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Effect of water imbibition on shale permeability and its influence on gas production Yinghao Shen, Hongkui Ge, Mianmo Meng, Zhenxue Jiang, and Xinyu Yang Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b00338 • Publication Date (Web): 05 Apr 2017 Downloaded from http://pubs.acs.org on April 9, 2017

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Effect of water imbibition on shale permeability and its influence on gas production Yinghao Shen a, b; Hongkui Gea; Mianmo Menga; Zhenxue Jianga;Xinyu Yanga a. China University of Petroleum, Beijing, 102249, China b. Texas Tech University, Texas, 79409. United States

ABSTRACT. Large volume water filtrates into the shale formation during multi-stage fracturing. The influence of water imbibition to shale permeability, which still needs further investigation, is crucial to gas production. This paper presents series of experiments to investigate the shale permeability change during water imbibition comparing with typical sandstone and volcanic samples using pulse-decay permeability technique. The permeability greatly fluctuates with the water imbibed into the shale, which is far different from that of sandstone and volcanic rock. Factors like water blocking, stress sensitivity and clay swelling are discussed. Five typical permeability changes are put forward, which are conducive to understand the effect of water imbibition on gas flow post- fracturing. The work indicates that the shale permeability enhancement by water imbibition may be one of the reasons getting shale gas well productivity increased by soaking after fracturing. The results and conclusions are fundamentally important to guide the industry practice, especially for the determination of soaking time after fracturing in a shale gas reservoir.

KEYWORDS. Shale; Permeability; Spontaneous imbibition; Hydraulic fracturing; Cracks

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1. INTRODUCTION Multi-stage horizontal wells with hydraulic fracturing have been widely used in developing tight reservoirs. Gas shale has been broadly acknowledged to have ultra-low porosity and permeability. For example, the typical porosity values of Mississippian Barnett Shale are 3–8% and permeability 0.06 to 1.23 micro-Darcy due to Loucks et al1. Large-scale hydraulic fracturing is an effective way to exploit shale formation2, 3. In view of the performance of hydraulic fracturing for most shale gas wells, a large amount of fracturing fluid (generally more than 50%), is retained in shale formation after flow-back4. The interaction between water and shale play an important role in gas production5-7. Thus, permeability decreases as water blocking the effective porosity and gas production is reduced at the start8, 9. However, shutting-in for a certain period (soaking time) after fracturing, production rate can be increased for a relevant portion of shale gas wells. The phenomenon which has gained extensive attention10-13. In previous studies, imbibition experiments have been widely used to investigate where the fracturing fluid goes14, 15. Some chemical materials were added into the fracturing fluid to reduce the filtration of fluid in different shale formations16, 17. A model has been proposed for hydraulic fracturing stimulation considering the osmotic and capillary effects. The model showed that micro-fracture generation is an important factor for high-gas production after shale reservoir shut-in for a certain period18, 19. Capillary affected water spontaneous imbibition of organic shale. The fractal geometry has been used to describe spontaneous imbibition20, 21. The Excess water spontaneous imbibition in shale was caused by the additional driving force from water adsorption by clay minerals and rock permeability increased by adsorption-induced microfractures22. Shale with a large amount of clay was prone to conducting micro-fractures and sample disintegration after sample imbibition. It also showed that samples with rich clays and

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micro-fractures have higher water adsorption and ion diffusion rate23. The previous study also suggested that the connected pore network of the intact samples is water-wet. However, majority of the rock, including poorly connected pores, is oil-wet24. Considering the permeability change during soaking may be one of important mechanisms soaking enhanced the productivity, this work aims to comprehend the effect of fluid filtration to shale permeability and subsequent influence to production. We used a pulse-decay permeability technique to examine the rock permeability change after certain imbibition time with one face contacting liquid, and two kinds of imbibition liquid have been used. We directly observed permeability change with spontaneous imbibition time. It can be found that the special characteristic of shale sample was closely related to three important factors, namely, stress sensitivity, water blocking, and crack expansion. The work provides a reference to determine the optimal soaking time. 2. SAMPLES AND EXPERIMENTAL PROCEDURES 2.1. Samples. The location of Longmaxi Shale Formation of Lower Silurian is in an extensive region with approximately 1.282×105 km2 in the Sichuan Basin. The thickness of these formations distributes from 60 m to 700 m. These formations usually are buried with a depth of 1500 m~4000 m25, 26. It is composed of a high proportion of quartz and low content of clay, which is the overall characters of Longmaxi Shale27. The porosity of the Longmaxi Formation in the Lower Silurian shale ranged from 3.65% to 18.26%. These rocks have bimodal pore-size distribution, which are composed of both micro-pores and nano-pores that vary from 30 µm to 60 µm and from 1.7 nm to 20 nm, respectively28. Clays are prone to swelling when rock contacts the liquid. High content quartz benefits the fragile of the formation, hence network fractures are

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produced easily and high-gas production has a greater likelihood of occurrence after hydraulic fracturing29. The parameters of the experiment samples are listed in Table 1. Shale samples were obtained from the Lower Silurian Longmaxi Formation in Chongqing, China; volcanic rock samples were taken from the Lower Cretaceous Yingcheng Formation in Jilin, China; sandstone samples were obtained from the Lower Cretaceous Denglouku Formation in Daqing, China. Sample pulsedecay permeability was determined under the conditions: temperature was 25 ºC; confining pressure was 8 MPa; pore pressure was 5 MPa. The different minerals of shale are listed as follow: illite is 23%; illite/smectite is 7%; chlorite is 12%; quartz is 44%; feldspar is 2%; calcite is 6%; dolomite is 3%; pyrite is 3%. The comparison of different samples are shown in Table 2. Table 1. Basic properties of rock samples Sample

Diameter (mm)

Length (mm)

Porosity (%)

Permeability (mD)

Sandstone S1

25.33

36.85

12.53

0.024

Sandstone S2

24.97

32.17

8.85

0.013

Sandstone S3

24.82

37.06

10.89

0.011

Volcanic rock L1

24.82

13.12

8.64

0.00068

Volcanic rock L2

24.84

23.79

6.37

0.00099

Volcanic rock L3

24.89

14.33

7.27

0.0014

Shale Y1

24.65

11.74

5.67

0.00074

Shale Y2

24.63

15.96

6.51

0.00063

Shale Y3

24.57

14.84

4.04

0.00055

Shale Y4

24.64

9.98

6.47

0.00065

Shale Y5

24.51

10.05

2.16

0.00050

Table 2. Mineral Concentration (wt %) of different samples determined by X-ray Diffraction Samples

Calcite

Quartz

Dolomite

pyrite

feldspar

Illite

Chlorite

illite/smectite

Sandstone

8

46

-

-

24

18

4

-

Volcanic rock

3

38

-

2

30

4

18

5

Shale

6

44

3

3

2

23

12

7

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Figure 1 shows shale backscattered electron images in which good connections are observed among organic matters. The pores in organic matter are well developed. Some inorganic pores also exist. Clay minerals are reflected by the red dash curve. Clay minerals are prone to adsorbing abundant liquid during hydraulic fracturing. They are likely to expand after water contacts these minerals. Many natural microcracks can also be observed.

Figure 1. Backscattered electron images of shale samples. The dark regions of the images are mainly organic matter; the regions surrounded by the red dash curve are mostly clay minerals. Some microcracks appear in the images. HV=accelerating voltage; det TLD=through the lens detector; WD=working distance; HFW=horizontal field width; mag=machine magnification; frame=time of image acquisition; Helios=Helios NanoLab 650. Figure 2 shows that shale sample has the behavior of strong stress sensitivity. The sample was continuously measured several times under a specific confining and pore pressure. Permeability was observed to decrease from 0.00038 mD to 0.00024 mD. The purpose of this experiment aims at showing the effect of stress on permeability with time, which is helpful to explain the factors influencing the permeability change with water imbibition.

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0.0004 Shale Permeability (mD)

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0.00035

Permeability

0.0003

0.00025

0.0002 0

2

4

6

8

10

12

Times(hours)

Figure 2. The permeability change of shale was continuously measured several times under a specific condition under constant confining pressure (8 MPa) and pore pressure (5 MPa). The experiment was conducted to investigate rock permeability stress sensitivity. 2.2. Pulse-decay permeability technique. The pulse-decay technique is one important method to determine shale permeability. It has been widely recognized because the permeability is too small to be accurately detected by a normal permeability apparatus permeability30-33. The schematic of the pulse-decay experimental apparatus is shown in Figure 3. The apparatus consists of an upstream reservoir of volume Vu and a downstream reservoir of volume Vd, and a cell that is capable of applying hydrostatic confining pressure pc and containing a cylindrical rock sample. The absolute pressure pd in the downstream reservoir is measured by a pressure transducer, and the pressure difference △p is determined by a differential pressure transducer. To control the initial differential pressure, a mini-osmotic valve 5 is used. The pressure differential can be plotted logarithmically versus time to obtain a straight line for late time data. Finally, sample permeability can be calculated according to the slope of a straight line.

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Figure 3. Schematic of pulse-decay permeability measurement. 2.3. Experimental Procedure. In the spontaneous imbibition experiment, distilled water and distilled water added with anionic surfactant (DWAS) were used as working liquids. The anionic surfactant is sodium dodecyl sulfate(SDS). Sandstones and volcanic rocks were used as comparison samples. Environmental temperature was 25 ºC. The sample contacts the liquid on one surface (Figure 4).

Figure 4. The schematic of the imbibition apparatus. Dry sample permeability and mass were measured as initial permeability and mass, respectively. Then, spontaneous imbibition was conducted on the rock. Rock permeability and mass were measured after sample imbibition at specific time intervals. The surface water of the

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sample was dried by adsorbent paper. The imbibition test and permeability test are carried out separately, hence the two tests’ time interval should as short as possible. By repeating the preceding steps, a series of permeability values versus imbibition liquid mass was obtained. The figure of the permeability versus time and the figure of imbibition water mass versus time were obtained in the same coordinate system. Here all the permeability are tested under the same boundary, with constant confining pressure (8 MPa) and pore pressure (5 MPa). This experiment simulates the permeability change of shale formation caused by fracturing fluid imbibition into the reservoir (Figure 5). In the process of hydraulic fracturing, hydraulic fractures were produced. Liquid filtrates into the formation, and permeability near the hydraulic fractures was affected significantly. With the fracturing fluid gradually enters into the matrix, the reservoir primitive permeability changed. The condition in our testing, where only one surface contacts the liquid, is similar to the environment of sample present in Figure 5.

Figure 5. Hydraulic fracturing fluid spontaneous imbibition during well shut-in. 3. RESULTS AND ANALYSIS 3.1. Pictures of samples before and after imbibition. Some of the experiment samples before and after imbibition are shown in Figure 6. Nearly no change was observed in sandstone and volcanic rock during the imbibition test, except for the color becoming darker. However, some cracks appeared on the surface of the shale samples after imbibition. A picture of the shale

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sample break can also be seen in the imbibition process because of crack expansion. This variation is consistent with the results of previous research34.

A1

B1

C1

A2

B2

C2

Figure 6. Pictures of samples before and after imbibition. A1, B1, and C1 represent sandstone, volcanic rock, and shale rocks before imbibition, respectively; A2, B2, and C2 represent sandstone, volcanic rock, and shale rocks after imbibition, respectively.

3.2. Sandstone sample imbibition with distilled water. The water is imbibed into sample under capillary pressure and osmosis pressure35, 36. The change in both permeability and water content with imbibition time in sandstones is shown in Figure 7. Permeability clearly decreases with imbibition time, while water content increases with time. This phenomenon can be attributed to the effect of capillary and clay swelling in the spontaneous imbibition process. Under the imbibition effect, incremental water blocked the effective flow channels; thus, permeability decreased. Besides, clay expansion for contacting the liquid also reduced the effective gas flow channels37, 38. Because no new gas flow channels were produced, gas permeability decreased again. Furthermore, stress sensitivity has certain effects on permeability reduction. The combined effect of these factors significantly damages sandstone permeability.

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0.024

0.09

0.015

0.3

Sandstone S1

Sandstone S2

0.012 Permeability Water content

0.008

0.03

0.01

0.2

Permeability Water content

0.005

0.1

Water content (g)

0.06

Permeability (mD)

0.016

Water content (g)

0.02 Permeability (mD)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.004 0

0 0

60

120 Time (min)

180

240

0

0 0

200 400 Time (min)

600

Figure 7. The change of permeability and water content with imbibition time in sandstones 3.3 Volcanic rocks imbibition with distilled water. Volcanic gas reservoir is one important tight gas reservoir in China39. The change in permeability and water content with imbibition time are shown in Figure 8. Water content clearly increases gradually. However, permeability decreases quickly in the early stages. The permeability of volcanic rocks reaches a constant value at about 100 min in the imbibition process. In the last period, permeability remains stable, while water content increases slowly. This phenomenon may be caused by the fluid entering into the pores that nearly have no contribution to rock permeability. Because nearly no new pores are developed during imbibition, the combined effect of water blocking, stress sensitivity and clay swelling is the reason of the decrease in permeability in volcanic rocks.

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0.0008

0.0012

0.3

0.2

Volcanic rock L1

Volcanic rock L2 0.24

0.16

Permeability

0.12

Water content

0.0008 0.12

0.08

Permeability 0.0004

Water content

Water content [g]

0.0004

Permeability (mD)

0.18

Water content (g)

0.0006 Permeability (mD)

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0.0002 0.06

0

0 0

100

200

300

0.04

0

0 0

Time (min)

100 200 Time [min]

300

Figure 8. Change of permeability and water content with imbibition time in volcanic rocks 3.4 Shale rocks imbibition with distilled water. The characteristics of shale rocks are very different with sandstones and volcanic rocks during imbibition. Water content of shale samples increases with imbibition time. However, permeability greatly fluctuates, as reflected in Figure 9. The trend of permeability change of shale Y1 has similar characteristics with the above two types of rocks in the early stage. This change may be due to the reduction of effective flow channels caused by water adsorption and stress sensitivity. However, an interval of tremendous fluctuation occurs between 130 min and 160 min. Within this period, permeability greatly increases. This behavior may be caused by the appearance of new effective cracks produced by clay swelling, and water has not occupied these new cracks simultaneously. Between the period of 160 min and 200 min, permeability greatly decreases. This behavior may be caused by stress sensitivity and water imbibition, which obviously reduces the effective gas flow channels.

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0.001

0.0008

0.1

0.09 Shale Y2

0.0006

0.06

0.0004

0.04

0.0006 0.06

0.0004 0.03

Water content (g)

0.08

Permeability (mD)

0.0008

Water content (g)

Permeability (mD)

Shale Y1

0.0002 0.02

Permeability Water content

0

Permeability Water content 0

0 0

100 200 Time (min)

0.0006

0 0

300

0.09

50

100 Time (min)

150

0.0008

0.1 Shale Y4

Shale Y3

0.08

0.06

0.0002

0.03

Permeability (mD)

0.0004

Water content (g)

0.0006

0.06 0.0004 0.04 0.0002

0.02 Permeability Water content

Permeability Water content 0

0 0

50

100

Water content (g)

0.0002

Permeability (mD)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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150

0

0 0

Time (min)

100 200 Time (min)

300

Figure 9. Change of permeability and water content with imbibition time in shale rocks. Shales Y2 and Y3 have similar characteristics. Water content continuously increases with imbibition time. However, permeability change has three different intervals. At the first stage, permeability significantly decreases probably because of the effect of the water that enters into the samples, which reduces effective gas flow channels. In the middle stage, permeability increases at a certain degree due to the appearance of new cracks caused by clay swelling. At the last stage, permeability gradually decreases, which may be due to stress sensitivity and water blocking. Shale Y4 has two intervals of permeability. Permeability linearly decreases at the first stage, which may be due to stress sensitivity and water imbibition into the effective gas channels. At

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approximately 140 min, a turning point occurs. Then, permeability increases almost linearly, which may be caused by the significant increase in new cracks. Finally, imbibition breaks the rock. Here we present the amount of water infiltration with time graphs for 3 types of rocks in the same plot (Figure 10). The ratio of imbibed volume to pore volume is presented to clearly show the water saturation change, ignoring the sample difference. It is shown that the ultra-imbibed volume are about in 20%~27%, where the shale sample has the highest ultra-imbibed volume, which may due to the higher capacity caused by the smaller pore diameter. 0.5 Sandstone S2

Imbibed volume(nPV)

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Volcanic rock L2

0.4

Shale Y1

0.3 0.2 0.1 0 0

100

200

300

400

500

600

Time (min)

Figure 10. Imbibed water volume (nPV) with imbibition time in different rocks. 3.5 Samples imbibition with water added with DWAS In Figure 11, three types of rocks are tested by using DWAS as imbibition liquid. The permeability change of sandstones with DWAS is very different with distilled water. Permeability linearly declines in the first period. In the last period, permeability becomes stable. The permeability change with DWAS has similar characteristics with distilled water in volcanic rocks. Shale Y3 was used in this experiment with anion surfactant after this sample went through the distilled water imbibition. So we can compare these two conditions. The permeability change of shale Y3 between distilled water and DWAS is similar. However, in the same imbibition time,

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such as approximately 150 min, the imbibition amount of DWAS (about 0.04 g) is far less than that of distilled water (about 0.085 g). However, the permeability of these two conditions has a value of 0.0003 mD at 150 min. The explanation for these different phenomena could be that the distilled water occupies the pores, which has no contribution to the rock permeability, is more than that of the DWAS. At the first stage, the permeability change of shale Y5 greatly fluctuates. After imbibition of 40 min, permeability change is slightly linear. At 150 min, permeability and liquid mass are approximately 0.035 mD and 0.064 g, respectively. Permeability decrease has a certain degree of correlation with stress sensitivity and water blocking. 0.0016

0.075

0.16 Volcanic rock L3

0.05 Permeability Liquid mass

0.004

0.025

0

Permeability [mD]

0.008

Liquid mass (g)

Permeability (mD)

Sandstone S3

50

100 150 Time (min)

200

0.12

0.0008

0.08

Permeability Liquid mass

0.0004

0.04

0

0 0

0.0012

0 0

250

Liquid mass (g)

0.012

40

80

120

160

Time [min]

0.0006

0.05 0.0008

Shale Y3

0.08

Shale Y5

0.02 0.0002 Permeability

0

0 50

100

0.06

0.0004

0.04

Permeability

0.0002

0.01

Liquid mass

0

0.0006

150

Liquid mass (g)

0.03

Permeability (mD)

0.0004

Liquid mass (g)

0.04 Permeability (mD)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.02

Liquid mass 0

0 0

Time [min]

50

100 Time [min]

150

Figure 11. Change of permeability and liquid mass with imbibition time for different rocks. The liquid used in this experiment is DWAS.

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4. DISCUSSION The permeability of sandstones and volcanic rocks decreases with imbibition time. However, the shale showed different behaviors in both distilled water and DWAS compared with the previous two kinds of samples. In most cases, permeability decreases with time at the first stage because of water blocking and stress sensitivity. Then, the middle stage shows an interval in permeability increase, which may be caused by incremental cracks induced by clay swelling. Subsequently, an interval of permeability reduction appears. This behavior should account for the combined action of stress sensitivity and water blocking. The sandstone and volcanic samples do have similar behaviour based on our samples. The difference between them is that the volcanic sample is that the pressure decrease is faster than that of sandstone. Permeability change of shale with DWAS had a similar behavior to distilled water. However, the liquid content adsorbed by the same shale rock was greatly different at the same imbibition time, while the permeability of these two conditions was almost similar. That could be that the invalid pores adsorbed by distilled water, which had no contribution to permeability, were more than those adsorbed by DWAS. The two shale samples tested with DWAS have a similar trend. Water blocking, stress sensitivity, and clay swelling are three important factors that influence the rock permeability change during imbibition. Water blocking and stress sensitivity generally exert damage on permeability. However, clay swelling has different effects on various types of rocks. For some natural crack abundant shales clay swelling can expand the natural cracks and enhance the connection. Thus, shale permeability increases significantly. Figure 12 lists five simplified type models related to rock permeability change with imbibition time based on series experiments in this work and our previous work40, 41. We classify these models mainly based on the capacity of crack expansion. These models contribute to the comprehension of rock permeability change with imbibition time. Using these models, we could

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deepen our understanding of fracturing fluid imbibition and its influence on shale formations as well as to determine the optimal soaking time after hydraulic fracturing. In future, high pressure and high temperature, fracturing fluids, shales from different regions could be considered to investigate the permeability change with imbibition more comprehensively. Type 1

This type of rock has the characteristic of permeability reduction until it reaches stability. No crack expansion appears in this type of rock; thus, it lacks the middle and last stages. Sandstone and volcanic rock belong to this type. If shale rock shows this behavior, the related reservoir may adopt fastback after fracturing. Type 2

This type of rock presents four stages. The middle stage is related to crack expansion. Shale rock with this type has the capacity for economic exploitation. The point that connects the middle and last stages is the optimal soaking time after fracturing.

Type 3

This type of rock has three stages. It has no first stage. This phenomenon may be due to the crack expansion that occurs rapidly at the start. Thus, the first stage is eliminated. The maximum permeability is usually larger than the original permeability. The largest permeability point is the optimal soaking time. Type 4

This type of rock has only two stages. At the end of the middle stage, the rock breaks. Therefore, it has no last stage and stable stage.

Type 5

This type of rock only has a middle stage. Permeability continuously increases until the sample breaks into several parts. This behavior occurs because the rock is full of natural cracks. When the induced cracks connect the natural cracks, permeability increases significantly.

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Figure 12. Types of permeability change with imbibition time. We classify permeability variation into five types and define four stages in the type. The first stage (combined effect of water blocking and stress sensitivity > effect of crack expansion); the middle stage (combined effect of water blocking and stress sensitivity < effect of crack expansion); the last stage (combined effect of water blocking and stress sensitivity > effect of crack expansion); the stable stage (imbibition stops and effect of different factors equilibrates). The solid line represents the permeability change of the special type and the dashed line represents the permeability change of the conventional rocks. 5. CONCLUSIONS This work studied the effect of water imbibition on permeability and the potential influence to gas production of Longmaxi shale in Chongqing, China through experiments on the water imbibition and permeability measurement. Water blocking, stress sensitivity and clay swelling were investigated. The sandstones and volcanic rocks were taken into comparison. The conclusions below can be drawn for the shale studied in this work. The thought, results and conclusions of this paper may provide guidance to other shales. (1) Clay swelling has different effects on various types of rocks. With sandstone and volcanic rock, clay swelling fills the pores and reduces the effective gas flow channels. Because the shale rock used is abundant of natural cracks, clay swelling stimulates the natural crack expansion and enhances the connection. Therefore, the effective gas flow channels increase. (2) The permeability change of sandstone and volcanic rock in this study is mainly controlled by three factors, namely, water blocking, stress sensitivity, and clay swelling. The combination of these three factors has an effect on damaging rock permeability.

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(3) The permeability change of shale rock is also mainly controlled by three factors, namely, water blocking, stress sensitivity and crack expansion. If the effect of crack expansion is less than the combined effect of the other two factors, permeability decreases. If not, permeability increases. (4) Five models are proposed to characterize permeability change with imbibition time for various types of rocks. The point that connects the middle and last stages could be the potential optimal soaking time. However, we should notice the laboratory data do not adequate since field conditions, workover expenses and field experiences are all required to be evaluated. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Phone: +1 (806)317-9429 Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (Grant No. 51604287, 51490652). REFERENCES (1) Loucks, R., Reed, R., Ruppel, S., Jarvie, D., Morphology, genesis, and distribution of nanometer-scale pores in siliceous mudstones of the Mississippian Barnett Shale. Journal of sedimentary research, 2009, 79(12), 848-861. DOI: 10.2110/jsr.2009.092. (2) Palisch, T. T.; Vincent, M.; Handren, P. J., Slickwater fracturing: food for thought. SPE Production & Operations 2010, 25 (03), 327-344. DOI: 10.2118/115766-PA.

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