# Experiments and Modeling on the Influence of Interfacial Tension on Imbibition Height of Low-Permeability Reservoir

Authors: Xiaoxia Ren; Aifen Li; Piyang Liu; Bingqing He
Journal: Geofluids
 (2020)

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## Abstract

Low-permeability reservoirs have tiny pores with winding and complicated pore throats. The oil recovery efficiency of low-permeability reservoirs can be enhanced through the displacement of reservoir oil through imbibition. In the present study, experiments were conducted to investigate variations in the imbibition height of hydrophilic and weakly-hydrophilic rock samples under different interfacial tensions. An imbibition model considering imbibition resistance and bending of pore throats was established based on fractal theory. According to the experimental results, variations in the imbibition height of low-permeability rock samples with time can be divided into three stages. In the first stage, the capillary force plays a dominant role, while the viscous force and gravity have very slight effects. The imbibition height first increases rapidly and then levels off to a constant rate. With the increase in interfacial tension, the imbibition rate in the first stage increases, the ultimate imbibition height increases initially and then decreases, and the contribution of the imbibition height in the first stage to the ultimate imbibition height becomes greater. There is an optimal interfacial tension that causes the ultimate imbibition height to reach its maximum. The calculated results obtained from the proposed imbibition model are consistent with the experimental results, indicating that the model can accurately reflect the change in the imbibition height in low-permeability reservoirs in the first stage.


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## Body

## 1. Introduction

Abundant oil and gas resources are present in low-permeability reservoir formations, which are widely distributed all over the world. Low-permeability reservoirs have small pore radii, few effective pores, and complicated pore throats with abundant microcracks [1]. It is difficult for traditional water-flooding to displace crude oil from capillary tubes in such a system [2]. Therefore, methods for the efficient development of low-permeability reservoirs have been widely studied in the petroleum industry [3].Research and practice indicate that the displacement of oil through imbibition can significantly enhance the recovery of fractured reservoirs [4]. Generally, imbibition refers to diffusion under the action of capillary force, as the wetting-phase fluid enters the pore throat of rock spontaneously and displaces the nonwetting-phase fluid in the pore [5]. Studying the imbibition characteristics of low-permeability reservoirs and understanding the mechanisms of imbibition displacement can provide an effective method to enhance the recovery of low-permeability reservoirs.According to current research findings, the interfacial tension of the fluid has a significant effect on the imbibition. Laboratory studies and field practices have yielded both positive and negative observations. The general principle is that surfactants trigger imbibition by either interfacial tension reduction or wettability alteration [6]. Standnes et al. [7] experimentally compared the oil recovery from oil-wet reservoir cores using different surfactant solutions and found that cationic surfactant C12TAB had high efficiency of imbibition. Alshehri et al. [8] found that a reduction in the interfacial tension can decrease the imbibition resistance of the oil-wet core and improve the imbibition effect. Sun et al. [9] reported that the reduction in interfacial tension could effectively decrease the work of adhesion, which is conducive to improving recovery through imbibition. Santanna et al. [10] performed static imbibition experiments and found that the imbibition rate and the oil recovery factor were higher when an ionic surfactant was used. It has been reported that the wettability change is the primary cause of improving the imbibition recovery rate using surfactants [11]. Babadagli [12] found that nonionic surfactant solution increased the recovery rate and ultimate recovery of heavy oil in water-wet sandstones. This effect was attributed to the change in wettability due to the addition of surfactant. Sun et al. [9] found that the imbibition effects of hydrophilic and lipophilic cores were both influenced by wettability, and the surfactant has a significant effect on the recovery of the lipophilic core through imbibition. Alvarez and Schechter [13] found that the addition of an appropriate surfactant can improve oil recovery by altering wettability and interfacial tension. Shen et al. [14] studied the surfactant solution imbibition in porous media with subnanometer and nanometer capillaries using a mechanical model, and the simulated calculation showed that the change in wettability contributes more to the imbibition recovery of tight oil than lower interfacial tension. Gao et al. [15] found that the imbibition degree of strong water-wet cores was large when water was used for imbibition. Liu et al. [16] found through experiments that the oil recovery through imbibition is improved as the rock wetness is shifted from oil-wet to water-wet. Saputra et al. [17] performed laboratory-scale imbibition experiments and found that oil recovery is enhanced by the addition of surfactant as a result of wettability alteration, and the reservoir properties have a significant effect on the results of surfactant-assisted spontaneous imbibition. Liu et al. [18] found that the addition of surfactant mainly recovered oils in the macropores, whose surfaces can be easily altered to be water-wet by surfactant, while it had less effect on the recovery of oil in the micropores. Kumar and Mandal [19] found that using surfactant solutions as the imbibing fluid can recover oils from sandstone and carbonate samples, mainly due to the alteration of wettability. However, Keijzer and de Vries [20] held that the addition of surfactants would result in extremely low interfacial tension. Consequently, the capillary force would not be sufficient to drive imbibition, thereby influencing the recovery through imbibition. Tang et al. [21] found that the injected surfactant did not improve the displacement effect through pulse imbibition.In order to analyze imbibition variations, researchers have also established different imbibition models, among which the Lucas-Washburn (LW) model, Terzaghi imbibition model, and Szekely model are frequently cited [22, 23]. The LW model (Equation (1)) can be used to calculate the relationship between imbibition height and imbibition time. However, in this model, the inertia force and gravity of the fluid during imbibition are ignored, thus affecting the accuracy in practical applications.(1)h=rσcosθ2ηt,where h denotes the imbibition height of the fluid in the capillary tubes, m; r denotes the radius of the capillary tube, with the unit of m; σ denotes the interfacial tension of fluid, N/m; θ denotes the wetting angle, degrees; t denotes imbibition time, s; and η denotes fluid viscosity, Pa·s.Zhmud et al. [24] found that the LW model was not suitable for describing the later stage of imbibition due to the influence of gravity and derived a long term model considering the gravitational acceleration:(2)h=2σcosθρgr1−e−ρ2g2r3/16ησcosθt,where g denotes the gravitational acceleration g (9.8 m/s2), ρ denotes the fluid density, with the unit of kg/m3.Based on the model proposed by Zhmud et al., Fries and Dreyer introduced the Lambert W function into the model and proposed a model with better accuracy [25]:(3)ht=ab1+W−e−1−b2/at,where a=σrcosθ/4η, and b=ρgr2/8η.These imbibition models based on LW model can directly calculate the imbibition height. In order to accurately describe the imbibition process, several studies have been performed to develop other kinds of improved models, which can predict the oil recovery rate or volume by imbibition. Cai et al. [26] derived analytical expressions for calculating the capillary rise of wetting liquid in a single tortuous capillary by introducing the tortuosity and fractal dimension for a tortuous capillary, respectively. After that, Cai et al. [27] discussed the effect of tortuosity on capillary imbibition in wet porous media based on the capillary model and fractal geometry and derived an analytical model for the time exponent for capillary imbibition in terms of fractal dimension for tortuous capillaries. Moreover, Cai et al. [28] proposed an analytical model for calculating the weight of wetting liquid imbibed into the porous media, and the predicting results are in good agreement with available experimental data published in the literature. Li and Zhao [29] derived a mathematical model using pore volume fractal dimension to predict the production rate by imbibition and found that there was almost no effect of wettability on the value of the fractal dimension function. Considering the effect of the high-permeability fracture network in the matrix rock, Andersen et al. [30] developed an imbibition model for oil recovery in fractured reservoirs. Considering the sizes and shapes of pores, the tortuosity of imbibition streamlines, and the initial wetting-phase saturation, Cai et al. [31] developed a more generalized spontaneous imbibition model, which can be used to characterize the imbibition behavior of different porous media. Subsequently, Andersen et al. [32] derived an exponential model for predicting the imbibition profile under the right conditions: disk with constant permeability, core with negligible capillary gradients, and constant capillary-pressure derivative in the saturation interval. Ashraf and Phirani [33] developed a one-dimensional model to predict the position of the fluid fronts during imbibition in a multilayer porous medium.In summary, the previous experimental studies on spontaneous imbibition mainly focused on the oil recovery volume and rate of imbibition. There are no unified conclusions regarding the effects of interfacial tension on the imbibition height of low-permeability reservoirs. The imbibition height can also directly indicate the imbibition characteristics and can help to understand the imbibition mechanism and select proper surfactants [34]. Therefore, it needs to be comprehensively studied. Actual low-permeability reservoirs have complicated characteristics [35], and the calculation results given by traditional imbibition models are quite different from the real cases. There is much room for improvement in the current models being applied in the field. Therefore, it is necessary to conduct experimental and modeling studies on the imbibition height to guide the enhanced oil recovery of low-permeability reservoirs.In this paper, the effect of the interfacial tension of imbibition fluids on the imbibition height of hydrophilic and weakly-hydrophilic low-permeability rocks was experimentally studied. Considering the imbibition resistance and bending of pore throat during imbibition, a prediction model for the imbibition height was established based on fractal theory.


## 2. Materials and Methods

### 2.1. Preparation of the Imbibition Solution and Measurement of Physical Properties

To study the influence of interfacial tension and salinity on the imbibition height, NaCl and distilled water were used to prepare brine with various salinities: 0, 2500, 25000, and 70000 mg/L. Surfactant TOF-1 (provided by Changqing Oil Field) was selected for testing. TOF-1 has been widely used in Changqing Oil Field as a commercial cleanup additive for fracturing fluids. It is a liquid product whose main ingredient is a cationic fluorocarbon surfactant. The critical micelle concentration of TOF-1 is3.5×10−2mol/L. TOF-1 was added to prepare the imbibition solution with various volume fractions: 0.001%, 0.005%, 0.05%, 0.5%, and 5%. Next, the density of imbibition solution was measured using a densimeter, the viscosity was measured using a capillary viscometer, and the interfacial tension was tested using TX-500C Spinning Drop Interface Tensiometer. The contact angle was measured using the contact angle analyzer (Kruss DSA100) according to Standard SY/T 5153-2017 [36].


### 2.2. Preparation of Rock Samples with Different Wettability and Measurement of Physical Properties

Low-permeability outcrop rocks were cut into several cuboid rock samples with cross-sections of2.5cm×2.5cm and cylindrical parallel rock samples with diameters of 2.5 cm. The samples were placed into a drying oven overnight. The porosity and permeability of the parallel rock samples were measured using a pulse porometer and pulse permeameter [37]. The porosities of prepared rock samples ranged from 14.2% to 14.5%, with an average of 14.3%. Permeability ranged from 0.170 to 0.175 mD, with an average of 0.172 mD. The maximum and minimum diameters of capillary tube (λmax and λmin), and the equivalent radius of the capillary tube (rc) were obtained by a high-pressure mercury injection capillary force curve [38].In this paper, rock samples with different degrees of wettability were used for the experiments. Several of the cut rock samples were calcined at 800°C for 24 h in a muffle furnace and denoted as Group A. The unprocessed rock samples were denoted as Group B. Water-wet indices of the parallel samples in Groups A and B were measured by spontaneous imbibition method according to Standard SY/T5153—2017 “Measurement of reservoir rock wettability” [36]. The water-wet index of the samples in Group A ranged from 0.86 to 0.90, with an average value of 0.883, indicating that they belong to hydrophilic rock samples. The water-wet index of Group B ranged from 0.60 to 0.63, so they can be categorized as weakly hydrophilic rock samples.


### 2.3. Methods for Imbibition Experiments

Through several exploratory experiments, a direct measurement method for imbibition height was established with the experimental equipment shown in Figure1. First, the dry rock sample was fixed in the support. To prevent water evaporation from occurring on the surface of the rock samples and affecting the experimental results, each rock sample was coated with a transparent hot-melt adhesive at 230°C before the experiment. After the samples cooled to room temperature, the bottom of the samples was then cut off so that the end of the sample was in direct contact with the imbibition solution. Imbibition solution was poured into a beaker until the surface of the liquid was in contact with the bottom end of the rock sample. This point was considered the beginning of the spontaneous imbibition process. During this spontaneous imbibition process, it was necessary to continuously add imbibition solution into the beaker to ensure that there was a solid-liquid contact surface. The entire process was recorded with a camera, and the imbibition height was analyzed. As the interface of the imbibed and unimbibed regions is usually not a horizontal line, the imbibition height was taken as the average of 3–5 height values.Figure 1
Flow chart of improved imbibition height experiment.


## 2.1. Preparation of the Imbibition Solution and Measurement of Physical Properties

To study the influence of interfacial tension and salinity on the imbibition height, NaCl and distilled water were used to prepare brine with various salinities: 0, 2500, 25000, and 70000 mg/L. Surfactant TOF-1 (provided by Changqing Oil Field) was selected for testing. TOF-1 has been widely used in Changqing Oil Field as a commercial cleanup additive for fracturing fluids. It is a liquid product whose main ingredient is a cationic fluorocarbon surfactant. The critical micelle concentration of TOF-1 is3.5×10−2mol/L. TOF-1 was added to prepare the imbibition solution with various volume fractions: 0.001%, 0.005%, 0.05%, 0.5%, and 5%. Next, the density of imbibition solution was measured using a densimeter, the viscosity was measured using a capillary viscometer, and the interfacial tension was tested using TX-500C Spinning Drop Interface Tensiometer. The contact angle was measured using the contact angle analyzer (Kruss DSA100) according to Standard SY/T 5153-2017 [36].


## 2.2. Preparation of Rock Samples with Different Wettability and Measurement of Physical Properties

Low-permeability outcrop rocks were cut into several cuboid rock samples with cross-sections of2.5cm×2.5cm and cylindrical parallel rock samples with diameters of 2.5 cm. The samples were placed into a drying oven overnight. The porosity and permeability of the parallel rock samples were measured using a pulse porometer and pulse permeameter [37]. The porosities of prepared rock samples ranged from 14.2% to 14.5%, with an average of 14.3%. Permeability ranged from 0.170 to 0.175 mD, with an average of 0.172 mD. The maximum and minimum diameters of capillary tube (λmax and λmin), and the equivalent radius of the capillary tube (rc) were obtained by a high-pressure mercury injection capillary force curve [38].In this paper, rock samples with different degrees of wettability were used for the experiments. Several of the cut rock samples were calcined at 800°C for 24 h in a muffle furnace and denoted as Group A. The unprocessed rock samples were denoted as Group B. Water-wet indices of the parallel samples in Groups A and B were measured by spontaneous imbibition method according to Standard SY/T5153—2017 “Measurement of reservoir rock wettability” [36]. The water-wet index of the samples in Group A ranged from 0.86 to 0.90, with an average value of 0.883, indicating that they belong to hydrophilic rock samples. The water-wet index of Group B ranged from 0.60 to 0.63, so they can be categorized as weakly hydrophilic rock samples.


## 2.3. Methods for Imbibition Experiments

Through several exploratory experiments, a direct measurement method for imbibition height was established with the experimental equipment shown in Figure1. First, the dry rock sample was fixed in the support. To prevent water evaporation from occurring on the surface of the rock samples and affecting the experimental results, each rock sample was coated with a transparent hot-melt adhesive at 230°C before the experiment. After the samples cooled to room temperature, the bottom of the samples was then cut off so that the end of the sample was in direct contact with the imbibition solution. Imbibition solution was poured into a beaker until the surface of the liquid was in contact with the bottom end of the rock sample. This point was considered the beginning of the spontaneous imbibition process. During this spontaneous imbibition process, it was necessary to continuously add imbibition solution into the beaker to ensure that there was a solid-liquid contact surface. The entire process was recorded with a camera, and the imbibition height was analyzed. As the interface of the imbibed and unimbibed regions is usually not a horizontal line, the imbibition height was taken as the average of 3–5 height values.Figure 1
Flow chart of improved imbibition height experiment.


## 3. Results and Discussion

### 3.1. Results on Imbibition Height Tests

At a constant temperature of 26°C, the rock samples of Groups A and B were placed in TOF-1 imbibition solution with different volume fractions, respectively, to analyze the change in imbibition height with time.


#### 3.1.1. Variation in Imbibition Height of Rock Samples in Group a

Figure2 shows the variations of imbibition height as a function of time for the imbibition solution with the salinity of 0 mg/L (deionized water). The imbibition process can be divided into three stages: Stage I is the initial stage of imbibition, in which imbibition height is positively correlated to time; Stage II is the metaphase of imbibition, in which imbibition height is still positively correlated to time, but the slope in this stage is less than that in the first stage and decreases continuously; in Stage III, imbibition height generally no longer changes, showing that the imbibition has essentially stopped. During the initial stage, the capillary force plays a dominant role and serves as the driving force, while the resistance effect of viscosity and gravity can be neglected [28]. Here, the imbibition height rapidly increases. After the imbibition lasts for several hours, gravity gradually increases, so the imbibition height increases at a lower rate. In the later stage, resistance has a more significant effect, and capillary force as a driving force for imbibition is no longer able to overcome resistance.Figure 2
Change in imbibition height with time under different interfacial tensions with a salinity of 0 mg/L.The ratio of imbibition height to the duration is defined as the growth rate of imbibition height during the first stage. It is abbreviated as imbibition rate. Table1 shows the imbibition rates and ultimate imbibition heights under different interfacial tensions with a salinity of 0 mg/L. The imbibition height during Stage I has a higher contribution to the overall imbibition height. Especially under higher interfacial tension (0.001% TOF-1), the imbibition height during Stage I is closer to the ultimate imbibition height. With the increase in interfacial tension, the duration of Stage I reduces, the imbibition height increases, and the imbibition rate accelerates. This is because with the increase in interfacial tension, capillary force increases, which then facilitates imbibition [39]. However, a higher imbibition rate results in the earlier effect of gravity in the imbibition process, and an overall shorter Stage I. With the increase in interfacial tension, the ultimate imbibition height first increases and then decreases. The reason is that the increase in interfacial tension increases the capillary force which serves as the driving force, thus facilitating imbibition. However, the increasing imbibition rate causes the fluid to flow through the small pore throats instead of the large ones [40], thus increasing imbibition resistance. Furthermore, a higher imbibition rate results in the earlier effect of gravity on the imbibition process, and the imbibition height increases more slowly. Therefore, for a hydrophilic reservoir formation, there exists an optimal interfacial tension which allows the ultimate imbibition height to reach its maximum and represents the best case for displacing oil in the reservoir.Table 1
Results of imbibition experiments under different interfacial tensions with a salinity of 0 mg/L.
Interfacial tension [mN/m]Time of stage I [min]Imbibition height in stage I [mm]Imbibition rate in stage I [mm/min]Ultimate imbibition height [mm]4.82556315.910.02849.628.61348023.520.04965.7515.23335940.780.11475.3921.71533954.920.16281.9330.14624667.540.27575.48The overall results of the experiments with a salinity of 2500 mg/L are essentially the same as those with a salinity of 0 mg/L (Figure3). With the increase in interfacial tension, the imbibition rate during stage I increases, and the ultimate imbibition height increases first and then decreases. The maximum is reached when the interfacial tension is 21.497 mN/m (0.005% TOF-1). Similarly, the experiments with salinities of 25000 mg/L and 70000 mg/L also show the same variation trend in imbibition, as shown in Figure 4. The ultimate imbibition height first increases and then decreases, and reaches its maximum when the interfacial tensions are 21.602 mN/m and 21.755 mN/m (0.005%TOF-1), respectively.Figure 3
Change in imbibition height with time under different interfacial tensions with a salinity of 2500 mg/L.Figure 4
Change of imbibition height with time under different interfacial tensions. (a) With a salinity of 25000 mg/L. (b) With a salinity of 70000 mg/L.
(a)(b)


#### 3.1.2. Variation in Imbibition Height of Rock Samples in Group B

In imbibition solutions with different TOF-1 volume fractions, the experimental results on the imbibition of rock samples in Group B are consistent with those in Group A, as shown in Table2. With a salinity of 0–70000 mg/L and interfacial tension in the range of 4–30 mN/m, the duration of Stage I decreases with an increase in the interfacial tension. Moreover, the imbibition rate increases and the ultimate imbibition height first increases and then decreases.Table 2
Results of imbibition experiments of rock samples in Group B.
Salinity (mg/L)TOF-1 volume fraction [%]Interfacial tension [mN/m]Time of Stage I [min]Imbibition height in Stage I [mm]Imbibition rate in Stage I [mm/min]Ultimate imbibition height [mm]054.8255353.980.00718.210.58.61346215.070.03319.920.0515.23319719.150.09722.510.00521.71517519.460.11124.030.00130.14614920.310.13622.15250054.96749212.750.02621.150.58.61337718.880.05030.380.0515.41117022.640.13343.150.00521.49713729.240.21355.010.00130.21412530.150.24135.072500054.33437324.750.06632.310.58.14535325.360.07244.480.0515.56720528.540.13965.400.00521.60212735.750.28180.590.00130.47111138.660.34870.277000054.72334733.990.09884.230.58.21225934.110.13297.160.0515.92422388.940.399135.280.00521.75515893.450.591151.510.00130.25610595.920.914148.78Thus, for either hydrophilic rock samples or weakly-hydrophilic rock samples, there exists an optimal interfacial tension to achieve the maximum imbibition height, which represents the best imbibition effect. In these experiments, the imbibition solution with a TOF-1 volume fraction of 0.005% (with the interfacial tension of 21.497–21.755 mN/m) has the best imbibition effect.According to the experimental results of the rock samples in Group A and Group B, the influence of the wettability of samples on imbibition height was further analyzed. The salinity of 2500 mg/L is taken as an example, as shown in Figure5. During Stage I of imbibition, the growth rates of the imbibition height of hydrophilic and weakly hydrophilic samples both increase with the increase in interfacial tension, and under the same experimental conditions, the imbibition height of the hydrophilic samples increases more rapidly. The more hydrophilic rock samples show greater imbibition heights during both Stage I and Stage III. According to the analysis, a more hydrophilic rock sample results in a smaller wetting angle of the liquid-solid interface [41]. Also, a greater capillary force results in a faster increase in imbibition height and greater overall imbibition height.Figure 5
Effect of rock sample wettability on the imbibition rate and imbibition height. (a) Imbibition rate in Stage I. (b) Imbibition height.
(a)(b)


### 3.2. Establishment of a Calculation Model for Imbibition Height

Flow resistance is an essential factor influencing the imbibition effect. However, the impact of flow resistance and the bending extent of pore throats are not adequately considered in the existing models. Thus, in this paper, a study was conducted to create a new imbibition model with consideration of flow resistance and the bending conditions of pore throats.


#### 3.2.1. Model Establishment

Figure6 shows a physical model for imbibition. The slender region in the middle is a capillary tube; the blue region is the imbibition solution; h denotes the imbibition height in the traditional capillary tube (Figure 6(a)); and hs denotes the imbibition height in the improved model (Figure 6(b)). Considering the bending of pore throats, the capillary tube is curved, and the streamline of the imbibition solution is also a curve. In this paper, the fractal theory is used to describe the curved streamline.Figure 6
Imbibition models. (a) Traditional model. (b) New model.
(a)(b)The fractal theory describes the complexity and diversity of a research object through mathematical methods from the perspective of the fractal dimension. Due to self-similarity and iterative generation between the parts and the whole of the fractal body, there is a scaling relationship between the physical quantityMε to be studied and the measurement scale ε of the object, as follows [42]:(4)Mε~εDf,where Df is the fractal dimension, which is an important parameter that describes the characteristics of a fractal body.Stratigraphic rocks are also fractal [43]. Hence, the pore fractures and fluid streamlines in rocks present fractal distributions [44], with the scaling relationship as follows:(5)hf=ε1‐DfhsDf,where hf denotes the imbibition length, with the unit of m; and hs denotes imbibition distance, with the unit of m.When the above scaling relationship is used to describe the fluid streamline in the stratum,ɛ denotes the equivalent pore diameter λc. Therefore, the scaling relationship can also be expressed as:(6)hf=λc1‐DfhsDf,Df denotes the property parameter of the fractal body [45]:(7)Df=d−lnφlnλmin/λmax,where d denotes spatial dimensions, with the constant of 2 or 3; φ denotes the porosity of the rock samples, dimensionless; λmax and λmin denote the maximum and minimum diameters of capillary tube, respectively, with a unit of m.According to Figure6(b), using the Bernoulli equation, the change in energy of the imbibition solution when moving from the initial position to the final position is:(8)p1+12ρvf12+ρgh1=p2+12ρvf22+ρgh2+ω,where P1 and P2 denote the pressures at the initial and final positions, respectively, with a unit of Pa; ρ is the density of the imbibition solution, with a unit of kg/m3; vf1 and vf2 denote the flow velocities of imbibition solution at the initial and final positions, respectively, with a unit of m/s; h1 and h2 denote the heights of the initial and final positions, with a unit of m; and w denotes the pressure loss in the flow process of imbibition solution, with a unit of Pa.The total energy loss of the fluid element in the imbibition process is:(9)E=2πrωdr,where r denotes the radius of the capillary tube, with a unit of mm.The energy loss caused by internal friction in the imbibition process of the fluid element can be obtained through the formula for Newtonian internal friction:(10)W=−2πhfηddrrdvfdr,where vf denotes the flow velocity of the fluid in the axial direction of the capillary tube, with a unit of m/s; and η denotes the viscosity of imbibition solution, with a unit of Pa·s.At the initial stage of imbibition, gravity has less effect on imbibition and is negligible. Therefore, the energy loss within the flow is caused by internal friction, that is,E=W.(11)2πrωdr=‐2πLfηddrrdvfdr.Calculating the derivative of the imbibition height, the following equation can be obtained:(12)1η∂ω∂hfrdr=‐ddrrdvfdr,where ∂ω/∂hf is the pressure gradient of the capillary tube along the direction of imbibition, with a unit of Pa/s.The boundary condition can be expressed as:(13)r=0,dvfdr=0r=rc,vf=0,where rc denotes the equivalent radius of the capillary tube, with the unit of m.After integrating Equation (13) through the boundary condition, the following equation can be obtained:(14)vf=14η∂ω∂hfrc2−r2.The derivative ofr can be obtained as:(15)dvfdr=−r2η∂ω∂hf.Using the average imbibition rate as the equivalent value of the flow velocity of the fluid along the axial direction of the capillary tube, the following equation can be obtained:(16)vf¯=1πrc2∫0rc2πrvfdr=1πrc2∫0rc14η∂ω∂hfrc2−r22πrdr=rc28η∂ω∂hf.Substituting Equation (16) into Equation (15), the following equation can be obtained:(17)dvfdr=−r2η⋅8ηvf¯rc2=−4rvf¯rc2.Substituting Equation (17) into the formula for Newtonian internal friction, the internal friction is calculated as follows:(18)τ=ηdvfdrr=rc=‐η×4rcvf¯rc2=‐4vf¯ηrc.Frictional forcef (i.e., imbibition resistance) on the side of imbibition part of the capillary tube is calculated as:(19)f=2πrchfτ=‐8πηhfvf¯.The average imbibition ratevf¯ is:(20)vf¯=dhfdt.Therefore, the differential equation for the length of the imbibition effect is:(21)ddtπrc2ρhfdhfdt=2πσrccosθ−8πηhfdhfdt,where t denotes imbibition time, with the unit of s; σ denotes interfacial tension of imbibition solution, with the unit of N/m; and θ denotes wetting angle, with the unit of °.After substituting Equation (6) into Equation (21), the formula for the change in imbibition height with time can be obtained by solving the equation:(22)hs=σrccosθηλc2−2Dft−rc28η1−e−8ηt/rc2ρ2Df.As the influence of gravity is not taken into consideration in the model derivation process,hs denotes the height of Stage I (gravity is negligible).


#### 3.2.2. Model Verification

The imbibition model was verified with the experimental data on the rock samples in Group A under different interfacial tensions, with a salinity of 0 mg/L taken as the example. The parameters of the imbibition solution and rock samples are shown in Table3, and the calculation results are shown in Figure 7. By taking the imbibition resistance and the bending condition of the pore throats into account, the new model is able to obtain calculated results that fit well with the experimental results. The coefficient of determination (goodness of fit) R2 was used to analyze the fitting effectiveness [46]. For solutions with the surface tensions of 4.825, 8.613, 15.233, 21.715, and 30.146 mN/m, the R2 values are 0.909, 0.927, 0.965, 0.954 and 0.933, respectively, indicating that the proposed model can predict the imbibition trend in the initial period of imbibition. Overall, during Stage I, with the increase in imbibition time, the imbibition height rapidly increases while the imbibition rate gradually decreases. Also, at the same imbibition time, larger interfacial tension results in greater imbibition height.Table 3
Basic parameters of cores and imbibition solution.
No.R [μm]Dfρ [kg/m3]σ [mN/m]cosθη [mPa·a]10.221.10510004.8250.803.828.6130.721.8315.2330.681.5421.7150.631.3530.1460.551.1Figure 7
Comparison between the experimental results and calculation results obtained by the improved capillary model under different interfacial tensions. The dotted line indicates experimental results, and the solid line represents the calculated results.


## 3.1. Results on Imbibition Height Tests

At a constant temperature of 26°C, the rock samples of Groups A and B were placed in TOF-1 imbibition solution with different volume fractions, respectively, to analyze the change in imbibition height with time.


### 3.1.1. Variation in Imbibition Height of Rock Samples in Group a

Figure2 shows the variations of imbibition height as a function of time for the imbibition solution with the salinity of 0 mg/L (deionized water). The imbibition process can be divided into three stages: Stage I is the initial stage of imbibition, in which imbibition height is positively correlated to time; Stage II is the metaphase of imbibition, in which imbibition height is still positively correlated to time, but the slope in this stage is less than that in the first stage and decreases continuously; in Stage III, imbibition height generally no longer changes, showing that the imbibition has essentially stopped. During the initial stage, the capillary force plays a dominant role and serves as the driving force, while the resistance effect of viscosity and gravity can be neglected [28]. Here, the imbibition height rapidly increases. After the imbibition lasts for several hours, gravity gradually increases, so the imbibition height increases at a lower rate. In the later stage, resistance has a more significant effect, and capillary force as a driving force for imbibition is no longer able to overcome resistance.Figure 2
Change in imbibition height with time under different interfacial tensions with a salinity of 0 mg/L.The ratio of imbibition height to the duration is defined as the growth rate of imbibition height during the first stage. It is abbreviated as imbibition rate. Table1 shows the imbibition rates and ultimate imbibition heights under different interfacial tensions with a salinity of 0 mg/L. The imbibition height during Stage I has a higher contribution to the overall imbibition height. Especially under higher interfacial tension (0.001% TOF-1), the imbibition height during Stage I is closer to the ultimate imbibition height. With the increase in interfacial tension, the duration of Stage I reduces, the imbibition height increases, and the imbibition rate accelerates. This is because with the increase in interfacial tension, capillary force increases, which then facilitates imbibition [39]. However, a higher imbibition rate results in the earlier effect of gravity in the imbibition process, and an overall shorter Stage I. With the increase in interfacial tension, the ultimate imbibition height first increases and then decreases. The reason is that the increase in interfacial tension increases the capillary force which serves as the driving force, thus facilitating imbibition. However, the increasing imbibition rate causes the fluid to flow through the small pore throats instead of the large ones [40], thus increasing imbibition resistance. Furthermore, a higher imbibition rate results in the earlier effect of gravity on the imbibition process, and the imbibition height increases more slowly. Therefore, for a hydrophilic reservoir formation, there exists an optimal interfacial tension which allows the ultimate imbibition height to reach its maximum and represents the best case for displacing oil in the reservoir.Table 1
Results of imbibition experiments under different interfacial tensions with a salinity of 0 mg/L.
Interfacial tension [mN/m]Time of stage I [min]Imbibition height in stage I [mm]Imbibition rate in stage I [mm/min]Ultimate imbibition height [mm]4.82556315.910.02849.628.61348023.520.04965.7515.23335940.780.11475.3921.71533954.920.16281.9330.14624667.540.27575.48The overall results of the experiments with a salinity of 2500 mg/L are essentially the same as those with a salinity of 0 mg/L (Figure3). With the increase in interfacial tension, the imbibition rate during stage I increases, and the ultimate imbibition height increases first and then decreases. The maximum is reached when the interfacial tension is 21.497 mN/m (0.005% TOF-1). Similarly, the experiments with salinities of 25000 mg/L and 70000 mg/L also show the same variation trend in imbibition, as shown in Figure 4. The ultimate imbibition height first increases and then decreases, and reaches its maximum when the interfacial tensions are 21.602 mN/m and 21.755 mN/m (0.005%TOF-1), respectively.Figure 3
Change in imbibition height with time under different interfacial tensions with a salinity of 2500 mg/L.Figure 4
Change of imbibition height with time under different interfacial tensions. (a) With a salinity of 25000 mg/L. (b) With a salinity of 70000 mg/L.
(a)(b)


### 3.1.2. Variation in Imbibition Height of Rock Samples in Group B

In imbibition solutions with different TOF-1 volume fractions, the experimental results on the imbibition of rock samples in Group B are consistent with those in Group A, as shown in Table2. With a salinity of 0–70000 mg/L and interfacial tension in the range of 4–30 mN/m, the duration of Stage I decreases with an increase in the interfacial tension. Moreover, the imbibition rate increases and the ultimate imbibition height first increases and then decreases.Table 2
Results of imbibition experiments of rock samples in Group B.
Salinity (mg/L)TOF-1 volume fraction [%]Interfacial tension [mN/m]Time of Stage I [min]Imbibition height in Stage I [mm]Imbibition rate in Stage I [mm/min]Ultimate imbibition height [mm]054.8255353.980.00718.210.58.61346215.070.03319.920.0515.23319719.150.09722.510.00521.71517519.460.11124.030.00130.14614920.310.13622.15250054.96749212.750.02621.150.58.61337718.880.05030.380.0515.41117022.640.13343.150.00521.49713729.240.21355.010.00130.21412530.150.24135.072500054.33437324.750.06632.310.58.14535325.360.07244.480.0515.56720528.540.13965.400.00521.60212735.750.28180.590.00130.47111138.660.34870.277000054.72334733.990.09884.230.58.21225934.110.13297.160.0515.92422388.940.399135.280.00521.75515893.450.591151.510.00130.25610595.920.914148.78Thus, for either hydrophilic rock samples or weakly-hydrophilic rock samples, there exists an optimal interfacial tension to achieve the maximum imbibition height, which represents the best imbibition effect. In these experiments, the imbibition solution with a TOF-1 volume fraction of 0.005% (with the interfacial tension of 21.497–21.755 mN/m) has the best imbibition effect.According to the experimental results of the rock samples in Group A and Group B, the influence of the wettability of samples on imbibition height was further analyzed. The salinity of 2500 mg/L is taken as an example, as shown in Figure5. During Stage I of imbibition, the growth rates of the imbibition height of hydrophilic and weakly hydrophilic samples both increase with the increase in interfacial tension, and under the same experimental conditions, the imbibition height of the hydrophilic samples increases more rapidly. The more hydrophilic rock samples show greater imbibition heights during both Stage I and Stage III. According to the analysis, a more hydrophilic rock sample results in a smaller wetting angle of the liquid-solid interface [41]. Also, a greater capillary force results in a faster increase in imbibition height and greater overall imbibition height.Figure 5
Effect of rock sample wettability on the imbibition rate and imbibition height. (a) Imbibition rate in Stage I. (b) Imbibition height.
(a)(b)


## 3.1.1. Variation in Imbibition Height of Rock Samples in Group a

Figure2 shows the variations of imbibition height as a function of time for the imbibition solution with the salinity of 0 mg/L (deionized water). The imbibition process can be divided into three stages: Stage I is the initial stage of imbibition, in which imbibition height is positively correlated to time; Stage II is the metaphase of imbibition, in which imbibition height is still positively correlated to time, but the slope in this stage is less than that in the first stage and decreases continuously; in Stage III, imbibition height generally no longer changes, showing that the imbibition has essentially stopped. During the initial stage, the capillary force plays a dominant role and serves as the driving force, while the resistance effect of viscosity and gravity can be neglected [28]. Here, the imbibition height rapidly increases. After the imbibition lasts for several hours, gravity gradually increases, so the imbibition height increases at a lower rate. In the later stage, resistance has a more significant effect, and capillary force as a driving force for imbibition is no longer able to overcome resistance.Figure 2
Change in imbibition height with time under different interfacial tensions with a salinity of 0 mg/L.The ratio of imbibition height to the duration is defined as the growth rate of imbibition height during the first stage. It is abbreviated as imbibition rate. Table1 shows the imbibition rates and ultimate imbibition heights under different interfacial tensions with a salinity of 0 mg/L. The imbibition height during Stage I has a higher contribution to the overall imbibition height. Especially under higher interfacial tension (0.001% TOF-1), the imbibition height during Stage I is closer to the ultimate imbibition height. With the increase in interfacial tension, the duration of Stage I reduces, the imbibition height increases, and the imbibition rate accelerates. This is because with the increase in interfacial tension, capillary force increases, which then facilitates imbibition [39]. However, a higher imbibition rate results in the earlier effect of gravity in the imbibition process, and an overall shorter Stage I. With the increase in interfacial tension, the ultimate imbibition height first increases and then decreases. The reason is that the increase in interfacial tension increases the capillary force which serves as the driving force, thus facilitating imbibition. However, the increasing imbibition rate causes the fluid to flow through the small pore throats instead of the large ones [40], thus increasing imbibition resistance. Furthermore, a higher imbibition rate results in the earlier effect of gravity on the imbibition process, and the imbibition height increases more slowly. Therefore, for a hydrophilic reservoir formation, there exists an optimal interfacial tension which allows the ultimate imbibition height to reach its maximum and represents the best case for displacing oil in the reservoir.Table 1
Results of imbibition experiments under different interfacial tensions with a salinity of 0 mg/L.
Interfacial tension [mN/m]Time of stage I [min]Imbibition height in stage I [mm]Imbibition rate in stage I [mm/min]Ultimate imbibition height [mm]4.82556315.910.02849.628.61348023.520.04965.7515.23335940.780.11475.3921.71533954.920.16281.9330.14624667.540.27575.48The overall results of the experiments with a salinity of 2500 mg/L are essentially the same as those with a salinity of 0 mg/L (Figure3). With the increase in interfacial tension, the imbibition rate during stage I increases, and the ultimate imbibition height increases first and then decreases. The maximum is reached when the interfacial tension is 21.497 mN/m (0.005% TOF-1). Similarly, the experiments with salinities of 25000 mg/L and 70000 mg/L also show the same variation trend in imbibition, as shown in Figure 4. The ultimate imbibition height first increases and then decreases, and reaches its maximum when the interfacial tensions are 21.602 mN/m and 21.755 mN/m (0.005%TOF-1), respectively.Figure 3
Change in imbibition height with time under different interfacial tensions with a salinity of 2500 mg/L.Figure 4
Change of imbibition height with time under different interfacial tensions. (a) With a salinity of 25000 mg/L. (b) With a salinity of 70000 mg/L.
(a)(b)


## 3.1.2. Variation in Imbibition Height of Rock Samples in Group B

In imbibition solutions with different TOF-1 volume fractions, the experimental results on the imbibition of rock samples in Group B are consistent with those in Group A, as shown in Table2. With a salinity of 0–70000 mg/L and interfacial tension in the range of 4–30 mN/m, the duration of Stage I decreases with an increase in the interfacial tension. Moreover, the imbibition rate increases and the ultimate imbibition height first increases and then decreases.Table 2
Results of imbibition experiments of rock samples in Group B.
Salinity (mg/L)TOF-1 volume fraction [%]Interfacial tension [mN/m]Time of Stage I [min]Imbibition height in Stage I [mm]Imbibition rate in Stage I [mm/min]Ultimate imbibition height [mm]054.8255353.980.00718.210.58.61346215.070.03319.920.0515.23319719.150.09722.510.00521.71517519.460.11124.030.00130.14614920.310.13622.15250054.96749212.750.02621.150.58.61337718.880.05030.380.0515.41117022.640.13343.150.00521.49713729.240.21355.010.00130.21412530.150.24135.072500054.33437324.750.06632.310.58.14535325.360.07244.480.0515.56720528.540.13965.400.00521.60212735.750.28180.590.00130.47111138.660.34870.277000054.72334733.990.09884.230.58.21225934.110.13297.160.0515.92422388.940.399135.280.00521.75515893.450.591151.510.00130.25610595.920.914148.78Thus, for either hydrophilic rock samples or weakly-hydrophilic rock samples, there exists an optimal interfacial tension to achieve the maximum imbibition height, which represents the best imbibition effect. In these experiments, the imbibition solution with a TOF-1 volume fraction of 0.005% (with the interfacial tension of 21.497–21.755 mN/m) has the best imbibition effect.According to the experimental results of the rock samples in Group A and Group B, the influence of the wettability of samples on imbibition height was further analyzed. The salinity of 2500 mg/L is taken as an example, as shown in Figure5. During Stage I of imbibition, the growth rates of the imbibition height of hydrophilic and weakly hydrophilic samples both increase with the increase in interfacial tension, and under the same experimental conditions, the imbibition height of the hydrophilic samples increases more rapidly. The more hydrophilic rock samples show greater imbibition heights during both Stage I and Stage III. According to the analysis, a more hydrophilic rock sample results in a smaller wetting angle of the liquid-solid interface [41]. Also, a greater capillary force results in a faster increase in imbibition height and greater overall imbibition height.Figure 5
Effect of rock sample wettability on the imbibition rate and imbibition height. (a) Imbibition rate in Stage I. (b) Imbibition height.
(a)(b)


## 3.2. Establishment of a Calculation Model for Imbibition Height

Flow resistance is an essential factor influencing the imbibition effect. However, the impact of flow resistance and the bending extent of pore throats are not adequately considered in the existing models. Thus, in this paper, a study was conducted to create a new imbibition model with consideration of flow resistance and the bending conditions of pore throats.


### 3.2.1. Model Establishment

Figure6 shows a physical model for imbibition. The slender region in the middle is a capillary tube; the blue region is the imbibition solution; h denotes the imbibition height in the traditional capillary tube (Figure 6(a)); and hs denotes the imbibition height in the improved model (Figure 6(b)). Considering the bending of pore throats, the capillary tube is curved, and the streamline of the imbibition solution is also a curve. In this paper, the fractal theory is used to describe the curved streamline.Figure 6
Imbibition models. (a) Traditional model. (b) New model.
(a)(b)The fractal theory describes the complexity and diversity of a research object through mathematical methods from the perspective of the fractal dimension. Due to self-similarity and iterative generation between the parts and the whole of the fractal body, there is a scaling relationship between the physical quantityMε to be studied and the measurement scale ε of the object, as follows [42]:(4)Mε~εDf,where Df is the fractal dimension, which is an important parameter that describes the characteristics of a fractal body.Stratigraphic rocks are also fractal [43]. Hence, the pore fractures and fluid streamlines in rocks present fractal distributions [44], with the scaling relationship as follows:(5)hf=ε1‐DfhsDf,where hf denotes the imbibition length, with the unit of m; and hs denotes imbibition distance, with the unit of m.When the above scaling relationship is used to describe the fluid streamline in the stratum,ɛ denotes the equivalent pore diameter λc. Therefore, the scaling relationship can also be expressed as:(6)hf=λc1‐DfhsDf,Df denotes the property parameter of the fractal body [45]:(7)Df=d−lnφlnλmin/λmax,where d denotes spatial dimensions, with the constant of 2 or 3; φ denotes the porosity of the rock samples, dimensionless; λmax and λmin denote the maximum and minimum diameters of capillary tube, respectively, with a unit of m.According to Figure6(b), using the Bernoulli equation, the change in energy of the imbibition solution when moving from the initial position to the final position is:(8)p1+12ρvf12+ρgh1=p2+12ρvf22+ρgh2+ω,where P1 and P2 denote the pressures at the initial and final positions, respectively, with a unit of Pa; ρ is the density of the imbibition solution, with a unit of kg/m3; vf1 and vf2 denote the flow velocities of imbibition solution at the initial and final positions, respectively, with a unit of m/s; h1 and h2 denote the heights of the initial and final positions, with a unit of m; and w denotes the pressure loss in the flow process of imbibition solution, with a unit of Pa.The total energy loss of the fluid element in the imbibition process is:(9)E=2πrωdr,where r denotes the radius of the capillary tube, with a unit of mm.The energy loss caused by internal friction in the imbibition process of the fluid element can be obtained through the formula for Newtonian internal friction:(10)W=−2πhfηddrrdvfdr,where vf denotes the flow velocity of the fluid in the axial direction of the capillary tube, with a unit of m/s; and η denotes the viscosity of imbibition solution, with a unit of Pa·s.At the initial stage of imbibition, gravity has less effect on imbibition and is negligible. Therefore, the energy loss within the flow is caused by internal friction, that is,E=W.(11)2πrωdr=‐2πLfηddrrdvfdr.Calculating the derivative of the imbibition height, the following equation can be obtained:(12)1η∂ω∂hfrdr=‐ddrrdvfdr,where ∂ω/∂hf is the pressure gradient of the capillary tube along the direction of imbibition, with a unit of Pa/s.The boundary condition can be expressed as:(13)r=0,dvfdr=0r=rc,vf=0,where rc denotes the equivalent radius of the capillary tube, with the unit of m.After integrating Equation (13) through the boundary condition, the following equation can be obtained:(14)vf=14η∂ω∂hfrc2−r2.The derivative ofr can be obtained as:(15)dvfdr=−r2η∂ω∂hf.Using the average imbibition rate as the equivalent value of the flow velocity of the fluid along the axial direction of the capillary tube, the following equation can be obtained:(16)vf¯=1πrc2∫0rc2πrvfdr=1πrc2∫0rc14η∂ω∂hfrc2−r22πrdr=rc28η∂ω∂hf.Substituting Equation (16) into Equation (15), the following equation can be obtained:(17)dvfdr=−r2η⋅8ηvf¯rc2=−4rvf¯rc2.Substituting Equation (17) into the formula for Newtonian internal friction, the internal friction is calculated as follows:(18)τ=ηdvfdrr=rc=‐η×4rcvf¯rc2=‐4vf¯ηrc.Frictional forcef (i.e., imbibition resistance) on the side of imbibition part of the capillary tube is calculated as:(19)f=2πrchfτ=‐8πηhfvf¯.The average imbibition ratevf¯ is:(20)vf¯=dhfdt.Therefore, the differential equation for the length of the imbibition effect is:(21)ddtπrc2ρhfdhfdt=2πσrccosθ−8πηhfdhfdt,where t denotes imbibition time, with the unit of s; σ denotes interfacial tension of imbibition solution, with the unit of N/m; and θ denotes wetting angle, with the unit of °.After substituting Equation (6) into Equation (21), the formula for the change in imbibition height with time can be obtained by solving the equation:(22)hs=σrccosθηλc2−2Dft−rc28η1−e−8ηt/rc2ρ2Df.As the influence of gravity is not taken into consideration in the model derivation process,hs denotes the height of Stage I (gravity is negligible).


### 3.2.2. Model Verification

The imbibition model was verified with the experimental data on the rock samples in Group A under different interfacial tensions, with a salinity of 0 mg/L taken as the example. The parameters of the imbibition solution and rock samples are shown in Table3, and the calculation results are shown in Figure 7. By taking the imbibition resistance and the bending condition of the pore throats into account, the new model is able to obtain calculated results that fit well with the experimental results. The coefficient of determination (goodness of fit) R2 was used to analyze the fitting effectiveness [46]. For solutions with the surface tensions of 4.825, 8.613, 15.233, 21.715, and 30.146 mN/m, the R2 values are 0.909, 0.927, 0.965, 0.954 and 0.933, respectively, indicating that the proposed model can predict the imbibition trend in the initial period of imbibition. Overall, during Stage I, with the increase in imbibition time, the imbibition height rapidly increases while the imbibition rate gradually decreases. Also, at the same imbibition time, larger interfacial tension results in greater imbibition height.Table 3
Basic parameters of cores and imbibition solution.
No.R [μm]Dfρ [kg/m3]σ [mN/m]cosθη [mPa·a]10.221.10510004.8250.803.828.6130.721.8315.2330.681.5421.7150.631.3530.1460.551.1Figure 7
Comparison between the experimental results and calculation results obtained by the improved capillary model under different interfacial tensions. The dotted line indicates experimental results, and the solid line represents the calculated results.


## 3.2.1. Model Establishment

Figure6 shows a physical model for imbibition. The slender region in the middle is a capillary tube; the blue region is the imbibition solution; h denotes the imbibition height in the traditional capillary tube (Figure 6(a)); and hs denotes the imbibition height in the improved model (Figure 6(b)). Considering the bending of pore throats, the capillary tube is curved, and the streamline of the imbibition solution is also a curve. In this paper, the fractal theory is used to describe the curved streamline.Figure 6
Imbibition models. (a) Traditional model. (b) New model.
(a)(b)The fractal theory describes the complexity and diversity of a research object through mathematical methods from the perspective of the fractal dimension. Due to self-similarity and iterative generation between the parts and the whole of the fractal body, there is a scaling relationship between the physical quantityMε to be studied and the measurement scale ε of the object, as follows [42]:(4)Mε~εDf,where Df is the fractal dimension, which is an important parameter that describes the characteristics of a fractal body.Stratigraphic rocks are also fractal [43]. Hence, the pore fractures and fluid streamlines in rocks present fractal distributions [44], with the scaling relationship as follows:(5)hf=ε1‐DfhsDf,where hf denotes the imbibition length, with the unit of m; and hs denotes imbibition distance, with the unit of m.When the above scaling relationship is used to describe the fluid streamline in the stratum,ɛ denotes the equivalent pore diameter λc. Therefore, the scaling relationship can also be expressed as:(6)hf=λc1‐DfhsDf,Df denotes the property parameter of the fractal body [45]:(7)Df=d−lnφlnλmin/λmax,where d denotes spatial dimensions, with the constant of 2 or 3; φ denotes the porosity of the rock samples, dimensionless; λmax and λmin denote the maximum and minimum diameters of capillary tube, respectively, with a unit of m.According to Figure6(b), using the Bernoulli equation, the change in energy of the imbibition solution when moving from the initial position to the final position is:(8)p1+12ρvf12+ρgh1=p2+12ρvf22+ρgh2+ω,where P1 and P2 denote the pressures at the initial and final positions, respectively, with a unit of Pa; ρ is the density of the imbibition solution, with a unit of kg/m3; vf1 and vf2 denote the flow velocities of imbibition solution at the initial and final positions, respectively, with a unit of m/s; h1 and h2 denote the heights of the initial and final positions, with a unit of m; and w denotes the pressure loss in the flow process of imbibition solution, with a unit of Pa.The total energy loss of the fluid element in the imbibition process is:(9)E=2πrωdr,where r denotes the radius of the capillary tube, with a unit of mm.The energy loss caused by internal friction in the imbibition process of the fluid element can be obtained through the formula for Newtonian internal friction:(10)W=−2πhfηddrrdvfdr,where vf denotes the flow velocity of the fluid in the axial direction of the capillary tube, with a unit of m/s; and η denotes the viscosity of imbibition solution, with a unit of Pa·s.At the initial stage of imbibition, gravity has less effect on imbibition and is negligible. Therefore, the energy loss within the flow is caused by internal friction, that is,E=W.(11)2πrωdr=‐2πLfηddrrdvfdr.Calculating the derivative of the imbibition height, the following equation can be obtained:(12)1η∂ω∂hfrdr=‐ddrrdvfdr,where ∂ω/∂hf is the pressure gradient of the capillary tube along the direction of imbibition, with a unit of Pa/s.The boundary condition can be expressed as:(13)r=0,dvfdr=0r=rc,vf=0,where rc denotes the equivalent radius of the capillary tube, with the unit of m.After integrating Equation (13) through the boundary condition, the following equation can be obtained:(14)vf=14η∂ω∂hfrc2−r2.The derivative ofr can be obtained as:(15)dvfdr=−r2η∂ω∂hf.Using the average imbibition rate as the equivalent value of the flow velocity of the fluid along the axial direction of the capillary tube, the following equation can be obtained:(16)vf¯=1πrc2∫0rc2πrvfdr=1πrc2∫0rc14η∂ω∂hfrc2−r22πrdr=rc28η∂ω∂hf.Substituting Equation (16) into Equation (15), the following equation can be obtained:(17)dvfdr=−r2η⋅8ηvf¯rc2=−4rvf¯rc2.Substituting Equation (17) into the formula for Newtonian internal friction, the internal friction is calculated as follows:(18)τ=ηdvfdrr=rc=‐η×4rcvf¯rc2=‐4vf¯ηrc.Frictional forcef (i.e., imbibition resistance) on the side of imbibition part of the capillary tube is calculated as:(19)f=2πrchfτ=‐8πηhfvf¯.The average imbibition ratevf¯ is:(20)vf¯=dhfdt.Therefore, the differential equation for the length of the imbibition effect is:(21)ddtπrc2ρhfdhfdt=2πσrccosθ−8πηhfdhfdt,where t denotes imbibition time, with the unit of s; σ denotes interfacial tension of imbibition solution, with the unit of N/m; and θ denotes wetting angle, with the unit of °.After substituting Equation (6) into Equation (21), the formula for the change in imbibition height with time can be obtained by solving the equation:(22)hs=σrccosθηλc2−2Dft−rc28η1−e−8ηt/rc2ρ2Df.As the influence of gravity is not taken into consideration in the model derivation process,hs denotes the height of Stage I (gravity is negligible).


## 3.2.2. Model Verification

The imbibition model was verified with the experimental data on the rock samples in Group A under different interfacial tensions, with a salinity of 0 mg/L taken as the example. The parameters of the imbibition solution and rock samples are shown in Table3, and the calculation results are shown in Figure 7. By taking the imbibition resistance and the bending condition of the pore throats into account, the new model is able to obtain calculated results that fit well with the experimental results. The coefficient of determination (goodness of fit) R2 was used to analyze the fitting effectiveness [46]. For solutions with the surface tensions of 4.825, 8.613, 15.233, 21.715, and 30.146 mN/m, the R2 values are 0.909, 0.927, 0.965, 0.954 and 0.933, respectively, indicating that the proposed model can predict the imbibition trend in the initial period of imbibition. Overall, during Stage I, with the increase in imbibition time, the imbibition height rapidly increases while the imbibition rate gradually decreases. Also, at the same imbibition time, larger interfacial tension results in greater imbibition height.Table 3
Basic parameters of cores and imbibition solution.
No.R [μm]Dfρ [kg/m3]σ [mN/m]cosθη [mPa·a]10.221.10510004.8250.803.828.6130.721.8315.2330.681.5421.7150.631.3530.1460.551.1Figure 7
Comparison between the experimental results and calculation results obtained by the improved capillary model under different interfacial tensions. The dotted line indicates experimental results, and the solid line represents the calculated results.


## 4. Conclusions

In low-permeability reservoir formations, the increase in imbibition height with imbibition time can be divided into three stages. During Stage I, the capillary force, which serves as the driving force, plays a dominant role. The resistance of viscous force and gravity are negligible. Therefore, the imbibition height rapidly increases, followed by a slow increase. Interfacial tension has a significant effect on imbibition height. With the increase in interfacial tension, the imbibition rate during Stage I increases. The ultimate imbibition height first increases and then decreases, and the contribution of the imbibition height during the first stage to the ultimate imbibition height becomes greater. There is an optimal interfacial tension that allows the ultimate imbibition height to reach its maximum. The newly-established imbibition model takes into account the imbibition resistance and the bending of the pore throats, so it can reflect the change in imbibition height with time during Stage I. The calculation results are consistent with the experimental results, indicating that the model is applicable for a low-permeability reservoir and would help predict the production results.



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