameters preventing accurate measurements in rock. This technique relies on the linear relation between the phase of the NMR signal and the imposed velocity-encoding gradient. But in rock the linearity is compromised as gradient increases [1], with its range becoming smaller at higher flow rates. Working at lower flow rates is not a solution, since the imparted phase shift is reduced, making the measurements prone to phase errors that can result in noise-dominated spatial velocity distributions.

We investigated the effect of several parameters on phase-shift velocimetry by performing spatially-resolved phase-shift and velocity measurements in Benthheimer sandstone samples using a bipolar Alternating Pulsed Gradient Stimulated Echo (APGSTE) pulse sequence in a 7T Bruker Instrument. Combining high magnetic field with a sequence that is less sensitive to T₂^* relaxation ensured minimal phase error, as the later is reciprocal of SNR. Our measurements show that the relation of phase-shift to gradient depends on the product of the flow rate (Q) to the observation time (Δ), that is proportional to the average proton displacement during Δ (Fig.1a). This allows explaining difficulties of previous studies to obtain accurate measurements at certain parameter ranges and provides with a simple way to calibrate the pulse sequence parameters so as to stay on the phase-to-gradient linearity region.
Once the linear region is identified for a given flow rate, one can find the minimum gradient making phase error negligible with respect to velocity distribution (Fig.1b). Our results suggest that it should then be possible to reproduce the same phase shift at any flow rate by keeping Q × Δ constant. We confirm this hypothesis and demonstrate the proposed methodology by acquiring accurate spatially resolved velocity maps in rock (2D and 3D) (Fig.2a) for a range of flow rates (Fig.2b).

Figure 1. (a)Phase-shift and (b) velocity image standard deviation against gradient for different flow rates and observation times.

Figure 2. (a)Velocity maps in the Bentheimer sandstone at Q = 2 ml min^-1. (c) Average and standard deviation of velocity in the obtained maps against flow rate for the optimised MR parameters.