Study of the correlation between magnetic susceptibility and NMR relaxation using T2-filtered and T1-weighted CPMG
Introduction
NMR is of great importance in the analysis of fluids in Porous Media, especially in the oil industry, where the NMR technique is applied both in the laboratory and for well-logging investigation (Dunn et al., 2002; George R. Coates, Lizhi Xiao, 1999). One of most important aspects of the NMR data obtained for the fluids in Porous Media is the relationship between the relaxation time and pore size, which allows obtaining information on micro and macroporosity. For example, for sandstones, the relaxation time distribution is divided into free fluid, irreducible water and clay bound water (Gonzalez et al., 2020).
Taking into account the relationship between NMR data and porous media features, much work has been done in the laboratory to investigate how other physical information can be accessed from relaxation time distribution, such as pore size, magnetic susceptibility, wettability and permeability. In general, this information is related to the morphology and mineralogical composition of the porous material (Howard, 1998; Jácomo et al., 2020; Lucas-Oliveira et al., 2021; Valori and Nicot, 2019).
Hürlimann (1998) developed a study about the effective magnetic field gradient that is generated inside porous due to magnetic susceptibility differences. He analyzed the magnetic field gradient using low field NMR (2 MHz, for protons), and concluded that for carbonates the effective internal gradient field are typically smaller than gradient of NMR well logging tools, and for sandstones it can be comparable or larger.
Keating and Knight (2010), studied the effects of the magnetite concentration on NMR relaxation rates. However, no correlation trend was found between the magnetic susceptibility and relaxation rate. One of the explanations they offered to understand the lack of correlation was that the strong magnetic field gradient is localized and does not reflect in the relaxation rate.
In order to understand the magnetic susceptibility effects on NMR measurements and correlate them with bulk magnetic susceptibility, the relaxation time distributions can be obtained by controlling some parameters of NMR experiments (d’Eurydice et al., 2016; Hürlimann et al., 2002).
Sun and Dunn (2002) published a method where, through a modification of the CPMG method, it is possible to determine the distribution of the internal magnetic field gradient. Later, Xiao et al. (Xiao et al., 2013) added a dimension related to T1 relaxation, allowing to combine T1, T2 and internal magnetic field. It should be noted that the method presented by Sun and Dunn is limited to pores associated with relaxation times greater than one fifth of the longest coding echo time.
The proposal of this work is to analyze whether micro and macro porosity are better related to magnetic susceptibility using relaxation filters before the CPMG pulse sequence (Carr and Purcell, 1954; Meiboom and Gill, 1958).
In this way, the relaxation time will be weighted by the total magnetization of the micro or macro porosity. Differently from what was proposed by Sun and Dunn, this work does not seek to determine the distribution of internal magnetic field gradients, but how the weight of T1 and T2 can influence the evolution of relaxation.
This study was performed for a set of 17 selected sandstones rocks, which magnetic susceptibilities ranging from to SI.
CPMG NMR experiments were performed using two strategies. One looking for longer relaxation times related to larger pores, and another to observe shorter relaxation times, which are associated with smaller pores. To observe longer relaxation times, T2-filtered technique (d’Eurydice et al., 2016) was used to suppress the short transverse relaxation times. To observe the signal weighted by the shortest transverse relaxation times, T1-weighted method was employed (Blümich et al., 1998).
In general aspects, the NMR transverse relaxation time T2 in Porous Media is well described in the literature (Dunn et al., 2002; Gonzalez et al., 2020) and can be related to three main contributions:where is the relaxation time due to the interaction of the fluid molecules with the pore surface, is the relaxation time due to the translational diffusion in the presence of magnetic field gradients, and is the bulk relaxation time in the absence of other relaxation mechanisms than the interaction between the fluid molecules.
The is described in detail by Brownstein and Tarr (1979). In the fast diffusion regime this relaxation time is proportional to the pore surface-to-volume ratio, given by:which shows that the relaxation time has no direct dependence on NMR pulse sequence parameters. is the transverse surface relaxivity. Several works associate this parameter with the porous medium composition (Benavides et al., 2017; Gonzalez et al., 2020; Washburn et al., 2017). On the other hand, relaxation time is related to the translational diffusion of the fluid molecules and, therefore, has a strong dependence on the echo time parameter found in the CPMG pulse sequence. Considering a constant magnetic field gradient , can be written as:In Eq. (3), is the bulk diffusion coefficient and γ is the nucleus gyromagnetic factor.
Hürlimann (1998) analyzed for both carbonates and sandstones the maximum magnitude of the internal magnetic field gradient in terms of the magnetic susceptibility difference of the fluid and porous materials , establishing the following expression:and, therefore, the relaxation time is expected to be proportional to the magnetic susceptibility difference . is the static magnetic field applied for the NMR experiments.
However, when the study is concentrated on a sample group saturated with a single fluid, the magnetic field gradient can be analyzed only as proportional to the bulk magnetic susceptibility of the rock matrix, since the fluid contribution would simply introduce a displacement relatively on the total magnetic susceptibility values.
Bearing in mind what was presented above about the relaxation in porous media and its dependence on internal magnetic field gradients, as well as the pore shape and mineralogical composition (Keating and Knight, 2012, Song, 2000), we proposed to study a set sandstone rocks with magnetic susceptibilities ranging from to SI, using two preparation methods preceding CPMG pulse sequence, Fig. 1a, which allow to observe, separately, short and long transverse relaxation times.
-filtered was used to suppress contributions from shorter relaxation times, in which a CPMG with fixed echo time () and number of echoes () is applied. Thus, only components with relaxation times longer than still contribute to the total magnetization, which is measured by a second CPMG, Fig. 1b. The magnetization evolution during the second CPMG is given by,where each has already evolved during the first CPMG and can be written as:
Here, is the total magnetization of each component of . Therefore, the shorter the relaxation time of the component, the lower it's the intensity for a fixed .
On the other hand, the -weighted method was used to partially suppress longer longitudinal relaxation times, where a train of pulses is applied to saturate the magnetization previously to CPMG sequence. After saturation, the total magnetization is zero in any direction. Consequently, each magnetization component begins to recover under its corresponding relaxation, in which the components with short also have shorter , that is, returning faster to the z-direction. Therefore, after an evolution period of time , which corresponds to the -weighting procedure (Blümich et al., 1998), a CPMG measurement is applied, and components with will be suppressed, Fig. 1c. In this case, the magnetization evolution during the CPMG can be written by Eq. (5), but is given by:where the shorter the longitudinal relaxation time of the component, the greater its intensity .
In general, the CPMG is used with short echo time to minimize the transverse relaxation time due to the translational diffusion effects in the presence of internal magnetic field gradients (Kleinberg and Horsfield, 1990). However, in order to evaluate the internal magnetic field gradients, the echo time is varied (Gonzalez et al., 2020). Thus, the relaxation time should be constant in the regime where the dominates and vary linearly with echo time squared in the regime where dominates. However, in complex porous media, is related to both translational diffusion and internal magnetic field gradient (Gonzalez et al., 2020).
The proposal here is to analyze the dependence of the relaxation times on the echo time squared, calculating , in the region where the following relationship is linear:where is the log-mean of the transverse relaxation time obtained by CPMG, and .
Another analysis performed was the measurement of the transverse relaxation times from the Hahn-Echo (Hahn, 1950), which is the first echo obtained in the CPMG sequence. Since the echo time of the CPMG sequence is varied, a curve of Hahn echoes is also obtained, which is known to be strongly influenced by the fluid molecules translational diffusion in the presence of an inhomogeneous magnetic field. Thus, a second relaxation time, , is obtained from the measurement of , the first echo on CPMG, for each sample.
Section snippets
Material and methods
This study was performed for a set of 17 selected sandstone rocks, whose bulk magnetic susceptibilities () vary from to SI. They were indexed from A to P, in order of increasing bulk magnetic susceptibility (Table 1).
The magnetic susceptibility was also weighted by the porosity of each sample, in order to obtain the magnetic susceptibility referring to the volume of the rock matrix instead of the total volume of the sample and these values will be used on the analyses.
Results and discussion
Fig. 4 shows the plots of vs obtained using the conventional CPMG method for all the samples, where one can observe a non-linear relationship (Fig. 4a). The linearity is expected in the regime where the magnetic field gradient can be approximated to a constant or average limit. It is also considered that the molecules have a free diffusion for the case of short echo times. A study of this non-linear behavior can be found in the work developed by Gonzales et al. (Gonzalez et al.,
Conclusion
The data observed for this set of 17 sandstone samples showed the expected relation, where the greater the magnetic susceptibility, the greater the relaxation rate (). However, even using low field NMR (2 MHz) and short echo time (150 μs), the log-mean relaxation time obtained shows to be proportional to the magnetic susceptibility. This may indicate that the relaxation time is dominated by the surface relaxivity for this set of samples, which preserves the proportionality between
Credit author statement
All authors provided critical feedback and helped shape the research, analysis and manuscript. Everton Lucas-Oliveira: Conceptualization, Investigation, Validation, Methodology, Formal analysis, Writing- Original draft preparation and Editing. Marta Henirques Jácomo: Conceptualization, Methodology, Formal analysis, Writing- Original draft preparation. Writing-Reviewing and Editing. Agide Gimenez Marassi: Conceptualization, Investigation, Validation, Methodology, Formal analysis,
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
Authors acknowledge the support of the following Brazilian Institutions: University of São Paulo (USP). T. J. Bonagamba acknowledges National Council for Scientific and Technological Development, Brazil (CNPq, 308076/2018–4) and São Paulo Research Foundation, Brazil (FAPESP, 2009/54880–6). Authors also acknowledge Eng. Edson L. G. Vidoto and Aparecido D. F. de Amorim for designing and assembling the NMR probes. The authors acknowledge also the National Agency of Petroleum, Natural Gas and
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