A simple liquid state 1H NMR measurement to directly determine the surface hydroxyl density of porous silica

Silica is widely used in industrial applications and its performance is partially decided by its surface hydroxyl density α_OH. Here we report a quick, simple liquid ¹H NMR method to determine α_OH using a benchtop ¹H NMR spectrometer. The results show excellent agreement with the literature with an α_OH range from 4.16 to 6.56 OH per nm².

Characterizing pore-scale structure-flow correlations in sedimentary rocks using magnetic resonance imaging

Quantitative, three-dimensional (3D) spatially resolved magnetic resonance flow imaging (flow MRI) methods are presented to characterize structure-flow correlations in a 4-mm-diameter plug of Ketton limestone rock using undersampled k- and q-space data acquisition methods combined with compressed sensing (CS) data reconstruction techniques. The acquired MRI data are coregistered with an X-ray microcomputed tomography (μCT) image of the same rock sample, allowing direct correlation of the structural features of the rock with local fluid transport characteristics. First, 3D velocity maps acquired at 35 μm isotropic spatial resolution showed that the flow was highly heterogeneous, with ∼10% of the pores carrying more than 50% of the flow. Structure-flow correlations were found between the local flow velocities through pores and the size and topology (coordination number) associated with these pores. These data show consistent trends with analogous data acquired for flow through a packing of 4-mm-diameter spheres, which may be due to the microstructure of Ketton rock being a consolidation of approximately spherical grains. Using two-dimensional and 3D visualization of coregistered μCT images and velocity maps, complex pore-scale flow patterns were identified. Second, 3D spatially resolved propagators were acquired at 94 μm isotropic spatial resolution. Flow dispersion within the rock was examined by analyzing each of the 331 776 local propagators as a function of observation time. Again, the heterogeneity of flow within the rock was shown. Quantification of the mean and standard deviation of each of the local propagators showed enhanced mixing occurring within the pore space at longer observation times. These spatially resolved measurements also enable investigation of the length scale of a representative elementary volume. It is shown that for a 4-mm-diameter plug this length scale is not reached.

Insights into Automotive Particulate Filters using Magnetic Resonance Imaging : Understanding filter drying in the manufacturing process and the effect of particulate matter on filter operation and fluid dynamics

Understanding the manufacture and operation of automotive emissions control particulate filters is important in the optimised design of these emissions control systems. Here we show how magnetic resonance imaging (MRI) can be used to understand the drying process, which is part of the manufacture of catalysed particulate filters. Comparison between a wall-flow particulate filter substrate and a flow-through monolith (FTM) has been performed, with MRI giving spatial information on the drying process. We have also used MRI to study the fluid dynamics of a gasoline particulate filter (GPF). Inlet and outlet channel gas velocities have been measured for a clean GPF and two GPF samples loaded with particulate matter (PM) to understand the effect of PM on the filter flow profiles and porous wall permeability as soot is deposited.

Identification of sampling patterns for high-resolution compressed sensing MRI of porous materials: 'learning' from X-ray microcomputed tomography data

There exists a strong motivation to increase the spatial resolution of magnetic resonance imaging (MRI) acquisitions so that MRI can be used as a microscopy technique in the study of porous materials. This work introduces a method for identifying novel data sampling patterns to achieve undersampling schemes for compressed sensing MRI (CS-MRI) acquisitions, enabling 3D spatial resolutions of 17.6 µm to be achieved. A data-driven learning approach is used to derive k-space undersampling schemes for 3D MRI acquisitions from 3D X-ray microcomputed tomography (µCT) datasets acquired at a higher spatial resolution than can be acquired using MRI. The performance of the new sampling approach was compared to other, well-established sampling strategies using simulated MRI data obtained from high-resolution µCT images of rock core plugs. These simulations were performed for a range of different k-space sampling fractions (0.125-0.375) using images of Ketton limestone. The method was then extended to consideration of imaging Estaillades limestone and Fontainebleau sandstone. The results show that the new sampling approach performs as well as or better than conventional variable density sampling and without need for time-consuming parameter optimisation. Further, a bespoke sampling pattern is produced for each rock type. The novel undersampling strategy was employed to acquire 3D magnetic resonance images of a Ketton limestone rock at spatial resolutions of 35 and 17.6 µm. The ability of the k-space sampling scheme produced using the new approach in enabling reconstruction of the pore space characteristics of the rock was then demonstrated by benchmarking against the pore space statistics obtained from high-resolution µCT data. The MRI data acquired at 17.6 µm resolution gave excellent agreement with the pore size distribution obtained from the X-ray microcomputed tomography dataset, while the pore coordination number distribution obtained from the MRI data was slightly skewed to lower coordination numbers. This approach provides a method of producing a k-space undersampling pattern for MRI acquisition at a spatial resolution for which a fully sampled acquisition at that spatial resolution would be impractically long. The approach can be easily extended to other CS-MRI techniques, such as spatially resolved flow and relaxation time mapping. LAY DESCRIPTION: Magnetic resonance imaging (MRI) is widely used to study the microstructure of, and fluid transport phenomena in porous media relevant for engineering applications. A major application is the study of water and hydrocarbon transport in porous sedimentary rocks, which typically have pore sizes smaller than 100 µm. The spatial resolution of routine MRI acquisitions, however, is limited to several hundred µm due to the relatively low sensitivity of the magnetic resonance method. Therefore, there exists a strong motivation to increase the spatial resolution of MRI by one to two orders of magnitude to be able to study these rocks at a pore scale. This work reports the initial step towards achieving this. Three-dimensional images of rock pore structure are acquired at both 35 and 17.6 µm spatial resolution. In ongoing work, these methods are now being incorporated into magnetic resonance velocity imaging methods, thereby enabling imaging of both pore structure and hydrodynamics at these much higher spatial resolutions than were hitherto possible. Although X-ray microcomputed tomography (µCT) produces high spatial resolution images, it is far more limited in being able to spatially map transport processes (i.e. flow) in porous media. This work reports a strategy for accelerating the image acquisition time such that sufficient signal-to-noise ratio (SNR) is achieved to increase the spatial resolution, that is, the voxel size within which there is sufficient SNR within the resulting image. To achieve this, a technique known as compressed sensing is used which exploits undersampling of the acquired data relative to the standard fully sampled image. In MRI, data are acquired in so-called k-space and Fourier transformed to yield the real space image. The challenge, when undersampling, is to optimise the specific points in k-space that are acquired because these will influence the quality of the resulting image. This work reports a straightforward, robust strategy for identifying the optimal sets of k-space points to acquire. The method introduced uses simulated MRI images calculated from high-resolution µCT images of the rocks of interest, from which optimised MRI sampling patterns are obtained. The method does not require any optimisation of parameters for its implementation, which is a significant advantage compared to other strategies. Moreover, we show that the pore space characteristics of the acquired MRI images are in excellent agreement with the same characteristics obtained from a high-resolution µCT image.

Optimising sampling patterns for bi-exponentially decaying signals

A recently reported method, based on the Cramér-Rao Lower Bound theory, for optimising sampling patterns for a wide range of nuclear magnetic resonance (NMR) experiments is applied to the problem of optimising sampling patterns for bi-exponentially decaying signals. Sampling patterns are optimised by minimizing the percentage error in estimating the most difficult to estimate parameter of the bi-exponential model, termed the objective function. The predictions of the method are demonstrated in application to pulsed field gradient NMR data recorded for the two-component diffusion of a binary mixture of methane/ethane in a zeolite. It is shown that the proposed method identifies an optimal sampling pattern with the predicted objective function being within 10% of that calculated from the experiment dataset. The method is used to advise on the number of sampled points and the noise level needed to resolve two-component systems characterised by a range of ratios of populations and diffusion coefficients. It is subsequently illustrated how the method can be used to reduce the experiment acquisition time while still being able to resolve a given two-component system.

Accelerating the estimation of 3D spatially resolved T2 distributions

Obtaining quantitative, 3D spatially-resolved T₂ distributions (T₂ maps) from magnetic resonance data is of importance in both medical and porous media applications. Due to the long acquisition time, there is considerable interest in accelerating the experiments by applying undersampling schemes during the acquisition and developing reconstruction techniques for obtaining the 3D T₂ maps from the undersampled data. A multi-echo spin echo pulse sequence is used in this work to acquire the undersampled data according to two different sampling patterns: a conventional coherent sampling pattern where the same set of lines in k-space is sampled for all equally-spaced echoes in the echo train, and a proposed incoherent sampling pattern where an independent set of k-space lines is sampled for each echo. The conventional reconstruction technique of total variation regularization is compared to the more recent techniques of nuclear norm regularization and Nuclear Total Generalized Variation (NTGV) regularization. It is shown that best reconstructions are obtained when the data acquired using an incoherent sampling scheme are processed using NTGV regularization. Using an incoherent sampling pattern and NTGV regularization as the reconstruction technique, quantitative results are obtained at sampling percentages as low as 3.1% of k-space, corresponding to a 32-fold decrease in the acquisition time, compared to a fully sampled dataset.

Optimising magnetic resonance sampling patterns for parametric characterisation

Sampling strategies are often central to experimental design. Choosing efficiently which data to acquire can improve the estimation of parameters and reduce the acquisition time. This work is focused on designing optimal sampling patterns for Nuclear Magnetic Resonance (NMR) applications, illustrated with respect to the best estimate of the parameters characterising a lognormal distribution. Lognormal distributions are commonly used as fitting models for distributions of spin-lattice relaxation time constants, spin-spin relaxation time constants and diffusion coefficients. A method for optimising the choice of points to be sampled is presented which is based on the Cramér-Rao Lower Bound (CRLB) theory. The method's capabilities are demonstrated experimentally by applying it to the problem of estimating the emulsion droplet size distribution from a pulsed field gradient (PFG) NMR diffusion experiment. A difference of <5% is observed between the predictions of CRLB theory and the PFG NMR experimental results. It is shown that CLRB theory is stable down to signal-to-noise ratios of ∼10. A sensitivity analysis for the CRLB theory is also performed. The method of optimizing sampling patterns is easily adapted to distributions other than lognormal and to other aspects of experimental design; case studies of optimising the sampling scheme for a fixed acquisition time and determining the potential for reduction in acquisition time for a fixed parameter estimation accuracy are presented. The experimental acquisition time is typically reduced by a factor of 3 using the proposed method compared to a constant gradient increment approach that would usually be used.

Validation of a low field Rheo-NMR instrument and application to shear-induced migration of suspended non-colloidal particles in Couette flow

Nuclear magnetic resonance rheology (Rheo-NMR) is a valuable tool for studying the transport of suspended non-colloidal particles, important in many commercial processes. The Rheo-NMR imaging technique directly and quantitatively measures fluid displacement as a function of radial position. However, the high field magnets typically used in these experiments are unsuitable for the industrial environment and significantly hinder the measurement of shear stress. We introduce a low field Rheo-NMR instrument (¹H resonance frequency of 10.7MHz), which is portable and suitable as a process monitoring tool. This system is applied to the measurement of steady-state velocity profiles of a Newtonian carrier fluid suspending neutrally-buoyant non-colloidal particles at a range of concentrations. The large particle size (diameter >200μm) in the system studied requires a wide-gap Couette geometry and the local rheology was expected to be controlled by shear-induced particle migration. The low-field results are validated against high field Rheo-NMR measurements of consistent samples at matched shear rates. Additionally, it is demonstrated that existing models for particle migration fail to adequately describe the solid volume fractions measured in these systems, highlighting the need for improvement. The low field implementation of Rheo-NMR is complementary to shear stress rheology, such that the two techniques could be combined in a single instrument.

Retaining both discrete and smooth features in 1D and 2D NMR relaxation and diffusion experiments

A new method of regularization of 1D and 2D NMR relaxation and diffusion experiments is proposed and a robust algorithm for its implementation is introduced. The new form of regularization, termed the Modified Total Generalized Variation (MTGV) regularization, offers a compromise between distinguishing discrete and smooth features in the reconstructed distributions. The method is compared to the conventional method of Tikhonov regularization and the recently proposed method of L₁ regularization, when applied to simulated data of 1D spin-lattice relaxation, T₁, 1D spin-spin relaxation, T₂, and 2D T₁-T₂ NMR experiments. A range of simulated distributions composed of two lognormally distributed peaks were studied. The distributions differed with regard to the variance of the peaks, which were designed to investigate a range of distributions containing only discrete, only smooth or both features in the same distribution. Three different signal-to-noise ratios were studied: 2000, 200 and 20. A new metric is proposed to compare the distributions reconstructed from the different regularization methods with the true distributions. The metric is designed to penalise reconstructed distributions which show artefact peaks. Based on this metric, MTGV regularization performs better than Tikhonov and L₁ regularization in all cases except when the distribution is known to only comprise of discrete peaks, in which case L₁ regularization is slightly more accurate than MTGV regularization.

Obtaining sparse distributions in 2D inverse problems

The mathematics of inverse problems has relevance across numerous estimation problems in science and engineering. L₁ regularization has attracted recent attention in reconstructing the system properties in the case of sparse inverse problems; i.e., when the true property sought is not adequately described by a continuous distribution, in particular in Compressed Sensing image reconstruction. In this work, we focus on the application of L₁ regularization to a class of inverse problems; relaxation-relaxation, T₁-T₂, and diffusion-relaxation, D-T₂, correlation experiments in NMR, which have found widespread applications in a number of areas including probing surface interactions in catalysis and characterizing fluid composition and pore structures in rocks. We introduce a robust algorithm for solving the L₁ regularization problem and provide a guide to implementing it, including the choice of the amount of regularization used and the assignment of error estimates. We then show experimentally that L₁ regularization has significant advantages over both the Non-Negative Least Squares (NNLS) algorithm and Tikhonov regularization. It is shown that the L₁ regularization algorithm stably recovers a distribution at a signal to noise ratio<20 and that it resolves relaxation time constants and diffusion coefficients differing by as little as 10%. The enhanced resolving capability is used to measure the inter and intra particle concentrations of a mixture of hexane and dodecane present within porous silica beads immersed within a bulk liquid phase; neither NNLS nor Tikhonov regularization are able to provide this resolution. This experimental study shows that the approach enables discrimination between different chemical species when direct spectroscopic discrimination is impossible, and hence measurement of chemical composition within porous media, such as catalysts or rocks, is possible while still being stable to high levels of noise.

Accelerating flow propagator measurements for the investigation of reactive transport in porous media

NMR propagator measurements are widely used for identifying the distribution of molecular displacements over a given observation time, characterising a flowing system. However, where high q-space resolution is required, the experiments are time consuming and therefore unsuited to the study of dynamic systems. Here, it is shown that with an appropriately sampled subset of the q-space points in a high-resolution flow propagator measurement, one can quickly and robustly reconstruct the fully sampled propagator through interpolation of the acquired raw data. It was found that exponentially sampling ∼4% of the original data-points allowed a reconstruction with the deviation from the fully sampled propagator below the noise level, in this case reducing the required experimental time from ∼2.8h to <7min. As a demonstration, this approach is applied to observe the temporal evolution of the reactive flow of acid through an Estaillades rock core plug. It is shown that 'wormhole' formation in the rock core plug provides a channel for liquid flow such that the remaining pore space is by-passed, thereby causing the flow velocity of the liquid in the remaining part of the plug to become stagnant. The propagator measurements are supported by both 1D profiles and 2D imaging data. Such insights are of importance in understanding well acidisation and CO₂ sequestration processes.

Fast imaging of laboratory core floods using 3D compressed sensing RARE MRI

Three-dimensional (3D) imaging of the fluid distributions within the rock is essential to enable the unambiguous interpretation of core flooding data. Magnetic resonance imaging (MRI) has been widely used to image fluid saturation in rock cores; however, conventional acquisition strategies are typically too slow to capture the dynamic nature of the displacement processes that are of interest. Using Compressed Sensing (CS), it is possible to reconstruct a near-perfect image from significantly fewer measurements than was previously thought necessary, and this can result in a significant reduction in the image acquisition times. In the present study, a method using the Rapid Acquisition with Relaxation Enhancement (RARE) pulse sequence with CS to provide 3D images of the fluid saturation in rock core samples during laboratory core floods is demonstrated. An objective method using image quality metrics for the determination of the most suitable regularisation functional to be used in the CS reconstructions is reported. It is shown that for the present application, Total Variation outperforms the Haar and Daubechies3 wavelet families in terms of the agreement of their respective CS reconstructions with a fully-sampled reference image. Using the CS-RARE approach, 3D images of the fluid saturation in the rock core have been acquired in 16min. The CS-RARE technique has been applied to image the residual water saturation in the rock during a water-water displacement core flood. With a flow rate corresponding to an interstitial velocity of vi=1.89±0.03ftday(-1), 0.1 pore volumes were injected over the course of each image acquisition, a four-fold reduction when compared to a fully-sampled RARE acquisition. Finally, the 3D CS-RARE technique has been used to image the drainage of dodecane into the water-saturated rock in which the dynamics of the coalescence of discrete clusters of the non-wetting phase are clearly observed. The enhancement in the temporal resolution that has been achieved using the CS-RARE approach enables dynamic transport processes pertinent to laboratory core floods to be investigated in 3D on a time-scale and with a spatial resolution that, until now, has not been possible.

Gravitational collapse of depletion-induced colloidal gels

We study the ageing and ultimate gravitational collapse of colloidal gels in which the interparticle attraction is induced by non-adsorbing polymers via the depletion effect. The gels are formed through arrested spinodal decomposition, whereby the dense phase arrests into an attractive glass. We map the experimental state diagram onto a theoretical one obtained from computer simulations and theoretical calculations. Discrepancies between the experimental and simulated gel regions in the state diagram can be explained by the particle size and density dependence of the boundary below which the gel is not strong enough to resist gravitational stress. Visual observations show that gravitational collapse of the gels falls into two distinct regimes as the colloid and polymer concentrations are varied, with gels at low colloid concentrations showing the onset of rapid collapse after a delay time. Magnetic resonance imaging (MRI) was used to provide quantitative, spatio-temporally resolved measurements of the solid volume fraction in these rapidly collapsing gels. We find that during the delay time, a dense region builds up at the top of the sample. The rapid collapse is initiated when the gel structure is no longer able to support this dense layer.

Do group 1 metal salts form deep eutectic solvents?

Group 1 salts were compared with quaternary ammonium chlorides for their ability to form deep eutectic solvents and it was found that while some formed liquids the sodium ions caused the liquids to become structured and increased their viscosity.

Characterising the rheology of non-Newtonian fluids using PFG-NMR and cumulant analysis

Conventional rheological characterisation using nuclear magnetic resonance (NMR) typically utilises spatially-resolved measurements of velocity. We propose a new approach to rheometry using pulsed field gradient (PFG) NMR which readily extends the application of MR rheometry to single-axis gradient hardware. The quantitative use of flow propagators in this application is challenging because of the introduction of artefacts during Fourier transform, which arise when realistic sampling strategies are limited by experimental and hardware constraints and when particular spatial and temporal resolution are required. The method outlined in this paper involves the cumulant analysis of the acquisition data directly, thereby preventing the introduction of artefacts and reducing data acquisition times. A model-dependent approach is developed to enable the pipe-flow characterisation of fluids demonstrating non-Newtonian power-law rheology, involving the use of an analytical expression describing the flow propagator in terms of the flow behaviour index. The sensitivity of this approach was investigated and found to be robust to the signal-to-noise ratio (SNR) and number of acquired data points, enabling an increase in temporal resolution defined by the SNR. Validation of the simulated results was provided by an experimental case study on shear-thinning aqueous xanthan gum solutions, whose rheology could be accurately characterised using a power-law model across the experimental shear rate range of 1-100 s(-1). The flow behaviour indices calculated using this approach were observed to be within 8% of those obtained using spatially-resolved velocity imaging and within 5% of conventional rheometry. Furthermore, it was shown that the number of points sampled could be reduced by a factor of 32, when compared to the acquisition of a volume-averaged flow propagator with 128 gradient increments, without negatively influencing the accuracy of the characterisation, reducing the acquisition time to only 3% of its original value.

Determining adsorbate configuration on alumina surfaces with 13C nuclear magnetic resonance relaxation time analysis

Relative strengths of surface interaction for individual carbon atoms in acyclic and cyclic hydrocarbons adsorbed on alumina surfaces are determined using chemically resolved ¹³C nuclear magnetic resonance (NMR) T₁ relaxation times.

Nuclear magnetic resonance measurements of velocity distributions in an ultrasonically vibrated granular bed

We report the results of nuclear magnetic resonance imaging experiments on granular beds of mustard grains fluidized by vertical vibration at ultrasonic frequencies. The variation of both granular temperature and packing fraction with height was measured within the three-dimensional cell for a range of vibration frequencies, amplitudes and numbers of grains. Small increases in vibration frequency were found--contrary to the predictions of classical 'hard-sphere' expressions for the energy flux through a vibrating boundary--to result in dramatic reductions in granular temperature. Numerical simulations of the grain-wall interactions, using experimentally determined Hertzian contact stiffness coefficients, showed that energy flux drops significantly as the vibration period approaches the grain-wall contact time. The experiments thus demonstrate the need for new models for 'soft-sphere' boundary conditions at ultrasonic frequencies.

Low-field permanent magnets for industrial process and quality control

In this review we focus on the technology associated with low-field NMR. We present the current state-of-the-art in low-field NMR hardware and experiments, considering general magnet designs, rf performance, data processing and interpretation. We provide guidance on obtaining the optimum results from these instruments, along with an introduction for those new to low-field NMR. The applications of lowfield NMR are now many and diverse. Furthermore, niche applications have spawned unique magnet designs to accommodate the extremes of operating environment or sample geometry. Trying to capture all the applications, methods, and hardware encompassed by low-field NMR would be a daunting task and likely of little interest to researchers or industrialists working in specific subject areas. Instead we discuss only a few applications to highlight uses of the hardware and experiments in an industrial environment. For details on more particular methods and applications, we provide citations to specialized review articles.

Grain sizing in porous media using Bayesian magnetic resonance

We introduce a Bayesian inference approach to analyze magnetic resonance data of granular solids. To characterize structure using magnetic resonance, it is usual to acquire data in k space which are then Fourier transformed to obtain an image. An alternative approach, adopted here, is to utilize the Rayleigh distribution observed in the signal intensity for a given k when a random selection of grains is measured in k space, to define a likelihood function for Bayesian analysis. This Bayesian likelihood function is used to noninvasively characterize grains within a porous medium on length scales below the practical resolution of magnetic resonance imaging. A pore size distribution is then calculated from the measured grain size distribution using a Monte Carlo approach. We demonstrate this general technique with specific examples of water-saturated rock cores.

Extending the use of Earth's Field NMR using Bayesian methodology: application to particle sizing

There is currently much interest in extending the use of low-field magnetic resonance measurements and in particular, to obtain spatial information from these data. Here, we demonstrate the application of a Bayesian magnetic resonance approach for the sizing of objects using low magnetic field measurement technology, where there is insufficient signal-to-noise to allow a conventional imaging approach for structural characterisation. The method is illustrated in application to the sizing of spheres, in this case of radius 9.5mm, using an Earth's Field Nuclear Magnetic Resonance (EFNMR) spectrometer with pre-polarisation. Numerical simulations of the measurement at different signal-to-noise ratios and implementation of different k-space sampling schemes are considered to identify the optimal experimental protocol. In this example, the determination of sphere radius is found to be accurate to ±1mm. We confirm that the posterior distribution provides an accurate estimate of the uncertainty in the measurement.

MRI technique for the snapshot imaging of quantitative velocity maps using RARE

A quantitative PGSE-RARE pulse sequence was developed and successfully applied to the in situ dissolution of two pharmaceutical formulations dissolving over a range of timescales. The new technique was chosen over other existing fast velocity imaging techniques because it is T(2) weighted, not T(2)(∗) weighted, and is, therefore, robust for imaging time-varying interfaces and flow in magnetically heterogeneous systems. The complex signal was preserved intact by separating odd and even echoes to obtain two phase maps which are then averaged in post-processing. Initially, the validity of the technique was shown when imaging laminar flow in a pipe. Subsequently, the dissolution of two drugs was followed in situ, where the technique enables the imaging and quantification of changes in the form of the tablet and the flow field surrounding it at high spatial and temporal resolution. First, the complete 3D velocity field around an eroding salicylic acid tablet was acquired at a resolution of 98×49 μm(2), within 20 min, and monitored over ∼13 h. The tablet was observed to experience a heterogeneous flow field and, hence a heterogeneous shear field, which resulted in the non-symmetric erosion of the tablet. Second, the dissolution of a fast dissolving immediate release tablet was followed using one-shot 2D velocity images acquired every 5.2 s at a resolution of 390×390 μm(2). The quantitative nature of the technique and fast acquisition times provided invaluable information on the dissolution behaviour of this tablet, which had not been attainable previously with conventional quantitative MRI techniques.

A Bayesian approach to characterising multi-phase flows using magnetic resonance: application to bubble flows

Magnetic Resonance (MR) imaging is difficult to apply to multi-phase flows due to both the inherently short T₂* characterising such systems and the relatively long time taken to acquire the data. We develop a Bayesian MR approach for analysing data in k-space that eliminates the need for image acquisition, thereby significantly extending the range of systems that can be studied. We demonstrate the technique by measuring bubble size distributions in gas-liquid flows. The MR approach is compared with an optical technique at a low gas fraction (∼2%), before being applied to a system where the gas fraction is too high for optical measurements (∼15%).

Magnetic resonance measurements of high-velocity particle motion in a three-dimensional gas-solid spouted bed

Magnetic resonance imaging has been used to measure particle velocities, exceeding 1 m s⁻¹ in a two-phase granular system, namely, a spouted bed. The measurements are complicated due to the high voidage, i.e., low particle density, in the region of the highest particle velocity. However, applying gradient shapes which allow fast switching and, thus, short encoding and observation times in combination with a short echo time enable these measurements. It was found that the profile of the particle velocity is nonparabolic. Based on these measurements it was possible to confirm observations made in numerical simulations that there must be a continuous momentum exchange between the annulus region and the spout along the entire length of the spout.

Reducing data acquisition times in phase-encoded velocity imaging using compressed sensing

We present a method for accelerating the acquisition of phase-encoded velocity images by the use of compressed sensing (CS), a technique that exploits the observation that an under-sampled signal can be accurately reconstructed by utilising the prior knowledge that it is sparse or compressible. We present results of both simulated and experimental measurements of liquid flow through a packed bed of spherical glass beads. For this system, the best image reconstruction used a spatial finite-differences transform. The reconstruction was further improved by utilising prior knowledge of the liquid distribution within the image. Using this approach, we demonstrate that for a sampling fraction of approximately 30% of the full k-space data set, the velocity can be recovered with a relative error of 11%, which is below the visually detectable limit. Furthermore, the error in the total flow measured using the CS reconstruction is <3% for sampling fractions > or = 30%. Thus, quantitative velocity images were obtained in a third of the acquisition time required using conventional imaging. The reduction in data acquisition time can also be exploited in acquiring images at a higher spatial resolution, which increases the accuracy of the measurements by reducing errors arising from partial volume effects. To illustrate this, the CS algorithm was used to reconstruct gas-phase velocity images at a spatial resolution of 230 microm x 230 microm. Images at this spatial resolution are prohibitively time-consuming to acquire using full k-space sampling techniques.

Nuclear magnetic resonance relaxation and diffusion in the presence of internal gradients: the effect of magnetic field strength

It is known that internal magnetic field gradients in porous materials, caused by susceptibility differences at the solid-fluid interfaces, alter the observed effective Nuclear Magnetic Resonance transverse relaxation times T2,eff. The internal gradients scale with the strength of the static background magnetic field B0. Here, we acquire data at various magnitudes of B0 to observe the influence of internal gradients on T2-T2 exchange measurements; the theory discussed and observations made are applicable to any T2-T2 analysis of heterogeneous materials. At high magnetic field strengths, it is possible to observe diffusive exchange between regions of local internal gradient extrema within individual pores. Therefore, the observed exchange pathways are not associated with pore-to-pore exchange. Understanding the significance of internal gradients in transverse relaxation measurements is critical to interpreting these results. We present the example of water in porous sandstone rock and offer a guideline to determine whether an observed T2,eff relaxation time distribution reflects the pore size distribution for a given susceptibility contrast (magnetic field strength) and spin echo separation. More generally, we confirm that for porous materials T1 provides a better indication of the pore size distribution than T2,eff at high magnetic field strengths (B0>1 T), and demonstrate the data analysis necessary to validate pore size interpretations of T2,eff measurements.

Quantitative single point imaging with compressed sensing

A novel approach with respect to single point imaging (SPI), compressed sensing, is presented here that is shown to significantly reduce the loss of accuracy of reconstructed images from under-sampled acquisition data. SPI complements compressed sensing extremely well as it allows unconstrained selection of sampling trajectories. Dynamic processes featuring short T2* NMR signal can thus be more rapidly imaged, in our case the absorption of moisture by a cereal-based wafer material, with minimal loss of image quantification. The absolute moisture content distribution is recovered via a series of images acquired with variable phase encoding times allowing extrapolation to time zero for each image pixel and the effective removal of T2* contrast.

Magnetic resonance velocity imaging of liquid and gas two-phase flow in packed beds

Single-phase liquid flow in porous media such as bead packs and model fixed bed reactors has been well studied by MRI. To some extent this early work represents the necessary preliminary research to address the more challenging problem of two-phase flow of gas and liquid within these systems. In this paper, we present images of both the gas and liquid velocities during stable liquid-gas flow of water and SF(6) within a packing of 5mm spheres contained within columns of diameter 40 and 27 mm; images being acquired using (1)H and (19)F observation for the water and SF(6), respectively. Liquid and gas flow rates calculated from the velocity images are in agreement with macroscopic flow rate measurements to within 7% and 5%, respectively. In addition to the information obtained directly from these images, the ability to measure liquid and gas flow fields within the same sample environment will enable us to explore the validity of assumptions used in numerical modelling of two-phase flows.

Rapid encoding of T(1) with spectral resolution in n-dimensional relaxation correlations

Nuclear magnetic resonance (NMR) T(1) relaxation times have been encoded in the second dimension of two-dimensional relaxation correlation and exchange experiments using a rapid "double-shot"T(1) pulse sequence. This technique also retains chemical shift information (delta) for short T(2)( *) materials. In this way, a spectral dimension can be incorporated into a T(2)-T(1)-delta correlation without an increase in experimental time compared to the conventional, chemically insensitive T(1)-T(2) correlation. Here, the T(2)-T(1)-delta pulse sequence is used to unambiguously identify oil and water fractions in a permeable rock. A novel T(1)-T(1)-(delta) relaxation exchange measurement is also introduced and used to observe diffusive exchange of water in cellulose fibres.

Determining NMR flow propagator moments in porous rocks without the influence of relaxation

Flow propagators, used for the study of advective motion of brine solution in porous carbonate and sandstone rocks, have been obtained without the influence of Nuclear Magnetic Resonance (NMR) relaxation times, T1 and T2. These spin relaxation mechanisms normally result in a loss of signal that varies depending on the displacement zeta of the flowing spins, thereby preventing the acquisition of quantitative propagator data. The full relaxation behaviour of the system under flow needs to be characterised to enable the implementation of a true quantitative measurement. Two-dimensional NMR correlations of zeta-T2 and T1-T2 are used in combination to provide the flow propagators without relaxation weighting. T1-zeta correlations cannot be used due to the loss of T1 information during the displacement observation time Delta. Here the moments of the propagators are extracted by statistical analysis of the full propagator shape. The measured displacements (first moments) are seen to correlate with the expected mean displacements for long observation times Delta. The higher order moments of the propagators determined by this method indicate those obtained previously using a correction were overestimated.

A rapid measurement of flow propagators in porous rocks

NMR flow propagators have been obtained for brine flowing through Bentheimer sandstone using the rapid DiffTrain pulse sequence. In this way, 8 flow propagators at different observation times Delta were acquired in 67 mins, compared to 7 h for the same measurements implemented with conventional pulsed field gradient (PFG) sequences. DiffTrain allows this time saving to be achieved through the acquisition of multiple displacement probability distributions over a range of Delta in a single measurement. If only the propagator moments are required, this experiment time can be further reduced to 9 mins through appropriate sparse sampling at low q values. The propagator moments obtained from DiffTrain measurements with dense and sparse q-space sampling are shown to be equivalent to those obtained from conventional PFG measurements.

Spatially resolved quantification of metal ion concentration in a biofilm-mediated ion exchanger

A bioremediation process to remove Co(2+) from aqueous solution is investigated in this study using a magnetic resonance imaging (MRI) protocol to rapidly obtain multiple 2D spatially resolved Co(2+) ion concentration maps. The MRI technique is described in detail and its ability to determine the evolution in both axial and radial concentration profiles demonstrated, from which total column capacity can be determined. The final ion exchange column design allows operation in the 'plug flow' regime, hence making use of its full capacity before breakthrough. Conventional techniques for such process optimization are either restricted to the analysis of the exchanger outlet, which provides no information on the spatial heterogeneity of the system, or are invasive and need a variety of sample points to obtain 1D concentration information. To the best of our knowledge, our results represent the first concentration maps describing the bioremediation of metal ion contaminated water.

Time-of-flight variant to image mixing of granular media in a 3D fluidized bed

This paper describes a variant of time-of-flight magnetic resonance (MR) imaging that provides a method of measuring the inherent mixing in a fluidized bed without the introduction of tracer particles. The modifications to conventional time-of-flight imaging enable the measurement of the axial mixing of a precisely controlled initial particle distribution, thereby providing measurements suitable for a direct comparison with models of solids mixing in granular systems. The imaging sequence is applied to characterize mixing, over time scales of 25-1000 ms, in a gas-fluidized bed of Myosotis seed particles; mixing over short timescales, inaccessible using conventional tracer techniques, is studied using this technique. The mixing pattern determined by this pulse sequence is used in conjunction with MR velocity images of the motion of the particles to provide new insight into the mechanism of solids mixing in granular systems.

Rapid two-dimensional imaging of bubbles and slugs in a three-dimensional, gas-solid, two-phase flow system using ultrafast magnetic resonance

Ultrafast magnetic resonance has been applied to measure the geometry of bubbles and slugs in a three-dimensional gas-solid two-phase flow. A bed of particles of diameter 0.5 mm were fluidized with gas velocities in the range of 0.08-0.26 m/s. Bubbles were imaged in transverse as well as vertical planes with an acquisition time of down to 25 ms and a spatial resolution down to 1.7 mm. Owing to the ultrafast character of these measurements, it is not only possible to evaluate correlations, e.g., for the bubble diameter, but also evaluate models of complex hydrodynamic phenomena, such as the splitting and coalescence of bubbles.

Modelling biofilm-modified hydrodynamics in 3D

A simulation-based study to predict the impact of biofilm growth on displacement distributions for flow of water through a supporting packed bed is presented. The lattice Boltzmann method and a directed random walk algorithm are used, and are applied to the system with and without biofilm being present. The aim of this simulation study is to model the anomalous transport dynamics induced by biofilm, as reported in the literature, and thus to study the impact of observation time, delta, on the shape of the displacement distributions (propagators). We believe that this is the first demonstration of a propagator simulation for flow through a complex porous structure modulated by biofilm growth. The propagator distributions undergo a transition from a pre-asymptotic to a Gaussian-shaped distribution with increasing delta. The propagators were simulated for a wide range of delta going up to 500 seconds. This transition occurs with and without biofilm, but is very significantly delayed when biofilm is present due to the consequential development of essentially stagnant regions. The transition can be classified into three stages: a diffusion-dominated stage, a "twin-peak" stage and an advection-dominated stage.

Real-time measurement of bubbling phenomena in a three-dimensional gas-fluidized bed using ultrafast magnetic resonance imaging

Ultrafast magnetic resonance imaging has been applied for the first time to measure simultaneously both the rise velocities and coalescence of bubbles, and the dynamics of the solid phase in a gas-solid two-phase flow. Here, we consider the hydrodynamics within a gas-fluidized bed of particles of diameter 0.5 mm contained within a column of internal diameter 50 mm; gas velocities in the range of 0.18-0.54 m/s were studied. The data are of sufficient temporal and spatial resolution that bubble size and the evolution of bubble size and velocity following coalescence events are determined.