Introduction

Accurate determination of water saturation \(\left({S}_{{\text{wirr}}}\right)\) is crucial for conducting a realistic evaluation of hydrocarbon reserves and is critical for reducing the economic risks associated with investments in the petroleum industry. Numerous models have been proposed in the industry to calculate \({S}_{{\text{wirr}}}\). The widely used saturation models, such as the Waxman-Smits-Thomas (WST) and dual-water (DW) models, which are primarily tailored to oil sandstone reservoirs, tend to lose their efficiency in more complex environments, such as carbonate gas reservoirs. Nuclear magnetic resonance (NMR) well logs provide a more precise representation of void space and the fluids residing within than traditional logs. Therefore, this research aims to present a well-defined equation for determining \({S}_{wirr}\) in carbonate gas reservoirs using NMR log data.

The NMR technique has proven to be a relatively straightforward and powerful tool for characterizing reservoir properties over the past few decades. NMR spectroscopy has been widely employed as a diagnostic tool for distinguishing pore fluids (Falong et al. 2016; Liu et al. 2018; Guo et al. 2020; Liao et al. 2021; Jin et al. 2023; Li et al. 2023). There is a wealth of research dedicated to estimating permeability through NMR relaxation measurements (Liu et al. 2017; Aghda et al. 2018; Mason et al. 2019; Masroor et al. 2022; Wu et al. 2022; Chen et al. 2023b). Numerous researchers have highlighted the advantage of the NMR technique in constructing imbibition and drainage curves (Liang and Wei 2008; Onuh and Ogbe 2019; Hosseinzadeh et al. 2020; Wu et al. 2021; Jin and Xie 2022; Heidary 2023). Recent efforts have primarily been focused on determining the wettability index through NMR measurements (Korb et al. 2018; Pires et al. 2019; Wang et al. 2019; Heidary 2021a; Alomair et al. 2023). The NMR technique has shown promise in a wide range of reservoir studies, including the analysis of pore structure and tortuosity (Wang et al. 2016; Liang et al. 2020; Deng et al. 2021; Elsayed et al. 2022; Xin et al. 2022; Zhang et al. 2023), characterization of unconventional shale reservoirs (Song and Kausik 2019; Lawal et al. 2020; Yang et al. 2020; Silletta et al. 2022; Chen et al. 2023a), evaluation of formation damage (Kamal et al. 2019; Adebayo and Bageri 2020; Alomair et al. 2022), determination of fluid movability (Li et al. 2020; Zhu et al. 2021), and detection of gas hydrate-bearing sediments (Bauer et al. 2015; Yang et al. 2017; Zhang et al. 2020, 2021, 2022; Shao et al. 2023).

One of the most frequently applied uses of NMR logs is the determination of fluid saturation. Numerous studies have explored the determination of fluid saturation based on NMR relaxation measurements (Dong et al. 2015; Mitchell et al. 2015; Newgord et al. 2020; Wang and Zeng 2020; Heidary 2021b; Gu et al. 2023). The process of using NMR relaxation data to determine \({S}_{wirr}\) involves deriving the transverse relaxation time \(\left({T}_{2}\right)\) cutoff from NMR measurements conducted on core samples. The measurement of the \({T}_{2}\) cutoff value, routinely performed in the laboratory, becomes impractical when considering the entire reservoir interval due to cost and time constraints. Considering the inherent drawbacks of traditional methods and the challenge posed by determining the \({T}_{2}\) cutoff value in the NMR-based approach, this research endeavor aims to present a rigorous method for determining \({S}_{{\text{wirr}}}\) in carbonate gas reservoirs by leveraging NMR log data and special core analysis (SCAL) tests. This study involves the development of a comprehensive model that relates \({S}_{{\text{wirr}}}\) to \({T}_{2}\) with the model parameter values being derived from SCAL tests. The methodology consists of the following steps:

  1. 1.

    Extraction of transverse relaxation time \(\left({T}_{2}\right)\) data from NMR well logging measurements.

  2. 2.

    Examination of the relationship between \({S}_{{\text{bw}}}\) and \({T}_{2}\) on a scatter plot.

  3. 3.

    Determination of the \({S}_{{\text{wirr}}}\) model.

  4. 4.

    Extraction of \({S}_{{\text{wirr}}}\) model parameter values from SCAL tests.

Materials and methods

This research focuses on developing a mathematical model for the determination of \({S}_{wirr}\) in carbonate gas reservoirs utilizing NMR log data and SCAL tests. The studied reservoirs (A, B, C) are natural-gas condensate reservoirs in the Persian Gulf. The apparent porosity has been adjusted for incomplete polarization and the low gas hydrogen index. SCAL tests have been carried out in all of these reservoirs. Figures 1, 2 and 3 depict the respective target reservoirs' drainage capillary pressure and relative permeability curves.

Fig. 1
figure 1

SCAL tests in reservoir A: a gas–water capillary pressure curve; b gas–water relative permeability curve

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Fig. 2
figure 2

SCAL tests in reservoir B: a gas–water capillary pressure curve; b gas–water relative permeability curve

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Fig. 3
figure 3

SCAL tests in reservoir C: a gas–water capillary pressure curve; b gas–water relative permeability curve

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The method presented in this study aims to develop a robust mathematical model for determining \({S}_{{\text{wirr}}}\) from NMR well logs and SCAL tests. The procedure is as follows:

  1. 1.

    Extraction of the \({T}_{2}\) value from the Carr–Purcell–Meiboom–Gill (CPMG) echo train at each depth.

  2. 2.

    Computation of bound water saturation \(\left({S}_{{\text{bw}}}\right)\) by applying various \({T}_{2}\) cutoff values to the \({T}_{2}\) distribution.

  3. 3.

    Investigation of the relationship between \({S}_{{\text{bw}}}\) and \({T}_{2}\) through scatter plots for different \({T}_{2}\) cutoff values, leading to the establishment of the \({S}_{{\text{wirr}}}\) model.

  4. 4.

    Calibration of the \({S}_{{\text{wirr}}}\) model parameters using SCAL tests.

The terms \({S}_{{\text{bw}}}\) and \({S}_{{\text{wirr}}}\) are, at times, used interchangeably. However, in this study, \({S}_{{\text{bw}}}\) pertains to the conventional method, whereas \({S}_{{\text{wirr}}}\) is used for the new NMR-based approach. Figure 4 provides an illustration of the workflow for this procedure. The \({S}_{{\text{wirr}}}\) model for the target reservoirs was established using this workflow.

Fig. 4
figure 4

Workflow for determination of \({S}_{{\text{wirr}}}\)

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Measurement of transverse relaxation time

NMR logging involves rapidly manipulating the hydrogen nuclei within rock formation pores (Luthi 2001). A quintessential multi-pulse sequence for moderately inhomogeneous magnetic fields, extensively employed in conventional NMR applications, is the Carr–Purcell–Meiboom–Gill (CPMG) sequence. The CPMG pulse sequence commences with a 90° pulse, succeeded by a series of 180° pulses. The initial pair of pulses are spaced apart by a time interval denoted as \(\tau\), while the subsequent pulses are separated by \(2\tau\). Echoes manifest themselves precisely midway between the 180° pulses at intervals of \(2\tau\), \(4\tau\), and so forth, up to \(2{\uptau } \times {\text{n}}\), where \(n\) signifies the echo order (refer to Fig. 5). These echoes materialize at the midpoint of successive refocusing pulses (Johns et al. 2015). In multiple-echo sequences, the envelope characterizing consecutive echoes diminishes exponentially over time, with a time constant denominated as \({T}_{2}\). This time constant \({T}_{2}\), governing the decay, is termed the transverse relaxation time. Consequently, \({T}_{2}\) can be determined from the magnitude of the consecutive echoes (Dunn et al. 2002):

$$M\left( t \right) = M_{0} {\text{exp}}\left( { - \frac{t}{{T_{2} }}} \right),$$
(1)
Fig. 5
figure 5

Basic CPMG pulse sequence

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In Eq. (1), \(M\left(t\right)\) and \({M}_{0}\) correspond to the magnetization at time t and time 0, respectively. \({M}_{0}\) can be calibrated to yield porosity information.

In NMR logging, \({M}_{0}\) and \({T}_{2}\) hold paramount significance, as they harbor valuable petrophysical and geological insights. A considerable level of noise often plagues NMR logging measurements. The initial step in extracting \({M}_{0}\) and \({T}_{2}\) entails noise elimination from the CPMG sequence. Heidary et al. (2019) introduced a proficient approach to managing the noisy spin-echo train at each depth using the wavelet analysis technique. This method was applied to the target reservoirs to derive \({M}_{0}\) and \({T}_{2}\) from the spin-echo train.

Generally, the \({T}_{2}\) value is affected by pore fluid type, fluid saturation, pore size, and wettability. Physical properties such as surface-to-volume ratio influence the \({T}_{2}\) value. The amount of \({T}_{2}\) signal below 33 ms is small for poorly consolidated sandstones and can be hard to resolve from the noise.

Bound water saturation

Reservoir rocks exhibit pores of varying sizes and contain multiple types of fluids. Consequently, the spin-echoes recorded during the CPMG measurement do not solely exhibit decay with a single \({T}_{2}\) value; instead, they display a distribution of \({T}_{2}\) value (Coates et al. 1999). The inversion of NMR log data yields the necessary information for computing bound water saturation \(\left({S}_{{\text{bw}}}\right)\). The \({T}_{2}\) distribution within rocks is a continuous function; however, to facilitate fitting the CPMG sequence, the inversion process employs a multi-exponential model. The primary focus of the inversion process is to identify relaxation times, \({T}_{2j}\), along with their corresponding porosity components, \({f}_{j}\), from the spin-echo decay data, \({g}_{i}\), while minimizing the error \({\varepsilon }_{i}\) (Dunn et al. 2002):

$$g_{i} = \frac{{M\left( {t_{i} } \right)}}{{M_{0} }} = \mathop \sum \limits_{j = 1}^{m} f_{j} e^{{ - \frac{{t_{i} }}{{T_{2j} }}}} + \varepsilon_{i} \;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\; i = 1, \ldots ,n$$
(2)

Figure 6 serves as a typical illustration of the outcome of the inversion process. The vertical cutoff in the \({T}_{2}\) distribution distinguishes between different pore sizes and quantifies the volume of bound water. Consequently, bound water saturation \(\left({S}_{{\text{bw}}}\right)\) is derived from the \({T}_{2}\) distribution function, \(f\left({T}_{2}\right)\), using the following Equation (Johns et al. 2015):

$$S_{{{\text{bw}}}} = \frac{{\mathop \smallint \nolimits_{{T_{{2{\text{min}}}} }}^{{T_{2} {\text{cutoff}}}} f\left( {T_{2} } \right){\text{d}}T_{2} }}{{\mathop \smallint \nolimits_{{T_{{2{\text{min}}}} }}^{{T_{{2{\text{max}}}} }} f\left( {T_{2} } \right){\text{d}}T_{2} }},$$
(3)

where \({T}_{2{\text{max}}}\) and \({T}_{2{\text{min}}}\) represent the maximum and minimum values of \({T}_{2}\), respectively. Determining the appropriate \({T}_{2}\) cutoff value is of utmost importance in calculating \({S}_{{\text{bw}}}\). This value is determined through NMR measurements on water-saturated core samples. The suitable \({T}_{2}\) cutoff value can vary from one reservoir to another. For carbonates, for instance, the \({T}_{2}\) cutoff value may range from 90 to 150 ms, with the default value set at 90 ms (Coates et al. 1999).

Fig. 6
figure 6

\({T}_{2}\) relaxation time distribution resulting from the inversion process

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\({S}_{{\text{bw}}}\) is contingent upon the chosen \({T}_{2}\) cutoff value. Sometimes, the \({T}_{2}\) cutoff value is variable. Using a single \({T}_{2}\) cutoff value to calculate \({S}_{bw}\) is not reasonable in a formation. Accurate determination of the \({T}_{2}\) cutoff values requires NMR laboratory measurements on many core samples. However, it is essential to note that NMR experiments can only be conducted on a limited number of core samples. In other words, \({T}_{2}\) cutoff values obtained from core NMR measurements cannot be extrapolated to the entire reservoir interval. Therefore, it is essential to develop a mathematical model independent of the \({T}_{2}\) cutoff value for precise determination of \({S}_{{\text{wirr}}}\). The \({S}_{{\text{wirr}}}\) model can be obtained by establishing the relationship between \({S}_{{\text{bw}}}\) and \({T}_{2}\). To achieve this, a scatter plot of \({S}_{{\text{bw}}}\) versus \({T}_{2}\) is employed to unveil this connection. By calibrating the parameters of the \({S}_{{\text{wirr}}}\) model with special core analysis (SCAL) tests, it becomes solely a function of the \({T}_{2}\) value. This calibration eliminates the necessity of measuring the \({T}_{2}\) cutoff value at each depth.

Results and discussion

The \({S}_{{\text{wirr}}}\) model has been developed for the target reservoirs using the proposed method, and the results obtained from each step are presented in this section.

CPMG sequence parameters

The CPMG sequence parameters, denoted as \(\left({M}_{0},{T}_{2}\right)\), were determined at various depths. The NMR porosity \(\left({\phi }_{{\text{NMR}}}\right)\) and \({T}_{2}\) logs for reservoirs A, B, and C are illustrated in Figs. 7, Fig. 8 to Fig. 9, respectively.

Fig. 7
figure 7

Logs in reservoir A: a \({\phi }_{{\text{NMR}}}\); b \({T}_{2}\)

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Fig. 8
figure 8

Logs in reservoir B: a \({\phi }_{{\text{NMR}}}\); b \({T}_{2}\)

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Fig. 9
figure 9

Logs in reservoir C: a \({\phi }_{{\text{NMR}}}\); b \({T}_{2}\)

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Correlation of CPMG sequence parameters

A robust correlation exists between \({\phi }_{{\text{NMR}}}\) and \({T}_{2}\) in water-wet rock that is fully saturated with water. In oil reservoirs, \({T}_{2}\) demonstrates a strong correlation with \({\phi }_{{\text{NMR}}}\). Consequently, the \({T}_{2}\) distribution proves to be a reliable indicator for determining \({S}_{{\text{bw}}}\) in oil reservoirs. However, in the target reservoirs, the correlation between \({\phi }_{{\text{NMR}}}\) and \({T}_{2}\) is insignificant due to the low hydrogen index of gas. The correlation coefficient \(\left({r}_{{\text{corr}}}\right)\) between \({\phi }_{{\text{NMR}}}\) and \({T}_{2}\) is calculated as follows (Shevlyakov and Oja 2016):

$$r_{corr} = \frac{{\mathop \sum \nolimits_{Z = 1}^{n} \left( {T_{2} \left( Z \right) - \overline{{T_{2} }} } \right)\left( {\phi_{{{\text{NMR}}}} \left( Z \right) - \overline{{\phi_{{{\text{NMR}}}} }} } \right)}}{{\sqrt {\mathop \sum \nolimits_{Z = 1}^{n} \left( {T_{2} \left( Z \right) - \overline{{T_{2} }} } \right)^{2} \left( {\phi_{{{\text{NMR}}}} \left( Z \right) - \overline{{\phi_{{{\text{NMR}}}} }} } \right)^{2} } }}$$
(4)

Here, \(\overline{{\phi }_{{\text{NMR}}}}\) and \(\overline{{T }_{2}}\) represent the mean values of \({\phi }_{{\text{NMR}}}\) and \({T}_{2}\) logs, respectively. The calculated values of \({r}_{{\text{corr}}}\) for reservoirs A, B, and C are − 0.03, − 0.14, and − 0.06, respectively. Figures 10a–c depict scatter plots of \({T}_{2}\) versus \({\phi }_{{\text{NMR}}}\) for reservoirs A, B, and C, respectively. Consequently, developing a comprehensive \({S}_{{\text{wirr}}}\) model, independent of \({T}_{2}\) cutoff, is crucial for the target reservoirs. Calibration of the derived model with experimental tests obviates the need for laboratory measurement of \({T}_{2}\) cutoff.

Fig. 10
figure 10

Scatter plot of \({T}_{2}\) versus \({\phi }_{{\text{NMR}}}\): a reservoir A; b reservoir B; c reservoir C

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Irreducible water saturation model

The calculation of \({S}_{{\text{bw}}}\) was carried out within the target reservoirs, considering various values of the \({T}_{2}\) cutoff. A scatter plot of \({S}_{{\text{bw}}}\) against \({T}_{2}\) reveals an exponential relationship, as depicted in Figs. 11a–c, for reservoirs A, B, and C, respectively. The regression coefficient \(\left({R}^{2}\right)\) is influenced by the chosen \({T}_{2}\) cutoff value. Consequently, the \({S}_{{\text{wirr}}}\) model is expressed as follows:

$$S_{{{\text{wirr}}}} = a \times {\text{exp}}\left( { - b \times T_{2} } \right)$$
(5)
Fig. 11
figure 11

Relationship between \({S}_{{\text{bw}}}\) and \({T}_{2}\); a reservoir A; b reservoir B; c reservoir C

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The parameter values \(\left(a,b\right)\) are unique to each reservoir. \({S}_{{\text{bw}}}\) is contingent upon the selected \({T}_{2}\) cutoff value at each depth. There is a specific \({T}_{2}\) cutoff value from which the resulting \({S}_{{\text{bw}}}\) log closely matches the \({S}_{{\text{wirr}}}\) log. This particular \({T}_{2}\) cutoff, referred to as the optimal one, yields the highest correlation coefficient between \({S}_{{\text{bw}}}\) and \({S}_{{\text{wirr}}}\). The determination of parameter values for the target reservoirs is crucial for the precise determination of \({S}_{{\text{wirr}}}\) at various depths. These parameter values are obtained through the calibration of Eq. (5) with SCAL tests, and the procedure for acquiring \(a\) and \(b\) is explained in the following section.

Model parameters

Determining the model parameter values requires establishing a relationship between \({T}_{2}\) and SCAL tests. The function of \({T}_{2}\) in porous media is analogous to resistivity. \({S}_{{\text{bw}}}\) decreases with increasing \({T}_{2}\) (or resistivity). Taking the natural logarithm of Eq. (5) yields:

$$\ln S_{{{\text{wirr}}}} = \ln a - b \times T_{2}$$
(6)

The \({S}_{{\text{wirr}}}\) model parameters are obtained by determining \({S}_{{\text{wirr}}}\) from SCAL tests at two different depths. Substituting the minimum and maximum values of \({S}_{{\text{wirr}}}\) \(\left({S}_{{\text{wirr}}}^{{\text{min}}},{S}_{{\text{wirr}}}^{{\text{max}}}\right)\) in Eq. (6) and solving for the \({S}_{{\text{wirr}}}\) model parameters yield:

$$\left\{ {\begin{array}{*{20}c} {\ln S_{{{\text{wirr}}}}^{{{\text{max}}}} = \ln a - b \times T_{2}^{{{\text{min}}}} } \\ {\ln S_{{{\text{wirr}}}}^{{{\text{min}}}} = \ln a - b \times T_{2}^{{{\text{max}}}} } \\ \end{array} } \right.$$
(7)
$$a = \left( {\frac{{S_{{{\text{wirr}}}}^{{{\text{max}}^{{T_{2}^{\max } }} }} }}{{S_{{{\text{wirr}}}}^{{{\text{min}}^{{T_{2}^{\min } }} }} }}} \right)^{{\frac{1}{{T_{2}^{\max } - T_{2}^{\min } }}}} \;\;\;{\text{and}}\;\;\;\;b = \frac{{\ln \frac{{S_{{{\text{wirr}}}}^{\max } }}{{S_{{{\text{wirr}}}}^{\min } }}}}{{T_{2}^{\max } - T_{2}^{\min } }},$$
(8)

where \({T}_{2}^{{\text{max}}}\) and \({T}_{2}^{{\text{min}}}\) are the maximum and minimum values of\({T}_{2}\), respectively. \({S}_{{\text{wirr}}}^{{\text{min}}}\) and \({S}_{{\text{wirr}}}^{{\text{max}}}\) correspond to \({T}_{2}^{{\text{max}}}\) and\({T}_{2}^{{\text{min}}}\), respectively. \({S}_{{\text{wirr}}}^{{\text{min}}}\) is determined from the capillary pressure curve measured on the core sample taken from a depth corresponding to \({T}_{2}^{{\text{max}}}\) (Figs. 1, 2 and 3). \({S}_{w}^{max}\) is associated with the residual gas saturation\(\left({S}_{{\text{gr}}}\right)\);\({S}_{{\text{wirr}}}^{{\text{max}}}=1-{S}_{{\text{gr}}}\). There is no SCAL test regarding the determination of \({S}_{{\text{gr}}}\) in the target reservoirs. Hence, this research resorted to the correlation relating \({S}_{{\text{gr}}}\) to petrophysical parameters. There are several correlations in the literature to estimate \({S}_{{\text{gr}}}\) in gas reservoirs. Agarwal developed a correlation for limestones in terms of porosity\(\left(\phi \right)\), absolute permeability\(\left(k\right)\), and initial gas saturation \(\left({S}_{{\text{gi}}}\right)\) as follows (Lee and Wattenbarger 1996):

$$S_{{{\text{gr}}}} = - 0.5348\phi + 0.03356\log k + 0.1546S_{{{\text{gi}}}} + 0.144$$
(9)

\({S}_{{\text{gi}}}\) can be obtained from the gas–water relative permeability curve (Figs. 1, 2 and 3). The values of \(a\) and \(b\) were calculated using Eq. (8) and SCAL test results for the target reservoirs. Table 1 shows the values of \(a\) and \(b\) for each reservoir.

Table 1 Values of \(a\) and \(b\) for each reservoir
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The \({S}_{{\text{wirr}}}\) model was established for the target reservoirs after calculating \(a\) and \(b\). Figures 1213 and 14 demonstrate the plot of \({S}_{{\text{wirr}}}\) and \({S}_{{\text{bw}}}\) versus depth for reservoirs A, B, and C, respectively. The \({S}_{{\text{bw}}}\) logs were obtained with the optimal \({T}_{2}\) cutoff value. The optimal \({T}_{2}\) cutoff value is less than the \({T}_{2}\) cutoff value at the depths where \({S}_{{\text{wirr}}}\) is greater than \({S}_{{\text{bw}}}\) \(\left({S}_{{\text{wirr}}}>{S}_{{\text{bw}}}\right)\). Similarly, the optimal \({T}_{2}\) cutoff value is greater than the \({T}_{2}\) cutoff value at the depths where \({S}_{{\text{wirr}}}\) is less than \({S}_{{\text{bw}}}\) \(\left({S}_{{\text{wirr}}}{<S}_{{\text{bw}}}\right)\).

Fig. 12
figure 12

Logs of \({S}_{{\text{wirr}}}\) and \({S}_{{\text{bw}}}\) in reservoir A

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Fig. 13
figure 13

Logs of \({S}_{{\text{wirr}}}\) and \({S}_{{\text{bw}}}\) in reservoir B

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Fig. 14
figure 14

Logs of \({S}_{{\text{wirr}}}\) and \({S}_{{\text{bw}}}\) in reservoir C

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Conclusions

In this study, a precise model has been successfully developed using transverse relaxation time \(\left({T}_{2}\right)\) and special core analysis (SCAL) tests to ascertain irreducible water saturation \(\left({S}_{{\text{wirr}}}\right)\) in carbonate gas reservoirs. The key findings of this study are as follows:

  1. 1.

    An exponential relationship exists between the \({S}_{{\text{wirr}}}\) and \({T}_{2}\) values derived from the recorded CPMG sequences in carbonate gas reservoirs.

  2. 2.

    The calibration of the \({S}_{{\text{wirr}}}\) model with SCAL tests obviates the necessity for determining \({T}_{2}\) cutoff in the novel methodology.

  3. 3.

    The \({S}_{{\text{wirr}}}\) model can be effectively utilized to determine the optimal \({T}_{2}\) cutoff value in gas reservoirs.

  4. 4.

    The recommended optimal \({T}_{2}\) cutoff values for reservoirs A, B, and C are 130 ms, 100 ms, and 110 ms, respectively.