Spatially Resolved Dynamic Longitudinal Relaxometry in Single-Sided NMR

Relaxation in nuclear magnetic resonance (NMR), both transverse and longitudinal, provides information on microscopic features of a wide variety of systems and may be used to monitor dynamic processes such as cementation, chemical reactions, gelatinization, and evaporation. Dynamic relaxometry, in combination with spatial resolution, is a useful technique that provides deep insight into complex systems evolution. In this work, we explore the range of applicability of single-sided NMR to determine the evaporation kinetics of fluid from porous media. We show that, due to technical experimental restrictions, the determination of the time-dependent amount of fluid in different voids as a function of the position is in general not feasible with transverse relaxation experiments. However, as opposed to common intuition, longitudinal relaxation experiments provide reliable and fast acquisition, compatible with the requirements needed to monitor a water evaporation process from a model oil-reservoir rock sample.


Introduction
Porous materials with voids that may be filled with fluids, liquid, or gas are present in a great variety of systems, such as soils, concrete, sponges, foodstuff, or oil-bearing rocks among others.Some of the most relevant parameters that define the exploitability of a porous media are the pore sizes, the number of pores, their connectivity, or the ability of a fluid to flow through the medium.The geometrical structure can be precisely determined by a variety of techniques such as gas sorption-desorption, mercury porosimetry, X-rays, neutron dispersion, and electronic microscopy.However, to fully understand the performance of a porous medium, the confined fluid dynamics must be determined.In this regard, nuclear magnetic resonance (NMR) stands out, as it directly detects the spin-bearing particles from the gas or liquid, without the need for tracer particles that may alter the flow or diffusion.It can detect fluids non-invasively in opaque media and may provide spatial and microscopic information.Spatial information is often obtained by magnetic resonance imaging (MRI), while microscopic information is obtained by the determination either of restricted diffusion coefficients or relaxation times.Transverse and longitudinal relaxation times will probe the system geometry due to magnetization loss by liquid/surface interactions [1,2].For systems changing with time, for instance, during an evaporation process, the term dynamic relaxometry is often used [3].A great variety of systems have been studied by probing changes in relaxation times, for instance, the cementation process [4][5][6][7], porosity, spore size distributions, and permeability in rock from oil wells among others [8][9][10][11].This approach can also be combined with MRI, often in one dimension.In this way, not only the total amount of fluid in a slice can be determined, but also its distribution between different environments [12][13][14][15].
Single-sided NMR englobes a family of devices where the magnetic field is used ex situ, where a strong magnetic field gradient is usually present.Examples are welllogging tools [16], the stray field of superconducting magnets [17,18] or tailored designed benchtop equipment [19,20] such as the MOLE (Mobile Lateral Explorer) [21,22], and the most developed by the industry, the NMR-MOUSE (Mobile Universal Surface Explorer) [23][24][25].The NMR-MOUSE presents the advantage of mobility and non-stringent sample sizes, with the penetration depth being the main geometric limitation.The technique has found a great variety of applications in the study of solid materials, polymers, cultural heritage [26][27][28], and liquids in oil rocks [29,30].Signals are acquired as a function of the penetration depth, namely, a 1D image or more commonly referred to as a profile.As the signals are detected with a train of echoes, different relaxation times will reflect changes in polymer crosslinking, fluid-restricted mobility, or changes in material composition.By acquiring multiple echoes and selecting different time intervals for signal addition, the contrast between different components is readily achieved [31].The main drawback with this strategy arises when the detected signals have relaxation times on the order of hundreds of milliseconds.As the influence of diffusion is in general to be avoided, short echo times are used leading to the application of thousands of radiofrequency pulses.This induces heating of the excitation/detection coil, which will eventually detune the resonant circuit, decreasing the performance of the system.
In this work, we analyze the limitation imposed by coil heating during the acquisition of transverse relaxation for liquids confined in porous media.We find that correctly determining the distribution of fluids in different void sizes requires an experimental time that may very well exceed the system's characteristic time in an evaporation process.We then analyze the approach of using longitudinal relaxation times to probe the fluid distribution.This counterintuitive approach is based on the boost in signalto-noise ratio that may be achieved by adding several echoes during a multi-pulse sequence used as a detection block.We have recently applied this approach to show that diffusion-relaxation maps may be acquired in a shorter time using D-T 1 rather

3
Spatially Resolved Dynamic Longitudinal Relaxometry in… than D-T 2 [32].We determine a set of parameters to be applied to a particular configuration of a PM25 NMR-MOUSE developed by Magritek GmbH and apply these conditions to the determination of spatial and microscopic information during the evaporation process of water from a model oil rock.

Profiles with Transverse Relaxation Times
Experiments were carried out in a single-sided NMR device PM25, from Magritek GmbH, which has a static gradient of 7 T/m and operates at a frequency of 12.99 MHz for 1 H, with detection volume 26.7 mm away from the magnets, and a sensitive area of approximately (4 × 4) cm 2  .The sample is positioned over a flat holder and the magnet position is moved with a precision lift; in this way, control of the position of the selective slice is achieved.The system may be set to five different configurations, setting the excitation/detection coil further from the magnet and closer to the sample, defining different penetration depths ranging from 5.4 to 25 mm.In this work, without loss of generality, we restrict our discussion to a penetration depth of 10.6 mm.The length of the radiofrequency (RF) pulses was t RF = 9.5 μs.The excited bandwidth for this con- figuration defines a maximum slice thickness of 350 μm [25,31].Due to the large static gradient, the magnetization is rapidly dephased after a 90° excitation pulse, in a time shorter than the typical receiver dead times; therefore, free induction decay (FID) cannot be acquired.This leads to the use of multi-pulse refocusing sequences that capture the evolution of multiple echoes.For liquids, a Carr-Purcell-Meiboom-Gill (CPMG) [33,34] pulse sequence is generally used to detect the magnetization evolution.
For the acquisition of proton densities as a function of the sample height, namely a profile, the addition of several echoes is used to enhance the signal-to-noise ratio (SNR).However, if the sample consists of different proton pools with varying transverse relaxation times, the acquisition of the whole echo train will provide information on the different proton pools as a function of the sensor´s position.Relaxation rates may be determined by fitting with multiple exponential decay functions or, more commonly, via a numerical data inversion, the so-called inverse laplace transform (ILT) where the most favored on-dimensional algorithm is CONTIN [35].The data acquired with a CPMG pulse sequence depend on molecular diffusion, which, in the presence of the strong magnetic field gradient, will render an apparent relaxation time, T 2D .The magnetization of a diffusing spin-bearing molecule in the presence of a constant magnetic field gradient during a CPMG sequence with echo time t E at the top of the m − th echo decays as [2,36,37]: where is the gyromagnetic ratio of 1 H and D the diffusion coefficient.T 2D is the apparent relaxation time driven both by diffusion and surface relaxation, T 2S , which, in the surface-limited relaxation regime, is expressed as [1,5,8]: where the bulk relaxation time of the fluid is T 2B , 2 is the surface relaxivity and d is the pore diameter.To reduce as much as possible the influence of diffusion in Eq. ( 1), echo times must be set as short as possible.This implies that thousands of radiofrequency pulses must be applied, leading to the heating of the surface coil.
The change in the coil's resistance produces a shift of the signal's phase and eventually a pulse drop.A straightforward solution to avoid coil heating beyond a critical level is to set the recycle delay between experiments to longer times than those usually applied in NMR, namely, 3-5 T 1 .For the PM25 NMR-MOUSE, considering the pulse of the CPMG as an on-period, and the recycle delay ( rd ) as an off-period, a duty cycle of 1.6 provides a good enough performance, with less than 3° dephasing.With this condition we did not detect changes in the sample temperature, which were measured with a thermocouple in a reference water sample.Therefore, the waiting time between experiments is set as rd = t RF * NE∕1.6 , where NE is the total number of echoes to be acquired.It is then clear that the total experimental time set by the relaxation time T 2D is defined by the diffusion coefficient and the echo time.
A second factor to consider is the spatial resolution of the profile.As all radiofrequency pulses are applied in the presence of the magnetic field gradient, the length of the pulse can be used to select a desired slice thickness.However, multiple points can be acquired at the top of each echo and the acquisition time can be set to acquire the desired field of view of a rectangular slice in the direction of the gradient, or in this case the desired resolution (Res) [25,38].Setting the acquisition bandwidth constant, in this case, with a dwell time of dw = 0.5 μs, the number of points to acquire in each echo are: For the used configuration, this corresponds to n = 19 for a resolution of 350 μm and defines a minimum echo time of t E = 67.5 μs.As the slice thickness decreases, the number of acquired data points will increase.For instance, for a slice of 100 μm, n = 67 and t E = 91 μs. Figure 1A shows the transverse relaxation time values cal- culated with Eq. ( 1) as a function of the resolution for an arbitrary value of T 2S = 500 ms and D = 1.10 -9 m 2 /s.As the slice becomes thicker, echo times are shorter; therefore, the effective relaxation time increases its value.If the acquisition is t acq = 3T 2D , the number of echoes increases from c.a. 4000 to 8500 for resolutions of 100 μm and 350 μm, respectively.Therefore, the recycle delay for each of these resolutions changes from 24 to 50 s, respectively.Considering a minimum of 4 phase cycled scans, which are required to remove unwanted coherence pathways [39][40][41][42], the total experimental time calculated to acquire a single profile is shown in Fig. 1B.The number of slices required to map the complete proton density of a 1 cm sample decreases with increasing slice thickness.Even though relaxation times for thinner slices are shorter, the experimental time is much longer than that required for thicker slices.However, for the worst resolution achievable in the configuration used with (2) Spatially Resolved Dynamic Longitudinal Relaxometry in… the NMR-MOUSE, more than 90 h is required for a single profile, a prohibitive time for monitoring dynamic processes.It is worth noting that these extreme values are found when considering pure water.In systems where shorter relaxation times are found, fewer echoes are required, and therefore less total experimental time.Nevertheless, longitudinal relaxation is often simpler to interpret, for instance when considering internal gradients, where the present approach may still be favorable.

Profiles with Longitudinal Relaxation Times
Implementation of a saturation-recovery [43] sequence for T 1 measurements is straightforward in single-side NMR, as the presence of the strong magnetic field gradient is used to destroy the magnetization at the beginning of each experiment by applying an odd number of multiple 90° pulses separated by increasing time delays Fig. 1 A Relaxation times calculated under the influence of diffusion as a function of the selected resolution for a surface-mediated relaxation time of T 2S = 500 ms.As the echo time decreases with a wider slice, the influence of diffusion is smaller, and thus the increase in the relaxation times.B Total acquisition time for a profile of 1 cm.As the slices become thicker, the number of slices decreases and consequently the experimental time [25].A given number of echoes, NE, can be accumulated to improve the SNR and are a key feature for obtaining shorter measurement times [32].For liquids confined in a porous media with a distribution of pore sizes, a distribution of transverse relaxation times will be present, following Eq.( 1).Addition of echoes inevitably introduces a contrast between the fluid confined in small and large cavities that could hinder a correct quantification of the smaller voids.A threshold value must then be set.Recently we proposed a penalty of 7% in the quantification of the populations associated with shorter relaxation times [32], giving rise to the relation: for the calculated amplitude of the shorter signal.T 2D,Sh is the shortest trans- verse relaxation time that depends both on the liquid/surface interactions and on diffusion and S 0,Sh is the real amplitude of the fast-decaying components.Figure 2 shows the number of echoes that can be added as a function of the shortest surfacedefined transverse relaxation time,T 2S,Sh , and of the selected resolution.As shown in Fig. 1A, for each value of T 2S,Sh the diffusion-relaxation time changes as a function of the selected resolution.This is the reason that for a given surface-defined relaxation time, the number of echoes to add decreases with increasing resolution.For instance, for T 2S,Sh = 60 ms, NE = 110 for a slice thickness of 350 μm and NE = 68 for a slice thickness of 100 μm.It is worth noting that the election of the threshold value depends on the percentual contribution of the fast-decaying components: for instance, for a contribution of 10% to the overall signal, the value calculated with Eq. ( 4) would be 9.3%, where the difference with the real value will most likely be within the measurement uncertainty.
Once a resolution is chosen, and the number of echoes is set to define the maximum SNR achievable, the encoding strategy of T 1 must be chosen.It is customary to sample the recovery curve to values of 3-5 T 1S,L , where T 1S,L stands for the longest surface-defined longitudinal relaxation time present in the sample.For the sake of speed, we use 3 T 1S,L in the present work.Several encoding data points, Np T1 , must be chosen with a compromise between numerical inversion accuracy and experimental time.Equally spaced recovery times in a logarithmic scale are considered, and the stability of the inversion procedure is discussed in the next section.Figure 2B shows the total experimental time for a single slice considering a fixed recycle delay of rd = 200 ms and four phase cycling steps which are needed to minimize the effect of unwanted coherence pathways.A sampling of 5-50 recovery times is considered.As an example, for T 1S,L = 800 ms, 93 s is required for Np T1 = 50, while 15 s is required for Np T1 = 5.

Sample Description
Bentheimer samples purchased from Kocurek Industries Inc. were used.Disks of 3.8 cm diameter and 8.7 mm height were cut and allowed to imbibe water by placing the submerged disks in a distiller under vacuum conditions for 48 h.Samples were tightly fit in a sample holder that enables evaporation through the top of the sample.Measurements were carried out at ambient pressure and temperature.The porosity of the samples is 23%, with a permeability of ~ 400 mD [11,44,45].

Determination of the Minimal Number of Points for T 1 Encoding
A CPMG pulse sequence with t E = 67.5 μs was used to determine the apparent transverse relaxation times.A fully saturated sample sealed with parafilm to avoid evaporation was used.The relaxation distribution is shown in Fig. 3A.The longest T 2D appears shifted to lower values as shown in Fig. 1A in agreement with recently reported data with this same sensor's configuration [30], where the shorter relaxation time does not change when measured with a PM25 sensor or in a homogeneous field at 3.9 MHz.A single penetration depth was used and a T 1 measurement was carried out with 50 recovery times and four steps of phase cycling of the radiofrequency pulses [25].The maximum recovery time was set to 2500.The detection of the signal was carried out by adding 50 echoes in a CPMG pulse sequence, as discussed in Fig. 2A.Data were inverted using a one-dimensional version of the FISTA algorithm [46].The mean value (see Fig. 3B) of the longest T 1 present is T 1S,L = 800 ms, while the mean value of the shortest one is 50 ms with a contribution of 90% and 10% to the total signal, respectively.By setting T 2S = 28 ms in Eq. ( 1), a value of T 2D = 26 ms is obtained.From the T 1S distribution, a ratio of T 1 /T 2 ~ 2 is obtained for the smaller pore.If a homogeneous relaxivity is considered throughout the sample, a value of T 2S ~ 400 ms is expected for the longer relaxation time, giving rise to T 2D = 172 ms calculated from Eq. 1, which is in very good agreement with Spatially Resolved Dynamic Longitudinal Relaxometry in… the data shown in Fig. 3A.That is, transverse magnetization is more affected by diffusion for water contained in the larger pores.
A series of experiments changing the number of recovery times was carried out.Figure 3C shows the percentual contribution of each population, obtained by taking the cumulative of the relaxation time distribution spectra, as a function of the number of recovery steps, which were equally spaced in a logarithmic scale in each case.The regularization parameter and inversion time domain were fixed for all data sets.The population associated with the shortest relaxation time is well described with more than ten recovery times.A profile experiment was then carried on to determine empirically the acquisition time between slices.A value of 30 s was found, which is in good agreement with the 24 s obtained from Fig. 2B, considering the additional time consumed by data transferring from the Kea2 console to the PC, and the movement of the lift to the position of the next slice.

Evaporation Monitoring
Evaporation of water from sandstones has been previously reported, under both diffusion and convection-driven evaporation conditions [12].The underlying physical processes involved and the information on the systems that may be obtained are out of the scope of this work.We will center on a description of the performance of the introduced methodology.A second fully saturated sample was placed tightly in a Teflon holder (no background signal was detected from the sample holder) and left to evaporate through the top face of the disk.Data acquisition was started immediately acquiring a total of 31 slices of 350 μm width using a saturation-recovery pulse sequence with ten encoding steps equally spaced in a logarithmic scale.Detection was carried out with a CPMG pulse sequence with acquisition of 50 echoes and four scans.The total acquisition time per profile was 15 min with 30 s per slice.A profile without contrast can be obtained by plotting the total cumulative value as a function of the position (black trace in Fig. 4A) or two profiles which describe the amount of water in each mean pore size (blue and red traces in Fig. 4A).Adding data points from 1.8 to 9.5 mm is equivalent to acquiring the signal from the whole sample on a homogeneous field equipment, where the sample is fully contained within the coil.In this way, the overall signal evaporation can be obtained, as shown in black dots in Fig. 4B.The initial period, c.a. 1 h, corresponds to a stabilization in the temperature of the system with a low drying rate [47,48] which is followed by a constant drying rate, depicted by the linear decay in the water content with time.This is a characteristic process of a funicular regime [49], where a hydraulic connection throughout the systems facilitates capillary flow, which moves water over to the evaporation front, mainly constant in position over the whole evaporation.The individual contribution from large and small pores can be extracted from the longitudinal relaxation distribution of each slice, added up to obtain the blue and red curves of Fig. 4B.Evaporation is produced only from the large pores during almost all the drying process, while water contained in the smaller pores does not evaporate until the latest stages, where the funicular regime gives place to a pendular regime [49] where the hydraulic connection is lost.
Figure 5A shows the total profiles obtained for each evaporation time.There is no visible receding of the evaporation front, that is, the whole sample loses water homogeneously until the last stages of evaporation, between 13 and 15 h, where a top-to-bottom evaporation is observed, that is, a receding evaporation front appears.Figure 5B shows that water contained in the larger pores evaporates homogeneously.When the smaller pores start becoming depleted (Fig. 5C), the evaporation starts from the top of the sample to the bottom part and this period corresponds to a pendular regime.It is interesting to note that the shorter relaxation times remain mainly constant (see Fig. 6B), with a medium value of 50 ms, until the last stages of evaporation, where T 1S,Sh is reduced.This is more noticeable in the longer relaxation times (large pores) as shown in Fig. 6A, where at the initial stages T 1S,L ~ 800 ms, while at 8 h it is reduced to ~ 500 ms.This is an expected behavior that corresponds to the presence of a vapor phase in the central part of the pores.Therefore, the fluid becomes a film of water covering the pore walls, and the available dimension for diffusion is smaller, and thus water molecules sense a smaller void.It is worth noting that a similar observation was carried out in porous polymer systems where spatial evaporation was determined with single-sided NMR, while the microscopic behavior was determined with a smaller sample in the homogeneous field of a 60 MHz magnet [50].The simultaneous determination presented in this work is comparable to those obtained by combining relaxation and MRI in homogeneous magnets, with more stringent requirements on sample geometry than the NMR-MOUSE.Finally, Fig. 4 A First profile obtained during a water evaporation process from a Bentheimer sandstone.Black symbols represent the total signal, while blue symbols correspond to the contribution from larger pores (longer T 1 s) and red symbols for smaller pores.B Evaporation dynamics integrating each profile.Water from smaller pores does not evaporate until the larger voids are fully depleted 1 3 Spatially Resolved Dynamic Longitudinal Relaxometry in… we note that the methodology is not restricted to monitoring evaporation processes but to any time-dependent process that can be discriminated by longitudinal relaxation, thus circumventing the inherent coil heating in this type of commercial devices by avoiding the acquisition of a full transverse relaxation decay [32].

Conclusions
We showed that the acquisition of profiles in a single-sided NMR sensor, combined with the determination of longitudinal relaxation time distributions, can be performed in an experimental time short enough to monitor time-dependent phenomena: in this case, the evaporation of water from a model oil-bearing rock.The acquisition and addition of multiple echoes during a CPMG pulse sequence provide a suitable signal-to-noise ratio, without the imposition of undesirable contrast between the different proton pools.As saturation-recovery sequences are implemented straightforwardly in the presence of large magnetic field gradients, which are used as magnetization crushers, the recycle delay between acquisitions depends only on the time necessary to dissipate heat from the superficial radiofrequency coil.Individual slices of 350 μm were acquired in 30 s, while the whole profile took 15 min plus the time required to reposition the sensor to the first slice, where the complete evaporation of water from the rock took around 15 h.Therefore, this approach is suitable for the application of dynamic relaxometry.

Fig. 2 A
Fig.2A Number of echoes that can be added in a T 1 measurement considering that the contrast of the shortest relaxation time, T 2D,Sh , is 93% from its zero-time value.As low resolution has a shorter echo time, a larger number of echoes can be acquired for a given relaxation time.B Total experimental time for the acquisition of a saturation-recovery experiment for a single slice as a function of the longest relaxation time in a sample and the number of encoding points

Fig. 3 A
Fig. 3 A T 2D distribution for water imbibed in a Bentheimer sandstone acquired with an echo time t E = 67.5 μs.B T 1S distribution obtained with ten recovery times encoded up to 3T 1S,L .C Percentual area of the short (red symbols) and long (black symbols) longitudinal relaxation times as a function of the relaxation delays acquired

Fig. 5 A
Fig. 5 A Total signal for each slice as a function of time for water imbibed in a Bentheimer sandstone.B Signals from larger pores and C signals from smaller pores.Large pores evaporate with a constant drying front while small pores evaporate from top to bottom after c.a. 13 h

Fig. 6 3
Fig. 6 Surface-mediated longitudinal relaxation times, T 1S , in a Bentheimer sandstone imbibed with water corresponding to: A the large pores (long relaxation time) and B the smaller pores (short relaxation times)