Introduction

With the conventional reservoirs, which contained high porosity and permeability, further developed, the hydrocarbon potential was decreased. The unconventional reservoirs, such as shale oil/gas and tight oil/gas, had become the hotspot in oil and gas exploration due to the unlimited potential (Zou et al. 2012; Jia et al. 2016; Li et al. 2020a, b). The Chang 8 Formation of Ordos Basin, which was a typical tight sandstone reservoir (Fig. 1a–c), was recently ardently developed. To improve the development quality, accurately reservoir identification and evaluation were of great importance (Gao and Hu 2018; Liu et al. 2019; Zhu et al. 2019; Li et al. 2020a, b). However, plenty of field researches illustrated that tight reservoir identification and evaluation of Chang 8 Formation are in-effective because it contained complicated pore structure. Many potential hydrocarbon-bearing formations were considered as dry layers or water saturated reservoirs (Xiao et al. 2011; Lis-Sledziona 2019). On the contrary, plenty of reservoirs, which were identified as water-bearing zones, were good hydrocarbon-bearing layers in production (Newberry et al. 1996; Ghafoori et al. 2009). To decrease these misdiagnoses, the pore structure of tight reservoirs should be first evaluated (Ge et al. 2014; Ge et al. 2015; Xiao et al. 2016a, b; Fournier et al. 2018; Puskarczyk et al. 2018; Krakowskal 2019; Zhang et al. 2019; Zahid et al. 2019; Xiao et al. 2020).

Fig. 1
figure 1

The core-derived porosities (a), permeabilities (b), and pore throat radius statistical histograms (c) of the Chang 8 Formation in Pengyang Region, west Ordos Basin in China. These three figures illustrated that the porosities of the Chang 8 Formation ranged from 4.36 to 19.20%, the permeabilities ranged from 0.008 to 5.38 mD, and the pore throat radius mainly distributed from 0.01 to 0.87 μm. These illustrated that the Chang 8 Formation of our target region was a typical tight sandstone reservoir

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Plenty of methods were proposed to evaluate formation pore structure (Volokitin et al. 2001; Looyestijn 2001; Ouzzane et al. 2006; Olubunmi and Chike 2011). The most effective method and material of effectively characterizing formation pore structure was mercury injection capillary pressure (MICP) curve (Xiao et al. 2016a, b). From the MICP curves of core samples, many valuable information, such as the pore throat size distribution, the average pore throat radius, the maximal pore throat radius, and the threshold pressure, can be acquired. Based on these parameters, formations can be classified into several types, and many hydrocarbon potential formations could be identified (Ge et al. 2015). Although the MICP curves were effective for pore structure evaluation, the quantity was limited because core samples were not consecutively drilled. Only limited core samples were drilled in the potential intervals. This led the pore structure cannot be consecutively characterized, and many un-cored wells also cannot be evaluated.

The nuclear magnetic resonance (NMR) data was another valuable material to evaluate formation pore structure (Coates et al. 2000; Dunn et al. 2002). From the NMR transverse relaxation time T2 distribution, the pore size and distribution can be qualitatively observed. However, the NMR data cannot be directly used to quantitatively characterize formation pore structure, because the pore throat size, but not the pore size, is the main factor that controls formation quality. To use the NMR data to characterize formation pore structure, many researchers pointed out that the NMR T2 distribution should be processed and transformed as the pore throat radius distribution, and then pseudo-capillary pressure (Pc) curve (Volokitin et al. 2001; Xiao et al. 2012). To transform the NMR T2 spectrum as pseudo Pc curve, several methods had been proposed. The most widely used methods contained linear scale function proposed by Volokitin et al. (2001) and nonlinear power function proposed by Xiao et al. (2016a, b). The linear scale function was easily used but the processed results were not reliable; the small pore throat size cannot be reappeared. Xiao et al. (2012) also proposed another method based on the J function; several restricted conditions made it cannot be well used in field application. In this study, we tried all these methods in field applications, and the nonlinear power function proposed by Xiao et al. (2016a, b) was verified to be the most optimal.

Method and model

Relationship between T2 time and Pc

Based on the principle of NMR logging, the transverse T2 relaxation time is constructed by three parts: the bulk relaxation time T2B, the surface relaxation time T2S, and the diffusion relaxation time T2D (Coates et al. 2000). The relationship among them can be expressed as follows:

$$ \frac{1}{T_2}\approx \frac{1}{T_{2\mathrm{B}}}+\frac{1}{T_{2\mathrm{S}}}+\frac{1}{T_{2\mathrm{D}}} $$
(1)

The relative importance of these three relaxation mechanisms depended on the property of saturated fluid in the pore spaces, pore size, surface relaxation strength, and rock surface wettability (Coates et al. 2000). Generally, for a water-wettable rock, and fully saturated with brine, the values of T2B and T2D were ignored, and T2 was dominated by T2S. The value of T2S was usually controlled by the pore size of rock. Hence, the T2 relaxation time can be simplified as follows:

$$ \frac{1}{T_2}\approx \frac{1}{T_{2\mathrm{S}}}={\rho}_2{\left(\frac{S}{V}\right)}_{\mathrm{por}} $$
(2)

where ρ2 was the proportionality constant between 1/T2 and surface to volume ratio of the pore; S was the surface area of rock pore; V was the volume of rock pore; and S/V was called as the surface relaxivity. The subscript por stood for rock pore.

If rock pore shape was assumed as regular, Eq. 2 can be rewritten as follows:

$$ \frac{1}{T_2}={\rho}_2{\left(\frac{S}{V}\right)}_{\mathrm{por}}={F}_{\mathrm{S}}\frac{\rho_2}{r_{\mathrm{por}}} $$
(3)

Based on the theory of capillary pressure, the relationship between capillary pressure and pore throat radius for air-mercury system can be expressed as follows (Purcell 1949):

$$ {P}_{\mathrm{c}}=\frac{0.735}{R_{\mathrm{c}}} $$
(4)

where Rc was the pore throat size in micrometer.

If we assumed that the pore throat radius was related to the pore size, it can be expressed as follows:

$$ {r}_{\mathrm{por}}=m\times {R}_c $$
(5)

Combining Eqs. 3 and 4, a derived equation could be written as follows:

$$ {P}_{\mathrm{c}}=C\times \frac{1}{T_2} $$
(6)

or

$$ {R}_{\mathrm{c}}={C}_{{\mathrm{T}}_2-{\mathrm{r}}_{\mathrm{c}}}\cdot {T}_2 $$
(7)

where \( C=\left(\frac{0.735n}{\rho_2\times {F}_s}\right) \) was the transformation coefficient of Pc and T2, and \( {C}_{{\mathrm{T}}_2-{\mathrm{R}}_{\mathrm{c}}}=\frac{\rho_2\times {F}_{\mathrm{s}}}{n} \) was the transformation coefficient of Rc and T2.

Equations 6 and 7 illustrated that the T2 relaxation time associated with Pc or Rc. The NMR T2 relaxation time could be transformed as the Pc curve or pore throat radius distribution once the transformation coefficient C was first determined.

Establishment of model for constructing pseudo Pc curve from NMR data

To calibrate the value of C involved in Eq. 6 or 7 to construct pseudo Pc curve from NMR data, 20 core samples, which were drilled from the Chang 8 Formation of Huangjiang Region, west Ordos Basin, were applied for NMR and MICP measurements. The experimental parameters were listed as follows: wait time (TW) 6.0 s, inter-cho spacing (TE) 0.2 ms, the number of echoes per echo train (NE) 4096, and the scanning number 128. The experimental results are listed in Table 1.

Table 1 The NMR and MICP experimental results of 20 core samples
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Swirr was the irreducible water saturation, Rmax stood for the maximal pore throat radius, Rm was the average pore throat radius, R50 was the median pore throat radius, Pd was the threshold pressure, and P50 stood for the median pressure.

Based on the experimental results, the NMR T2 distributions and MICP curves of the 20 core samples were acquired and displayed in Figs. 2 and 3, separately. These two figures illustrated that their pore structures were complicated. It was difficult to obtain a single function to establish the relationship between NMR and Pc curves. To establish a reliable relationship to relate Pc with T2, we tried to classify these core samples into three types based on the difference of rock physical properties. Three types of NMR T2 spectra and the corresponding Pc curves are displayed in Figs. 4a–c and 5a–c, respectively.

Fig. 2
figure 2

The NMR T2 distributions of 20 core samples. This figure illustrated that the small pore sizes were dominated in the Chang 8 Formation of Pengyang Region, west Ordos Basin, China

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Fig. 3
figure 3

The MICP curves of 20 core samples. The threshold pressures ranged from 0.08 to 7.0 MPa, and the maximal mercury injection saturations ranged from 61.50 to 94.30%

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Fig. 4
figure 4

Three types of NMR distributions. From the first type to the third type of rocks (a–c), the position of the main peaks moved to the left

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Fig. 5
figure 5

Three types of MICP curves. From the first type to the third type (a–c), the represented rock pore structures became poorer and poorer

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Figures 4 and 5 illustrated that the shapes of T2 distributions and MICP curves for every type of core samples were similar after they were classified into three types. In this condition, we could calculate an average T2 distribution and MICP curve to represent all core samples within the same type.

In Figs. 6 and 7, we displayed the average T2 spectra and MICP curves for the three types of core samples. From these two figures, we observed that the shapes and positions were correlated. For the first type of core samples, which represented the best pore structure, the NMR T2 distribution lied to the right, and the large pore was dominated; the corresponding Pc curve lied to the bottom. As the pore structure deteriorated, the NMR T2 distributions moved to the left, and the Pc curves raised to the top. This meant the core samples were reasonably classified. These three average T2 spectra and Pc curves can be used to represent all 20 core samples.

Fig. 6
figure 6

Three types of average NMR distribution. From the first to the third type of rock, the average T2 distribution became narrow, the position of the main peak moved to the left, and the shape of the T2 spectrum became unimodality from bimodality

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Fig. 7
figure 7

Three types of average Pc curve. From the first to the third type of rock, the pore structure became poorer, and the morphologic trend was the same as the NMR T2 spectra of the three types of rock

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To establish the transformational relation between NMR data and Pc curves, the NMR T2 distributions were first processed and normalized to acquire the inverse cumulative curves, which contained similar shapes with the Pc curves. Figure 8 shows the three types of average inverse cumulative curves, those corresponded to the NMR distributions displayed in Fig. 6. Comparing Figs. 7 and 8, we observed that the shapes of the three types of average inverse cumulative curves were similar with those of the average Pc curves. However, their order of magnitudes were absolutely different. To transform the NMR T2 distributions as pseudo Pc curves, relationships between Figs. 7 and 8 should be established, specifically.

Fig. 8
figure 8

Three types of average NMR inverse cumulative curves. The shapes of the average NMR inverse cumulative curves were similar with those of the average Pc curves displayed in Fig. 7, but the y-axis was absolutely different. The purpose of this study was to determine the optimal C to make Figs. 7 and 8 completely coincided

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To establish the relationship between Pc and T2 time, the equisaturation principle was used to extract the T2 time that corresponded to every Pc value under the same non-wettable saturation. Figure 9 displayed the principle of the equisaturation method.

Fig. 9
figure 9

Principle of acquiring the T2 time that corresponded to every Pc. Based on the equisaturation method, a given non-wettable saturation was first determined. Next, a line, which passed the give non-wettable saturation and was perpendicular to the x-axis, was drawn. This line would both cross to the Pc and NMR inverse cumulative curve; the corresponded Pc and 1/T2 would be extracted

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By using the method displayed in Fig. 9, three types of average NMR inverse cumulative curves were processed, the those T2 times corresponded to every Pc value were acquired and cross plotted with the Pc values, and the results illustrated that good power function relation existed among them. Meanwhile, based on the relationship between Pc and Rc displayed in Eq. 4, the relationship between Pc and T2 was converted to Rc and T2, as is shown in Fig. 10.

Fig. 10
figure 10

Relationship between T2 time and Rc for every type of formation

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By using the relationships displayed in Fig. 10, the NMR data can be processed and transformed as pseudo Pc curve. Once they were extended into field applications, field NMR logging can be processed, and consecutive pseudo Pc curves could be acquired in the intervals with which field NMR logging was acquired.

In Fig. 11, we displayed a field example of constructing pseudo Pc curves from field NMR logging in the Chang 8 Formation, Huangjiang Region, west Ordos Basin. In the first three tracks in Fig. 11, we displayed the conventional well logging curves; they were used for identifying effective formation, estimating porosity, and indicating hydrocarbon-bearing formation. In the fourth track, we displayed the NMR T2 distributions; the NMR data were acquired from Schlumberger’s CMR-Plus tool. In the fifth and sixth tracks, the constructed pseudo Pc curves and pore throat radius distributions were displayed. We displayed the comparison of permeabilities predicted from three different methods in the seventh track. PERMSWAN was the predicted permeability based on the Swanson parameter model established by Xiao et al. (2014). PERM was the estimated permeability curve from porosity based on the routine crossplot of core-derived porosity and permeability, and CPERM was the core-derived permeability. This comparison illustrated that the PERMSWAN was coincided with the core-derived permeability much well, while the permeability estimated by using the routine method was underestimated. The reason was that the PERMSWAN was predicted based on formation pore structure characterization (Xiao et al. 2017; Daigle et al. 2020). This meant that pore structure should be precisely characterized to predict permeability in tight sandstone reservoirs. In the last two tracks, we compared the predicted maximal pore throat radius (RMAX) and the median pore throat radius (R50) from pseudo Pc curves and derived from laboratory experimental results of core samples (CRMAX and CR50, separately). The results illustrated that these two formation pore structure evaluation parameters were consistent. This meant that the constructed pseudo Pc curves were accurate and indirectly verified the reliability of the proposed method in this study.

Fig. 11
figure 11

A field example of constructing pseudo Pc curves and characterizing tight sandstone reservoir pore structure from NMR logging data by using the proposed method

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Conclusions

Tight sandstone reservoirs cannot be directly estimated by using conventional methods due to the complicated pore structure. To improve tight sandstone reservoir evaluation, the pore structure should be first evaluated. The NMR logging was effective in evaluating formation pore structure once it was first transformed as the pseudo Pc curve.

The relationship between the T2 time and the pore throat radius Rc was not a simple linear function, but a non-linear power function. To acquire reliable Pc curves from NMR logging, the classified power function method should be used. In the method, formation was first classified into three types, and for every type of rock, the power function was used to establish the relationship of T2 time and pore throat radius.

In this study, 20 core samples recovered from the Chang 8 Formation in Huangjiang Region, west Ordos Basin, were used to calibrate the involved parameters in the proposed model, and a well was processed to verify the reliability. The result illustrated that the proposed method was available; it can be used to consecutively construct pseudo Pc curves from NMR logging in the intervals with which field NMR logging data was acquired.