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 <X²(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^.     

Time Correlations in Fluid Transport Obtained by Sequential Rephasing Gradient Pulses

We present a basic experiment by which the evolution of the displacement probability density (propagator) of static or flowing fluid inNsuccessive time intervals is obtained by single labeling, coupled with multiple rephasing events during the course of a pulsed field-gradient sequence. We term this type of sequence SERPENT: SEquential Rephasing by Pulsed field-gradients Encoding N Time-intervals. Realizations of the SERPENT experiment for the caseN= 2 which include spin echo, stimulated echo, and Carr–Purcell pulse sequences are suggested. They have in common a spatial spin-labeling of the initial magnetization by a gradient of area q₀, followed by successive rephasing via gradients q₁and q₂at timest= Δ₁andt= Δ₂, respectively, where q₀+ q₁+ q₂= 0. A two-dimensional Fourier transform with respect to q₁and q₂gives directly the joint probability densityW₂(R₁, Δ₁; R₂, Δ₂) for displacements R₁and R₂in times Δ₁and Δ₂, respectively. q₁and q₂may be in arbitrary directions. Assuming R₁R₂, the correlation coefficient ρ_R1,R2then reflects the time-history of the fluctuating velocities. The behavior of the cross moment R₁(Δ₁) · R₂(Δ₂) can be obtained from either a full two-dimensional or a set of one-dimensional SERPENT measurements. Experimental results are presented for water flowing through a bed of packed glass beads. While Δ₁is appropriately chosen to sample the short-time velocity field within the system, increasing Δ₂clearly shows the loss of correlation when the average fluid element displacement exceeds the bead diameter.     

Wavelet Filter Functions,the Matrix Completion Problem,and Projective Modules Over <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Bold">T</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We discuss how one can use certain filters from signal processing to describe isomorphisms between certain projective C(T ⁿ )-modules. Conversely, we show how cancellation properties for finitely generated projective modules over C(T ⁿ ) can often be used to prove the existence of continuous high pass filters, of the kind needed for multivariate wavelets, corresponding to a given continuous low-pass filter. However, we also give an example of a continuous low-pass filter for which it is impossible to find corresponding continuous high-pass filters. In this way we give another approach to the solution of the matrix completion problem for filters of the kind arising in wavelet theory.     

Investigation of two phase flow and phase trapping by secondary imbibition within Fontainebleau sandstone

Pulsed magnetic field gradient stimulated echo NMR is used to investigate the simultaneous flow of two phases (an aqueous phase and an hydrocarbon phase) within a strongly water-wet sample of Fontainebleau sandstone. The Fontainebleau sandstone is prepared in increasing steady-state water saturations by a secondary imbibition process. The increase in the water saturation causes an increasing fraction of the oil phase (non-wetting phase) to become trapped within the sample. The stimulated echo dependence on the gradient pulse area, q, is used to derive the displacement probability, PX, for a fixed observation time. These displacement probabilities clearly show the progressive trapping of the hydrocarbon phase with increasing steady-state water saturations. Quantitative measurements of the fraction of the oil phase trapped were made from the echo attenuation function Edelta(q), both as a function of water saturation and observation time.     

Magnetic resonance in porous media: forty years on

A personal view of the field of magnetic resonance in porous media is presented in which an attempt is made to survey the current status and achievements, to highlight some of the contributions made by my group over the years and, at the end, to try and identify where further effort and growth points may be perceived. All this is done with the knowledge that the first and last sections are certain to be partial, incomplete and wrong, at least in part, and that the middle section describes work carried out by some of the many excellent students, post-doctoral researchers and other colleagues with whom it has been a pleasure to collaborate over a forty year research career.     

1H spin-lattice relaxation and observation of interfacial material in solid polyethylenes

The ¹H NMR laboratory- and rotating-frame (LF,RF) spin-lattice relaxation (SLR) behaviour in solid polyethylenes is reviewed in light of the two region spin diffusion coupled model. Highly crystalline high-density materials show single T¹ values indicative of the fast diffusion limit whilst RFSLR is close to the slow diffusion limit, allowing direct determination of the dimensions of the crystalline and disordered regions. The LFSLR and RFSLR of the HDPE's suggest the existence of an interfacial material which the SLR behaviour includes as part of the crystalline region in contrast to XRD and DSC. ¹³C CPMAS measurements of ¹H RFSLR in a LLDPE provide an additional insight into this problem. The ¹H RFSLR is best represented by a superposition of two exponentials, measured by either ¹H or ¹³C NMR. The ¹H RFSLR, measured at different chemical shifts in the ¹³C NMR spectrum, provides direct evidence of the interfacial material. The possible use of spin-locking at angles other than 90° in the rotating frame to scale the spin diffusion rates is reviewed and its possible utility for LLDPE materials explored.     

Essential Oil Composition and Stable Isotope Profile of Cultivated Ocimum campechianum Mill. (Lamiaceae) from Peru

Ocimum campechianum Mill. (Peruvian basil) is an essential oil-bearing plant of the Lamiaceae family. Volatile oil produced through steam distillation of Peruvian basil was examined to establish the aromatic and stable isotope profiles of samples (n = 9) from three different cultivated plots in Peru. The resulting essential oils were analyzed by GC/FID, GC/MS, and GC/IRMS. In accordance with findings from other researchers, multiple chemotypes, defined by the most abundant aromatic compounds, exist within these populations. Overall, 55% of samples are the eugenol chemotype (values ranging 15.4–30.2%), 33% are the methyl eugenol chemotype (values ranging 68.1–68.7%), and a single sample is a mixture of both chemotypes, containing high levels of both eugenol (38.1%) and methyl eugenol (8.6%). Stable isotope ratios, δ²H and δ¹³C, performed on prominent compounds provide supporting data for distinguishing chemotypes. Complete aromatic profiles, stable isotope ratios, and essential oil yield are established for each sample. This study confirms the existence of multiple chemotypes and, for the first time, to the author’s best knowledge, establishes stable isotope ratios for O. campechianum essential oil, which proves a useful tool in further investigating plant metabolism and determining essential oil authenticity.     

Nuclear magnetic resonance and X-ray evidence of crystal structure for acetates of calcium,strontium, and barium

High-resolution ¹³C spectra of calcium acetate hemihydrate (CAH), strontium acetate hemihydrate (SAH), and barium acetate monohydrate (BAM) have been obtained using cross-polarization (CP), high-power decoupling, and magic-angle spinning (MAS). The results indicate the existence in each case of four nonequivalent crystallographic sites for the acetate groups. These findings have been confirmed for SAH and BAM by single-crystal X-ray diffraction studies. The carboxyl chemical shifts for CAH cover a range of 9.3 ppm and the lines are remarkably sharp, so that this material is suggested as a suitable standard for checking the performance of CP/MAS spectrometers