## Abstract

The low-field nuclear magnetic resonance (NMR) technique is widely used as a noninvasive method to characterize the water content of subsurface porous media, such as aquifers and hydrocarbon reservoirs, but the quantitative correlation between the water saturation and the NMR relaxation signal has not been fully addressed. We conducted a laboratory study to measure the NMR signals of sandstone samples with different water saturations and to develop an empirical model for estimating the water saturation. The partially saturatinthe irreducible water saturationg states were derived by a high-speed centrifuge. The result shows that the water saturation is proportional to the geometric mean of the transverse relaxation time and can be fitted through a power function. Moreover, it has been found that the fitting parameters vary with the porosity and exhibit similar behaviors with the parameters of the classical Archie equation. The water saturation as well as its mobility state can be estimated with the NMR signals and porosity data. The proposed method has the potential to be applied to detect and quantify the water content in vadose zones, phreatic aquifers, permafrost regions, and gas hydrate reservoirs.

## 1. Introduction

The increased consumption and shortage of water resource pose a great pressure on the whole planet and require more efficient methods to explore the groundwater and estimate its content. There are many traditional geophysical techniques used for groundwater investigations, including the electrical conductivity, dielectric permittivity, induced polarization, and ground penetrating radar. However, the responses of these methods are influenced by many factors, including the mineralogical composition, grain size, and water salinity.

The low-field nuclear magnetic resonance (NMR) method is featured as a noninvasive technique to measure the signal of hydrogen in pore fluids directly. It has been developed as a powerful way to characterize the pores and fluids in geological studies. Compared with other geophysical methods, NMR data are selective with respect to water molecules [1]. There are two major types of geophysical NMR measurements so far, the surface NMR, which is mostly used in the groundwater studies, and the downhole NMR, which is popular in oil and gas exploration. The mathematical relationships between the water content and its spatial distribution and the surface NMR relaxations are well established [2-4], but the accuracy and the resolution are limited by the electromagnetic noise during the measurement [5, 6]. In addition, a considerable part of water cannot be detected by surface NMR tool because of the instrumental dead time [7-9]. Moreover, the signal is commonly nonexponential and depends both on the pore geometry and the formation property, leading to significant errors and overestimation of water content by standard processing approaches [10]. Furthermore, the pulse sequence and the acquisition model are simpler than the downhole NMR measurement [11], and even the combination of free induction decay (FID) and the spin-echo data and adiabatic pulses are proposed to overcome some limitations [12, 13].

There is a growing interest to use the downhole NMR method for the investigation of water in vadose zones or aquifers [14-18] with the advantages of high vertical resolution and low data acquisition time. Moreover, the measurement can be performed in the existing pumping and observation wells cased with polyvinyl chloride pipes [16, 19]. The downhole NMR measurement adopts the Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence to excite and record the NMR signals. The pulse sequence can compensate for the dephasing effect of the FID data in surface NMR measurements [20]. Additionally, the measurement can provide pore-scale and lithology-independent information, such as porosity, permeability, saturation, and pore size distribution [21-23].

The initial amplitude of the measured signal is linearly proportional to the water content of the excited volume [10, 12, 24]. Unfortunately, due to the instrument’s dead time, the initial amplitude is not measured directly but by extrapolating the recorded signal backward in time, leading to significant errors in the predicated water content [10]. Additionally, the predicted result may deviate from the true value if no core calibration data are available.

Given the challenges mentioned above, the main objective of this study is to develop an effective model for estimating the water saturation and to assist the calibration of the borehole NMR logging data. To reach this aim, we conduct a laboratory experiment to investigate the NMR responses of partially saturated samples. These measurements are performed by the 2-MHz low-frequency benchtop NMR spectrometer at the ambient condition, and the saturation degrees are achieved by the high-speed centrifuge. Generally, there are two types of conventional methods of using the NMR data to infer the fluid saturation. The first one is based on the amplitude of the spectrum or the initial echo. The second one is based on the displacement experiment, which is commonly used in laboratories to get the irreducible water saturation. Our approach followed the second one and concentrates on the relationship between the water saturation and the relaxation parameters and elaborates on the spectrum approach. In this article, we only discuss the water-gas two-phase system where the nonwetting fluid does not deliver a relaxation signal. Other influential factors, such as the wettability and the bulk relaxation of the nonwetting fluid, are not considered.

## 2. Methods and Experiments

### 2.1. The Low-Field NMR Basis

The transverse relaxation for a porous media is composed of the surface relaxation, bulk relaxation, as well as diffusion relaxation, which can be expressed as [22, 25],

where $T2s$, $T2b$, and $T2d$ are the surface transverse relaxation time, bulk transverse relaxation time, and diffusion transverse relaxation time; $S$ is the surface area of pore space; $V$ is the pore volume; $\rho 2$ is the transverse surface relaxivity; $D$ is the diffusion coefficient of fluid; γ is the gyromagnetic ratio; $TE$ is the echo spacing of the CPMG pulse sequence; $G$ is the average magnetic field gradient. The diffusion term is mainly neglected since the magnetic gradient of the tool is low.

In laboratory NMR spectrometers and borehole NMR logging instruments, the time evolution of the magnetization signal is commonly excited by the CPMG pulse sequence. The discrete expression between the magnetization and the transverse relaxation time is given by [21, 22],

where $M(ti)$ and $\epsilon (i)$ are the magnetization and the noise of the *i*th echo; $T2,j$ is the predefined *j*th transverse relaxation time; $fj$ is the amplitude of $T2,j$; $m$ and $n$ are the number of echoes and the predefined transverse relaxation times, respectively.

Transforming the decaying curve into the distribution of transverse relaxation time is usually called the ill-posed inversion. There are many successful algorithms to solve this problem, including BRD (Butler-Reeds-Dawson), non-negative least squares, truncated singular value decomposition, and maximum entropy [26-29]. In this experiment, the BRD algorithm is implemented to obtain the *T*_{2} distribution.

### 2.2. Experimental Detail

Sixteen core samples drilled in the Ordos Basin of the western China were selected to conduct the laboratory measurements under different water saturations. They are drilled, reshaped, and polished into plunger samples with 2.54 cm in diameter and 3–5 cm in length. The remnants of water and the drilling mud were removed by the oven at a temperature of 95°C. The gas-filled porosity and permeability were obtained before the water-saturating process. These dried samples were then put into a saturator container for 48 hours under a confining pressure of 20 MPa, ensuring most pores are fully saturated with the drinking water.

After the measurement of the fluid-filled porosity, they were moved to the 2 MHz MARAN DRX2 (Oxford Instruments) for the NMR measurements. The water was then displaced through a high-speed centrifuge gradually. Therefore, we obtained the NMR spectrums of core samples at different water saturations. The measurement temperature was 35°C, and the echo spacing and the waiting time were 0.2 ms and 6 seconds, respectively. The receiving gain and the number of echoes were 80% and 4096, respectively. After the NMR measurements, the decay signals were inverted into the *T*_{2} spectrums by the BRD algorithm. The centrifugal pressures were 0.345, 0.689, 1.379, 2.069, 2.758, and 3.448 MPa, respectively. To sum up, the workflow for the experiment is shown in Figure 1. The water saturation is computed by the gravimetric method, which is expressed as

where $W$ is the weight of the core sample after the centrifugation; $Wwet$ and $Wdry$ are the weights of the core sample at the dry state and the fully saturated state, respectively.

## 3. Results and Discussions

### 3.1. The NMR Response of Partially Saturated Samples

Figure 2 shows the *T*_{2} spectrums of a representative core sample under different water saturations. It is observed that the area of the spectrum is inverse to the water saturation. Moreover, the left part of the spectrum remains unchanged with the increase of the centrifugal pressure, indicating the irreducible water. The *T*_{2} cutoff value of the irreducible water is approximately 1 ms. Figure 3 shows the comparison of the water saturation calculated by different methods. It is obvious that the water saturation obtained by the gravimetric method agrees well with the value obtained by the NMR method [30-32]. However, it is still difficult to get the water saturation directly from the NMR spectrum since the initial echo amplitude needs to be calibrated to the porosity of the real rock. To make this goal, we can obtain the porosity from the fully water-saturated NMR data or determine it by other geophysical measurements, such as acoustic, neutron, and density logging.

### 3.2. The Improved Water Saturation Model

To develop a practical method to quantify the water content directly from the NMR data, we investigated the relationship between the *T*_{2} geometric mean and the water saturation. As shown in Figure 4, the water saturation is positively correlated with the *T*_{2} geometric mean, which can be expressed as [33],

where $T2gm$ is the geometric mean of the *T*_{2} spectrum at arbitrary water saturation; $T2gm,w$ is the geometric mean of the *T*_{2} spectrum for the water-saturated core sample; $a$ and $b$ are the empirical fitting parameters.

According to the previous study [34], since the nonwetting fluid does contribute to the relaxation signal, the total transverse relaxation can be expressed as

Combing Equations (5) and (4), we can obtain the following relationship,

where $\rho 2w=\rho 2a(T2gm,wT2gm)b$ is defined as the surface relaxivity of the partially saturated sandstone.

The proposed relation provides a practical way of transforming the relaxation time to the surface to volume at given water saturation.

Table 1 summarized the basic petrophysical information and fitting results of these samples. The porosity is measured by the gas porosimeter, and the permeability is measured by the steady-state gas permeameter. Moreover, the irreducible water saturation (Swir) is obtained with the 1-ms cutoff value from the raw water-saturated *T*_{2} spectrum. It is seen that the coefficient $a$ is nearly invariant, with an average value of 1. However, the exponent $b$ has a relatively wide range and varies from 0.609 to 0.927. This relationship is similar to the Archie equation used for the electrical data processing. Figure 5 depicts the relationship between $T2gm,w$ and the porosity. Generally, $T2gm,w$ is proportional to the porosity and can be expressed as

where $\varphi $ is the porosity; $c$ and $d$ are the fitting parameters. Consequently, the *T*_{2} geometric mean of the reservoir rock with 100% water saturation can be obtained by the porosity data, which is easy to be computed by the density, the neutron, and the acoustic well logging data.

Figure 6 reveals the relationship between the exponent $b$ and the porosity. Interestingly, it is negatively correlated with the porosity when the porosity is smaller than 9% and keeps invariant when the porosity is larger than 9%. Therefore, the exponent $b$ is expressed as

Assuming $a$ as 1, the water saturation can be expressed by the combination of Equations (4) and (7),

By introducing a transforming parameter, Equation (9) can be reshaped as,

Moreover, the irreducible water saturation is found to bear a favorable correlation with the porosity, as is shown in Figure 7. The irreducible water saturation is then expressed by,

where $Swi$ is the irreducible water saturation.

Then, the movable water saturation can be computed by the following equation,

where $Swm$ is the movable water saturation.

Therefore, the total water saturation, the movable water saturation, as well as the irreducible water saturation can be computed conveniently when the NMR measurement and conventional porosity logs are available. However, it should be noted that fitting parameters in Equations (4) to (10) may vary with the properties of porous rocks, such as lithology, cementation degree, porosity, permeability, and pore size distribution. Core calibrations are necessary for better estimation of the water saturation for partially saturated aquifers.

Figure 8 showed the comparison between the measured water saturation and the predicted water saturation by the proposed method. Figure 9 depicted the relative error of the predicted water saturation. It is observed that the correlation factor between the predicted water saturation and the measured water saturation is 0.976. Moreover, the relative error is distributed from 0.04% to 6.71%, and its average value is 1.74%.

## 4. Conclusions

This work demonstrates low-field NMR measurements for representative tight sandstones under different water saturations and provides an empirical model to estimate the water safturation and the mobility state with the aid of NMR and conventional porositylogging data. Based on our investigations, the main conclusions are as follows:

The water saturation is exponentially increased with the geometric mean of the transverse relaxation time, which is similar to the classical Archie equation.

The fitting coefficient is nearly a constant of 1. The fitting exponent changes inversely with the porosity but keeps invariant when the porosity is larger than 9%.

The

*T*_{2}geometric mean of the cores at 100% water saturation is proportional to the porosity and can be regressed by the power function model. Meanwhile, the irreducible water saturation is negatively correlated with the porosity and can be predicted with similar methods.

Our measurements are conducted for consolidated sandstones with low porosity and permeability, which may differ from low-consolidated or unconsolidated aquifers. The established relationships and fitting parameters may be affected by many geological factors, such as the grain size, the porosity and permeability, the pore distribution, as well as the consolidation degree.

We suggest to generalize the proposed equation for other porous media when the pore space is occupied by two-phase fluids. The laboratory NMR and the related petrophysical measurements are necessary to recognize these empirical parameters and their relationships with the reservoir properties.

## Acknowledgments

This work was supported by the State Key Laboratory of Shale Oil and Gas Enrichment Mechanisms and Effective Development, the National Natural Science Foundation of China (42174142), CNPC Innovation Found (2021DQ02-0402), and CNPC Science and Technology Project (2021DJ3804).

## Conflicts of Interest

The authors declare that they have no conflicts of interest.

## Data Availability

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy restrictions.