Experimental Study on the Distribution and Height of Spontaneous Imbibition Water of Chang 7 Continental Shale Oil

Abstract

After multi-stage volume hydraulic fracturing in a shale oil reservoir, massive amounts of water can be imbibed into the matrix pores. One of the key imbibition characteristics of a shale reservoir is the imbibition water and its height distribution. Based on high pressure mercury injection (HPMI) experiments and nuclear magnetic resonance (NMR) analyses, this study quantitatively evaluated the pore-size distribution of Chang 7 continental shale oil reservoirs in Yanchang Formation, Ordos Basin. The pores could be divided into three types as micropores (≤0.1 μm), mesopores (0.1–1.0 μm), and macropores (>1.0 μm), while the volume of micropores and mesopores accounted for more than 90%. This demonstrated that there were strong heterogeneity and micro–nano characteristics. According to the spontaneous imbibition (SI) experiments, the cumulative proportion of imbibition water content was the largest in micropores, exceeding 43%, followed by mesopores around 30%, and that of macropores was the lowest, and basically less than 20%. The negative values of stage water content in the macropore or mesopore indicated that these pores became a water supply channel for other dominant imbibition pores. Additionally, combining the fractal theory with the NMR T₂ spectrum, the relative imbibition water and actual height were calculated in different pores, while the height distribution varied with cores and shale oil. The shorter the core, the higher was the relative height, while the radius of macropores filled with imbibition water was reduced. This indicates that the height distribution was affected by the pore structure, oil viscosity, and core length.

1. Introduction

Spontaneous imbibition (SI) is a two-phase flow process by which a wetting fluid displaces a non-wetting fluid in porous media, capillary and/or gravity-driven. It is a common natural phenomenon and plays an important role in oil and gas recovery from tight or shale reservoirs [1,2,3,4,5,6,7]. Many reports indicate that when micro- and nano-scale pores and throats are developed in shale reservoirs, the imbibition effect is enhanced [6,8,9,10]. Therefore, the characteristics of imbibition water distribution and height will be significantly affected by the pores and throat scale.

Recently, the experimental study of the imbibition of shale reservoirs has undergone great development. K. Makhanov et al. [11] revealed that the imbibition rate in the direction parallel to the bedding plane was higher than that in the perpendicular direction, and that of the aqueous phases was also significantly higher than the oleic phases, which was caused by water adsorption of clay minerals, although the experimental samples were oil-wet in the case of the Horn River shale. Zhou et al. [12] conducted imbibition experiments using the Horn River, Woodford, and Niobrara shale cores and illustrated that the imbibition process was dominated by both capillary and osmosis pressure, and the dominance changed with the imbibition water saturation. Eslahati et al. [13,14] investigated imbibition experiments to show that the SI behavior was affected by the initial water saturation conditions and mineral compositions. Sun et al. [15] illustrated that the imbibition rate was a function of the time of the liquid intake in Marcellus shale and the SI process was divided into two stages based on the difference in imbibition rate, and found that the surfactant worked to reduce the imbibition capacity of the shale. Jiang et al. [16] conducted spontaneous water imbibition experiments with Longmaxi shale core; the results showed that water imbibition was related to both capillary pressure and additional hydration stress generated by clay swelling. Liu et al. [17] analyzed the effects of wettability alteration and interfacial tension (IFT) reduction on the SI process of shale rocks of the Sichuan Basin in China. The results showed that the more water-wetness, the higher was the recovery, and with the reduction of IFT, the final oil recovery increased, and the imbibition rate decreased. Lyu et al. [18] conducted SI experiments under five types of boundary conditions in the tight sandstone outcrop from the Yanchang formation, Ordos Basin. The imbibition rate of all-face-open was faster than that under closed ends. Zhu Y. et al. [19] studied the imbibition law to use the Longmaxi formation shale in the Pengshui area. The results showed that the shale imbibition process could be divided into three stages and that the SI characteristic was affected by the pore structure. Research by Fang et al. [20] showed that the imbibition velocity is closely related to the clay mineral content and pore structure characteristics of the shale core. The maturity of organic matter and the TOC in shale oil reservoirs affect pore connectivity and wettability, which in turn affect the shale imbibition rate [21,22,23]. Li et al. [24] proclaimed that the most influential factors of imbibition capability were fracture size, porosity, clay content, and sample size through imbibition experiments using three different shale reservoir rocks.

Moreover, previous researchers also used numerous mathematical models and numerical simulations to explicate the mechanisms and influence factors of imbibition in-depth. Cai et al. [25,26,27,28] developed the Lucas–Washburn (LW) model based on the fractal characteristics of imbibition streamlines and investigated the SI law of water into gas-saturation porous media considering the gravity effect, while the imbibition process was assumed to be piston-like. Shi et al. [29] further developed a modified SI model with the assumption of non-piston-like imbibition. In addition, researchers have deduced many new imbibition equations based on the consideration of influencing factors like viscosity, IFT, gravity, boundary conditions, surface roughness, slip length, dynamic contact angle, pore shape, etc. [30,31,32,33,34]. Meng et al. [35] used the phase-field method to simulate the counter-current imbibition, and the simulation results showed that the imbibition oil recovery was very small in homogeneous porous media but more in heterogeneous porous media, while the difference was closely related to pore structure and size. Diao et al. [36] simulated the SI in heterogeneous porous media micromodels by a quasi-three-dimensional color-gradient lattice Boltzmann model to study the influence of viscosity ratio, tortuosity, and mixed wettability. Results showed that the influence of tortuosity was less than that of the viscosity ratio, and the mixed wettability significantly affected the stability of imbibition displacement. Qin et al. [37] established an image-based dynamic pore-network model to quantitatively predict SI processes in geological formations and bridge the gap between pore-scale flow dynamics and the SI Darcy theory. Dutta et al. [38] used computed tomography (CT) scans to investigate the SI in low-permeability sandstones and indicated that the distribution of water saturation along the core length changed with time and that the rock heterogeneity played an important role in the spreading and final saturation of the imbibition front. Wang et al. [39] revealed through CT scans that imbibition water distribution was governed by clay particle distribution.

In order to characterize the position of the imbibition front in porous media and analyze imbibition variations in the imbibition process, many reports on imbibition height have been investigated by many scholars. Zhmud et al. [40] took gravity into account and further developed the Lucas–Washburn equation to obtain the asymptotic solution of the rising height of the liquid–gas interface changing with time. Fries and Dreyer [41] researched the fact that gravity must be considered in the later imbibition stage of capillary rise and derived an analytic solution for imbibition height with a Lamber W function containing gravity. Shen et al. [42] reported that the fixed sticky layer decreased the capillary radius and caused a lower imbibition height than the classical LW model. Amadu et al. [43,44,45] used a fractal analytical model to characterize the SI height relating to the parameters of microstructured porous media and wetting liquid properties such as porosity, fractal dimensions, pore size and viscosity, surface tension, and liquid–solid interactions. Zhang et al. [46] analyzed the dynamic characteristics of the contact angle and its effects on the imbibition length during gas–liquid SI in microchannels by the pseudopotential multiphase flow lattice Boltzmann method (LBM). However, the previous studies mostly focused on the solid and fluid properties affecting the imbibition rate or oil and gas recovery variation characteristics under the imbibition effect [47,48,49,50,51,52], with little research on the content and height distribution variations of imbibition water in the imbibition process. This would directly influence the imbibition law in shale oil reservoirs.

In this study, four cores drilled from three wells were selected to investigate the SI in the Chang 7 continental shale oil cores. The pore size distribution of shale samples was characterized through HPMI and NMR tests. NMR T₂ spectra were used to quantitatively monitor the changes of imbibition water distribution in different pores. The imbibition water in the matrix pores is called the imbibition water shield (IWS), which affects water imbibing and blocks the shale oil flowing out. Then the geometric size of IWS was characterized based on the fractal approach and NMR T₂ spectrum; the height distribution of IWS revealed the change in imbibition water-front position, and the IWS height was influenced by the shale core length, pore structure, and fluid viscosity. It was thus found helpful to understand further the water distribution of the imbibition process and the mechanism of SI in continental shale oil reservoirs.

2. Theory and Method

2.1. Conversion between the T₂ Spectrum and Pore Radius

According to the fundamental principle of NMR, for porous media, the transverse relaxation time T₂ obtained by detecting the NMR signals of oil and water in pores is related to the size and shape of the pore space, and the relationship can be expressed as follows:

$$\frac{1}{T_2}={\rho }_2\frac{S}{V}={\rho }_2\frac{F_s}{r}$$

(1)

where T₂ is the total pore fluid transverse relaxation time, ms; ρ₂ is the transverse surface relaxivity, μm/ms; (S/V) is the specific surface area of the pores, μm²/μm³, S is the pore surface area (μm²), and V is the pore volume (μm³); F_s is the pore shape factor, dimensionless (for spherical pores, F_s = 3; for cylindrical pores, F_s = 2); r is the pore-throat radius, μm. A large number of previous experimental studies [53] have displayed that the relationship between the T₂ value and pore-throat radius r is a power function as follows:

$$\frac{1}{T_2}={\rho }_2\frac{F_s}{r^n}$$

(2)

where n is a power exponent, dimensionless. By defining a constant C = 1/(ρ₂·F_s), Equation (2) can be simplified as the following equation:

$$r=C{T}_2^{1/n}$$

(3)

Equation (3) expresses the relationship between the pore-throat radius r and transverse relaxation time T₂. However, the constant C cannot be measured directly in practice [54]. Considering the pore-throat size distribution (PSD) obtained through high-pressure mercury injection (HPMI) and the T₂ spectrum measured by NMR for oil or water in pores, the constants C and n can be calculated.

2.2. Relationship between T₂ and the Imbibition Water Content

The NMR T₂ spectrum is the relationship curve between transverse relaxation time T₂ and NMR signal amplitude. The NMR signal amplitude reflects the amount of fluid corresponding to the T₂ value in pores, as follows:

$$m={\displaystyle \sum m_i}={\displaystyle \sum (\gamma A_{T_{2i}}}+a)$$

(4)

where m is the total fluid quality in porous media, g; T_2i is the ith transverse relaxation time T₂ in the T₂ spectrum, ms; m_i and A_T2i are the NMR signal amplitude and fluid mass corresponding to T_2i, respectively, g and dimensionless; γ and a are the constants of detected fluid by experiment. Then the difference of NMR signal amplitude A_T2i can be used to characterize the mass change of fluid contained in the pores at different moments. In the process of this imbibition experiment, the detected fluids were n-dodecane, or 7# white oil in the pores. The amplitude of the pores is maximum at oil-saturation and decreases with imbibition, which means the water content increases. Therefore, the imbibition water saturation in the whole shale cores can be expressed as follows:

$$S_w(t)=\frac{V_w(t)}{V_p}=\frac{\Delta V_o(t)}{V_p}=\frac{\raisebox{1ex}{${\displaystyle \sum \gamma A_{T_{2i}}(t_0})-{\displaystyle \sum \gamma A_{T_{2i}}(t})$}\!\left/ \!\raisebox{-1ex}{${\rho }_o$}\right.}{\raisebox{1ex}{${\displaystyle \sum \gamma A_{T_{2i}}(t_0})$}\!\left/ \!\raisebox{-1ex}{${\rho }_o$}\right.}=\frac{{\displaystyle \sum A_{T_{2i}}(t_0})-{\displaystyle \sum A_{T_{2i}}(t})}{{\displaystyle \sum A_{T_{2i}}(t_0})}$$

(5)

With the pore radius r_i corresponding to T_2i calculated according to Equation (3), the imbibition water saturation in the pore is the following:

$$S_{iw}(t)=\frac{A_{T_{2i}}(t_0)-A_{T_{2i}}(t)}{A_{T_{2i}}(t_0)}$$

(6)

where S_w is the imbibition water saturation of the core, fraction; V_w, V_p, and ΔV_o are the total volume of imbibition water, pores, and oil reduced, respectively, mm³; ρ_o is the oil density, g/cm³; t₀ and t are separately the imbibition beginning and the imbibition time, s. Further analysis can be drawn on the cumulative and stage distribution of water content in different scale pores with imbibition. The formulas are as follows:

$$C_{iw}(t)=\frac{A_{T_{2i}}(t_0)-A_{T_{2i}}(t)}{{\displaystyle \sum A_{T_{2i}}(t_0)}-{\displaystyle \sum A_{T_{2i}}(t)}}$$

(7)

$$S{t}_{iw}(t)=\frac{A_{T_{2i}}(t-1)-A_{T_{2i}}(t)}{{\displaystyle \sum A_{T_{2i}}(t-1)}-{\displaystyle \sum A_{T_{2i}}(t)}}$$

(8)

where C_iw is the cumulative water content ratio of different pores, %; t − 1 represents the previous imbibition time of t; St_iw is the stage water content ratio of different pores, %.

2.3. Imbibition Height Based on Fractal Theory and NMR T₂ Spectrum

Fractal theory has been widely used in the research fields of PSD characterization and flow simulation in porous media. Based on the Brooks–Corey model [55,56], assuming that the shale rock pore space is composed of a series of tortuous capillary bundles (Figure 1) and that the relationship between the number of pores and the pore sizes can be characterized by fractal theory [57,58], the cumulative number of capillaries is as follows:

$$N_t(\ge r_{\max })={(\frac{r_{\max }}{r_{\min }})}^{D_f}$$

(9)

where N_t is the total number of pores; r_max and r_min are the maximum and minimum capillaries radii respectively; D_f is the pore fractal dimension, (in the two-dimensional space, 1 < D_f < 2). For a shale core with diameter d and porosity ϕ, the number of pores lying between radius r and r + dr in the cross section can be expressed as in [59] as follows:

$$-d{N}_t=\frac{(2-D_f)\varphi d^2}{4(1-\varphi )r_{\max }^{2-D_f}}{r}^{-(D_f+1)}dr$$

(10)

where the minus sign (−) on the left indicates that the number of pores decreases with pore radius.

Furthermore, the tortuosity is a basic parameter which quantitatively characterizes the tortuous pores in reservoir rocks and also has fractal characteristics. Then the fractal dimension of pore D_f and tortuosity D_T can be calculated as follows [60,61]:

$$D_f=d_E-\frac{\ln \varphi }{\ln (r_{\min }/r_{\max })}$$

(11)

$$D_T=1+\frac{\ln {\tau }_{av}}{\ln (L_s/2r_{av})}$$

(12)

where d_E is the dimension of Euclidian space, d_E = 2 and 0 < D_f < 2 (two-dimensional space), d_E = 3 and 0 < D_f < 3 (three-dimensional space); D_T is the dimension of tortuosity, in the two-dimensional space, 1 < D_T < 2, the pore is a straight line with D_T = 1 and tortuous extreme with D_T = 2; τ_av and r_av are the average tortuosity and radius of pores, respectively; L_s is the length of core. The average tortuosity and radius of pores can be calculated as follows [62,63]:

$${\tau }_{av}=1+B\ln (1/\varphi )$$

(13)

$$r_{av}=\frac{D_f{r}_{\min }}{D_f-1}[1-{(\frac{r_{\min }}{r_{\max }})}^{D_f-1}]$$

(14)

where B is an empirical constant, 0.5 < B < 1.0, (for spherical particles, B = 0.41; for cube particles, B = 0.63). And the relationship between the actual length of tortuous pores and pore radius can be expressed as [64,65] below:

$$L_f(r)=L_s^{D_T}{(2r)}^{1-D_T}$$

(15)

$${\tau }_f(r)=\frac{L_f(r)}{L_s}$$

(16)

where τ_f(r) is the tortuosity of pore with radius r. Inserting Equation (3) into Equations (10), (11), (14) and (15) yields:

$$-d{N}_t=\frac{(2-D_f)\varphi d^2}{4(1-\varphi )n{C}^2{T}_{2\max }^{\frac{2-D_f}{n}}}{T}_2^{-(\frac{D_f}{n}+1)}d{T}_2$$

(17)

$$D_f=d_E-\frac{\ln \varphi }{\frac{1}{n}\ln (T_{2\min }/T_{2\max })}$$

(18)

$$r_{av}=\frac{D_f}{D_f-1}·C{T}_{2\min }^{\frac{1}{n}}[1-{(\frac{T_{2\min }}{T_{2\max }})}^{\frac{D_f-1}{n}}]$$

(19)

$$L_f(r)=L_s^{D_T}{(2C{T}_2^{1/n})}^{1-D_T}$$

(20)

where T_2min and T_2max are the value of NMR transverse relaxation time T₂ corresponding to the minimum and maximum radius of the core, respectively. Based on Equation (4), the length of remaining oil in the pore of radius r is the following:

$$l(r)=\frac{\raisebox{1ex}{$m(r)$}\!\left/ \!\raisebox{-1ex}{${\rho }_o$}\right.}{\pi r^2·(-d{N}_t(r))}=\frac{\gamma A_{T_2}(r)+a}{{\rho }_o}/\pi r^2·(-d{N}_t(r))$$

(21)

where −dN_t(r) is the number of pores with radius r; m(r) and l(r) are the remaining oil mass and length of pore with radius r, respectively. Then the imbibition water height after imbibition time t is as follows:

$$\begin{array}{c}\Delta l[r(t)]=l[r(0)]-l[r(t)]\\ =\frac{\gamma A_{T_2}[r(0)]+a}{{\rho }_o}/\pi r^2·[-d{N}_t(r)]-\frac{\gamma A_{T_2}[r(t)]+a}{{\rho }_o}/\pi r^2·[-d{N}_t(r)]\\ =\frac{\gamma }{{\rho }_o\pi r^2·[-d{N}_t(r)]}\left\{A_{T_2}[r(0)]-A_{T_2}[r(t)]\right\}\end{array}$$

(22)

where Δl[r(t)] represents the imbibition water height of pore radius r with imbibition time t; l[r(0)] and l[r(t)] are the oil length of pore radius r before imbibition and after imbibition time t, separately; A_T2[r(0)] and A_T2[r(t)] are the oil NMR signal amplitude of pore with radius r before imbibition and after imbibition time t, separately. The relative height Δδ of imbibition water can be expressed as below:

$$\Delta \delta =\frac{\Delta l[r(t)]}{l[r(0)]}=\frac{A_{T_2}[r(0)]-A_{T_2}[r(t)]}{A_{T_2}[r(0)]}$$

(23)

when the core is saturated, the actual length of pore with radius r is equal to the oil length before imbibition, as the following:

$$l[r(0)]=L_f(r)$$

(24)

Inserting Equation (24) into Equations (10) and (16) yields:

$$\begin{array}{c}\Delta l[r(t)]=\Delta \delta ·{\tau }_f(r)·L_s\\ =\frac{A_{T_2}[r(0)]-A_{T_2}[r(t)]}{A_{T_2}[r(0)]}·{\tau }_f(r)·L_s\end{array}$$

(25)

Therefore, Equations (23) and (25) are formulas of the relative and actual height of imbibition water based on fractal theory and the NMR T₂ spectrum, respectively.

3. Experiment

3.1. Materials

The Yanchang Formation of Upper Triassic is the main oil-bearing formation of the Ordos Basin in China, which can be divided into 10 oil layers from Chang 1 to Chang 10 [53]. In this study, four cylindrical cores were taken from the Chang 7 Formation, which is a set of source rock series that are mainly composed of continental shale, belonging to lacustrine sedimentation. To be specific, Core #H1 was drilled from Well H38, Core #Z11 was drilled from Well Z55, and Cores #B14 and #B15 were drilled from Well B20. After the four shale samples were cleaned and dried, porosity and permeability tests were carried out with nitrogen.

Table 1. Basic properties of core samples.

Well#	Core#	Diameter, cm	Length, cm	Porosity, %	Permeability, 10−3 μm2
H38	H1	2.501	4.132	9.81	0.222
Z55	Z11	2.500	3.002	7.31	0.223
B20	B14	2.499	3.422	6.99	0.0476
	B15	2.495	3.774	8.96	0.0486

displays the petrophysical parameters of the core samples. Their permeabilities of Cores #H1 and #Z11 are almost equal and much larger than those of Cores #B14 and #B15. However, there is little difference in porosity among them.

For pore-size testing by NMR, the cores were saturated with distilled water (DW). In SI experiments, because the signals between hydrogen protons in oil and water cannot be distinguished by NMR, the experimental fluid was heavy water (HW, without any hydrogen signal), and the oils used in this study were #7 white oil (WO7) and n-dodecane (C₁₂), both produced from TOMA Lubricating Oil (Beijing) Co. Ltd. The viscosities and densities of the fluids are listed in

Table 2. Viscosities and densities of fluids (20 °C, 1 atm).

Fluid	Density (g/cm3)	Viscosity (mPa·s)
DW	0.998	1.002
HW	1.105	1.095
WO7	0.830	5.477 (40 °C)
C12	0.7847	1.508

. The viscosity of WO7 is the largest, followed by C₁₂, and there is a little difference between DW and HW. Additionally, the calibration curves between distilled water, #7 white oil, n-dodecane, and NMR signal amplitude are shown in Figure 2, and there is a positive linear relationship among them.

3.2. Procedure and Setup

To investigate the pore radius distribution, HPMI tests and NMR-detecting shale cores with saturated distilled water were carried out. In addition, in the SI experiment, the microscopic spontaneous imbibition process could be analyzed by the NMR T₂ spectra of residual oil at different imbibition times. The main experimental procedures and setup used are as follows:

Core preparation: the shale cores, approximately 25 mm in diameter, were prepared using a linear cutting machine. The specimens were subsequently cleaned for 28 days and dried at 120 °C in an oven for about 3 days until the mass changes were less than 0.005 g and the initial mass of the cores was recorded. Finally, we measured the samples’ basic physical parameters, such as length and diameter, with a vernier caliper, while the permeability and porosity were tested with helium.

HPMI tests: the AutoPore IV 9505 used for HPMI measurements was produced by Micromeritics Co., Norcross, GA, USA. The measurement of pressure is from 1 atmosphere to 228 MPa, and the pore diameter range is from 5 nm to 6 µm. Three short cores with a length of about 1 cm cut from the downhole samples were chosen for the HPMI tests. During the test, the mercury saturations for various mercury injection pressures were determined to infer the pore and throat distribution radius of the core sample.

NMR tests: the NMR spectrometer shown in Figure 3 was produced by Suzhou Niumag Analytical Instrument Co., Ltd., Suzhou, China. The magnetic field strength is 0.3 ± 0.05 T, and the main frequency is 10 MHz. The dried cores were put into the vacuum saturation device at −0.1 MPa for 72 h. Next, the specimens were soaked with distilled water in a high-pressure container at 30 MPa for 96 h to completely saturate. Moreover, the T₂ spectra of the samples saturated with distilled water were measured by an NMR apparatus; the pore size distribution could be obtained according to the relationship between the T₂ value and the pore radius r tested by HPMI.

Imbibition experiment

• The four cores were dried at 120 °C in an oven until the mass changes were less than 0.005 g again and vacuumized with a vacuum pump for 48 h.
• A high-pressure container was placed and filled with #7 white oil and another filled with n-dodecane in a thermotank where the temperature was set at 60 °C for 24 h. Then Cores #H1 and #Z11 were put into the former container, and Cores #B14 and #B15 were placed into the other container, and then soaked at 30 MPa and 60 °C for about 5 days until the mass showed little change.
• The saturated oil cores were taken out and the side covered in Teflon tape, exposing only the end face for 2–3 mm. Next, the whole samples were immersed in a beaker containing heavy water to carry out the imbibition experiments at 60 °C. The cores were taken out of the beaker at different times and the NMR T₂ spectrum curves measured. The measuring time should be limited to within 2 min. The SI experiments are illustrated in Figure 3.

4. Results and Discussion

4.1. Pore Radius Distribution

Figure 4 provided the intrusion and extrusion curves of HPMI of three Chang 7 continental shale samples, representing the pore structure characteristics of Well H38, Well Z55, and Well B20, respectively. The capillary pressure curves showed some differences among them. The capillary pressure curve of Well H38 was distributed to the lower left, and the curve was relatively gentle. Compared with Well H38, the curve of Well Z55 rose, and the gentle section became shorter while the curve of well B20 showed a significant rise in distribution and a shorter horizontal length, with a tendency towards a steep slope. The HPMI results are shown in

Table 3. Pore-throat characteristics of HPMI results.

Well#	Displacement Pressure, MPa	Maximum Connected Pore Radius, μm	Maximum Mercury Intrusion Saturation, %	Mercury Snap-Off, %	Extrusion Efficiency, %
H38	0.676	1.087	95.346	72.731	23.719
Z55	0.676	1.087	94.187	71.950	23.609
B20	2.736	0.269	65.002	21.840	66.401

. It indicates that the radii of connected pores were smaller due to the larger displacement pressure, and the maximum connected pore radius of Well B20 was the smallest, about 0.269 μm. The maximum mercury intrusion saturation reflected that pore volumes with a radius below 3.7 nm accounted for less than 5% of Wells H38 and Z55, while that of Well B20 accounted for 33.6%. Meanwhile, for Wells H38 and Z55, the large intrusion saturation and low extrusion efficiency revealed a large pore-throat discrepancy, resulting in 72.73% and 71.95% of mercury snap-off in pores, and the lower intrusion saturation and higher extrusion efficiency of Well B20 indicated that the pore-throat size was relatively uniform at the nano-scale level. This plays an important role in imbibition.

As shown in Figure 5a, the cumulative frequency curves of PSD obtained from HPMI tests were first plotted with the NMR T₂ spectrum on a single graph using the same horizontal axis. Second, the values of T₂(i) and pore-throat radius r(i) corresponding to the same cumulative frequency S(i) were calculated by the interpolation method. Finally, the parameter C and n can be obtained by fitting the curves of T₂(i) and r(i) according to the linear least squares principle (Figure 5b). The values of C and n were calculated and are listed in

Table 4. The value of C and n and pore volume ratio of different sizes of core samples.

Core#	C	1/n	R2	Pore Volume Ratio, %
				Micropores	Mesopores	Macropores
H1	0.0295	0.7343	0.972	46.56	48.91	4.53
Z11	0.0388	0.7241	0.8805	57.49	32.90	9.61
B14	0.0388	0.6175	0.7720	61.25	29.81	8.94
B15	0.0375	0.5695	0.7915	53.26	37.19	9.55

. Therefore, the NMR T₂ spectra of shale cores with water saturation (Figure 6a) were converted into PSD curves (Figure 6b) and showed a certain difference in their distribution. Table 4 exhibits the shale pores divided into three types and their ratios of pore volume [66,67]. Pores with radii less than 1 μm account for more than 90%, but there were ratio differences with micropores and mesopores among the cores, and the difference between Core #H1 and Core #B14 was close to 20%. As a result, shale cores demonstrated visibly smaller pores in contrast with conventional formation. This characteristic which is mainly composed of micro-and mesopores but fewer macropores, increased the capillary pressure and promoted imbibition flow in some way. The pore classification was consistent with the research of Zhang et al. [10].

4.2. NMR T₂

Because it is tight and the volume of saturated oil is very low for shale core, it was difficult to accurately measure the oil production in core experiments by conventional methods. However, NMR technology could provide a way to quantitatively determine the production variation of the SI process at pore-scale, and a series of NMR tests were carried out. Figure 7 presents a series of SI T₂ spectra obtained from four shale cores with oil-saturated at different imbibition times between 0.5 and 200 h at 60 °C and normal pressure. Then, the SI characteristics and water distribution of shale oil cores can be analyzed, wherein Cores #H1 and #Z11 were saturated with WO7 and Cores #B14 and #B15 were saturated with n-dodecane. The SI T₂ spectra of WO7-saturated cores have two obvious peaks (Figure 7a,b). At the beginning of SI, the peaks decline largely and quickly, then gradually slow down, and most of the curve segments overlap after 80 h. This indicated that the SI of Cores #H1 and #Z11 first occurred in micropores and macropores and then transitioned to mesopores. However, for Core #B14 and #B15 saturated with n-dodecane, as shown in Figure 7c,d, their SI T₂ spectra have three peaks, and the left peak is higher than the others. At the beginning of SI, the peaks also decline largely and rapidly and gradually slow down, and most curve segments overlap after 48 h. Further analysis shows that the T₂ spectrum peak of the micropores initially decreased the fastest and largest, followed by the macropores and finally the mesopores. The whole drop in the magnitude of the T₂ spectrum in the micropores and macropores was greater than that of the mesopores, indicating that the order of imbibition of shale samples in Well B20 was micropores, macropores, and mesopores. Meanwhile, it can be seen that the capillary pressure in shale pores was very high, and imbibition was strong based on the characteristics of the SI T₂ spectra of four rock samples. According to the differences in the sequence of SI and the strength of imbibition in different pores, it could be determined that the capillary force, as the driving force of imbibition, was extremely strong in micropores, while the fluid viscosity force, as the resistance of imbibition, was relatively small in macropores. While in mesopores, the capillary force is relatively small and the viscosity force is relatively large, this results in a greater difference between the capillary pressure and the viscosity force in micropores and macropores, and a smaller difference in mesopores, which ultimately causes the differences in the sequence and intensity of imbibition in different pores.

In addition, the T₂ spectra variations of oil-saturated and SI processes among the shale samples were mainly caused by the differences of shale pore structure and oil properties. The micropore volume proportions of Cores #Z11, #B14, and #B15 were significantly higher than those of mesopores and macropores, and their left peak values of T₂ spectra for saturated oil and SI were also significantly higher than those of the middle and right peaks. However, there was little difference between the micropore and mesopore volume proportions of Core #H1, and since the molecule diameter and viscosity of WO7 are relatively large and some extreme micropores may block the flow of WO7, it was difficult to saturate some of the micropores, resulting in the left peak values being significantly lower than those of the middle and right peaks. Moreover, due to the oil properties, the oil-saturated T₂ spectra of Cores #H1 and #Z11 differed from those of saturated water without the obvious three-peak features.

4.3. Imbibition Water Saturation Characteristics

The SI water saturation of four samples is shown in Figure 8. After 200 h of imbibition, the water saturation of Core #Z11 was the highest, reaching 28.78%, while that of Core #H1 was the lowest, at 20.54%. Core #B14 and #B15 were 25.39% and 24.84%, respectively. It was found that the final water saturation was inversely proportional to the core length. According to the characteristics of the imbibition water saturation curves, the imbibition of oil-saturated shale can be divided into three stages: the initial stage of rapid imbibition, the middle stage of transitional imbibition, and the late stage of equilibrium imbibition. The duration of the initial stage was relatively short, and the slopes of the curves decreased significantly after approximately 8 h. Due to the oil properties, the water saturations of Cores #B14 and #B15 were 20.37% and 19.86% at the first inflection point, respectively, which were higher than the 14.82% of Core #H1 and 17.35% of Core #Z11. However, after about 80 h, the imbibition of Core #H1 and #Z11 reached an equilibrium stage, and the duration of their transition stage was longer than that of Cores #B14 and #B15, whose second inflection points were about 48 h. This reflects again the influence of oil properties and pore structure characteristics on shale oil imbibition.

4.4. Distribution Characteristics of Imbibition Water in Different Pores

Since the pore size span of shale cores was large, the distribution of water content demonstrated great differences in different pores. Figure 9 shows the cumulative distribution of pore water content, which was characterized by the water content proportion curves of different pores under different imbibition times. The proportion of imbibition water content was the largest in micropores, reaching 64.36% for Core #B14 and 43.62% for Core #H1. Next were mesopores, with the final water content proportion reaching 38.32% in Core #H1 and being less than 30% in other samples. The proportion of macropores was the lowest and was actually less than 20%. Further analysis of the above curves found that the proportion of imbibition water content in micropores was initially the largest, then rapidly decreased, and then gradually increased again, with a “V” shape. The proportion of imbibition water content in macropores was initially large, then gradually decreased to a stable value. In contrast, the proportion of imbibition water content in mesopores was initially small, then gradually increased to an equilibrium value. Except for Core #H1, the order of initial water content proportion was micropores, macropores, and mesopores. This also reflected that imbibition occurred first in micropores and macropores and then gradually expanded to mesopores.

During the SI process, the imbibition of different pores influenced each other due to the discrepancy in pore structure. Figure 10 shows the curves of stage imbibition water content in different pores, which indicated that the proportion of stage imbibition water content was fluctuating in different pores over time, particularly in the initial stage, and the curves of different pores intersected without any significant characteristics. The overall trend of water content followed the order of micropores, mesopores, and macropores with imbibition time (except for Core #B15, where although the proportion of macropores was larger than that of mesopores within the experimental timeframe, the water content of mesopores would still be larger than that of macropores according to the trend of the curve). Meanwhile, there was also a phenomenon of negative stage imbibition water content in the macropores or mesopores, revealing that the macropores or mesopores had become a water supply channel for other dominant imbibition pores in this stage [68]. When the imbibition front in the macropore reached the junction with the smaller pores, the imbibition water was imbibed into the micropores, resulting in the water content decreasing in the macropores and increasing in the other pores.

4.5. Distribution Characteristics of Imbibition Water Height

After imbibition of an oil-saturated shale core, the geometric size distribution of IWS for different pores can be obtained according to the imbibition results based on the NMR test. The cross-sectional size of the IWS was determined by the pore radius, which was calculated by conversion of Equation (3). The height of IWS for different scale pores can be determined according to Equations (23) and (25). Figure 11 exhibits the relative and actual height distribution characteristics of spontaneous IWSs in saturated oil shale samples. Four experimental shale cores were taken separately from three wells. Cores #H1 and #Z11 were saturated with WO7, and Cores #B14 and #B15 were saturated with n-dodecane. For the height distribution of IWS in Cores #H1 and #Z11, the heights of micropores and macropores were generally higher than that of mesopores, and the relative height of IWS in a few macropores was equal to 1, while others were less than 1, including that of the micropores and mesopores. Additionally, compared with Cores #H1 and #Z1, the height distribution of IWS in Core #B14 and #B15 demonstrated a multimodal shape. HPMI and NMR tests revealed that the cores drilled from Well B20 had smaller pore sizes and a more complex structure. Moreover, the height of IWS in mesopores of the two cores even showed significant negative values during the imbibition process, reflecting that new pores may have been generated in the core under the SI process. In addition, the peak value of the height was close to 70 mm where the pore radius appeared in the range of 0.05–0.1 μm, which was much larger than that of Cores #H1 and #Z11 at this pore radius range. These results indicated that pore structure and shale oil viscosity had an important and complex influence on the distribution of IWS.

Figure 12 shows the graph of the height distribution of IWS after SI 200 h. For Cores #H1 and #Z11, apart from a few micropores with diameters ranging from 0.03 to 0.09 μm, the relative height of the IWS of Core #Z11 was higher than that of Core #H1, and the radius of macropores filled with imbibition water was smaller in the former core. According to the intrusion and extrusion curves of the mercury injection capillary pressure of shale cores taken from Wells H38 and Z55 shown in Figure 4, the difference in pore structure between the two samples was relatively small. However, the length of Core #H1 was 1.38 times that of Core #Z11, indicating that the shale core length had a significant impact on the distribution of the IWS—as the shorter the core length, the higher the relative height of the IWS, and the smaller the radius of macropores filled with imbibition water.

5. Conclusions

Based on HPMI experiments and NMR analyses, this paper quantitatively evaluated the pore-size distribution of Chang 7 continental shale oil reservoirs in Yanchang Formation, Ordos Basin. Furthermore, SI experiments were conducted to study the imbibition law and distribution of imbibition water in Chang 7 continental shale, and the microscopic occurrence characteristics of imbibition water height were analyzed by fractal theory. The main conclusions were as follows:

• According to HPMI experiments, the displacement pressures of samples drilled from the three wells were 0.676, 0.676, and 2.736 MPa, and the higher the displacement pressure, the smaller the maximum connected pore radius, such that Well B20 was the smallest at about 0.269 μm. Meanwhile, there were three peaks for the NMR T₂ spectra with saturated distilled water, and the overall pore distribution of shale was divided into three types: micropores (<100 nm), mesopores (100–1000 nm), and macropores (>1000 nm), with the volume of micropores and mesopores accounting for 90%, indicating that Chang 7 continental shale pore-fracture had strong heterogeneity and distinct micro–nano characteristics.
• The NMR T₂ spectra of the remaining oil in four shale samples decreased significantly with time, reflecting a strong imbibition effect in the shale pore. The peaks of T₂ spectra in micropores and macropores initially declined faster and finally dropped more than those in mesopores, showing that there was a bigger gap between imbibition driving force and imbibition resistance in micro- and macropores, but a smaller difference in mesopores. It ultimately led to the above differences in sizes.
• After SI experiments, the cumulative proportion of imbibition water content was the largest in micropores, exceeding 43%, followed by mesopores around 30%, and that of macropores was the lowest, basically less than 20%. The cumulative proportion curves of micropores were basically a “V” shape and kept decreasing to a stable level in macropores and increasing to an equilibrium state in mesopores. This reflects that imbibition first occurred in micropores and macropores, then expanded to mesopores. Additionally, the stage proportion fluctuated in different pores during the SI process, and the negative values of stage water content in the macropore or mesopore indicated that these pores became a water supply channel for other dominant imbibition pores;
• Based on the fractal theory and NMR T₂ spectrum, the relative imbibition water and actual height were calculated in different pores. There were significant differences in the shape of the height distribution curve for samples drilled from different wells and saturated with different oils, and the shorter the core, the higher the relative height. This indicates that the height distribution was affected by the pore structure, oil viscosity, and core length.
• Moreover, after hydraulic fracturing, the imbibition process will be affected by the change in pressure and temperature of the shale oil reservoir, and then the distribution of imbibition fluid and its height will also change. Therefore, carrying out experiments with changes in pressure and temperature can investigate further the distribution and height of the IWS.