A new model for predicting irreducible water saturation in tight gas reservoirs

The irreducible water saturation (Swir) is a significant parameter for relative permeability prediction and initial hydrocarbon reserves estimation. However, the complex pore structures of the tight rocks and multiple factors of the formation conditions make the parameter difficult to be accurately predicted by the conventional methods in tight gas reservoirs. In this study, a new model was derived to calculate Swir based on the capillary model and the fractal theory. The model incorporated different types of immobile water and considered the stress effect. The dead or stationary water (DSW) was considered in this model, which described the phenomena of water trapped in the dead-end pores due to detour flow and complex pore structures. The water film, stress effect and formation temperature were also considered in the proposed model. The results calculated by the proposed model are in a good agreement with the experimental data. This proves that for tight sandstone gas reservoirs the Swir calculated from the new model is more accurate. The irreducible water saturation calculated from the new model reveals that Swir is controlled by the critical capillary radius, DSW coefficient, effective stress and formation temperature.


Introduction
The irreducible water saturation (S wir ) is one of the key parameters of gas-water two-phase flow in tight gas reservoirs. It plays an important role in reservoir evaluation, productivity prediction and hydrocarbon reserve estimation in tight sandstone gas reservoirs. S wir is the ratio of the volume of immobile water adsorbed on the rock surface, remained in micropores and trapped in the dead-end pores to the total pore volume of the reservoir rocks (Cheng et al. 2017).
At present, S wir can be obtained from the nuclear magnetic resonance (NMR) experiment (Borgia et al. 1996), weighing method (Coskun and Wardlaw 1995;Taktak et al. 2011), mercury injection capillary pressure method (Bear et al. 2011;Zhang et al. 2017) and semi-permeable diaphragm capillary pressure method (Coskun and Wardlaw 1995;Taktak et al. 2011). However, these methods have some disadvantages. The applied interpretation method of NMR data is to split the signal into a fast relaxing part and a slow relaxing part. The first is interpreted as bound water, and the later as free fluid (Looyestijn 2001). Generally, T 2cutoff is the split between the movable fluid and the immovable fluid in the NMR T 2 distributions. The T 2cutoff values obtained by different ways will lead to different calculation results of S wir in NMR experiments. It is very difficult to choose a suitable centrifugal force for low-permeability tight core samples in the weighing method. Similarly, the appropriate choice of the lower limit of capillary radius is the key to the success in mercury injection capillary pressure experiment. The test time of semi-permeable diaphragm experiment is too long. In addition, due to the limited bearing capacity of Edited by Yan-Hua Sun the separator, this method cannot obtain a complete capillary pressure curve. Pore network modeling and other methods can also be used to research the S wir (Goral et al. 2018;Zhu et al. 2019); however, none of the above experimental methods can obtain S wir under reservoir conditions. Previous studies suggested that the irreducible water was simply considered as water in sub-capillary pores and water film in theoretic models. The water in sub-capillary pores is unable to flow because the displacement pressure cannot overcome the capillary force. Assuming the rock surface is hydrophilic, the water film (WF) is the water trapped on the pore wall by molecular forces. The thickness of the water film is not constant which is mainly affected by the radius of pore throat, fluid properties, pressure difference, temperature and other conditions . In actual tight gas reservoir conditions, the structure of the porous media is complex, especially in the heterogeneous reservoir. Civan pointed out that fluids in some disconnected pores including small corners and dead-end pores may be trapped in dead or stationary fluid regions (Civan 2001). Therefore, some immovable water called the dead or stationary water (DSW) remained in the special pores and it cannot be able to flow under flow conditions. A schematic figure of the capillary model of flow in the porous media considering the effects of dead-end pores is presented in Fig. 1. The dead-end pore system is completely filled with immobile water due to detour flow effect and special pore structure. This part of the immobile water cannot flow due to it is located in disconnected pores, and there no channel to allow it flow; however, it can be squeezed less under the effective stress condition. At present, there is no model available which incorporates the impact of special immobile water of DSW.
A new S wir model was established by deriving tube capillary model based on the fractal theory under stress effect and considering the influence of dead or stationary water, water film, pressure gradient and temperature. This proposed model is suitable for tight rocks for achieving S wir accurately under real reservoir conditions.

The new model
In order to obtain the new irreducible water saturation model for tight sandstone gas reservoirs, the following methods are adopted. First, the fractal scaling theory is modified by considering stress sensitivity in the capillary model. Second, a new capillary model in a tight gas reservoir is established by considering DSW, water film, pressure gradient and temperature. Third, the impacts of stress dependency are integrated into the model to simulate the change of effective stress in the actual tight sandstone gas reservoir. Figure 2a manifests that the flow paths in the reservoir are composed by a bunch of capillaries with different crosssectional areas, and the size, length and tortuosity satisfy the statistically fractal scaling characteristics. The capillary diameter is equivalent and smooth in the conventional bundle models; however, there are many dead-end pores in the actual porous medium. In order to describe the characteristics of porous media, the small cones are used to describe dead-end pores on the capillary tubes, as shown in Fig. 2b. In addition, the mechanism of capillary length variation under different effective stresses has been taken into the new model.
For the simultaneous gas-water flow, as shown in Fig. 2c, there is a gas flow region with a radius of r g , a movable water region with a thickness of r − r g and an immovable water region including water film and DSW. Generally, there is a critical radius r c for the capillary bundle which is determined by the capillary pressure. While r s > r c , the flow pattern is as shown in Fig. 2c. On the contrary r s ≤ r c , the capillary tubes are fully occupied by wetting water. Therefore, there is a critical water saturation under a certain displacement pressure.
In Fig. 2, L 0 is the representative length of a capillary, m; P o is the overburden pressure, MPa; r 2 is the external radius of a certain pore, m; r s is the inner radius of a certain pore under effective stress, m; r g is the radius of the pores occupied by the gas, m; r δ is the radius of the pores without water film, m.
Assuming the wall of the capillary is hydrophilic, the gas would occupy the center of the tube and the water would stick with the wall. The water in small cones cannot be moved because of losing momentum. The gravity is neglected during the process of gas and water flow in tubes simultaneously (Feng and Serizawa 1999;Israelachvili 2015;Kawahara et al. 2002;Wu et al. 2014), as illustrated in Fig. 2c. The water in sub-capillary pores, dead or stationary water and water film constitute the irreducible water in porous media.  The first part of the immobile water is the water trapped in the micro-capillaries because of insufficient displacement force. Ignoring the change of interfacial tension and rock wettability, the critical capillary radius r c can be obtained as (Cheng et al. 2017): (1) r c = 2 cos Δp where r c is the critical radius, m; σ is the gas-water interfacial tension, N/m; Δp is the drawdown pressure, MPa; θ is the wetting angle, degree. The dead water or stationary water is the second part of the immobile water phase. It is the phenomenon that dead-end pores are filled with water and the water cannot be removed due to flow around effect and complex pore structure (He and Hua 2005;Melrose 1990;Li et al. 2008). In this study, we assumed that multiple small cones are distributed on a single capillary and the volume of total cones and the volume of the capillary have a significantly where V cone is the volume of total cones, m 3 ; V tube is the volume of capillary, m 3 . We also defined η as the DSW coefficient, which is the ratio of the cone volume to the capillary volume. Considering the impact of the volume of cones, the porosity of the capillary model based on the fractal theory needs to be modified. Hence, the relationship between the porosity under real conditions and the porosity in the fractal scaling model is shown as follows: Based on Eq.
(3) will be further modified as: where ϕ is porosity of the capillary; ϕ real is the real porosity, and it can be tested in the laboratory. It is noted that the decrease in pore pressure can induce the change of effective stress during the development process of a gas reservoir (David et al. 1994). The radius r of the capillary after deforming can be rewritten as: (Gangi 1978;Lei et al. 2018a) where r s is the inner radius of a certain pore under effective stress, m; r is the inner radius of a certain pore, m; P eff is the effective stress, P eff = P o − p , MPa; P o is the overburden pressure, MPa; p is the pore pressure, MPa; E is the rock elastic modulus, MPa; β is the power law index which is affected by the pore structure, dimensionless; α is the effective stress constant, dimensionless; ν is Poisson's ratio, dimensionless.
The change of a single capillary in porous media will induce the change of fractal parameters and the calculation of fractal parameter considering stress sensitivity are shown in "Appendixes 1 and 2." The stress may affect the radius and length of the capillary tube and the cone shape on the capillary tube. We assumed that the volume changes of the capillary tube and cones are consistent in porous media. Thus, the DSW coefficient is a constant and it can be expressed as follows: where V ′ cone is the volume of the total cones considering the effective stress, m 3 ; V ′ tube is the volume of the capillary considering the effective stress, m 3 .
The water film is the third part of the immobile water in the capillaries. It is assumed to adhere to the macropore wall due to the molecular forces making the water molecules bounded on the wall. Previous studies have confirmed that the thickness of the water film is related to the pressure gradient (Mo et al. 2015;Tian et al. 2014Tian et al. , 2015Yu and Cheng 2002;Yu and Li 2001). Based on the micro-tube experiment, Tian et al. (2014) presented the relationship between the pressure gradient, water viscosity and the thickness of the water film as: where δ is the thickness of the water film, μm; ∇p = Δp/L 0 is the pressure gradient, MPa/m; and μ w is the water viscosity, mPa s. The water viscosity is mainly affected by the temperature, and it can be expressed as (Cheng et al. 2017;Li 2006): where T is the formation temperature, °C.
Based on the above assumption of the gas-water distribution patterns, the volume of the immobile water and the total pores are defined as: where V wir is the volume of the immobile water, m 3 ; V total is the volume of the total pores, m 3 ; N is the number of pores in a unit cell; L s is the tortuous flow path under effective stress, m; f s is the probability density function for pore size-distribution under effective stress; r cs is the critical pore radius under effective stress, m; r maxs is the maximum pore radius under effective stress, m; r mins is the minimum pore radius under effective stress, m; By the definition of the irreducible water saturation, we can obtain the irreducible water saturation S wir : Inserting Eqs. (9) and (10) into Eq. (11) yields: Based on the "Appendixes 1 and 2," Eq. (12) can be simplified as follows: where D f ′ is the stress-dependent fractal dimension, D T ′ is the stress-dependent fractal dimension for tortuosity.

Model validation
A total of 10 low permeability sandstone samples were selected from the Xujiahe Formation in the central Sichuan Basin, and the basic parameters of the wax-sealed core samples (Cheng et al. 2017) are shown in Table 1.
First, the S wir values of core samples were calculated from the Cheng et al. model under reservoir conditions in which the formation temperature was determined by geothermal gradient formula (Xu et al. 2011) and the critical capillary radius was taken as 0.1 μm. Then, in order to calculate S wir values of 10 core samples using the new proposed model in this paper, the DSW coefficient was in the range from 0.01 to 0.02 and the other parameters were consistent with the Cheng et al. model. The relationship curve between the mean absolute error and the DSW coefficient is shown in Fig. 3. It was obvious that the mean absolute error was smallest with = 0.0135 . Finally, the values of S wir calculated by these two models are contrasted with the measured vales of wax-sealed core samples, and the detailed parameters are presented in  Tables 3 and 4. The irreducible water saturation tests were conducted on all cores using the continuous gas drive method at normal temperature and low pressure (2.07 MPa). The experimental apparatus is composed of a pressure control system, a core holding system and a measurement system, as shown in Fig. 5. The injection fluids are synthesized formation water (  Fig. 3 The relationship between the absolute error and the DSW coefficient

Fig. 4 The irreducible water saturation calculated by different methods
In order to verify the reliability of the model proposed in this paper, different models were used to calculate the irreducible water saturation and the predicted results are shown in Fig. 7. The Ofagbohunmi et al. model (2012) (Ofagbohunmi et al. 2012) is an empirical relationship between the S wir and the normalized porosity during compaction. In Li et al. model (Li et al. 2010), the irreducible water saturation simulations were conducted on tight sandstone samples with the initial porosity (ϕ) of 6.17%, initial permeability (K) of 0.0951 × 10 −3 μm 2 and the initial  Fig. 5 The flow diagram of the apparatus for irreducible water saturation measurements   Fig. 7 The comparison between the newly developed model and other models Li et al. 2010;Ofagbohunmi et al. 2012) irreducible water saturation of 9.8%. In Lei et al. model (2015)  , the initial porosity ϕ = 6.17%, the Euclidean dimension d = 2, the initial irreducible water saturation S wc0 = 9.8%, rock Poisson's ratio ν = 0.25, and the rock elastic modulus E = 1.05 × 10 3 MPa. In addition, the initial maximum and minimum radii, r max and r min , were 1.836 and 0.016 μm, respectively, which can be consistent with the initial permeability K = 0.0951 × 10 −3 μm 2 .
The calculated results based on Eq. (13) and the relevant parameters applied are listed in Table 6. It can be observed from Fig. 7 that the S wir increases with an increase in the effective stress and the new model predictions are in good consistency with other models.

Results and discussion
It can be observed from Figs. 8 and 9 that the S wir increases with an increase in the effective stress and the changing trend is consistent with the result of model validation. The reason is that the pore radius is changed, which is caused by the stress effect. A further explanation is that the increase in the effective stress means the cross-sectional area of capillary bundles decreasing and results in the radius of more capillaries less than the critical capillary radius. It can also be observed from Fig. 8 that the higher S wir corresponds to the lower rock elastic modulus.  The reason for this result is that the lower rock elastic modulus is, the lower resistance to contract capillary bundles and the easier changing the rock microscopic pore structure. Figure 9 also demonstrates that the S wir is higher when Poisson's ratio is higher. The phenomenon can be explained that the higher Poisson's ratio is, the higher elongation of capillary bundles and more pore space contacting with water. However, the changing range of S wir with Poisson's ratio is not very large. The transverse strain of porous media is very small in actual reservoirs, so the proposed model is in agreement with the actual physical situation.
The increasing extent of the S wir increases with the increase in the DSW coefficient, as shown in Fig. 10. It can be explained by that the more complicated the structure of porous media, the more dead-end pores are filled with immobile water, which results in the increase of S wir .
When the r max has a constant value of 1.68 μm, the ratio r min /r max is changed. (The result of S wir shown in Fig. 11.) At a certain maximum displacement pressure, the S wir increases with the decrease in the ratio r min /r rmax . The increase in S wir value indicates that when the r min becomes smaller, the pore radius decreases, and more immobile water will be adsorbed on the surface of porous media on account of the increase in the specific area of the porous media. In addition, the radius of more capillaries is less than the critical capillary radius with an increase in the ratio r min /r max . Therefore, the S wir will be increased. Figure 12 shows that the S wir decreases with the increase in the formation temperature if the maximum displacement pressure remains constant. Theoretical analysis shows that different wettabilities may result in different distributions of irreducible water on the rock surface (She and Sleep 1998;Zhang et al. 2016Zhang et al. , 2017. As the temperature increases, the water viscosity decreases and the water film thickness becomes thinner. The model proposed in this paper is based on the hydrophilic rock model. Therefore, S wir decreases with increasing temperature.

Conclusions
In this paper, a new theoretical model is derived for calculating S wir in tight gas reservoirs, which is based on the capillary model and fractal theory. This new model takes into account the effects of the DSW, water film, stress and the formation temperature. The accuracy and applicability of this new model for tight sandstone reservoirs are verified by comparing the results of previous studies and experimental data.
The theoretical study of S wir under reservoir conditions shows that the S wir is related to the effective stress, rock elastic modulus, pore radius, DSW coefficient, formation temperature and other microstructural parameters. Based on the proposed model, the parametric study shows that the S wir increases with the increases in the effective stress and Poisson's ratio of rocks and a decrease in the elastic modulus of rocks. For a given maximum displacement pressure in a hydrophilic reservoir, the lower ratio r min /r rmax , higher DSW coefficient, and lower formation temperature yield higher S wir .
The proposed model is developed considering the effects of DSW. However, the method for describing the effect of DSW is still idealized in this paper. The gas-water twophase fluid flow in a tight gas reservoir under the condition

Appendix 1: Fractal scaling theory
Previous studies have suggested that natural porous media have the fractal characteristics (Hansen and Skjeltorp 1988;Katz and Thompson 1985;Ozhovan 1993;Yu and Liu 2004); in addition, the microstructures and porous size distribution of porous media undoubtedly have fractal characters (Yu and Cheng 2002;Yu and Li 2001). Based on the fractal scaling theory, the cumulative pore size distribution can be reasonably expressed by the following equation (Majumdar and Bhushan 1990;Xu et al. 2013): where r and r max are the radius of a certain pore and the maximum pore, respectively; ξ is the length scale; N is the number of pores; D f is the fractal dimension for pore size distribution. For tight sandstones, the ratio r min /r max in porous media can be obtained through the following equation: which indicates that the tight sandstones truly have fractal characters. Fractal dimension can be determined by porosity ϕ, and the ratio r min /r max in the natural porous media (Yu and Li 2001): where ϕ is the porosity; d is the Euclidean dimension; r min is the minimum pore radius. In addition, the number of pores from the minimum size r min to the maximum size r max can be determined by the following equation: Generally, it is no doubt that the pores in the porous media are numerous in Eq. (4). After differentiating Eq. (4), the number of pores from the range r to r + dr can also be expressed as (Yu and Cheng 2002;Yu and Li 2001): where −dN > 0, which implies that the number of capillary bundles decreases with the increase in the capillary diameter.
Then, the probability density function for pore size distribution can be determined as (Yu and Cheng 2002): where f(r) is the probability density function for pore size-distribution in the rock.
Due to the tortuous nature of those capillaries, the actual length of that L > L 0 , in which L 0 is the representative length of a capillary (Xu et al. 2013). The length L(r) of a tortuous flow path is related to the capillary radius and the capillary length based on the self-similar fractal law (Bonnet et al. 2001): where D T is the tortuosity fractal dimension; L(r) is the tortuous flow path, m; L is the actual length of the capillary; and L 0 is the representative length of these capillary. The fractal dimension for tortuosity D T can be determined as follows (Yu 2005;Lei et al. 2015;Xu and Yu 2008;Zheng et al. 2013): where the average tortuosity is given by: where τ ave is the average tortuosity; r ave is the average pore radius. The total pore area on the cross section of the porous media, as suggested by Xu et al. (2013), can be obtained as: The representative length L 0 can be written as (Yu and Cheng 2002): where A is the cross-sectional area of the sample, m 2 .