Effective Gradients in Porous Media Due to Susceptibility Differences

In porous media, magnetic susceptibility differences between the solid phase and the fluid filling the pore space lead to field inhomogeneities inside the pore space. In many cases, diffusion of the spins in the fluid phase through these internal inhomogeneities controls the transverse decay rate of the NMR signal. In disordered porous media such as sedimentary rocks, a detailed evaluation of this process is in practice not possible because the field inhomogeneities depend not only on the susceptibility difference but also on the details of the pore geometry. In this report, the major features of diffusion in internal gradients are analyzed with the concept of effective gradients. Effective gradients are related to the field inhomogeneities over the dephasing length, the typical length over which the spins diffuse before they dephase. For the CPMG sequence, the dependence of relaxation rate on echo spacing can be described to first order by a distribution of effective gradients. It is argued that for a given susceptibility difference, there is a maximum value for these effective gradients,g_max, that depends on only the diffusion coefficient, the Larmor frequency, and the susceptibility difference. This analysis is applied to the case of water-saturated sedimentary rocks. From a set of NMR measurements and a compilation of a large number of susceptibility measurements, we conclude that the effective gradients in carbonates are typically smaller than gradients of current NMR well logging tools, whereas in many sandstones, internal gradients can be comparable to or larger than tool gradients.     

Xenon NMR measurements of permeability and tortuosity in reservoir rocks

In this work we present measurements of permeability, effective porosity and tortuosity on a variety of rock samples using NMR/MRI of thermal and laser-polarized gas. Permeability and effective porosity are measured simultaneously using MRI to monitor the inflow of laser-polarized xenon into the rock core. Tortuosity is determined from measurements of the time-dependent diffusion coefficient using thermal xenon in sealed samples. The initial results from a limited number of rocks indicate inverse correlations between tortuosity and both effective porosity and permeability. Further studies to widen the number of types of rocks studied may eventually aid in explaining the poorly understood connection between permeability and tortuosity of rock cores.     

Probing four orders of magnitude of the diffusion time in porous silica glass with unconventional NMR techniques

The combined use of two unconventional NMR diffusometry techniques permits measurements of the self-diffusion coefficient of fluids confined in porous media in the time range from 100 microseconds to seconds. The fringe field stimulated echo technique (FFStE) exploits the strong steady gradient in the fringe field of a superconducting magnet. Using a standard 9.4 T (400 MHz) wide-bore magnet, for example, the gradient is 22 T/m at 375 MHz proton resonance and reaches 60 T/m at 200 MHz. Extremely short diffusion times can be probed on this basis. The magnetization grid rotating frame imaging technique (MAGROFI) is based on gradients of the radio frequency (RF) field. The RF gradients not necessarily need be constant since the data are acquired with spatial resolution along the RF gradient direction. MAGROFI is also well suited for unilateral NMR applications where all fields are intrinsically inhomogeneous. The RF gradients reached depend largely on the RF coil diameter and geometry. Using a conic shape, a value of at least 0.3 T/m can be reached which is suitable for long-time diffusion measurements. Both techniques do not require any special hardware and can be implemented on standard high RF power NMR spectrometers. As an application, the influence of the tortuosity increasing with the diffusion time is examined in a saturated porous silica glass.     

Swelling/deswelling of polyacrylamide gels in aqueous NaCl solution: Light scattering and macroscopic swelling study

Swelling kinetics of water-swollen polyacrylamide (PAAm) hydrogels (WSG) was investigated in various concentrations of aqueous NaCl by macroscopic swelling measurements. For lower concentration of NaCl, WSG showed exponential swelling whereas at higher concentration of NaCl it underwent deswelling at short times and exponential swelling at long times. From these studies, collective diffusion coefficient, D, of the polymer network and polymer?Csolvent interaction parameter, ??, were calculated and found to decrease with increase in [NaCl]. Collective diffusion coefficients measured from dynamic light scattering (DLS) and that obtained from macroscopic swelling measurements are found to agree well. Measured ensemble-averaged dynamic structure factor f(q,t) for WSG and salt-swollen gels (SSG) showed an initial decay followed by a plateau at long times and it can be described by harmonically bound Brownian particle (HBBP) model. Enhanced scattering intensity at low scattering angles using static light scattering (SLS) measurements revealed the presence of inhomogeneities in PAAm gels. The reasons for increased scattering intensity of SSG over WSG gel and the linear decrease of D with increase in NaCl concentration are explained.     

Water signal attenuation in diffusion-weighted 1H NMR experiments during cerebral ischemia: influence of intracellular restrictions,extracellular tortuosity,and exchange

The “concept of restricted intracellular water diffusion at permeable boundaries,” which was recently used to model diffusion-weighted ¹H NMR experiments on glioma cells, was applied to measurements on the rat brain in vivo. Combined with the “concept of extracellular tortuosity,” various physiological states of the brain were simulated. Hereby, a variable intracellular volume fraction, intracellular exchange time, and extracellular tortuosity factor were considered for young, adult, and ischemic rat brains. The model simulated the cytotoxic shift of extracellular water, changes in membrane permeability and tissue morphology, and was able to explain the diffusion time dependence as well as the non-monoexponentiality of the diffusion attenuation curves. Preliminary diffusion time dependent experiments on the healthy rat brain (¹H NMR imaging) agreed well with the theoretical concept. Hereby, the intracellular water signal was separated from extracellular signal contributions by large diffusion weighting. It showed the characteristic of restricted diffusion as well as a signal decay due to the exchange of intracellular water across the plasma membrane. A map of the mean intracellular exchange time for water in living animal brain was determined, and the upper limit in rat brain was evaluated to 15 ms. The presented methods can be applied to correlate local differences in a map of exchange times with tissue morphology and to detect pathological deviations of the exchange time, e.g., during ischemia.     

Internal Magnetic Field Gradients in Heterogeneous Porous Systems: Comparison Between Spin-Echo and Diffusion Decay Internal Field (DDIF) Method

Two techniques used for evaluating internal magnetic field gradient (G _i), spin-echo (SE) and diffusion decay internal field (DDIF), were investigated at 9.4 T and compared in porous systems characterized by different pores size ranging from 4 to 96 μm with magnetic susceptibility difference between solid and liquid phase, \(\Delta \chi\)  ≈ 1.6 ppm. Since diffusion of a fluid in a solid porous matrix plays a role in both SE and DDIF methods, we investigated these two different methods by highlighting their dependence on characteristic parameters and length scales used to describe diffusion behavior of fluids in porous systems. Therefore, G _i behavior as a function of the dephasing length (l _g), diffusion length (l _d) and pores size (l _s) was obtained. Moreover G _i was evaluated by using both free diffusion and measured apparent diffusion coefficient of water, to quantify diffusion effect in different porous samples. This study gives more insight into the physical dynamics process to explain contrast mechanisms recently exploited by DDIF and SE applications for cancellous bone quality measurements.     

Pulsed-Field-Gradient Measurements of Time-Dependent Gas Diffusion

Pulsed-field-gradient NMR techniques are demonstrated for measurements of time-dependent gas diffusion. The standard PGSE technique and variants, applied to a free gas mixture of thermally polarized xenon and O₂, are found to provide a reproducible measure of the xenon diffusion coefficient (5.71 × 10⁻⁶m²s⁻¹for 1 atm of pure xenon), in excellent agreement with previous, non-NMR measurements. The utility of pulsed-field-gradient NMR techniques is demonstrated by the first measurement of time-dependent (i.e., restricted) gas diffusion inside a porous medium (a random pack of glass beads), with results that agree well with theory. Two modified NMR pulse sequences derived from the PGSE technique (named the Pulsed Gradient Echo, or PGE, and the Pulsed Gradient Multiple Spin Echo, or PGMSE) are also applied to measurements of time dependent diffusion of laser polarized xenon gas, with results in good agreement with previous measurements on thermally polarized gas. The PGMSE technique is found to be superior to the PGE method, and to standard PGSE techniques and variants, for efficiently measuring laser polarized noble gas diffusion over a wide range of diffusion times.     

Restricted diffusion and exchange of water in porous media: average structure determination and size distribution resolved from the effect of local field gradients on the proton NMR spectrum

NMR Pulsed field gradient measurements of the restrained diffusion of confined fluids constitute an efficient method to probe the local geometry in porous media. In most practical cases, the diffusion decay, when limited to its principal part, can be considered as Gaussian leading to an apparent diffusion coefficient. The evolution of the latter as a function of the diffusion interval yields average information on the surface/volume ratio of porosities and on the tortuosity of the network. In this paper, we investigate porous model systems of packed spheres (polystyrene and glass) with known mean diameter and polydispersity, and, in addition, a real porous polystyrene material. Applying an Inverse Laplace Transformation in the second dimension reveals an evolution of the apparent diffusion coefficient as a function of the resonance frequency. This evolution is related to a similar evolution of the transverse relaxation time T2. These results clearly show that each resonance frequency in the water proton spectrum corresponds to a particular magnetic environment produced by a given pore geometry in the porous media. This is due to the presence of local field gradients induced by magnetic susceptibility differences at the liquid/solid interface and to slow exchange rates between different pores as compared to the frequency differences in the spectrum. This interpretation is nicely confirmed by a series of two-dimensional exchange experiments.     

Relationship between susceptibility induced field inhomogeneities, restricted diffusion, and relaxation in sedimentary rocks

Low field relaxation and diffusion measurements have become essential tools to study the pore space of sedimentary rocks with important practical applications in the field of well logging and hydrocarbon extractions. Even at Larmor frequencies below 2 MHz, diffusion measurements are often affected noticeably by internal field inhomogeneities. These field inhomogeneities are induced by susceptibility contrast between the rock and the fluid and are evident in most sandstones. Using sets of two-dimensional diffusion-relaxation measurements in applied and internal gradients, we study in detail the correlation between the field inhomogeneities, restricted diffusion, and relaxation time in three rocks of different susceptibility. We find that in the sandstone cores, the field inhomogeneities in large pores can be described by a local gradient that scales inversely with relaxation time above 250 ms. At shorter relaxation times, the extracted internal gradients deviate from this scaling relationship and we observe a dependence on diffusion time. This demonstrates that in this case, the internal field has structure on a length scale of a few microns.     

Dependence of temporal diffusion spectra on microstructural properties of biological tissues

The apparent diffusion coefficient (ADC) measured using magnetic resonance imaging methods provides information on microstructural properties of biological tissues, and thus has found applications as a useful biomarker for assessing changes such as those that occur in ischemic stroke and cancer. Conventional pulsed gradient spin echo methods are in widespread use and provide information on, for example, variations in cell density. The oscillating gradient spin echo (OGSE) method has the additional ability to probe diffusion behaviors more readily at short diffusion times, and the temporal diffusion spectrum obtained by the OGSE method provides a unique tool for characterizing tissues over different length scales, including structural features of intracellular spaces. It has previously been reported that several tissue properties can affect ADC measurements significantly, and the precise biophysical mechanisms that account for ADC changes in different situations are still unclear. Those factors may vary in importance depending on the time and length scale over which measurements are made. In the present work, a comprehensive numerical simulation is used to investigate the dependence of the temporal diffusion spectra measured by OGSE methods on different microstructural properties of biological tissues, including cell size, cell membrane permeability, intracellular volume fraction, intranucleus and intracytoplasm diffusion coefficients, nuclear size and T₂ relaxation times. Some unique characteristics of the OGSE method at relatively high frequencies are revealed. The results presented in the paper offer a framework for better understanding possible causes of diffusion changes and may be useful to assist the interpretation of diffusion data from OGSE measurements.     

Probing porous media with gas diffusion NMR.

We show that gas diffusion nuclear magnetic resonance (GD-NMR) provides a powerful technique for probing the structure of porous media. In random packs of glass beads, using both laser-polarized and thermally polarized xenon gas, we find that GD-NMR can accurately measure the pore space surface-area-to-volume ratio, S/V rho, and the tortuosity, alpha (the latter quantity being directly related to the system's transport properties). We also show that GD-NMR provides a good measure of the tortuosity of sandstone and complex carbonate rocks.     

Effects of finite-width pulses in the pulsed-field gradient measurement of the diffusion coefficient in connected porous media

We analytically compute the apparent diffusion coefficient D(app) for an open restricted geometry, such as an extended porous medium, for the case of a pulsed-field gradient (PFG) experiment with finite-width pulses. In the short- and long-time limits, we give explicit, model-independent expressions that correct for the finite duration of the pulses and can be used to extract the pore surface-to-volume (S/V) ratio as well as the tortuosity. For all times, we compute D(app) using a well-established model form of the actual time-dependent diffusion coefficient D(t) that can be obtained from an ideal narrow-pulse PFG. We compare D(app) and D(t) and find that, regardless of pulse widths and geometry-dependent parameters, the two quantities deviate by less than 20%. These results are in sharp contrast with the studies on closed geometries [J. Magn. Reson. A 117 (1995) 209], where the effects of finite gradient-pulse widths are large. The analytical results presented here can be easily adapted for different pulse protocols and time sequences.     

Tortuosity for streamlines in porous media

An analysis of tortuosity for streamlines in porous media is presented by coupling the circle and square models. It is assumed that some particles in porous media do not overlap and that fluid in porous media is incompressible. The relationship between tortuosity and porosity is attained with different configurations by using a statistical method. In addition, the tortuosity fractal dimension is expressed as a function of porosity. Those correlations do not include any empirical constant. The percolation threshold and tortuosity fractal dimension threshold of porous media are also presented as: c = 0.32, D T c = 1.07. The predicted correlations of the tortuosity and the porosity agree well with the existing experimental and simulated results.     

Longitudinal dispersion of solutes in porous media solely by advection

The purpose of this work is to predict the transport of non-sorbing solutes through water flow in the subsurface. We derive what we consider to be the first reliable calculations of the entire distribution of arrival times, W(t), for non-sorbing solutes in advective flow in strongly disordered porous media. Solutes treated can be contaminant plumes from any source or radioactive tracers, both experimentally and naturally generated. Our approach is microscopic and based on effects of disorder. It generates longitudinal dispersion (in the direction of flow) in the absence of diffusion. Effects on dispersion from a single capillary tube velocity distribution, known to produce long-tailed arrival time distributions, are also neglected. On the other hand, our calculations are based on effects generated from real porous media, such as wide pore-size distributions and complex connectivity. In particular, the calculation of the distribution of arrival times is based on a distribution of conserved fluxes and the known tortuosity of the associated geometrical paths. The results are found to be predictive when compared with simulations of two-dimensional flow on percolation structures, and appear to have relevance for experiments as well.     

Displacement propagators of brine flowing within different types of sedimentary rock

This paper explores the correlation between different microstructural characteristics of porous sedimentary rocks and the flow properties of a Newtonian infiltrating fluid. Preliminary results of displacement propagator measurements of brine solution flowing through two types of sedimentary rock cores are reported. The two types of rocks, Bentheimer and Portland, are characterized by different porosities, pore-size distributions and permeabilities. Propagators have been measured for brine flow rates of 1 and 5 ml/min. Significant differences are seen between the propagators recorded for the two rocks, and these are related to the spatial distribution of porosity within these porous media.     

Transient-state method for coupled evaluation of Soret and Fick coefficients,and related tortuosity factors,using free and porous packed thermodiffusion cells: Application to CuSO4 aqueous solution ( 0.25M)

The measurement of Soret coefficients in liquids is not easy and usually not very precise because the resulting concentration gradient is small and moreover can be perturbed by undesired convection currents. In order to suppress, or to drastically reduce these convection currents, the use of a porous medium is sometimes suggested. The question arises as to whether the Soret coefficient is the same in free fluid and in porous medium. This is the aim of this paper. To this end, for a given liquid mixture, the time evolution of the vertical concentration gradient is experimentally measured in the same thermodiffusion cell filled first with the free liquid and next with a porous medium followed by saturation by the liquid mixture. Both the isothermal diffusion (Fick) coefficient and the Soret coefficient can be deduced, providing that a correct working equation is used. The proposed equation results from integration of the general mass conservation equation with realistic boundary conditions (zero mass flux at the boundaries) and some simplifying assumptions rendering this equation more tractable than the one proposed some decades ago by Bierlein (J.A. Bierlein, J. Chem. Phys. 23, 10 (1955)). The method is applied here to an electrolytic solution (CuSO4, 0.25 M) at a mean temperature of 37^°C. The Soret coefficients in free and porous medium (zircon microspheres in the range of 250- 315 ^. 10^-6m) may be considered to be equal ( S_T = 13.2±0.5 ^. 10^-3 K^-1) and the tortuosity factors for the packed medium are the same relative to thermodiffusion and Fick coefficients ( = 1.51±0.02).     

Porous structure of membranes of an acrylonitrile copolymer. Porosity, 1H-NMR permeability

Nanoporous polymer membranes (porosity ) used for dialysis are studied from NMR relaxation times of water confined in the pore space. Fast interpore water diffusion is observed. Two structural parameters are evidenced: i) a reduced NMR relaxation time, , which reflects the width of the pore-size distribution; ii) the average polymer-grain size of the solid matrix deduced from NMR experiments performed on membranes partially filled by water. A relation is found between the ratio , where k is the permeability to water and the porosity. This relation is in qualitative agreement with numerical simulations reported in the literature on low-porosity systems and with experimental results obtained for sedimentary rocks and for fused glass model systems. It supports the idea that is the relevant structural parameter to describe convective transport in a wide class of porous systems. Received 8 July 1999     

Tortuosity measurement and the effects of finite pulse widths on xenon gas diffusion NMR studies of porous media.

We have extended the utility of NMR as a technique to probe porous media structure over length scales of approximately 100-2000 microm by using the spin 1/2 noble gas 129Xe imbibed into the system's pore space. Such length scales are much greater than can be probed with NMR diffusion studies of water-saturated porous media. We utilized Pulsed Gradient Spin Echo NMR measurements of the time-dependent diffusion coefficient, D(t), of the xenon gas filling the pore space to study further the measurements of both the pore surface-area-to-volume ratio, S/V(p), and the tortuosity (pore connectivity) of the medium. In uniform-size glass bead packs, we observed D(t) decreasing with increasing t, reaching an observed asymptote of approximately 0.62-0.65D(0), that could be measured over diffusion distances extending over multiple bead diameters. Measurements of D(t)/D(0) at differing gas pressures showed this tortuosity limit was not affected by changing the characteristic diffusion length of the spins during the diffusion encoding gradient pulse. This was not the case at the short time limit, where D(t)/D(0) was noticeably affected by the gas pressure in the sample. Increasing the gas pressure, and hence reducing D(0) and the diffusion during the gradient pulse served to reduce the previously observed deviation of D(t)/D(0) from the S/V(p) relation. The Pade approximation is used to interpolate between the long and short time limits in D(t). While the short time D(t) points lay above the interpolation line in the case of small beads, due to diffusion during the gradient pulse on the order of the pore size, it was also noted that the experimental D(t) data fell below the Pade line in the case of large beads, most likely due to finite size effects.     

Photothermal deflection measurement of effective gas diffusion coefficient of a porous medium

In this work, an effective gas diffusion coefficient of a porous medium was measured using photothermal deflection (PD) technique. An in-house made Loschmidt diffusion cell with a photothermal-deflection probe were employed to measure the effective gas diffusion coefficient of a gas diffusion layer (GDL) with a porosity ε ≈ 0.7. The concentration evolutions of CO₂ in O₂ with and without the GDL were measured, respectively, using a transverse normal PD technique. The concentration variations were used to deduce the gas diffusion coefficients in the presence and absence of the GDL by solving mass diffusion equations. The effective gas diffusion coefficient of the GDL was calculated from the diffusion coefficients using a model of an equivalent resistance to diffusion and found to be 4.39 × 10^-6 m²s^-1, demonstrating that PD technique can be employed to determine the effective gas diffusion coefficient of a porous medium.     

Deduction of tortuosity and porosity from acoustic reflection and transmission measurements on thick samples of rigid-porous materials

An acoustic method for obtaining the tortuosity, and porosity of thick samples of rigid porous materials consisting of large (>1 mm) grains or fibres is proposed. The method uses pulses with central frequencies close to 12 kHz and an approximate bandwidth of between 3 and 20 kHz. In this frequency range, inertial rather than viscous or scattering effects dominate sound propagation in large pores. This allows application of the high frequency limit of the “equivalent fluid” model. Both reflected and transmitted signals are used in the measurements. Tortuosity is deduced from the high frequency limit of the phase speed (obtained from transmission data) and porosity is obtained from the high frequency limit of the reflection coefficient once the tortuosity is known. The method is shown to give good results in the cases where significant scattering does not occur until frequencies much higher than the upper limit of the pulse bandwidth. Apart from its applicability to samples with several centimetres thickness, the method needs only one set of measurements with the sample to deduce both tortuosity and porosity. In principle the method can be used also to estimate characteristic lengths. However, the errors are found to be larger and the results less consistent than for tortuosity.