Experimental Study on the Forcible Imbibition Law of Water in Shale Gas Reservoirs

Abstract

Water imbibition is a key factor affecting the flowback regime of shale gas wells after volume fracturing. In this study, a set of experimental apparatus and corresponding test and evaluation methods were developed to analyze the laws of forcible imbibition of water in a shale reservoir, characterize the initiation time of microfractures induced by shale hydration quantitatively, and optimize the shut-in time of shale gas wells; the imbibition depths of different pore types are quantitatively calculated based on the multiple pore imbibition analytical model. The experimental results show that: according to imbibition saturation growth rate, the shale forcible imbibition can be divided into three periods, imbibition diffusion, imbibition transition, and imbibition balance. Among them, the imbibition diffusion period is the main period for imbibition capacity rise. The reason for this phenomenon is that due to the fluid pressure difference effect, the shale fills its large pores and microfractures rapidly in the early stage, and in the percolation transition period, the percolation rate decreases continuously due to the gradual increase of fluid saturation. Due to the Jamin effect, it is difficult for the fluid to enter the small pores and the fluid fills the pore roar channel, the seepage saturation tends to stabilize, and the seepage equilibrium period appears. In the early period of shut-in, the imbibition capacity of shale increases significantly under the action of fluid pressure, providing a large amount of imbibition fluid for the spontaneous imbibition later. The imbibition depth of a clay pore was much greater than that of a brittle mineral pore and an organic pore. The reservoir confining pressure has prohibition on shale imbibition, but even under reservoir confining pressure, imbibition can still improve the fracturing effect of the reservoir, resulting in an increase in porosity of 0.42–1.63 times and an increase in permeability of 17.6–67.3 times. Under the experimental conditions, the initiation time of induced microfractures is 98.5 h on average and is in negative correlation with imbibition capacity. On this basis, the optimized shortest shut-in time of a shale gas well is 5 days. The study results can provide a scientific basis for the optimization of the flowback regime of shale gas reservoirs.

1. Introduction

Volume fracturing in horizontal wells is the key technology for economic development of shale gas [1,2,3,4]. In recent years, drawing on experience in North America, many oilfield companies in China have carried out continuous research and field tests on shale gas development technologies, and the technologies have enhanced production effectively [5,6,7]. Statistics on production data after fracturing show many shale gas wells have a special flowback feature after fracturing; the low flowback ratio is generally less than 50% and even less than 3% in some wells [8,9]. With a large proportion of fracturing fluid retained underground, the wells gave high shale gas production after shut-in for some time. This phenomenon has drawn wide attention from researchers [10,11]. To explain this special flowback phenomenon of shale gas wells, the mechanism of water imbibition in shale must first be studied. In the initial stage of shale gas well shut-in, fracturing fluid would imbibe forcibly into the reservoir under the joint effect of confining pressure, temperature, fluid pressure in fracture, etc., [12]. This stage has a crucial impact on the law of water imbibition in shale and in turn affects the flowback ratio.

A lot of research on imbibition laws in porous rocks has been conducted before, including imbibition experiments [13,14] and theoretical models [15,16,17,18]. In imbibition experiments, the test methods used commonly include direct weighing, indirect weighing, and volume methods [19,20]. Among them, the weighing method is used most commonly and is simple in principle, easy in operation, and high in precision. Most of the experiments have focused on spontaneous imbibition under normal temperature and pressure, that is the imbibition of fracturing fluid to the shale matrix under the effect of capillary pressure and osmotic pressure, etc., when the pressure of fluid in the fracture reaches a balance with the reservoir pressure. However, in fact, shale is under the effect of stresses in three directions and different temperatures and there is the pressure of the fluid in the fracture in the initial stage of shut-in. Only a few researchers have examined the effect of confining pressure. Roshan et al. concluded that water imbibition in shale can produce induced fractures even under confining pressure [21]. In respect to an imbibition theoretical model, many researchers have proposed imbibition models considering mainly capillary pressure, including Lucas–Washburn (LW) [22], Handy [23], Terzaghi [24], and fractal [25] models. However, these models do not consider the effects of reservoir temperature, confining pressure, fluid pressure, etc., on the laws of water imbibition in shale. In addition, as shale often has multi-scales of pores from micron to nanometer sizes, so the interaction between fracturing fluid and shale is very complex, and the existent imbibition models have many assumed conditions and cannot characterize the imbibition laws of water in shale under reservoir confining pressure and temperature.

The water imbibition of shale is influenced by several factors; previous studies have indicated that high temperature decreases the amount of fluid absorbed by shale and that the peritectic pressure severely reduces the percolation rate parallel to the laminae direction. Additionally, the permeability of shales decreases when they absorb water [26]. The water absorption characteristics of shale are influenced by the combination of temperature, surrounding pressure, and other factors, and the current model at room temperature and without considering the surrounding pressure is not applicable. Therefore, we have developed an imbibition experiment device considering the pressure of fluid in fracture, reservoir confining pressure, and temperature, etc., and worked out a test method. On this basis, we analyzed the forcible imbibition laws in the initial stage of shale gas well shut-in and calculated the shortest shut-in experiment time. The research results can guide the study of seepage laws of fracturing fluid in shale gas reservoirs.

2. Experimental Study

As we know, in the initial stage of shut-in after the fracturing of shale gas wells, the fluid in hydraulic fractures will imbibe into the shale matrix and natural fractures under the effect of fluid pressure and reservoir temperature and pressure, the imbibition in this stage is called forcible imbibition [27]. When the pressure of the fluid in the hydraulic fracture reaches a balance with the reservoir balance, the fracturing fluid will imbibe into the shale matrix under the effect of capillary pressure and osmotic pressure; the imbibition in this stage is called spontaneous imbibition.

2.1. Experimental Apparatus

To find out the forcible imbibition law of water in shale, we designed an experimental apparatus and corresponding test method able to reflect the effects of the pressure of the fluid in fracture, confining pressure, and temperature, etc. The apparatus can quantitatively test the imbibition capacity of water in shale with time under various reservoir conditions. The apparatus includes a constant rate and pressure pump, a reactor, a heating jacket, a core holder, and a confining pressure pump, etc., (Figure 1). The apparatus can work at a maximum temperature of 200 °C and a maximum pressure of 70 MPa. The constant rate and pressure pump have a measurement accuracy of over 0.001 mL/min (Figure 2).

We also designed an experimental apparatus to test the spontaneous imbibition of water in shale under no confining pressure and constant temperature. This apparatus is made up of a high precision scale, a computer, and a beaker. The scale has a measurement range of 220 g and a measurement accuracy of 0.0001 g. The principle of the experiment is to measure the variation in the shale sample weight with imbibition time to obtain the curve of imbibition quantity with time. Then, the law of water imbibition in shale can be analyzed.

2.2. Experimental Method

Standard shale samples of 2.5 cm in diameter and 5 cm long were prepared from shale cores using an ELK-2 ultra-low permeameter with a pulse decay method for the decay characteristics and pressure changes during the ultra-low permeability test, calculating the permeability before and after the experiment, and using an HXK-III helium porosimeter by Boyle’s law, the product of pressure and volume of the gas as a constant was used to measure the porosity before and after the experiment under the same temperature. According to the geologic parameters and fracturing parameters of the shale reservoir, the confining pressure, temperature, and fluid pressure in the experiments were set. The quantity of fracturing fluid injected by the constant rate and pressure pump was recorded by the computer, and the cumulative flow at each time point was taken as the forcible imbibition quantity of the shale at this point. Based on this, the law of forcible imbibition was analyzed.

2.3. Experimental Samples

When fracturing the Longmaxi Formation reservoir, the shale absorbs water and leads to a low fracturing fluid rejection rate. It is important to select shale rock samples from the Longmaxi Formation and analyze their forcible imbibition characteristics. The experimental samples were taken from Longmaxi Formation cores collected from the southern Sichuan area. The reservoir, the major part in Longmaxi Formation, from which the cores were collected, has burial depths of 2400–2450 m. The black carbonaceous shale, black carbonaceous shale interbedded with dark gray laminated siltstone, gray to gray-green mud siltstone, and gray siltstone are developed from the bottom to the top, the organic carbon and silica content gradually decreases, and the color gradually becomes lighter. There is a pore pressure of 46 MPa, a maximum horizontal stress of 48 MPa, a minimum horizontal stress of 40 MPa, a vertical stress of 43 MPa, temperatures of 70.5–90.5 °C, and an average temperature of 80 °C.

The shale samples, 7 in total, were coded S1–S7 (Figure 3). They were dried in an oven at 100 °C to a constant weight. The permeabilities and porosities of the 7 samples before and after the experiment were tested with an ELK-2 ultra-low permeameter and an HXK-III helium porosimeter, respectively. The test results are shown in

Table 1. Basic parameters of shale samples.

Shale No.	Length /cm	Diameter /cm	Imbibition Area /cm2	Permeability /mD	Porosity /%
S1	5.0	2.5	4.9	0.0042	5.7
S2	5.0	2.5	4.9	0.0045	6.0
S3	5.0	2.5	4.9	0.0040	5.4
S4	5.0	2.5	4.9	0.0027	4.9
S5	5.0	2.5	4.9	0.0024	4.6
S6	5.0	2.5	4.9	0.0026	5.0
S7	5.0	2.5	4.9	0.0032	5.3

.

The composition of the samples was tested with an XPert PRO X-ray diffractor and is shown in Figure 4. It can be seen from this figure that the 7 shale samples are mainly composed of quartz, mineral calcite, clay minerals, and feldspar, which is mainly composed of sodium feldspar and potassium feldspar and higher in the content of clay minerals (52.2% at maximum and 32.0% on average).

To investigate the effects of fluid pressure and confining pressure on forcible imbibition in shale, two kinds of experiments were conducted; the results were compared with the results of spontaneous imbibition experiments under ambient temperature and pressure and are shown in

Table 2. Designed forcible imbibition experiments on shale samples.

Sample No.	Influencing Factor	Fluid Pressure /MPa	Confining Pressure /MPa	Temperature/°C
S1	Fluid pressure	1	16	80
S2		2	16	80
S3		3	16	80
S4	Confining pressure	2	12	80
S5		2	14	80
S6		2	16	80
S7	/	0	0	25

. The fluid used in the experiments was deionized water. In the experiments, the end face of the shale sample contacted with deionized water. The experiments lasted for 5 days. Physical parameters of some of the shale samples after forcible imbibition were tested to evaluate the permeability improvement in the shale samples caused by imbibition.

2.4. Evaluation Method

2.4.1. Imbibition Capacity

Based on physical parameters and tested dynamic imbibition quantity of the shale sample, imbibition saturation was defined to quantitatively characterize imbibition capacity, that is the percentage of imbibition quantity to pore volume of the sample I, which is expressed as:

$$I=\frac{V}{\phi AL}\times 100\%$$

(1)

where $V$ is the dynamic imbibition quantity, cm³; $\phi$ is the porosity, %; $A$ is the imbibition area, cm²; and $L$ is the core length, cm.

2.4.2. Initiation Time of the Microfracture

Makhanov et al. [28] defined the curve of imbibition quantity in unit shale area with the square root of imbibition time to characterize the imbibition capacity. The slope of the curve is the imbibition rate of the porous medium and can effectively compare the imbibition rates of shale samples with different imbibition areas. Its calculation equation is as follows:

$$R=\frac{V}{A}=\sqrt{\frac{2P_C\phi K{S}_{wf}}{\mu }}\sqrt{t}$$

(2)

where $R$ is the imbibition capacity per unit area, cm³/cm²; $P_c$ is the capillary pressure, Pa; $K$ is the permeability of the sample, mD; $S_{\mathit{wf}}$ is the front saturation, decimal; $\mu$ is the viscosity of the imbibition liquid, mPa·s; and $t$ t is the imbibition time, h.

During the imbibition, the time that induced microfractures start to generate due to hydration is defined as the initiation time of the microfracture. This value can be obtained by calculating the slope of Equation (2). As shown by Equation (3), the slope of Equation (2) represents the imbibition rate of the liquid. Violent fluctuation points on the curve means that there are new fractures generated and the imbibition rate increases. Moreover, it can be seen from Equation (3) that the imbibition rate is proportional to the capillary pressure, porosity, and permeability and that the occurrence of microfractures can further improve the micropore structure and physical properties of the rock, so variation of the slope can be used to characterize the variation in capillary force and the variation of pore size during imbibition.

$$v_i=\sqrt{\frac{2P_c\phi K{S}_{wf}}{\mu }}$$

(3)

where $v_i$ is the imbibition rate, cm/h.

Based on the imbibition experiment data of sample S7, the initiation time of the induced microfracture was obtained, as shown in Figure 5. Points on the imbibition curve with violent fluctuations are where induced microfractures occurred, for example, point A. Meanwhile, the unit area imbibition quantity (A’) corresponding to point A also fluctuated. After a period, the imbibition rate fluctuated widely (at points B, C, E, and F). It can be seen the shale hydration is a dynamic equilibrium process, during which microfractures come about intermittently; as a result, the shale has increasing flow channels, decreasing flow resistance, and significantly increasing imbibition capacity. It can be seen from the variation amplitude of the imbibition rate curve that the curve has a very high fluctuation frequency, reflecting the heterogeneity of the shale.

3. Analysis of Experimental Results

3.1. Laws of Forcible Imbibition

The spontaneous imbibition curve of sample S7 and forcible imbibition curve of sample S4 were compared (Figure 6) (the experimental conditions are shown in Table 2). According to the variation pattern of imbibition saturation with time, the forcible imbibition curve can be divided into three stages: imbibition diffusion, imbibition transition, and imbibition equilibrium, of which the imbibition diffusion stage is the main stage contributing to the imbibition capacity. Compared with the spontaneous imbibition curve, shale forcible imbibition under reservoir temperature and pressure condition has considerable delay at all the stages, longer imbibition diffusion and transition stages, a longer unbalanced state, and does not reach equilibrium quickly as it does during spontaneous imbibition under normal temperature and pressure. Such a pattern is also observed in the static percolation experiments for Ordos core; this is because the rate of water entering large pores and microfractures is initially relatively fast and then tends to slow down gradually as the rock samples are gradually saturated. Most of the percolation experiments have a pattern of water absorption rate from fast to slow before finally leveling off.

The relationship curves between imbibition capacity and time for the six shale samples under different experimental conditions are shown in Figure 7. The curves show that the samples differed widely in imbibition capacity due to different experimental conditions and heterogeneity. At an imbibition time of 5 d, they had a maximum forcible imbibition capacity between 46.6 and 166.5% (112.4% on average). In the early stage of imbibition, the forcible imbibition capacity increased quickly. For example, at an imbibition time of 2 d, S3 and S4 had a maximum imbibition capacity of 116.7% and 77.1%, respectively. In the conventional spontaneous imbibition experiment of sample S7, the sample S7 quickly reached a state of equilibrium and had an imbibition capacity of 158.6% at an imbibition time of 1 d. In contrast, under the conditions of reservoir temperature and pressure, the other six samples did not reach an equilibrium state after quite a long time into the experiment. Clearly, at the early stage of shut-in the fracturing fluid would imbibe forcibly into the reservoir quickly, providing imbibition fluid for subsequent spontaneous imbibition to the shale matrix pores and natural fractures and further stimulating the shale reservoir through hydration.

Figure 8 shows the relationship curves of imbibition quantity per unit area and square root of imbibition time for the six samples. At an imbibition time of 5 d, the six samples under reservoir temperature and pressure conditions had an imbibition quantity per area range between 0.152 cm/h^0.5 and 0.564 cm/h^0.5 and a maximum imbibition quantity per area of 0.564 cm/h^0.5, showing strong imbibition capacity. The curve slope denotes the forcible imbibition rate, which reflects the fluid flowability in the pore throat. After 3 d to 5 d of imbibition the curves decreas very slowly in slope, showing that the samples were in a dynamic equilibrium state. The fluctuation of the imbibition curve slope indicates that there were new seepage channel(s) appearing, and consequently, the imbibition quantity and rate increased noticeably. The curves of some samples have multiple fluctuations in slope, indicating continuous modification of the shale reservoir by imbibition. The point at which the slope changes is defined as the generation point of microfracture (Figure 8). Take sample S4 as an example: after 3 to 4 days of imbibition this sample had an imbibition quantity per unit area increasing from 0.412 cm/h^0.5 to 0.553 cm/h^0.5 and an imbibition capacity increasing from 120.6% to 160.5% (a 33.1% increase). It can be seen that hydration imbibition under reservoir temperature and pressure can give rise to microfractures in the shale, improving the imbibition capacity of the reservoir.

The results of the mineral composition of rock samples (Figure 2) indicate that the rock samples of the Longmaxi Formation contain 10.2−46.7% clay minerals, 21.2−62.1% quartz, and 4.3−37.2% carbonate minerals, and the rock samples have large differences in mineral composition content with strong non-homogeneity. The results of water imbibition experiments indicate that rock samples S4 and S5 (with high clay content) have strong imbibition capacity, S3 rock samples do not have poor imbibition capacity because of the high quartz content, and S1 rock samples (with the lowest total clay mineral and quartz content) show the worst imbibition capacity overall. Clay mineral content is the main control factor affecting the imbibition capacity characteristics of shale, whereas quartz content also has a non-negligible influence on the imbibition capacity characteristics of shale. Therefore, the imbibition capacity of shale is a combination of clay mineral and quartz content, which is consistent with Zhao’s study [29].

3.2. Different Fluid Pressures

To investigate the influence of fluid pressure in the fractures after volume fracturing on the imbibition law, we conducted forcible imbibition experiments at different fluid pressures under the reservoir temperature and pressures shown in Table 2. Overall, the results show that the fluid pressure has a positive correlation with the imbibition capacity. At the initial stage of shut-in, the samples increased significantly in imbibition capacity due to the effect of fluid pressure. For example, sample S3 under a loading pressure of 3 MPa reached an imbibition capacity of 70.11% after 3 days of forcible imbibition. Figure 9b shows the curves of forcible imbibition capacity per unit area with square root of time of the samples under different fluid pressures. The slopes of the curves reflect the imbibition rates, and the curves show a variation pattern similar to the imbibition capacity curves of the samples.

3.3. Different Confining Pressures

The test results of imbibition capacity and imbibition quantity per area under different confining pressures are shown in Figure 10. With an increase in confining pressure the imbibition capacity and imbibition rate drop sharply, and the imbibition capacities and imbibition rates under different confining pressures differ widely. The reason for this is that the increase in confining pressure inhibits the generation of induced fractures, whereas induced microfractures are easier to created under low confining pressure conditions. For example, the imbibition curve of sample S4 had a sharp fluctuation point in the slope 2 days into imbibition, whereas sample S6 had a smaller fluctuation in the slope 4 days into imbibition (Figure 10b), which indicates that with an increase in confining pressure, the imbibition rate drops considerably and it takes much more time to induce microfracture by hydration.

4. Imbibition Models of Multiple Types of Pores

4.1. Physical Model

The shale samples from the Longmaxi Formation in southern Sichuan were observed under a field emission electron microscope (FESEM) at different magnifications to analyze their microscopic pore structures; over 300 representative pictures were obtained, covering shale mineral forms and contact relations, organic matter forms, and mineral composition, etc. The minerals are divided into brittle minerals, clay minerals, and organic matter according to their origin (Figure 11).

The brittle minerals of Longmaxi shale samples are mainly composed of quartz, feldspar, and carbonate (Figure 11a). Based on the microscopic pore structures of the shale samples, the pores in the shale samples from the Longmaxi Formation in southern Sichuan are divided into two categories: non-clay pore and clay pore. The non-clay pores include brittle mineral pores and organic matter pores. Zeng et al. established a petrophysical characterization model for brittle mineral pores, clay pores, and organic matter pores based on the occurrence components of shale pores (Figure 12) [16].

4.2. Model Establishment

(1)

Imbibition model for non-clay pore

The model assumes that only single-phase flow meets laminar flow, the fluid is incompressible, and the capillary is simplified to an eclipse tube.

According to the form characteristics of the brittle mineral pores and organic matter pores in shale, the eclipse was adopted to characterize the capillary form. The physical flow model of monophasic fluid in the eclipse tube is shown in Figure 13.

For the laminar flow of incompressible viscous fluid inside the eclipse tube, the Navier–Stokes equation is simplified as:

$$\frac{{\partial }^{2}u}{{\partial }^{2}y}+\frac{{\partial }^{2}u}{{\partial }^{2}z}=\frac{1}{\mu }\frac{dp}{dx}$$

(4)

As the flow velocity is the fastest and the flow velocity gradient is zero in the center of the eclipse, then:

$$\{\begin{array}{l}y=0,z=0,\frac{\partial u}{\partial y}=0\\ y=0,z=0,\frac{\partial u}{\partial z}=0\end{array}$$

(5)

The flow velocity is zero at the eclipse tube wall, so the border condition is:

$$\{\begin{array}{l}y=\pm 0.5b,z=0,u=0\\ z=\pm 0.5a,y=0,u=0\end{array}$$

(6)

Based on the eclipse polar coordinate transformation and trigonometric integrals, the flow equation can be obtained [26]:

$$q=\frac{\pi a^3{b}^3}{64\mu (a^2+b^2)}\frac{\mathrm{\Delta }p}{L}$$

(7)

where $a$ and $b$ are the long and short axes of the elliptical tube, m, respectively; $\mathrm{\Delta }p$ is the imbibition dynamics, MPa; and $L$ is the depth of imbibition, cm.

In light of the shape features of the brittle mineral pores and organic pores of the shale, the dynamics of imbibition mainly consists of capillary force. At the same time, the capillary tube is simplified as an eclipse tube, and the corresponding capillary force model of imbibition dynamics is as below:

$$p_c=2\sigma \cos \theta (\frac{1}{a}+\frac{1}{b})$$

(8)

where $\theta$ is the wetting contact angle of water phase, °; $\sigma$ is the surface tension, mN/m; and $\epsilon$ is the number of ions after solute ionization, dimensionless.

The partial differential relation of fluid flow in the eclipse tube and time is:

$$q=\frac{\pi ab}{4}\frac{dL}{dt}$$

(9)

The imbibition depth model of a single eclipse tube of a non-clay pore is obtained based on the eclipse tube flow equation (Equation (7)), capillary force model of eclipse tube (Equation (8)), and partial differential relation of fluid flow tube and time (Equation (9)):

$$L=\frac{\sqrt{ad(a+b)\sigma \cos \theta }}{2\mu (a^2+b^2)}\sqrt{t}$$

(10)

(2)

Imbibition model for clay pore

The model assumes that the clay pores are parallel flat capillaries and that the single-phase fluid is incompressible.

Based on the shape features of the clay pores, the clay pore was described as a parallel flat capillary and the fluid was uncompressible and vicious; therefore, the Navier–Stokes equation was simplified as:

$$\mu \frac{d^{2}u}{d{y}^2}=\frac{dp}{dx}=-\frac{\mathrm{\Delta }p}{L}$$

(11)

Assuming that there is no glide, the flow velocity is zero at the boundaries of the top and bottom border and the boundary condition is:

$$\{\begin{array}{l}y=0,u=0\\ y=w,u=0\end{array}$$

(12)

The equation of fluid flow in the parallel flat capillary is:

$$q=\frac{B{w}^3}{12\mu }\frac{\mathrm{\Delta }p}{L}$$

(13)

where $w$ is the width of clay pore fracture (length of short axis), m; and $B$ is a constant term, dimensionless.

The shale formation water is generally high in salinity. Statistics show flowback fluids from part of shale gas wells in southern Sichuan have a salinity range between 23,450 and 35,840 mg/L. Slick water, currently commonly used in the fracturing of shale reservoirs, has a lower salinity of about 500 to 1000 mg/L. Meanwhile, shale reservoirs in the study area have higher clay contents and clay minerals have the characteristic of a semi-permeable diaphragm, which, together with the big concentration difference between the formation water with high salinity and the fracturing fluid with low salinity, results in high osmotic pressure that facilitates imbibition of the fracturing fluid toward the reservoir matrix and natural fractures. In practical application, the van ‘t Hoff equation is usually used to calculate the osmotic pressure [26]:

$$p_{\pi }=\epsilon E_{\pi }RT(C_{sh}-C_f)$$

(14)

where $E_{\pi }$ is the efficiency of semi-permeable film, which characterizes the ratio between the actual pressure difference and the ideal osmotic pressure, 0–1; $R$ is the gas constant, the value is 0.00008206 (L·Pa)/(mol·K); $T$ is the formation temperature, K; and $C_{sh}$ and $C_f$ are the molar concentrations of the formation water phase and fracturing fluid water phase, respectively, mol/L.

As the clay pore is subjected to the combined action of capillary force and osmotic pressure, its imbibition dynamics model is:

$$\mathrm{\Delta }p=\frac{2\sigma \cos \theta }{w}+\epsilon E_{\pi }RT(C_{sh}-C_f)$$

(15)

Based on flow equation (Equation (13)) and imbibition dynamics model (Equation (12)) of the parallel flat capillary of clay, the imbibition depth model of a single eclipse clay capillary is obtained:

$$L=\sqrt{\frac{\sigma w\cos \theta }{3\mu }+\frac{\epsilon E_{\pi }RT(C_{sh}-C_f)w^2}{6\mu }}\sqrt{t}$$

(16)

4.3. Model Verification

When the long and short axes of the eclipse are equal (a = b), the imbibition model of non-clay pore (Equation (10)) degenerates into the classical LW imbibition model, as shown in Equation (17). The validity of the flow model (13) and imbibition dynamics model (15) of clay pore has been demonstrated in the relevant literature. The deduction process of the non-clay pore imbibition model, completely similar to that of clay pore, is rigorous theoretically, so the non-clay pore imbibition model is correct theoretically. Therefore, the models of multiple types of pores established in this study have certain accuracy and reliability and can be used to predict the imbibition depths of different types of pores quantitatively.

$$L=\sqrt{\frac{d\sigma \cos \theta }{\mu }}\sqrt{t}$$

(17)

The shale samples were subjected to imbibition experiments and the experimental results were compared with the calculated results of the theoretical model, which indicated that the multiple types of pores model established in this study was in good agreement with the experimental results and proved the usability of the theoretical model (Figure 14).

4.4. Analysis of Results

The viscosity, molar concentration, and other parameters of the liquid of the Longmaxi group were tested using measuring instruments such as a fluid viscometer, the contact angles of advance and retreat were measured based on the seat-drop method, and the surface tension after equilibrium was measured using an optical tensiometer (

Table 3. Basic parameters of shale samples and testing fluid.

Parameter	Value
	Organic Matter Pore	Brittle Mineral Pore	Clay Pore
Long axis a (nm)	89.2	163.3	81.7
Short axis b (nm)	46.7	61.7	22.4
Wetting contact angle of water phase θ (°)	80.4	16.2	11.5
Viscosity of imbibition fluid μ (mPa·s)	1
Surface tension σ (mN/m)	74.1
Semi-permeable film efficiency $E_{\pi }$ (dimensionless)	0.5775
Mole concentration of formation water phase Csh (mol/L)	0.525
Mole concentration of fracturing fluid water phase Cf (mol/L)	0.017
Gas constant R [L·Pa/(mol·K)]	8314.5
Experimental temperature T (K)	293

). We calculated the imbibition depths of different types of pores with Equations (7) and (13) to evaluate the contribution of different types of pores to the imbibition capacity. It can be seen from Figure 15 that different types of pores differ widely in their imbibition depth. The imbibition depths of the clay pore, brittle pore, and organic pore at an imbibition time of 96 min calculated by the analytical models proposed in this paper are 2.02 cm, 0.68 cm, and 0.16 cm, respectively. The imbibition depth of clay pore is much greater than that of brittle mineral pore and organic matter pore. Through analysis, this is closely related to the different sizes, imbibition dynamic forces, and wettabilities of these types of pores.

5. Inspiration of Imbibition to Engineering

In order to analyze whether water imbibition can improve the flowing capacity of the shale reservoir, the porosities and permeabilities before and after imbibition of some of the shale samples were compared to evaluate the flowing capacity of the shale samples after imbibition and answer the question as to whether the shale gas well needs to be shut-in objectively. Figure 16 shows the physical parameters of some of the shale samples before and after forcible imbibition under the formation temperature and pressure. It can be seen that all the samples have a significant increase in these physical parameters. At an imbibition time of 5 days, the three samples increased by 0.42 to 1.63 times in porosity and 17.6 to 67.3 times in permeability. S4 has the most clay mineral content and S6 has the least clay mineral content. The porosity and permeability of the three groups of rock samples did not differ much before water imbibition. However, with water imbibition in the shale, the porosity and permeability increased to different degrees, and the increase had a positive correlation with the clay mineral content. The composition of the samples was tested with an XPert PRO X-ray diffractor and the results are shown in Figure 3. The key factors controlling the hydration of shale are the contents and types of clay minerals, which directly affect the hydration capacity of the shale. The higher the clay content is, the stronger the hydration ability is. The test results of the shale sample from the Longmaxi Formation of the Sichuan Basin indicate that the microcracks induced by shale hydration are mainly lamellation with obvious directionality, as shown in Figure 15. The clay contents of the shale samples S4, S5, and S6 are 52.2%, 40%, and 35%, respectively. The increases in porosity and permeability are positively correlated with the clay content.

It can also be seen from Figure 17 that sample S4 had several induced macroscopic microfractures, mainly parallel bedding fractures, developing on the surface after the experiment. Therefore, increasing imbibition time under reservoir conditions can improve the reservoir flowing capacity and shut-in of shale gas wells after volume fracturing can improve the effect of reservoir stimulation. The mineral composition of the induced microfracture areas in the picture was tested. The results shows that clay minerals dominate, with an average content of 50.1%, which indicates that the microfractures induced by clay hydration play a leading role in the expansion of fractures. The induced microfractures can provide new channels for imbibition and improve the physical properties and imbibition capacity of the shale further. Tao et al. drew the same conclusions from the imbibition and hydration experiments on Longmaxi Formation shale cores from southern Sichuan province [30].

We investigated the relationship between the initial fracturing time and imbibition capacity of some of the shale samples based on the calculation method of microfracture initiation time of shale due to imbibition (Figure 18). The initiation time is in negative correlation with imbibition capacity, the higher the imbibition capacity, the earlier the fracture initiation time will be, and the microfractures further enhance the imbibition capacity and hydration. The fracture initiation time of the samples is between 60.2 and 118.5 h and is 98.5 h on average. Therefore, according to the experimental results, the shortest shut-in time of shale gas wells after volume fracturing is 5 days.

6. Conclusions

In this study, a new experimental device for spontaneous imbibition of fracturing fluid under the conditions of formation temperature and confining pressure was designed. A set of experimental apparatus and corresponding test and evaluation methods were developed to analyze the laws of forcible imbibition of water in shale reservoirs. Clay minerals are the main controlling factors of shale hydration. We reach the following conclusions through the study:

• According to imbibition saturation, the shale forcible imbibition of Longmaxi Formation shale reservoirs in southern Sichuan can be divided into three periods, imbibition diffusion, imbibition transition, and imbibition balance.
• The contribution degree of different pore water imbibition depths of shale is clay pore, brittle mineral pore, and organic pore in turn. The mineral composition of the induced microfracture areas was tested. The results show that clay minerals dominate, with an average content of 50.1%, which indicates that clay minerals are the main controlling factors of microfractures induced by shale hydration.
• The reservoir confining pressure has a prohibitive effect on shale imbibition but even under reservoir confining pressure, imbibition can still improve the fracturing effect of the reservoir, resulting in an increase in porosity of 0.42–1.63 times and an increase in permeability of 17.6–67.3 times. Under the experimental conditions, the initiation time of induced microfractures is 98.5 h on average and is in negative correlation with imbibition capacity. On this basis, the optimized shortest shut-in time of shale gas wells is 5 days.
• In this study, the pore network development characteristics and connectivity were not considered. The pore network characteristics can be subsequently characterized by CT scan to study the effect of pore network characteristics on the water imbibition effect of shale rock samples.