Correlation between pressure gradient and phase saturation for oil-water flow in smooth- and rough-walled parallel-plate models

Transcription

1 WATER RESOURCES RESEARCH, VOL. 44, W02418, doi: /2007wr006043, 2008 Correlation between pressure gradient and phase saturation for oil-water flow in smooth- and rough-walled parallel-plate models R. C.-K. Wong, 1 X. Pan, 1 and B. B. Maini 1 Received 15 March 2007; revised 16 October 2007; accepted 30 October 2007; published 13 February [1] The present study is aimed at an investigation of the pressure gradient/phase saturation correlation in the flow of an oil-water mixture through a parallel-plate (single fracture) model. It was found that the correlation depends on the interaction between the two flowing fluids, or flow pattern, which is in turn governed by the injection method, water flow rate, viscosity ratio, fracture aperture, and fracture surface roughness. Three distinct flow patterns were identified, namely, channel, dispersed, and mixed flows. Measurements of pressure gradient and phase saturation suggest that the widely used Romm s relative permeability relationship is applicable to channel flow in which oil and water phases are continuous in the fracture. For dispersed or mixed flow in which either the oil or water phase is discontinuous, a Lockhart-Martinelli type correlation developed for gas-liquid flow in pipes should be valid at low viscosity ratio but not at high viscosity ratio and water flow rate. Citation: Wong, R. C.-K., X. Pan, and B. B. Maini (2008), Correlation between pressure gradient and phase saturation for oil-water flow in smooth- and rough-walled parallel-plate models, Water Resour. Res., 44, W02418, doi: /2007wr Schulich School of Engineering, University of Calgary, Calgary, Alberta, Canada. Copyright 2008 by the American Geophysical Union /08/2007WR W Introduction [2] Immiscible two-phase flows in fractured media are encountered in many engineering processes such as recovery of oil and gas, exploitation of geothermal energy, and contamination of groundwater by immiscible chemicals. Fluid flow in fractured media is very complex. The highly permeable pathways formed by fracture networks often dominate fluid flow. To have a better understanding of fluid transport in complicated fracture networks, one has to understand the physics of fluid flow in a single fracture first. Previous studies [e.g., Louis, 1969; Witherspoon et al., 1980; Barenblatt et al., 1990; Zimmerman et al., 1991; Brush and Thomson, 2003] have demonstrated that the cubic law is applicable for single-phase flow through synthetic smooth- and rough-walled, loosely mated and tight fractures. However, Konzuk and Kueper [2004] observed that the cubic law applied locally might overpredict the observed flow rate in stress-induced fractures due to the effect of fracture surface undulation and abrupt aperture change. They concluded that further studies of the cubic law applied at the single-fracture scale for different rock types are required. [3] The air-water flow system in a fracture represents a particular extreme of two-phase immiscible flow systems characterized by high density and viscosity ratio. Pyrak- Nolte et al. [1990] and Persoff and Pruess [1995] conducted experiments on air-water flow in natural rough-walled rock fractures. They observed that a very stable flow pattern occurred in tests in which the capillary forces controlled phase distribution. In situations where viscous forces dominated fluid flow, unstable flow regimes were induced, in which the interfaces between the discontinuous and continuous phases were constantly changing and moving. Fourar and Bories [1995] and Chen et al. [2004] also observed different flow structures in air-water flow experiments in both smooth-walled and rough-walled fractures. They found that the summation of air and water relative permeability values was less than unity. [4] Romm [1966] conducted tests on immiscible flow of water and kerosene in a parallel-plate fracture in which strips of polyethylene film were inserted to control the wettability at different parts of the fracture surface. He reported a linear dependence of relative permeability on saturation, and the sum of two-phase relative permeability values equal to unity. This implies that neither phase interferes with the flow of the other. Owing to the lack of experimental data, Romm s linear correlation has been widely used in fractured reservoir simulations. Results from recent laboratory experiments of oil-water flow in parallelplate models [Merril, 1975; Pan et al., 1998] indicated that under a viscous force controlled condition, the interference between two phases flowing concurrently in a fracture could be strong at high flow rates. At intermediate saturations, the sum of wetting and nonwetting phase relative permeability values was found to be much less than unity. In addition, in high water saturations and flow rates, the oil phase became discontinuous and unstable with its structures changing with the continuous flowing water phase. Application of the relative permeability concept based on Darcy s law to the discontinuous flowing phase becomes questionable. [5] The behavior of two-phase flows in pipes has been studied in applications of chemical engineering. Compared with gas-liquid flows, few studies in liquid-liquid systems are available. Lockhart and Martinelli [1949] were probably the first researchers to recognize that there exist unique semiempirical correlations between pressure gradient and 1of16

2 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 1. Schematic diagrams showing (a) cross section of the fracture model and (b) experimental setup of fracture flow. phase saturation for gas-liquid flows in pipes. Fourar et al. [1993] studied gas-liquid flows in rough-walled fractures using porous medium and pipe flow models. Charles and Lilleleht [1965] and Stapelberg and Mewes [1994] used the Lockhart-Martinelli (L-M) type correlation to study twophase liquid-liquid flows in rectangular pipes. They found that the L-M type correlation should be valid at high viscosity ratios and at low input fractions of the more viscous phase. Chakrabarti et al. [2005] used the energy dissipation principle for analysis of pressure drops in liquidliquid flow in horizontal pipes. They reported that an L-M type correlation could not predict pressure drop well for liquid-liquid flows. [6] Two-phase flows in fractures are very different from those in porous media because the representative element volume is difficult to define in flows in fractures. Experimental studies on two-phase liquid-liquid flows in fractures are limited. Virtually no work has been reported to predict the pressure gradient and phase saturations in fractures. Very few studies have been carried out regarding flow patterns and their evolution. The present study is an experimental study on the characteristics of two-phase liquid-liquid flow in a parallel-plate model. The main objectives of the present work are to determine important parameters which affect steady state two-phase liquid-liquid flow in a fracture, and to study the interaction between immiscible fluids flowing in a fracture. The study also attempts to investigate the conditions under which the relative permeability concept and L-M type correlations are applicable. The first part of the paper describes the experimental setup details and testing procedures. The second part presents test results along with interpretation, and is followed by concluding remarks. 2. Experimental Program 2.1. Fracture Models and Fluids Used [7] The experiments reported herein were conducted in parallel-plate fracture models with smooth and rough surfaces. The model (Figure 1a) was constructed with two Plexiglass plates 38 mm in thickness with an adjustable aperture of zero to about 0.4 mm. The aperture of the fracture was controlled by inserting a shim stock of known thickness in the periphery of the fracture sealed by an O-ring. Three pairs of differential pressure measurement manifolds were installed in the bottom plate distributed evenly across the width of the fracture. Both the upstream and downstream pressure manifolds were installed at a distance of 100 mm away from the fluid inlet and outlet to avoid the end effect. The inlet and outlet could be equipped with removable sintered metal strips which were imbedded into the bottom plate so that the tops of the sintered metal strips were leveled with the surface of the bottom plate. These porous strips with a nominal permeability of 10 darcies were 99 mm in length, and 1 mm shorter than the width of the fracture. This feature allows different fluid mixing and injection methods for different flow patterns. The fracture horizontally placed has a flow area of 100 (width) 550 (length) mm 2 measured between the inlet and outlet. The effective testing area between the differential pressure ports was mm 2. 2of16

3 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 2. Picture showing a layer of glass beads of 0.22 mm nominal diameter glued to the fracture surface. [8] Two high-pressure liquid chromatography (HPLC) constant-rate pumps were used to inject oil and water into the fracture. After passing through a T-joint, oil and water were divided into four tubes which were connected to the inlet at the bottom plate. The fluids were collected at the outlet connected to another set of four tubes. A schematic diagram showing the connections between the fracture and the peripheral equipment is presented in Figure 1b. More details are given by Pan [1999]. [9] For the rough-walled fracture model, the basic structure is similar to that of the smooth-walled fracture model. The surfaces of the plates within the flow area were covered by a layer of glass beads with a nominal diameter of 0.22 mm. The beads, cleaned with hydrochloric acid solution and deionized water, were sprinkled by a sieve and glued to the plate surface using a thin layer of epoxy. A picture of part of one surface of the rough-walled fracture is shown in Figure 2. [10] Dyed distilled water was used throughout the experiments to facilitate visualization of the interaction between water and oil. Three mineral oils with different viscosities were used in the experiments. Oil-1 was a mixture of 50% Kaydal and 50% Deo-base (vol/vol) with a measured viscosity of 9.6 mpa s at room temperature. Oil-2 was a mixture of 95% Kaydal and 5% Deo-base (vol/vol) with a measured viscosity of mpa s. Oil-3 was a mixture of 85% Kaydal and 15% Deo-base (vol/vol) with a measured viscosity of 68.2 mpa s. Oil-3, a less viscous fluid, was used in some rough-walled fracture flow tests when the pressure buildup exceeded the equipment capacity. The physical properties of the fluids are given in Table Testing Procedure [11] Three test series (I, II, and RI; Table 2) were performed on three different fracture models. In test series I, the sintered metal strips were removed at the inlet and outlet and the fluids were injected into the smoothwalled fracture through the tubes directly. In test series II, the sintered metal strips were inserted at the inlet and outlet, and the fluids were injected into the smooth-walled fracture through the strips. This injection method simulates the field conditions in which the fluids enter into the fracture through a porous medium (sintered metal). In test series RI, the test setup was similar to that in test series II, except that the rough-walled fracture model was used. Each test series involved sets of flow tests at a constant different fracture aperture and viscosity ratio. Each set, in turn, consisted of single- and two-phase flow tests with different flow rates. [12] For each test set, the fracture was initially saturated with dyed water. First, a single-phase (water) flow test was conducted by injecting dyed water at a constant rate into the fracture. After steady state had been reached, the pressure drop and flow rate were measured three times and average values were recorded. The pressure drop and flow rate data were used to calculate the hydraulic aperture of the fracture from the cubic law. [13] Following the single-phase test, a two-phase oilwater flow test was initiated by injecting both fluids at the same time. The water injection rate was maintained constant throughout one test set, while the oil injection rate was increased from zero to a maximum designed rate in eight to 10 increments. Consequently, a wide range of saturation value was covered. During the tests, differential pressure was measured continually at the three pairs of pressure ports, and oil and water production rates were measured once a steady state had been reached. The evolving flow pattern was then videotaped, and pictures were taken with a camera. The steady state was diagnosed when the average of the pressure differentials did not change over a period of 30 min. Then, the experiment was continued at higher oil injection rates. The last test in each set was the single-phase oil flow test. The result was used to verify that of the singlephase water flow tests. [14] New sets of flow tests were conducted with different oil fluids and at different fracture apertures. Table 2 lists the details of test series and sets in three different fracture models. [15] Since the fracture was transparent and the fracture surfaces were separated by a constant width, we were able to determine saturation from measuring the area occupied by each phase from the photographs taken during experiments. Image analysis was used for this purpose, i.e., the saturation of that phase was calculated from dividing the measured value by the total area of the photograph. The effectiveness of this method is largely dependent on the resolution of the photograph and flow pattern. In our experiments, the measurement was more accurate at higher water phase saturation because the fluid interfacial edge Table 1. Physical Properties of the Fluids Used Viscosity, mpa s Density, kg/m 3 Interfacial Tension, mn/m Water Advancing Angle Dyed water Oil Oil Oil Water/oil Water/oil Water/oil of16

4 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Table 2. Details of the Flow Tests Test Set/Series Hydraulic Aperture, mm Water Injection Rate, ml/s Oil Injection Rate, ml/s Oil-Water Viscosity Ratio Reynolds Number, Re a Capillary Number Ca, b 10 4 Maximum Water Velocity, c 10 3 m/s Smooth-Walled Fracture IA : IB : IC : ID : IE : IIA : IIB : IIC : IID : IIE : IIF : IIG : IIH : Rough-Walled Fracture RIA : RIB : RIC : RID : RIE : RIF : RIG : RIH : RIJ : RIK : a Re = 2bv w r w /m w. b For two-phase flow, Ca = v w r w /IFT or krp/ift; the first definition was used herein because the second one yielded higher Ca values. c Water velocity estimated at single-phase flow. was vertical and clear. With increasing oil saturation, the flow pattern became chaotic and sometimes phase stratification occurred. This resulted in poor phase saturation measurements. [16] As an alternative to the photographic method and to calibration of phase saturation measurement in roughwalled fractures, the electrical resistivity method was used in the tests. The electrical resistance of the immiscible fluid was monitored using two brass strips installed on one of the parallel plates (see Figure 1a). This method is based on an established linear correlation between water phase saturation and resistivity on a log-log scale. The resistivity index is defined as the ratio between resistivity of the two-phase fluid and water only. Since this relies on the presence of an electrolyte, the dyed water was replaced by a 2% sodium chloride solution. At higher water saturation, a very stable and linear correlation between logarithm values of water saturation and resistivity index was obtained for all of the tests [Pan, 1999]. At low water saturation, however, the resistivity measurements became unstable due to the constantly changing oil/water interface and sometimes water fingering. An average value was taken to calculate the resistivity index. 3. Results and Discussion 3.1. Single-Phase Flow [17] The most commonly used model for flow in a single fracture is the cubic law, in which the fracture is treated as two parallel plates separated by a constant aperture. The equation for one-dimensional single-phase flow through a parallel-plate fracture is given as Q ¼ b3 12m rp; where Q is the volumetric flow rate per unit width; b is the aperture of the fracture; rp is the pressure gradient; and m is the dynamic viscosity of the fluid. Comparison between equation (1) and Darcy s law reveals that the absolute permeability of a parallel-plate fracture, k, is equal to b 2 /12. [18] A general dimensionless expression of equation (1) can be obtained by introducing two dimensionless parameters, Reynolds number (Re) and friction factor (f), as defined as follows: Re ¼ rvd h m ð1þ ð2þ f ¼ D h 1 2 rv2 rp; ð3þ where v is the velocity, D h is the hydraulic perimeter, and r and m are density and dynamic viscosity of the fluid, respectively. For a smooth-walled fracture, D h becomes 2b, and equation (1) can be written in terms of Re and f: f ¼ 96 Re : ð4þ 4of16

5 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 3. Comparison between theoretical and experimental results of friction factor versus Reynolds number for smooth-walled and rough-walled fractures with different fluids. The physical meaning of the Reynolds number is the ratio of inertial to viscous forces in the flow system. Therefore the Reynolds number clarifies the flow regime. As Re increases, the inertial forces become more significant, and the friction factor starts deviating from the above relationship. This deviation is in the direction of the friction factor becoming higher than that predicted by equation (4). As turbulence develops, the decrease of friction factor with increasing Reynolds number becomes much slower than equation (4). It is well known that the transition from laminar to turbulent flow in a pipe takes place at a Reynolds number of about 2100 [Rott, 1990]. The critical Reynolds number defining the transition from laminar to turbulent flow in a smooth-walled fracture is about 500 [Romm, 1966]. For single-phase flows in porous media, the noninertial flow regime to inertial flow regime occurs at about Re = 1[Greenkorn, 1983]. [19] The smooth-walled parallel-plate model can be considered only as a qualitative description of a natural fracture. Natural fracture surfaces are rough with some discrete contact areas that increase the resistance to the flow. Louis [1969] introduced a surface relative roughness parameter to account for the reduction of flow rate due to surface roughness. Equation (4) for smooth-walled fracture flow is modified as follows: " f ¼ 96 # Re 1 þ 17:0 e 1:5 ; ð5þ where e is the absolute roughness of the surface. [20] Figure 3 presents single-phase flow test results from test sets IIE to IIG with a smooth-walled fracture aperture of mm. The fracture opening was measured based on the volume of fluid trapped in the fracture. Reynolds numbers and the friction factor were calculated from equations (2) and (3), respectively. Excellent agreement was found for the D h tests with three fluids (water, oil-1, and oil-2). Figure 3 also includes test results from the rough-walled fracture flow test series of RI with a mean aperture of mm. A relative roughness parameter e/d h is to account for the effect of the surface roughness. The parameter e is equivalent to half of the bead diameter (0.11 mm used in this study). As shown in Figure 3, there is a very good agreement between the experimental data and equation (5). However, it is also observed from Figure 3 that the friction factor starts to deviate slightly from the linear relationship at Re > 1. However, this minor deviation is in the opposite direction of what would be expected from any contribution of the inertial forces, and is most likely due to measurement errors. Thus it can be suggested that the inertial forces did not become significant in these experiments. The Reynolds number in our experiments varied from 1 to 5 (Table 2). Konzuk and Kueper [2004] reported that the measured flow rates in stress-induced fractures deviated significantly from those predicted by cubic law when the flow regime was beyond Re > 10. They suggested that the deviation might be due to the tortuosity effect induced by fracture surface undulation and an abrupt aperture change in natural rock fractures Two-Phase Flow Smooth-Walled Fracture Model: Test Series I [21] In this test series, oil and water were injected into the smooth-walled fracture model through separate tubes (with the sintered metal strips at inlet and outlet removed). It was observed that in this test series, oil and water were flowing in channels. Some oil droplets might be formed in water channels. The width of the oil channel is large enough so that half of the oil flows through its own channels. A typical picture of a channel flow pattern is shown in Figure 4a. Light and dark areas represent water and oil phases, respectively. 5of16

6 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 4. Pictures of (a) channel flow, IA, (b) dispersed flow with oil droplets, IIA, (c) dispersed flow with oil islands, IID, (d) dispersed flow with encapsulated oil, IIG, (e) mixed flow with oil in water, IIB, (f) mixed flow with water in oil, IIC, and (g) mixed stratified flow, IIE (light and dark areas denote water and oil phases, respectively; scale: vertical dimension = 90 mm). [22] Since the dominant flow patterns observed in this test series are channel flows, the relative permeability concept could be used to study the steady state flow of each continuous phase in a fracture. The conventional relative permeabilities to water and oil, k rw and k ro, respectively, can be defined by the extended Darcy s law: v w ¼ b3 k rw 12m w rp v o ¼ b3 k ro 12m o rp; where v w, m w,v o, and m o are velocity and viscosity values of water and oil, respectively; b is the hydraulic aperture of the fracture; and rp is the time-averaged pressure gradient of two-phase flow. Relative permeability and saturation data for test series I are plotted in Figure 5, along with Romm s relationship in which the sum of relative permeability values to oil and water (k ro and k rw ) equals unity. Figure 5 ð6þ ð7þ illustrates that each phase relative permeability increases nonlinearly with its increasing saturation. Relative permeabilities of oil and water in the smooth-walled fractures are not equal to the individual saturation of each phase as predicted from Romms relationship. The summation of k ro and k rw is less than unity, which is due to phase interference during the concurrent oil-water channel flow. The phase interference observed in this test series should not be due to capillary action or immobility of the trapped fluids between fracture walls. The capillary number in our experiments varied from 10 4 to 10 3 (Table 2). The capillarity in porous medium or rock fracture is insignificant when the capillary number is of the order of 10 4 or larger [e.g., Melrose and Brandner, 1974; Longino and Kueper, 1999; Amili and Yortsos, 2006]. Thus the phase interference observed in this test series is due to the presence of another fluid affecting the flow of the other. The greater the extent of mixing between two fluids, the more significant would be the interference. [23] Lockhart and Martinelli [1949] introduced several parameters for correlation of flow test data for isothermal 6of16

7 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 4. (continued) two-phase flow in pipes. These L-M parameters, f o, f w, and X, are defined as follows: f 2 o ¼ rp TP rp o f 2 w ¼ rp TP rp w ð8þ ð9þ X 2 ¼ Q a o m b o r c o ; ð10þ Q w m w r w where rp TP is the pressure gradient of two-phase flow in the pipe; rp o is the pressure gradient of single-phase oil in the pipe; rp w is the pressure gradient of single-phase water in the pipe; Q o /Q w (= r) is the ratio of oil to water flow rate; m o /m w (= l) is the ratio of oil to water viscosity; r o /r w is the ratio of oil to water density; and a, b, and c are empirical correlation constants. In our flow experiments, three pairs of differential pressure measurement manifolds were used to monitor the average pressure gradient of the flow in the fracture. The total pressures were measured at the manifold ports, and no capillary barrier device was used to measure pressures of discontinuous and continuous phases separately. On the basis of Laplace s equation, the capillary pressure could be as high as up to about 0.9 kpa for the test series of IE with a 45-mm fracture aperture (assuming that the oil droplet radius is half of the fracture aperture). The measured minimum total pressure in this test series was about 14 kpa. Thus the pressure in the oil phase is higher than that in the water phase by about 6%. [24] The oil and water densities are comparable in this study, and thus the X-parameter is assumed to be a function of r and l only. Saturation and pressure gradient data of test series I are presented in Figures 6, 7, and 8. Correlations show that the exponent coefficients, a and b, fall in a narrow range of independent of viscosity ratio, flow rate ratio, and aperture. In this case, it is possible to combine flow rate and viscosity ratios into the X-parameter. Pan [1999] analytically derived L-M type correlation equations for channel flow in a smooth-walled parallel-plate model. The equations can be approximated by S w ¼ 1 1 þ X m ð11þ f 2 o ¼ 1 þ 1 X n : ð12þ 7of16

8 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 5. Relative permeability curves for oil and water in smooth-walled fractures (test series I; open and solid symbols represent tests with low- and high-viscosity oils, respectively). The theoretical values of m and n are about 1.7 and 2.0, respectively, which are comparable to those of correlations shown in Figures 6 and 7. This illustrates that an L-M type correlation could be valid for channel flow in smoothwalled fracture Smooth-Walled Fracture Model: Test Series II [25] The testing conditions in this test series were similar to those in test series I, except that oil and water were injected into the smooth-walled fracture model through the separate tubes and sintered metal strip at inlet. This injection Figure 6. Correlation between water saturation and oil-water flow rate ratio in smooth-walled fractures (test series I; open and solid symbols represent tests with low- and high-viscosity oils, respectively; numbers in legend indicate fracture aperture in mm; r = oil-water flow rate ratio, l = viscosity ratio). 8of16

9 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 7. Correlation between oil pressure gradient and oil-water flow rate ratio in smooth-walled fractures (test series I). method simulates the fluid flow through and from a porous medium (sintered metal) into the fracture. Different flow patterns (Figure 4) and evolution (Figure 9) were observed in this test series as compared with those observed in test series I in which channel flow is dominant. [26] Dispersed flow is a flow pattern in which the oil flows predominantly in the form of small droplets and/or large islands dispersed in the continuous water phase. Oil droplet flow occurred in all series II tests at low oil saturation (Figure 9). Droplets in sizes of several millimeters were formed within the continuous flowing water phase at high velocity (Figure 4b). In many situations, they were found in coexistence with other flow patterns, usually in continuous water flow pathways. Because of their size, Figure 8. Correlation between water pressure gradient and oil-water flow rate ratio in smooth-walled fractures (test series I). 9of16

10 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 9. Correlation between oil pressure gradient and oil-water flow rate ratio in smooth-walled fractures (test series II). oil droplets were more stable than their large counterparts. The shape of the oil droplets, especially those with high viscosity (oil-2), was round and did not change along their flow paths. Oil droplet flow accounted for almost the entire oil production in this flow pattern. [27] Oil island flow is characterized by the preponderance of large and disconnected oil blobs with the shape of islands dispersed in the continuous water phase, as shown in Figure 4c. It is similar to oil droplet flow in every respect except they are substantially larger in size and flowing slower than the droplet, i.e., mobility of discontinuous oil phase is size-dependent. This flow pattern was first observed and reported by Merril [1975]. Oil island flow dominated in tests (test sets IIA, IIB, IIE, and IIF) with less water flow rate and at low to intermediate oil saturation, especially with low viscous oil (oil-1). We did not observe large oil islands at high water flow rate and high oil viscosity. The shape and size of some oil islands changed during the flow as they were either split by water or collided with other oil islands or droplets to merge into even larger oil islands (test sets IIE and IIF, Figure 9). All of the oil islands were constantly moving, contributing remarkably to the flow of oil. [28] Encapsulated oil droplet flow is similar to the droplet flow in every respect except that the oil droplets are separated from the fracture walls by a water film so that they move faster than in oil droplet flow (see Figure 4d). It is like oil slug or droplet flow in a water-wet capillary tube in which a water film separates the oil from the wall. The existence of this flow pattern is supported by the observation that some oil droplets (less than 2 mm) moved at or close to the velocity of water (test sets IIC and IID; Figure 9). These droplets were formed from breakup of large oil droplets or islands by fast flows. The encapsulated oil droplets were stable in the flow, and the development of larger droplets or islands through collision or coalescence was not observed. A combination of high water flow rate, intermediate to high flow rate ratio, and high viscosity ratio favored the development of this flow pattern (test sets IIG and IIH; Figure 9). It coexisted with droplet flow at intermediate to high oil saturation for high water flow rate and high viscosity ratio. In such cases, the droplet sizes were comparable to the aperture size. [29] Mixed flow is a flow with an ill-defined and complex interface geometry. Oil-in-water mixed flow is marked by an irregularly shaped oil phase dispersed by water fingers, but still disconnected by the continuous water phase (Figure 4e). This pattern evolved either from oil droplet or island flow as the oil flow rate has increased (test sets IIA, IIB, and IIF; Figure 9), and dominated at intermediate saturation in all tests. Although there were some oil droplets and/or islands in this flow pattern, the predominant feature of it was the frequent splitting and coalescing of the disconnected oil blobs as well as the constantly changing shapes. [30] Water-in-oil mixed flow denotes a pattern in which a continuous water phase becomes disconnected and dispersed in the continuous oil phase (Figure 4f). This pattern happened at high oil saturation only and with oil of high viscosity (test sets IIC, IID, and IIH; Figure 9). Not all of the water phase was dispersed in the oil phase. At a sufficiently high oil flow rate and low water flow rate, water would be completely dispersed in continuous oil phase. In addition, there were many tiny oil droplets flowing along with the disconnected water phase. The water phase dispersed in the oil phase still had a higher mobility than the oil surrounding it, as evidenced by the tiny oil droplets flowing at the back of the water blobs. [31] Stratified mixed flow is characterized by the existence of a water film separating oil from the wall(s) by 10 of 16

11 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 10. Correlation between water saturation and oil-water flow rate ratio in smooth-walled fractures (test series II; open and solid symbols represent tests with low- and high-viscosity oils, respectively; numbers in legend indicate fracture aperture in mm; r = oil-water flow rate ratio, l = viscosity ratio). flow beneath or both on top of and beneath the oil phase (Figure 4g). This pattern was observed in the tests under the combined conditions of low water flow rate and intermediate to high oil saturation for both oils (test sets IIE and IIG; Figure 9). The stratified flow pattern was observed only in oil channels or paths where very large oil islands flowed. This flow pattern was clearly visible in large-aperture fractures, but not in small-aperture fractures. [32] Water and oil phases were not flowing in continuous channels in this test series. Thus the relative permeability concept is not applicable, and an L-M type correlation will be used for data interpretation. Saturation and pressure Figure 11. Correlation between water pressure gradient and oil-water flow rate ratio in smooth-walled fractures (test series II). 11 of 16

12 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 gradient data of test series II are plotted against the flow rate ratio in Figures 9, 10, and 11. Correlations show that the exponent coefficients, a and b, are much less than those determined in test series I. In addition, coefficient a is less than coefficient b so that expressing the rate (r) and viscosity (l) ratios in terms of the X-parameter is not possible. From Figure 10, saturation data yield two correlation curves for low and high viscosity ratios, respectively. The effects of water flow rate and aperture display an insignificant effect on saturation curve. [33] Oil pressure gradient data plotted in Figure 9 clearly illustrate that test series II data deviate from test series I data. The flow pattern and evolution observed in test series II included in Figure 9 are very different from those observed in test series I. For flow tests with low oil viscosity, the f o values in test series II are higher than those in test series I, implying that more energy is dissipated in discontinuous flow than in continuous channel flow. However, the differences are small, though the flow patterns in the two test series are different. The fracture aperture and flow rate have a minor enhancing effect on the pressure loss. The pressure drop in the fracture becomes lower when the aperture and water flow rate increase. For flow tests with high oil viscosity, the f o values in test series II are lower than those in test series I. They even drop below unity in high flow rate ratios where the dispersed flow pattern evolved from dispersed flow into an encapsulated and mixed flow pattern. In such flow patterns the flowing oil phase was sandwiched or stratified by water films on fracture walls, and these water films acted as a lubricant. The effect of lubrication increases with increasing water flow rate. Such a lubrication phenomenon was reported in studies on two-phase flows in porous media, fracture, and pipe studies [e.g., Odeh, 1959; Charles and Redberger, 1962; Rose, 1988; Kalaydjian, 1990; Zimmerman et al., 1991; Avraam and Payatakes, 1995; Pan, 1999]. The lubrication phenomenon or the remarkable increase of the mobility of oil is due to viscous drag or coupling in the concurrent immiscible flow system. The physical meaning behind viscous coupling can be understood by considering the flow characteristics that occurred in an idealized case of sandwich flow. Because of the continuity of velocities at the interface, the flow of water near the walls is actually helping the flow of oil at the center plane by imparting its maximum velocity to the oil. If there is no pressure gradient in oil, i.e., oil is in a discontinuous phase, it will still flow at the same velocity as that at the oil-water interface, which is the maximum velocity of the water. If, however, the oil forms a continuous phase, the existence of a pressure gradient in the oil induces a flow in itself so that the velocity within the oil phase is greater than, or at least equal to, the velocity at the oil-water interface. As a result, the oil flows faster than it would if it was alone in the fracture. [34] Water pressure gradient data of test series II shown in Figure 11 fall within a regime bounded by the two correlation curves for test series I. The pressure drop increases with increasing oil viscosity, fracture aperture, and water flow rate. [35] In our experiments the evolution of different flow patterns is largely dependent on two parameters, i.e., water flow rate and saturation (Figure 9). Different viscosity ratios may result in a shift of flow pattern boundaries, but will not lead to the evolution of new flow patterns. The basic mechanism behind the development of different flow patterns appears to be viscous dissipation. It has been postulated that the immiscible flow of the two liquids chooses an interface that minimizes viscous dissipation for a given flow rate maximizing volume flux for the applied pressure gradient [Joseph et al., 1984]. As a result, the less viscous liquid may break up the more viscous one encapsulating it in regions where shearing is greatest. The viscous dissipation principle may explain the transition from large oil islands to small droplets at fast flow rate, as well as the formation of a lubricated stratified flow at high oil saturation leading to the lubrication phenomenon mentioned above. However, it does not explain the situation in which the less viscous liquid is dispersed in a more viscous liquid as in the case of the water-in-oil flow pattern observed in our experiments. In this situation the frequent collision between oil islands and/or droplets dominates over the dispersion effect of water Rough-Walled Fracture Model: Test Series RI [36] The testing conditions in this test series were similar to those in test series II, except that the rough-walled plate model was used. In addition, two very different aperture sizes were used: 100 and 400 mm. Test sets with 100-mm aperture size (RIA to RIF) were conducted to evaluate the effect of surface roughness on flow characteristic as test series II had a comparable aperture size. Test sets with 400-mm aperture size (RIG to RIK) were used to investigate the effect of aperture size on the flow. [37] Saturation and pressure gradient data of test sets RIA to RIF are presented in Figures 12, 13, and 14. The surface roughness exerts an important influence on the flow pattern as well as on the saturation and pressure gradient. Different flow patterns and evolution (partially and fully mobilized droplet flow) were observed in rough-walled fracture experiments (test sets RIA to RIF) as compared with those observed in test series II. Partially mobilized droplet flow was the prevailing pattern at low oil injection rate regardless of water flow rate and viscosity ratio. Oil droplets of mm in size were stranded in some areas while mobile ones were restricted to high-permeability pathways. With increasing oil injection rate, all of the oil droplets started to flow, but at different velocities. [38] The effect of surface roughness has an influence not only on the flow pattern, but also on saturation and the pressure gradient. From Figure 12, the saturation measured in test sets RIA to RIF are higher than those measured in test series II (smooth-walled model). The saturation increases with decreasing viscosity ratio and increasing water flow rate. Though the flow patterns in smooth- and rough-walled experiments are different, the correlations between f o and Q o /Q w for both test series are comparable, particularly in low viscosity ratio (Figure 13). The lubrication mechanism was also observed in test sets with high viscosity ratio, but to a lesser extent as compared with those observed in smoothwalled models. This might be because the surface roughness reduces the thickness and uniformity of the thin water film, thereby slowing down the oil flow. [39] The water pressure drops in rough-walled models are much higher than those in smooth-walled models (Figure 14). It is interesting to note that all water pressure gradient data from test sets RI to RF fall in a narrow range, 12 of 16

13 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 12. Correlation between water saturation and oil-water flow rate ratio in rough-walled fractures (test series RIA-RIF; open and solid symbols represent tests with low- and high-viscosity oils, respectively; numbers in legend indicate fracture aperture in mm; lw, mw, and hw denote low, medium, and high water injection rates, respectively; r = oil-water flow rate ratio, l = viscosity ratio). implying that the water flow rate and the viscosity ratio have a minor effect on water pressure drop. [40] Saturation and pressure gradient data of test sets RIG to RIK of large aperture size are presented in Figures 15, 16, and 17. With a large aperture, the flow patterns in the roughwalled fracture became similar to those in the smoothwalled fracture (Figure 16). This is because the effect of fracture surface roughness on flow resistance becomes insignificant with increasing aperture. At large aperture, the water flow rate and viscosity ratio have an individual Figure 13. Correlation between oil pressure gradient and oil-water flow rate ratio in rough-walled fractures (test series RIA-RIF). 13 of 16

14 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 14. Correlation between water pressure gradient and oil-water flow rate ratio in smooth-walled fractures (test series RIA-RIF). Figure 15. Correlation between water saturation and oil-water flow rate ratio in rough-walled fractures of large apertures (test series RIG-RIK; open and solid symbols represent tests with low- and highviscosity oils, respectively; numbers in legend indicate fracture aperture in mm; lw, mw, and hw denote low, medium, and high water injection rates, respectively; r = oil-water flow rate ratio, l = viscosity ratio). 14 of 16

15 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 Figure 16. Correlation between oil pressure gradient and oil-water flow rate ratio in rough-walled fractures of large apertures (test series RIG-RIK). influence on the saturation (Figure 15) and pressure gradient (Figures 16 and 17) responses, i.e., higher variation was observed in the L-M type correlation. 4. Conclusions [41] Steady state two-phase oil-water flows were studied in smooth- and rough-walled parallel-plate models with aperture sizes of mm. Primary conclusions that can be drawn from the study are as follows: [42] 1. Flow pattern and evolution depend on fluid injection method, phase flow rate, viscosity ratio, fracture aperture, and roughness. [43] 2. Channel flow was observed only in tests using the separate injection method and the smooth-walled model. Dispersed flow (oil droplet, island, and encapsulated oil) Figure 17. Correlation between water pressure gradient and oil-water flow rate ratio in smooth-walled fractures of large apertures (test series RIG-RIK). 15 of 16

16 W02418 WONG ET AL.: PRESSURE GRADIENT-SATURATION CORRELATION IN FRACTURE W02418 and mixed flow occurred in other tests using the mixed injection method (through a porous medium). [44] 3. Viscosity ratio plays a more important role in saturation and pressure gradient behavior in two-phase flow as compared with other factors such as flow rate, flow rate ratio, fracture aperture, and surface roughness. [45] 4. The relative permeability concept could be applied in tests with a channel flow pattern in which oil and water phases are continuous. For other flow patterns with one discontinuous flowing phase, a Lockhart-Martenilli type correlation should be valid. However, this type of correlation would yield high variations at high viscosity ratio and high water injection rate. [46] 5. The fracture surface roughness affects the oilwater flow pattern, and thus the saturation-pressure gradient correlation. However, this effect becomes insignificant with increasing aperture. 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Bodvarsson (1991), Lubrication theory analysis of the permeability of rough-walled fractures, Int. J. Rock Mech. Min. Sci., 28(2), B. B. Maini, X. Pan, and R. C.-K. Wong, Schulich School of Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4. (rckwong@ucalgary.ca) 16 of 16