Experimental study of liquid-gas flow structure effects on relative permeabilities in a fracture



[1] Two-phase flow through fractured media is important in geothermal, nuclear, and petroleum applications. In this research an experimental apparatus was built to capture the unstable nature of the two-phase flow in a smooth-walled fracture and display the flow structures under different flow configurations in real time. The air-water relative permeability was obtained from experiment and showed deviation from the X curve behavior suggested by earlier studies. Through this work the relationship between the phase channel morphology and relative permeability in fractures was determined. A physical tortuous channel approach was proposed to quantify the effects of the flow structure. This approach could replicate the experimental results with a good accuracy. Other relative permeability models (viscous coupling model, X curve model, and Corey curve model) were also compared. Except for the viscous coupling model, these models did not interpret the experimental relative permeabilities as well as the proposed tortuous channel model. Hence we concluded that the two-phase relative permeability in fractures depends not only on liquid type and fracture geometry but also on the structure of the two-phase flow.

1. Introduction

[2] Multiphase flows through fractured porous media are of great importance in several domains, such as petroleum recovery, geothermal energy, and environmental engineering. In the case of two-phase flow in a single fracture this importance extends to several industrial processes, such as compact heat exchangers and nuclear engineering. Despite this importance few theoretical and experimental studies have been devoted to establishing the laws governing these flows. Thus the results presented in the literature seem to be in contradiction. So far, the mechanisms of two-phase flows in a fracture are not well understood, and a general model for describing these flows has still not been determined.

[3] The approach used commonly to describe two-phase flow in a fracture is the relative permeability concept, which is based on a generalization of the Darcy equation. Three models for the relative permeabilities are presented in the literature: the X model (relative permeability equals saturation; straight lines), the Corey model, and the viscous coupling model. The experimental results presented in the literature also show different behavior for the relative permeabilities. Some results are in accordance with the X model, whereas other results are in accordance with the Corey model or the viscous coupling model.

[4] Although the flow structures are of a great importance in modeling two-phase flows, the three models do not take into account these effects. The purpose of this paper is to examine the effects of the flow structures on the relative permeabilities during two-phase flow in a single fracture. The paper is organized as follows: in section 2 we present a review of the theoretical and experimental approaches for two-phase flow in a fracture. In section 3 we describe the experimental apparatus and the measurements of pressure drops, flow rates, and saturations. In section 4 we describe the observed flow structures and calculated relative permeabilities. In section 5 we propose a tortuous channel approach and discuss the flow structure effects on relative permeabilities. Finally, in section 6 we outline the main conclusions of this study.

2. Background

[5] In this section we present the main theoretical models for relative permeabilities in a single fracture. Then, we discuss some experimental results from the literature.

2.1. Theoretical Approaches

[6] The commonly used equations to model steady state laminar two-phase flow in a single fracture are the generalized Darcy equations. For the water phase,

equation image

To take the compressibility effect of the gas into account, the gas phase equation must be written in the following form [Scheidegger, 1974]:

equation image

where subscripts l and g stand for liquid and gas, respectively, pi and po are the pressures at the inlet and the outlet of the fracture, u is the superficial velocity or the Darcy velocity (flow rate per unit of cross-section area), μ is the dynamic viscosity, L is the fracture length, kabs is the absolute permeability, and krl and krg are the relative permeabilities of the liquid and the gas, respectively.

[7] The absolute permeability of a smooth-walled fracture is related to the fracture aperture b by [Yih, 1969, equation (341); Witherspoon et al., 1980]

equation image

The concept of the relative permeability provides us a means to quantify the relative resistance or interference between phases. For liquid-gas flow the sum of krl and krg indicates the extent of phase interference: the lower the sum of the relative permeabilities below 1, the greater the phase interference. The critical point of the generalized Darcy model is the determination of the relative permeabilities, which are usually supposed to be functions only of the saturation. When the pressure loss due to the interaction between phases is negligible against the pressure loss due to the flow of each fluid, the relative permeabilities can be modeled by the X curves:

equation image
equation image

where Sl and Sg are the liquid and gas saturation, respectively.

[8] According to the X model, the sum of krl and krg equals 1, which means the absence of phase interference. Physically, this implies that each phase flows in its own path without impeding the flow of the other. In fractures, if each phase flows via perfect straight channels along the flow direction with negligible capillary pressure and wetting phase stratified flow, then X curve behavior is possible. However, for two-phase flows through a real fracture the surface contact between the two fluids can be important, and consequently, the interference between the two fluids may be significant. As the fracture flow can be considered either as a limiting case of a flow in a porous medium [Pruess and Tsang, 1990] or as a limiting case of a pipe flow [Fourar et al., 1993], there are mainly two approaches for modeling relative permeabilities.

[9] In the porous media approach the fracture is treated as a connected two-dimensional porous medium where the pore space occupied by one phase is not available for the flow of the other phase. A phase can move from one position to another only upon establishing a continuous flow path for itself. The competition for pore occupancy is controlled by the capillary pressure. On the basis of this approach several numerical studies were performed [Murphy and Thomson, 1993; Rossen and Kumar, 1992; Mendoza and Sudicky, 1991; Pyrak-Nolte et al., 1992]. The main result is that the sum of kr is less than 1, and consequently, the X model is not suitable to describing the numerical calculations of the relative permeabilities as functions of the saturation. However, these studies do not establish theoretical relationships for kr. The shape of the curves obtained was more similar to the curves obtained in classical porous media, namely, the Corey curves [Corey, 1954]:

equation image
equation image

In these equations, S* is the saturation, defined by

equation image

where subscript r refers to residual saturation. The Corey model represents strong phase interference in comparison with the X model.

[10] The pipe flow approach is based on the observation that flow structures observed in a fracture show more similarity to the structures observed in a pipe than to those expected for a porous medium [Fourar and Bories, 1995]. These flow structures show strong interference between the two fluids flowing simultaneously. However, the real mechanisms are very difficult to model because the geometry of the interface between the fluids is unknown and one of the phases is generally discontinuous. Fourar and Lenormand [1998] assumed that the complexity of the real flow can be modeled, in a first approximation, by viscous coupling between the fluids. The fracture is then modeled by two parallel planes with a small aperture. The two fluids are flowing simultaneously, and the interface is assumed to be planar. Fluid l is considered as the wetting fluid and therefore is in contact with the walls, and fluid g (nonwetting) flows in between. The viscous coupling between fluids is derived by integrating Stokes' equation for each stratum. Identification of the established equations and the generalized Darcy equations leads to

equation image
equation image

where μr = μgl is the viscosity ratio.

[11] These equations show that the relative permeability of the nonwetting fluid depends on the viscosity ratio μr, and krg can be larger than unity when μr > 1 (lubrication effects). However, for gas and liquid two-phase flows, μr ≪ 1, and consequently, the second term on the right-hand side of equation (10) is negligible.

2.2. Experimental Approaches

[12] Several studies on two-phase flows in a fracture have been performed. Romm [1966] studied kerosene and water two-phase flow through an artificial fracture by using parallel glass plates. The surface of each plate was lined with strips of polyethylene or waxed paper. The strips divided the entire fracture into 10–20 narrow parallel fractures (2–3 mm width) with alternate wettability.

[13] Persoff et al. [1991] and Persoff and Pruess [1995] also performed experiments on gas and water flow through rough-walled fractures using transparent casts of natural fractured rocks. The study of Persoff et al. [1991] showed strong phase interference similar to that in porous media. In the case of rough-walled fracture at small flow rates, flow of a phase was characterized by having a localized continuous flow path that undergoes blocking and unblocking by the other phase.

[14] Diomampo [2001] performed experiments of nitrogen and water flow through smooth-walled artificial fractures. She also observed the intermittent phenomenon in her experiments. Furthermore, her results conform mostly to the Corey type of relative permeability curve. This suggests that flow through fractures can be analyzed by treating it as a limiting case of porous media flow and by using the relative permeability approach.

[15] Fourar et al. [1993] and Fourar and Bories [1995] studied air-water two-phase flow in a fracture constituted of two parallel glass plates (1 m × 0.5 m) with an opening equal to 1 mm. The injector consisted of 500 stainless steel tubes of 1 mm outside diameter and 0.66 mm inside diameter. Air and water were injected through alternating capillary tubes to achieve uniform distribution of the inlet flow. For all experiments, air was injected at a constant pressure, and its volumetric flow rate was measured by a rotameter and corrected to the standard pressure. Water was injected by a calibrated pump. At the outlet of the fracture the gas escaped to the atmosphere and the water was collected.

[16] The fracture was initially saturated with water, which was injected at a constant flow rate for each experiment. Air injection was then started and increased stepwise. When the steady state was reached for each flow rate, the pressure drop and the saturation were measured. The pressure drop was measured by a transducer, and the saturation was measured by using a balance method. Then, the fracture was resaturated with water, and the experiment was repeated several times at different liquid flow rates. This study has been extended to fractures constituted by bricks made of baked clay (30 cm × 14 cm) with different apertures (0.54, 0.40, and 0.18 mm).

[17] Results obtained in these various studies are presented in Figure 1 in terms of krg versus krl. As can be seen, experimental results obtained by Romm [1966] are well described by the X model; that is, the relative permeability of each phase equals its saturation. On the other hand, the relative permeablities obtained by Fourar and Bories [1995] are rather in accordance with the viscous coupling model. However, the results obtained by Diomampo [2001] are well described by the Corey model. Last, the results of the rough-walled experiments obtained by Persoff and Pruess [1995] show more interaction between phases than the other results. It is obvious that these previous studies show a diversity of behavior of relative permeabilities in fractures. Furthermore, less attention has been paid to study the relationship between flow structures and relative permeabilities. Although the flow structure from the experimental study performed by Romm [1966] is not reported, it is likely that it was a channel flow because of the fracture configuration (alternate wetting and nonwetting stripes). In this case, the interaction between phases is minimum, and consequently, the results are described by the X model. The experiments performed by Persoff and Pruess [1995] showed that the flow is characterized by having a localized continuous flow path that is undergoing blocking and unblocking by the other phase at certain flow ratios. This behavior is related to capillary effects, which are dominant in classical porous media. Consequently, the flow of one phase is strongly impeded by the flow of the other phase. In the experiments performed by Fourar and Bories [1995], several flow structures were observed: bubbles, fingering bubbles, drops and films. It seems that these flows are dominated by viscous forces.

Figure 1.

Compendium of previous measurements of relative permeabilities in fractures (Romm [1966] results are from kerosene-water flow; the rest were air-water or nitrogen-water flow).

[18] The purpose of this study is to visualize and discuss the two-phase flow behavior in a fracture. The obtained gas-water relative permeabilities are analyzed by examining the flow structures. Finally, a tortuous channel approach is developed to reproduce experimental results and to show the effect of the flow structures on the relative permeabilities.

3. Experimental Apparatus and Measurements

[19] In this study, gas-water flow experiments were conducted at 24°C. The fluids were pure nitrogen and deionized water. The water injection was controlled by a meter pump (Dynamax, SD-200, rate of 0.1–200 mL/min). Gas injection was controlled through a flow regulator (Brooks Instrument, Flow Controller Model 0151E), which was connected to a gas meter (Brooks Instrument, Flowmeter model 5850E, maximum rate of 200 mL/min). All measurements were electronic and digitized by using high-speed data acquisition system (DAQ, National Instrument, SCSI-1000 with PCI 6023E A/D board) and digital video recording system (Sony Digital-8 560X with Pinnacle Studio DV IEEE 1394 image capture card). The whole experimental system is illustrated in Figure 2, which shows the fluid supply, the fracture apparatus, data acquisition system, and digital image recording. In this section we first describe the fracture apparatus. Then we describe the pressure, flow rates, saturation measurements, and data processing method.

Figure 2.

Process flow diagram for air-water experiment.

3.1. Fracture Apparatus Description

[20] The fracture is created by a smooth glass plate on top of an aluminum plate confined by a metal frame bolted to the bottom plate. The aperture of the fracture was set to 130 μm by using stainless steel shims along the flow boundaries. The schematic view and photograph of the fracture apparatus are shown in Figure 3. An O-ring (Viton 1/8″ thick 2-272) was placed in between the glass and aluminum plate as a seal. The fluids enter into the fracture through two separate canals. Each canal has several ports drilled so that they align on the surface (see Figure 3). The surface of the fracture apparatus was designed such that there is an available 30.48 cm by 10.16 cm space for flow. Throughout this flow area, tiny pressure ports were drilled so as to minimize surface discontinuity. Four pressure ports were drilled along the fracture as shown in Figure 3. The two-phase fluid exits through a single outlet. The apparatus was designed to be of sufficient length to minimize the capillary end effects.

Figure 3.

Schematic diagram and picture of fracture apparatus.

3.2. Pressure Measurements

[21] Low-range differential transducers were used to measure the pressure difference through the fracture as well as the intermediate pressure and the two-phase outlet pressure. Two liquid differential transducers (Validyne Transducer, model DP-15, range 0–2 psig) were attached to four pressure ports inside the fracture to measure the pressure difference through the fracture. Another transducer (Validyne Transducer, model DP-15, range 0–5 psig) was attached to the middle point of the fracture. The fourth transducer (Validyne Transducer, model DP-15, range 0–5 psig) was attached to the two-phase outlet of the fracture apparatus. These transducers send electrical signals to the data acquisition system, which was monitored using the LabView® programmable virtual instrument software. The complete measurement configuration in the fracture apparatus is shown in Figure 2.

3.3. Flow Rates Measurements

[22] Aside from the known input rates, to obtain the instantaneous flow rates, a fractional flow ratio detector (FFRD) was designed and constructed as shown in Figure 4a. The FFRD was used to measure the outlet gas and water fractional flows, fg and fw.

equation image
equation image

where Qout,g is the outlet gas flow rate, Qout,w is the outlet water flow rate, and Qout,t is the outlet total flow rate. The principle of the FFRD is that different phases will have different refractive indices. A phototransistor (NTE 3038, NPN-Si, Visible) was installed inside the FFRD, producing different voltages when sensing different strengths of light. The water phase produces a higher voltage when flowing through the FFRD, as shown in Figure 4b. The gas and water phase flow ratios are obtained by determining the ratio of the number of gas and water signals. Once fg and fw are obtained at steady state condition, it is easy to calculate Qout,g in equation (11) by assuming that water flow rate remains constant from inlet to outlet of the fracture. This assumption can hold if the wetting phase water, flows in a continuous channel, and the water viscosity are much larger than the nonwetting phase. The calibration of the FFRD demonstrated a good linearity between measurement and actual values (Figure 5).

Figure 4.

Fractional flow ratio detector (FFRD) (a) schematic and (b) detected gas and water signal corresponding to different gas and water segments inside FFRD tubing.

Figure 5.

FFRD calibration (fluids are water and nitrogen gas; FFRD tubing inside diameter is 1.0 mm).

3.4. Saturation Measurements

[23] Still images were extracted from digital video recorded during the experiments. The photographs were processed in a Matlab® program. The program does quadratic discriminant analysis (QDA) to divide the pixels of the picture into three groups: water phase, gas phase, and the frame. The grouping is based on color differences. Saturation is calculated as total pixels of the liquid group over the sum of the gas and liquid groups. Figure 6 is a comparison of the gray scaled image produced by the QDA program and the original photograph from the digital camcorder. Pan et al. [1996] also used a similar technique for measurement of saturation. Their study noted that the source of errors in this technique is the quality of the photographs and the water film adsorbed on the surfaces of the plates with the latter being of minimal effect. Good quality photographs are the ones with clear distinction between the gas and liquid phase. Good lighting is necessary so that the colors in the image come out clearly. The lighting should also be positioned in a way that it does not produce shadow on the flow area. The program will mistakenly take the shadow as gas phase even if there is liquid. From Figure 6 it can be said that the program has good accuracy.

Figure 6.

Comparison between the true color image of the fracture flow and gray scale image from the Matlab quadratic discriminant analysis program used in measuring saturation.

3.5. Data Processing

[24] The experiment contains several runs with designated input rates of gas and water. According to the air-water experiments of earlier studies, these fracture flow experiments are not expected to reach a perfect steady state. Instead, they are unsteady by nature. There are significant pressure fluctuations accompanied by saturation changes when the gas and liquid flow rates vary. Because of this behavior, instantaneous outlet rates of gas were also measured by the FFRD device for the relative permeability calculation and flow rate comparison. Digital video was taken in each run when the flow reached steady state or repeated similar fluctuating behaviors. The continuous images of each run were extracted from the video every 0.33 s. These hundreds of images in each run were then input into the computer programs for the saturation calculation, flow structure recognition, and characterization of the stability of the two-phase flow. The data acquisition task requires frequent gathering of instantaneous and synchronized pressure, flow rate, and saturation values. Instantaneous gathering of data was accomplished by the use of the high–speed data acquisition system and digital video camcorder. Video shots were taken of the pressure, time, and saturation data displayed all at the same time. Pressure data and related time were displayed by the LCD monitor connected to the computer, which also ran the data acquisition system. The methodology used to integrate all the data and signals and then calculate the gas-water relative permeability is illustrated in the flowchart in Figure 7.

Figure 7.

Data and signal processing flowchart.

4. Experimental Results

[25] We first describe the flow structures observed during the experiment and then present the relative permeabilities calculated at 1-s frequency in each run and their averaged values. Finally, we interpret the relative permeabilities using models from earlier studies.

4.1. Description of the Flow Structures

[26] The use of a transparent glass plate allows us to observe and videotape the two-phase flow structures. The recorded digital video was then transformed to a series of snapshots with the frequency of 3 images/s (0.33-s period). More than 3000 still images were extracted from the digital video and used for the flow structure characterization and saturation calculation. Several flow structures depending on the flow rates were observed. Generally, when the water saturation decreases, one observes successively bubbles, slugs, and channels (Figure 8).

Figure 8.

Photographs of flow structures in the smooth-walled fracture. Each set contains four continuous images. Gas is dark, and water is light. Flow direction was from left to right.

[27] The bubble flow is observed only at extremely high values of the water saturation Sw. Because of the small fracture aperture (130 μm), large gas bubbles propagate either in bullet-like or ellipse-like forms with greater longitudinal length (Figure 8a). As the gas flow rate increases or water flow rate decreases, gas moves in narrow slugs and flows discontinuously when the gas rate is still relatively low, as shown in Figure 8b. The higher pressure difference occurs mostly when a slug tries to break through the water region. Once the slug reaches the outlet of the fracture, the pressure difference decreases. This type of slug movement is seen frequently for high values of Sw. With the appearance of bubbles and slugs the two-phase flow is fairly unstable. When the longitudinal size of slugs increases as the gas rate increases, the moving slugs tend to form stable gas channels and reach a steady condition. However, the steadiness collapses either because water breaks the thinnest throat of the short-lived channel or because of the insufficient gas supply. This causes continuous fluctuations in the saturation, fractional flow, and pressure difference.

[28] As the gas rate increases further, the short-lived channel becomes more stable. The gas channel meanders through the fracture with branches and junctions because the viscous force is insufficient to break through intermediate water islands. We define this kind of channel as a tortuous channel, as shown in Figure 8c. The lifetime of these channels seems to depend on the complexity and meander of their structure. Some tortuous channels might exist just for a few seconds, whereas others exist for long periods. The Sw, fw, and pressure difference histories plotted in Figure 9 are obtained from a tortuous channel–dominated run. Clearly, even though constant gas and water rates are injected into the fracture, the Sw keeps fluctuating in the fracture, and the instantaneous fw sensed from the FFRD follows this saturation fluctuation as shown in Figure 9 (top). The corresponding pressure response was also recorded as shown in Figure 9 (bottom). Most of the peaks in these two plots are due to the collapse, reconstruction, and reconfiguration of channels and some intrusions from other minor flow structures (bubbles and slugs). It is also observed that the more tortuous the channel, the larger the pressure difference along the fracture for the same water saturation.

Figure 9.

Relationship among water saturation (Sw), water fractional flow (fw), and pressure difference along the fracture in a highly tortuous channel flow.

[29] On the other hand, straighter channels were observed more frequently in high fg situations shown in Figure 8d. In this situation, almost all of the gas flows solely in a fairly straight channel, while most of the water flows above and below the gas path. Except for a small amount of immobile water inside the central gas channel, the gas path is more uniform and less tortuous in comparison with the tortuous channel flow. Three major factors may affect the morphology of gas channels. These factors are fg (viscous force), gas-water viscosity ratio (viscous force), and the interfacial tension (capillary force). Therefore, for the same Sw, flows with the viscosity ratio closer to 1 or the smaller interfacial tension have straighter channels.

[30] Figure 10 shows a flow structure map for the experiment. The corresponding Sw is also provided. It is easy to see that the channel flow spans most of the Sw range and major bubble and gas slug flows only exist in high Sw situations. The shaded areas in the flow maps indicate transitional flow, which means the flow structure is a combination of the two neighboring structures. The transition between the slug flow and channel flow is defined as unstable channel flow. In this region some large fingering slugs were able to bridge the two ends of the fracture. However, the bridged slug collapsed in a short time because of its unstable structure, water intrusion, and the insufficient gas supply. The flow structure map is similar to the flow map (glass fracture) presented by Fourar and Bories [1995].

Figure 10.

Flow structure map for air-water flow in the smooth-walled fracture.

4.2. Relative Permeability Calculations

[31] The absolute permeability of the smooth-walled fracture was first measured in single-phase water flow. The absolute permeability of the fracture deduced from measurements by using the Darcy equation is 1.51 × 10−9 m2 (1510 darcies). The gap width (aperture) of the fracture is 130 μm, and the estimated permeability from equation (3) is then 1.41 × 10−9 m2 (1410 darcies). As can be seen in Figure 11, measured values are in a good agreement with the theoretical value of the absolute permeability.

Figure 11.

Absolute permeability of the fracture (aperture is 130 μm).

[32] Air-water relative permeabilities were calculated by using equations (1) and (2) at 1-s frequency in each run. The instantaneous flow rate was obtained from the FFRD. Comprehensive air-water relative permeabilities obtained from thousands of data points are plotted in Figure 12. Data points in Figure 12 show acceptable correlation. Comprehensive water relative permeabilities are scattered at high water saturation owing to the slug and unstable channel flows in the gas phase as shown in Figures 8a and 8b. The vertically scattered effect observed with the gas phase relative permeability under extremely low values of water saturation may be associated with either the pressure fluctuation due to the moving water slugs or the difficulty in sensing the instantaneous fw from the FFRD at low fw.

Figure 12.

Comprehensive air-water relative permeabilities in the smooth-walled fracture calculated from equations (1) and (2).

4.3. Relative Permeabilities Interpretation

[33] The averaged values of the relative permeabilities and the saturation with their standard deviation error bars for each pair of liquid and gas flow rates are presented in Figure 13. Also plotted in Figure 13 are the curves of the X model (equations (4) and (5)), the Corey model (equations (6) and (7)) with Srl = 0 and Srg = 0, and the viscous coupling model (equations (9) and (10)). It appears that the X model does not fit the experimental data. For a given value of Sw the krl and krg are below the straight lines. Although gas phase experimental values are closer to the Corey curve, water phase values are much higher than the water phase Corey curve. Therefore neither the Corey curve nor the X curve can fit the experimental result. On the other hand, the viscous coupling model is in good agreement with experimental results, especially for the water phase. For the gas phase the viscous coupling model shows acceptable fitting, but it seems to be less accurate at high water saturation. The photographs in Figure 8 provide some clue about this less accurate result. At high values of Sw most of the water channels extend from longitudinal boundaries of the fracture space and flow straighter and more continuously. On the other hand, the active gas flows as fingering slugs and channels with amorphous shapes. These changing shapes generate the corresponding pressure, saturation, and flow rate changes illustrated in Figure 9. The relative permeability may be affected by these changes that are not considered by the previous models (X, Corey, and viscous coupling). Further study of flow structure effects on relative permeabilities will be presented in section 5.

Figure 13.

Comparison of average experimental relative permeability with the Corey-curve, X curve, and viscous-coupling models. The error bars mean ± standard deviation of kr and Sw.

5. Effects of Flow Structures on Relative Permeabilities

[34] Figure 10 shows that the channel flow is the dominant flow structure in our experiments. We might expect that these flows would be described by the X model. Instead, they are rather in accordance with the viscous coupling model. This can be explained by characterizing the morphology of phase channels and quantifying the magnitude of the “tortuosity” in this flow structure. A new approach will be discussed here to include the tortuosity created by the phases.

5.1. Tortuous Channel Approach

[35] Considering a simplified fracture space with a dimension of L (length) × W (width) × b (aperture) as shown in Figure 14, the fluids, water and gas, form ideal and straight channels through the fracture. From fundamentals of multiphase flow the X curve behavior is readily derived from Darcy's law and the cubic law by assuming negligible capillary pressure, gas slippage, inertia effect, and wetting phase stratified flow.

Figure 14.

A simple model of a straight gas channel in a smooth-walled fracture.

[36] Consequently, the straight phase channel in the smooth-walled fracture will yield the X curve relative permeability as shown by Romm [1966]. However, as shown in this study, the actual flow structures in fractures seldom reach the ideal straight channel even though the fracture is smooth walled. Most of the flow structures are either fingering or tortuous channels instead, and the relative permeability obtained from this experiment shows a deviation from X curve as presented in Figure 13. As a result, a physical, tortuous channel approach to modify X curve and take into account the channel tortuosity is required.

[37] The concept of the tortuous channel approach is based on the relationship between the fluid tortuosity and the relative permeability. As the channel flow observed in our study is tortuous (Figure 8c), we define a tortuosity coefficient for the channel flow τc to characterize its morphology. The definition of this apparent parameter is based on the area of the channel and the smallest bounding rectangle that covers the whole channel for a specific phase. As shown in Figure 15, the binary images processed from continuous true-color images were input into another image-processing program, which allows recognition and separation of the different flow structures. Then, the channel area Ac (pixels2) and length and width of the smallest bounding box, Lx and Ly (pixels), were computed. The channel tortuosities for gas and water are then defined as

equation image

As for the relative permeability, coefficient τc for each phase varies between 1 and equation image. This coefficient is related to the interfacial area and, consequently, allows quantifying the shear stress at the interface between the two fluids.

Figure 15.

Evolution of channel tortuosity algorithm.

[38] In this study, thousands of continuous images extracted from the video of the experiment with a 0.33-s period were analyzed, and the tortuosities of gas and water were deduced for the channel flow configuration. Illustrations of these calculations are presented in Figure 16. It appears that water channels are less tortuous than gas channels. This may be due to the fact that the water viscosity is higher than the gas viscosity and that water channels tend to adhere to longitudinal boundaries of the fracture, while gas channels flow in between.

Figure 16.

Representative images and corresponding processed gas channel and water channel images extracted from the Channel's Area and Aspect Recognition (CAAR) image processing. In the original images, gas is white and water is black; in the phase channel images the phase interested is white. Rectangles indicate the bounding boxes of the phase channels.

[39] In order to characterize the deviation from the X model we propose the following relationships:

equation image
equation image

These relationships assume that the channel tortuosity is directly proportional to the deviation of X curve behavior. In the case where each phase flows in the form of homogeneous strata (no dispersed phase), τcw = 1 and τcg = 1 and equations (14) and (15) reduce to the X model.

5.2. Reproduction of Relative Permeabilities

[40] After deducing both water and gas tortuosities from the extracted images, the comprehensive water phase and gas phase relative permeabilities obtained from equations (14) and (15) are shown in Figure 17. The original experimental results (Figure 12) are also provided. The results from the tortuous channel approach show good reproduction of the experimental results in both water and gas phases. Thus the tortuous channel approach leads to low scatter in comparison with the experimental relative permeabilities. This is because some unstable phenomena, bubbles and slugs, were excluded and there are almost no sources of measurement errors in the tortuous channel approach, except the image processing errors.

Figure 17.

Relative permeabilities from tortuous channel approach using phase tortuosities obtained from the processing of continuous images and its comparison with the original result: (a) gas phase and (b) water phase.

[41] The mean tortuosity of gas and water channels was then computed by averaging all tortuosities in each run. Table 1 presents the mean channel tortuosity in each run for the experiments. The average phase tortuosity (reciprocal) versus the phase saturation is shown in Figure 18. Values of tortuosity reciprocal approach 1.0 when the channels increase straightness. A tortuosity value (or reciprocal) of 1.0 means perfect straight phase channels. From these two plots it is evident that the higher the phase saturation, the less tortuously those phase channels behave. The phase tortuosity values can be expressed by their phase saturations in second-order approximations. The relative permeabilities calculated by using the average tortuosity in Table 1 and equations (14) and (15) are presented in Figure 19 (squares and diamonds). They are in a good agreement with average experimental relative permeabilities for both gas and water phases. Aside from the generally matching performance, the tortuous channel approach not only approximates the experimental result but closely reproduces some experimental points, especially the portion where the viscous coupling model performs poorly.

Figure 18.

Reciprocal of average phase channel tortuosity versus phase saturation in the air-water experiment: (a) gas phase and (b) water phase.

Figure 19.

Comparison of the experimental relative permeability with the tortuous channel approach (averaging from Figure 17), tortuous channel model, and viscous coupling model for the air-water experiment.

Table 1. Averages of Tortuous Channel Parameter Obtained From Channel's Area and Aspect Recognition (CAAR) Image Processing Program and the Relative Permeability Values for the Tortuous Channel Approach and Experiment
SaturationAverage Channel TortuositiesExperiment ResultsTortuous Channel Approach
  • a

    No channel detected.


5.3. Comparison With Models

[42] To compare the proposed tortuous-channel approach with other theoretical and empirical models mentioned previously, simple curve fits to express the tortuosity of phase channels mathematically were done in this preliminary stage. The optimal equations for fitting phase tortuosities as shown in Figure 18 are

equation image
equation image

This leads equations (14) and (15) to a tortuous channel model:

equation image
equation image

This tortuous channel model was compared with the best model in Figure 13, the viscous coupling model. As shown in Figure 19, these two models perform equally accurate for the water phase. For the gas phase the tortuous-channel model fits the experimental data better in high Sw range. On the other hand, the viscous coupling model shows better performance in low Sw range. This model transition may indicate that the dominant force shifts from the x-z plane shear stress to x-y plane shear stress, using the coordinate system as illustrated in Figure 14. Considering the gas phase, except in the low Sw range, most of the force contributed by the flow is applied to exploring and sustaining the gas region horizontally (in the x-y plane). Therefore the magnitude of the x-y plane shear stress was relatively small and negligible. This leads to the suitability of the tortuous channel approach. However, when most of the fracture space was occupied by the gas at low Sw, less force was used to sustain the gas region horizontally (x-y plane) since the gas channels were wider and straighter. Instead, most of the force was contributed by the x-y plane shear force because of the importance of the vertical (x-z) stratified flow at this stage. This will be close to the viscous coupling behavior.

[43] To compare these models visually and quantitatively, the comparison between measured and predicative water saturation and relative permeabilities was made using equations (4)–(10) and (18) and (19). The result is shown in Figure 20. The statistical analysis of the absolute errors and standard deviations between experimental and modeling relative permeabilities and saturations is also presented in Table 2. Apparently, both the viscous coupling model and tortuous channel model can interpret and predict the relative permeability and water saturation with good accuracy, whereas both the X curve and Corey curve models show worse performance. From the statistical mean absolute error and standard deviation the tortuous channel model performs slightly better than the viscous coupling model in predicting both relative permeability and water saturation. To sum up, from validations of both the comprehensive and average results and model comparisons, the proposed tortuous channel approach can interpret the relative permeabilities at the air-water experiment successfully. This tells us that the relative permeability is flow structure-dependent. At the moment, the flow structure prediction is a complicated and immature subject.

Figure 20.

Comparison between measured and calculated (a) water saturation and (b) relative permeability.

Table 2. Statistical Analysis Descriptors of Relative Permeabilities and Water Saturation Between Experiment and Models
X curve0.16330.09200.16330.0920
Corey curve [Corey, 1954]0.12070.11470.13130.1008
Viscous coupling modelc0.04960.03770.07290.0653
Tortuous channel model (this work)0.04850.03840.05050.0325

6. Conclusion

[44] Experimental results of air-water flow through a smooth-walled fracture were presented. Using a high-speed data acquisition system and digital image processing technology, the experimental apparatus was able to characterize the unstable nature of the flow.

[45] Different flow configurations were visualized: bubbles, slugs, and channels. Experimental results of pressure drop, flow rates, and saturation were interpreted by using the relative permeability concept and compared to three models from the literature: the X model, the Corey model and the viscous coupling model. The theoretical viscous coupling model provides the better estimate of the gas and the water relative permeabilities. However, this model does not take into account the real flow structures.

[46] By introducing the tortuous channel concept we showed that the deviation from the X model is due to the flow configuration. The proposed tortuous channel approach has reproduced both original and average experimental relative permeabilities. After applying mathematical expressions of phase tortuosity coefficients, the preliminary tortuous channel model performed best among other models. Therefore the concept of the tortuosity coefficient may allow modeling two-phase flow in a fracture whatever the flow configuration. To this end, the coefficient τc must be related physically to the shear stress at the surface between the two fluids and to the interface area. Unfortunately, up to now, there has not been a model that allows us to predict flow structures in a fracture. It was also noticed that the values of the gas phase relative permeability from the experiment tend to shift from the proposed tortuous-channel model to the viscous-coupling model in the low water saturation range, which may be because the vertical stratified flow is no longer negligible in this situation. It must be mentioned that the feasibility of the tortuous channel model was demonstrated by using experimental results obtained with smooth-walled fractures only. Consequently, more experiments are required, particularly with rough-walled fractures, in order to generalize the use of this approach. To this end, some tomography techniques have to be utilized to image the two phase flow field, which are believed to be a major challenge for the future research. Nevertheless, the proposed tortuous channel model may provide an upper bound of the air-water relative permeability at room temperature for simulations of fractured reservoirs because of their smooth surfaces, negligible capillary effects, and absence of fracture-matrix interaction. On the other hand, the X curve behavior was not achieved, even in smooth-walled fractures.


[47] Financial support for this research was provided by the U.S. Department of Energy under contract DE-FG07-02ID14418. The authors would like to thank Kewen Li for helpful discussions.