EXPERIMENTAL STUDY OF TWO-PHASE FLOW STRUCTURE EFFECTS ON RELATIVE PERMEABILITIES IN A FRACTURE

Transcription

1 PROCEEDINGS, Tenty-Ninth Workshop on Geothermal Reservoir Enineerin Stanford University, Stanford, California, January 6-8, 4 SGP-TR-75 EXPERIMENTAL STUDY OF TWO-PHASE FLOW STRUCTURE EFFECTS ON RELATIVE PERMEABILITIES IN A FRACTURE Chih-Yin Chen, Roland N. Horne and Mostafa Fourar. Department of Petroleum Enineerin, Stanford University, Stanford, California 9435, USA alnchen@stanford.edu, horne@stanford.edu. Ecole Nationale Supérieure des Mines de Nancy LEMTA, Parc de Saurupt, 544 Nancy Cedex France fourar@mines.inpl-nancy.fr ABSTRACT To-phase flo throuh fractured media is important in eothermal, nuclear, and petroleum applications. Hoever, the knolede of the relationship beteen flo behavior and relative permeabilities is limited. In this research, an experimental apparatus as built to capture the unstable nature of the to-phase flo in a smooth-alled fracture and display the flo structures under different flo confiurations in real time. The air-ater relative permeability as obtained from experiment and shoed deviation from the X-curve behavior suested by earlier studies. Throuh this ork the relationship beteen the phase-channel morpholoy and relative permeability in fractures as determined. A physical tortuouschannel approach as proposed to quantify the effects of the flo structure. This approach could replicate the experimental results ith a ood accuracy. Other relative permeability models (viscous-couplin model, X-curve model and Coreycurve model) ere also compared. Except for the viscous-couplin model, these models did not interpret the experimental relative permeabilities as ell as the proposed tortuous-channel model. Hence, e concluded that the to-phase relative permeability in fractures depends not only on liquid type and fracture eometry but also on the structure of the to-phase flo. INTRODUCTION Multiphase flos throuh fractured porous media are of reat importance in several domains such as petroleum recovery, eothermal enery and environmental enineerin. In the case of to-phase flo in a sinle fracture, this importance extends to several industrial processes such as compact heat exchanes and nuclear enineerin. Despite this importance, fe theoretical and experimental studies have been devoted to establishin the las overnin these flos. Thus, the results presented in the literature seem to be in contradiction. So far, the mechanisms of to-phase flos in a fracture are not ell understood and a eneral model for describin these flos is still not determined. The approach used commonly to describe to-phase flo in a fracture is the relative permeability concept, hich is based on a eneralization of the Darcy equation. Three models for the relative permeabilities are presented in the literature: the X-model (straiht lines), the Corey model, and the viscous-couplin model. The experimental results presented in the literature also sho different behavior for the relative permeabilities. Some results are in accordance ith the X-model hereas other results are in accordance ith the Corey-model or the viscous-couplin model. Althouh the flo structures are of a reat importance in modelin to-phase flos, the three models do not take into account these effects. The purpose of this paper is to examine the effects of the flo structures on the relative permeabilities durin to-phase flo in a sinle fracture. The paper is oranized as follos: in Section, e present a revie of the theoretical and experimental approaches for to-phase in a fracture. In Section 3, e describe the experimental apparatus and the measurements of pressure drops, flo rates, and saturations. In Section 4, e describe the observed flo structures and calculated relative permeabilities. In Section 5, e propose a tortuous-channel approach and discuss the flo structure effects on relative permeabilities. Finally, in Section 6, e outline the main conclusions of this study. BACKGROUND In this section, e present the main theoretical models for relative permeabilities in a sinle fracture. Then, e discuss some experimental results from the literature.. Theoretical Approaches The commonly used equations to model steady-state laminar to-phase flo in a sinle fracture are the eneralized Darcy equations:

2 kabskrl ( pi po ) l ul = () µ L u l k absk r ( pi po ) = () µ L here subscripts l and stand for liquid and as, respectively; p i and p o are the pressures at the inlet and the outlet of the fracture; u is the superficial velocity or the Darcy velocity (flo rate per unit of section area); µ is the dynamic viscosity; L is the fracture lenth; k abs is the absolute permeability and k rl and k r are the relative permeabilities of the liquid and the as, respectively. To take the compressibility effect of the as into account, Equation () must be reritten in the folloin form (Scheideer, 974): u kabsk r ( p po ) = (3) µ Lp i The absolute permeability of a smooth-alled fracture is related to the fracture aperture, b, by (Yih, 969: Eq. 34; Witherspoon et al., 98): b o k abs = (4) The concept of the relative permeability provides us a means to quantify the relative resistance or interference beteen phases. For liquid-as flo, the sum of k rl and k r indicates the extent of phase interference: the loer the sum of the relative permeabilities belo, the reater the phase interference. The critical point of the eneralized Darcy model is the determination of the relative permeabilities, hich are usually supposed to be only functions of the saturation. When the pressure loss due to the interaction beteen phases is neliible aainst the pressure loss due to the flo of each fluid, the relative permeabilities can be modeled by the X- curves: k rl = S l (5) k r = S (6) here S l and S are the liquid and as saturation respectively. Accordin to X-model, the sum of k rl and k r equals, hich means the absence of phase interference. Physically this implies that each phase flos in its on path ithout impedin the flo of the other. In fractures, if each phase flos via perfect straiht channels alon the flo direction ith neliible capillary pressure and ettin-phase stratified flo, then X-curve behavior is possible. Hoever, for tophase flos throuh a real fracture, the surface contact beteen the to fluids can be important and, consequently, the interaction beteen the to fluids may not be neliible. As the fracture flo can be considered either as a limitin case of a flo in a porous medium (Pruess and Tsan, 99) or as a limitin case of a pipe flo (Fourar et al., 993), there are mainly to approaches for modelin relative permeabilities. In the porous media approach, the fracture is treated as a connected to-dimensional porous medium here the pore space occupied by one phase is not available for the flo of the other phase. A phase can move from one position to another only upon establishin a continuous flo path for itself. The competition for pore occupancy is controlled by the capillary pressure. Based on this approach, several numerical studies ere performed (Murphy and Thomson, 99; Rossen et Kumar 994, Mondoza et Sudick, 99, Pyrak-Nolte et al. 99). The main result is that the sum of k r is less than and, consequently, the X-model is not suitable to describin the numerical calculations of the relative permeabilities as functions of the saturation. Hoever, these studies do not establish theoretical relationships for k r. The shape of the curves obtained as more similar to the curves obtained in classical porous media, namely the Corey curves (Corey, 954): *4 k rl = S (7) k r * * = ( S ) ( S ) (8) In these equations, S * is the saturation defined by: S * = ( S S ) /( S S ) (9) l here subscript r refers to residual saturation. The Corey model represents stron phase-interference in comparison ith the X-model. The pipe flo approach is based on the observation that flo structures observed in a fracture sho more similarity to the structures observed in a pipe than to those expected for a porous medium (Fourar and Bories, 995). These flo structures sho stron interference beteen the to fluids floin simultaneously. Hoever, the real mechanisms are very difficult to model because the eometry of the interface beteen the fluids is unknon and one of the phases is enerally discontinuous. Fourar and Lenormand (998) assumed that the complexity of the real flo can be modeled, in a first rl rl r

3 approximation, by viscous couplin beteen the fluids. The fracture is then modeled by to parallel planes ith a small aperture. The to fluids are floin simultaneously and the interface is assumed to be planar. Fluid l is considered as the ettin fluid and therefore is in contact ith the alls, and fluid (nonettin) flos in beteen. The viscous couplin beteen fluids is derived by interatin Stokes' equation for each stratum. Identification of the established equations and the eneralized Darcy equations leads to: k k rl r Sl = (3 Sl ) () 3 3 = ( Sl ) + µ rsl ( Sl )( Sl ) () here µ r = µ /µ l is the viscosity ratio. These equations sho that the relative permeability of the nonettin fluid depends on the viscosity ratio µ r. k r can be larer than unity hen µ r > (lubrication effects). Hoever, for as and liquid to-phase flos, µ r << and, consequently, the second term in the riht-hand-side of Equation () is neliible.. Experimental Approaches Several studies on to-phase flos in a fracture have been performed. Romm (966) studied kerosene and ater to-phase flo throuh an artificial fracture by usin parallel lass plates. The surface of each plate as lined ith strips of polyethylene or axed paper. The strips divided the entire fracture into to narro parallel fractures (-3 mm idth) ith alternate ettability. Persoff et al. (99) and Persoff and Pruess (995) also performed experiments on as and ater flo throuh rouh-alled fractures usin transparent casts of natural fractured rocks. The study of Persoff et al. (99) shoed stron phase interference similar to that in porous media. In these experiments, flo of a phase as characterized by havin a localized continuous flo path that underoes blockin and unblockin by the other phase. Diomampo () performed experiments of nitroen and ater flo throuh smooth-alled 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 suests that flo throuh fractures can be analyzed by treatin it as a limitin case of porous media flo and by usin the relative permeability approach. Fourar et al. (993) and Fourar and Bories (995) studied air-ater to-phase flo in a fracture constituted of to parallel lass plates (m x.5 m) ith an openin equal to mm. The injector consisted of 5 stainless steel tubes of mm outside diameter and 6 mm inside diameter. Air and ater ere injected throuh alternatin capillary tubes to achieve uniform distribution of the inlet flo. For all experiments, air as injected at a constant pressure and its volumetric flo rate as measured by a rotameter and corrected to the standard pressure. Water as injected by a calibrated pump. At the outlet of the fracture, the as escaped to the atmosphere and the ater as collected. The fracture as initially saturated ith ater hich as injected at a constant flo rate for each experiment. Air injection as then started and increased stepise. When the steady state as reached for each flo rate, the pressure drop and the saturation ere measured. The pressure drop as measured by a transducer and the saturation as measured by usin a balance method. Then, the fracture as resaturated ith ater and the experiment as repeated several times at different liquid flo rates. This study has been extended to fractures constituted by bricks made of baked clay (3 cm x 4 cm) ith different apertures (.54 mm, mm and mm). Results obtained in these various studies are presented in Fiure in terms of k r versus k rl. As can be seen, experimental results obtained by Romm (966) are ell described by the X-model, that is the relative permeability of each phase equals to its saturation. On the other hand, the relative permeablities obtained by Fourar and Bories (995) are rather in accordance ith the viscous-couplin model. Hoever, the results obtained by Diomampo () are ell described by the Corey model. Lastly, the results obtained by Persoff and Pruess (995) sho more interaction beteen phases than the other results. It is obvious that these previous studies sho a diversity of behavior of relative permeabilities in fractures. Furthermore, less attention has been paid to study the relationship beteen flo structures and relative permeabilities. Althouh the flo structure from the experimental study performed by Romm (966) is not reported, it is likely that it as a channel flo due to the fracture confiuration (alternate ettin and nonettin stripes). In this case, the interaction beteen phases is minimum and, consequently, the results are described by the X-model. The experiments performed by Persoff et al. (995) shoed that the flo is characterized by havin a localized continuous flo path that is underoin blockin and unblockin by the other phase. This behavior is related to capillary effects, hich are dominant in classical porous media. Consequently, the flo of 3

4 one phase is stronly impeded by the flo of the other phase. In the experiments performed by Fourar and Bories (995), several flo structures ere observed: bubbles, finerin bubbles, drops and films. It seems that these flos are dominated by viscous forces. The purpose of this study is to visualize and discuss the to-phase flo behavior in a fracture. The obtained as-ater relative permeabilities are analyzed by examinin the flo structures. Finally a tortuous-channel approach is developed to reproduce experimental results and to sho the effect of the flo structures on the relative permeabilities. kr. X-curve Viscous-couplin model Corey model Romm (966), smooth-alled ith strips Fourar and Bories (995), smooth-alled Diomampo(), smooth-alled Persoff and Pruess (995) Expt D, Rouh-alled Persoff and Pruess (995) Expt C, Rouh-alled. Fiure. Compendium of previous measurements of relative permeabilities in fractures (Romm (966) results from kerosene-ater flo, the rest ere air-ater or nitroen-ater flo). 3 EXPERIMENTAL APPARATUS AND MEASUREMENTS In this study, as-ater flo experiments ere conducted at 4 o C. The fluids consisted of pure nitroen and deionized ater. The ater injection as controlled by a meter pump (Dynamax, SD-, krl rate:.- ml/min). Gas injection as controlled throuh a flo reulator (Brooks Instrument, Flo Controller Model 5E), hich as connected to a as meter (Brooks Instrument, Flo Meter model 585E, max. rate: ml/min). All measurements ere electronic and diitized by usin hih-speed data acquisition system (DAQ; National Instrument, SCSI- ith PCI 63E A/D board) and diital video recordin system (Sony Diital-8 56X ith Pinnacle Studio DV IEEE 394 imae capture card). The hole experimental system is illustrated in Fiure, hich shos the fluid supply, the fracture apparatus, data acquisition system, and diital imae recordin. In this section, e first describe the fracture apparatus. Then e describe the pressure, flo rates, saturation measurements, and data processin method. 3. Fracture Apparatus Description The fracture is created by a smooth lass plate on top of an aluminum plate confined by a metal frame bolted to the bottom plate. The aperture of the fracture as set to 3 µm by usin stainless steel shims alon the flo boundaries. The schematic vie and photoraph of the fracture apparatus are shon in Fiure 3. An O-rin (Viton /8" thick #-7) as placed in beteen the lass and aluminum plate as a seal. The fluids enter into the fracture throuh to separate canals. Each canal has several ports drilled in a ay that they alin on the surface (see Fiure 3). The surface of the fracture apparatus as desined such that there is an available inch by 4 inch space for flo. Throuhout this flo area, tiny pressure ports ere drilled so as to minimize surface discontinuity. Four pressure ports ere drilled alon the fracture as shon in Fiure 3. The to-phase fluid exits throuh a sinle outlet. The apparatus as desined to be of sufficient lenth to minimize the capillary end effects. Nitroen as supply Gas reulator Gas rate controller Data acquisition system Transducer Transducer Diital camcorder Transducer computer N Relief valve Air bath Glass plate Transducer Water supply Aluminum plate Fracture apparatus FFRD DI ater reservoir Meter pump 5 psi relief valve Fiure. Process flo diaram for air-ater experiment. Leend: Electrical irin Process pipin Water collection bin 4

5 bolts detail Inlet port Pressure port Gas inlet canal Liquid inlet canal Top vie Side vie Glass plate To phase outlet Aluminum frame Pressure port Fiure 3. Schematic diaram and picture of fracture apparatus. 3. Pressure Measurements Lo-rane differential transducers ere used to measure the pressure difference throuh the fracture, as ell as the intermediate pressure and the tophase outlet pressure. To liquid differential transducers (Validyne Transducer, model DP-5, rane - psi) ere attached to four pressure ports inside the fracture to measure the pressure difference throuh the fracture. Another transducer (Validyne Transducer, model DP-5, rane -5 psi) as attached to the middle point of the fracture. The fourth transducer (Validyne Transducer, model DP- 5, rane -5 psi) as attached to the to-phase outlet of the fracture apparatus. These transducers send electrical sinals to the data acquisition system, hich as monitored usin the LabVie prorammable virtual instrument softare. The complete measurement confiuration in the fracture apparatus is shon in Fiure. 3.3 Flo rates Measurements Aside from the knon input rates, to obtain the instantaneous flo rates, a fractional flo ratio detector (FFRD) as desined and constructed as shon in Fiure 4a. The FFRD as used to measure the outlet as and ater fractional flos, f and f. f q out, = and qout, t f q out, = () qout, t installed inside the FFRD, producin different voltaes hen sensin different strenths of liht. The ater phase produces a hiher voltae hen floin throuh the FFRD as shon in Fiure 4b. The as and ater phase flo ratios are obtained by determinin the ratio of the number of as and ater sinals. Once f and f are obtained at steady-state condition, it is easy to calculate q out, in Equation () by assumin that ater flo rate remains constant from inlet to outlet of the fracture. This assumption can hold if the ettin phase, ater, flos in a continuous channel, and the ater viscosity is much larer than the nonettin phase. The calibration of the FFRD demonstrated a ood linearity beteen measurement and actual values (Fiure 5). Transparent mm" lass tubin Hih-temperature black rubber (a) Volt (b) LED Photo transistor Gas response Water sement Gas sement 5 ohm resistor L = 3 cm Schematic of flo inside FFRD tubin Gas Water Connect to data acquisition system Water response Time (.5sec) Connect to outlet of fracture apparatus Fiure 4. Fractional flo ratio detector (FFRD) (a) schematic (b) detected as and ater sinal correspondin to different as and ater sements inside FFRD tubin. f detected y =.9x +.7 R = f real Fiure 5. FFRD calibration (Fluids: ater and nitroen as; FFRD tubin ID:.mm). q out, t qout, + qout, = (3) here q out, is the outlet as flo rate, q out, is the outlet ater flo rate, and q out,t is the outlet total flo rate. The principle of the FFRD is that different phases ill have different refractive indices. A phototransistor (NTE 338, NPN-Si, Visible) as 3.4 Saturation Measurements Still imaes ere extracted from diital video recorded durin the experiments. The photoraphs ere processed in a Matlab proram. The proram does quadratic discriminant analysis (QDA) to divide the pixels of the picture into three roups: ater phase, as phase and the frame. The roupin is 5

6 based on color differences. Saturation is calculated as total pixels of the liquid roup over the sum of the as and liquid roups. Fiure 6 is a comparison of the ray-scaled imae produced by the QDA proram and the oriinal photoraph from the diital camcorder. Pan et al. (996) 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 photoraphs and the ater film adsorbed on the surfaces of the plates ith the latter bein of minimal effect. Good quality photoraphs are the ones ith clear distinction beteen the as and liquid phase. Good lihtin is necessary so that the colors in the imae come out clearly. The lihtin should also be positioned in a ay that it does not produce shado on the flo area. The proram ill mistakenly take the shado as as phase even if there is liquid. From Fiure 6, it can be said that the proram has ood accuracy. Water Frame Fiure 6. Comparison beteen the true color imae of the fracture flo and ray scale imae from the Matlab QDA proram used in measurin saturation. 3.5 Data Processin The experiment contains several runs ith desinated input rates of as and ater. Accordin to the airater experiments by earlier studies, these fracture flo experiments are not expected to reach a perfect steady state. Instead, they are unsteady by nature. There are sinificant pressure fluctuations accompanied by saturation chanes hen the as and liquid flo rates vary. Due to this behavior, instantaneous outlet rates of as ere also measured by the FFRD device for the relative permeability calculation and flo rate comparison. Diital video as taken in each run hen the flo reached steady state or repeated similar fluctuatin behaviors. The continuous imaes of each run ere extracted from the video every.3 seconds. These hundreds of imaes in each run ere then input into the computer prorams for the saturation calculation, flo structure reconition and characterization of the stability of the to-phase flo. The data acquisition task requires frequent atherin of instantaneous and synchronized pressure, flo rate and saturation values. Instantaneous atherin of data as accomplished by the use of the hih speed data acquisition system and diital video camcorder. Video shots ere taken of the pressure, time and saturation data displayed all at the same time. Pressure data and related time ere Gas displayed by the LCD monitor connected to the computer, hich also ran the data acquisition system. The methodoloy used to interate all the data and sinals and then calculate the as-ater relative permeability is illustrated in the flo chart in Fiure 7. Fiure 7. Data and sinal processin flochart. 4 EXPERIMENTAL RESULTS We first describe the flo structures observed durin the experiment, and then present the relative permeabilities calculated at one-second frequency in each run and their averaed values. Finally, e interpret the relative permeabilities usin models from earlier studies. 4. Description of Flo Structures The use of a transparent lass plate allos us to observe and videotape the to-phase flo structures. The recorded diital video as then transformed to continuous snapshots ith the frequency of 3 imaes/sec (.33 seconds period). More than 3 still imaes ere extracted from the diital video and used for the flo structure characterization and saturation calculation. Several flo structures dependin on the flo rates ere observed. Generally, hen the ater saturation decreases, one observes successively bubbles, slus and channels (Fiure 8). 6

7 (a) Bubble Flo (b) Slu Flo (c) Tortuous Channel Flo (d) Straiht Channel Flo Fiure 8. Photoraphs of flo structures in the smooth-alled fracture. Each set contains four continuous imaes. Flo direction as from left to riht. f psi Time (sec) Time (sec) Fiure 9. Relationship amon ater saturation (S ), ater fractional flo (f ) and pressure difference alon the fracture in a hihly tortuous channel flo. 7 f S Pressure f S f

8 The bubble flo is observed only at extremely hih values of the ater saturation, S. Due to the small fracture aperture (3 µm), lare as bubbles propaate either in bullet-like or ellipse-like forms ith reater lonitudinal lenth (Fiure 8a). As the as flo rate increases or ater flo rate decreases, as moves in narro slus and flos discontinuously hen the as rate is still relatively lo as shon in Fiure 8b. The hiher pressure difference occurs mostly hen a slu tries to break throuh the ater reion. Once the slu reaches the outlet of the fracture, the pressure difference decreases. This type of slu movement is seen frequently for hih values of S. With the appearance of bubbles and slus, the to-phase flo is fairly unstable. When the lonitudinal size of slus increases as the as rate increases, the movin slus tend to form stable as channels and reach a steady condition. Hoever, the steadiness collapses either because ater breaks the thinnest throat of the short-lived channel or because of the insufficient as supply. This causes continuous fluctuations in the saturation, fractional flo and pressure difference. As the as rate increases further, the short-lived channel becomes more and more stable. The as channel meanders throuh the fracture ith branches and junctions because the viscous force is insufficient to break throuh intermediate ater islands. We define this kind of channel as a tortuous channel as shon in Fiure 8c. The lifetime of these channels seems to depend on the complexity and meander of their structure. Some tortuous channels miht exist just for a fe seconds, hereas others exist for lon periods. The S, f and pressure difference histories plotted in Fiure 9 are obtained from a tortuouschannel-dominated run. Clearly, even thouh constant as and ater rates are injected into the fracture, the S keeps fluctuatin in the fracture and the instantaneous f sensed from the FFRD follos this saturation fluctuation as shon in the top plot of Fiure 9. The correspondin pressure response as also recorded as shon in the bottom plot of Fiure 9. Most of the peaks in these to plots are due to the collapse, reconstruction and reconfiuration of channels, and some intrusions from other minor flo structures (bubbles and slus). It is also observed that the more tortuous the channel, the larer the pressure difference alon the fracture for the same ater saturation. On the other hand, straihter channels ere observed more frequently in hih f situations shon in Fiure 8d. In this situation, almost all of the as flos solely in a fairly straiht channel, hile most of the ater flos above and belo the as path. Except for a small amount of immobile ater inside the central as channel, the as path is more uniform and less tortuous in comparison ith the tortuous channel flo. Three major factors may affect the morpholoy of as channels. These factors are the f (viscous force), as-ater viscosity ratio (viscous force) and the interfacial tension (capillary force). Therefore for the same S, flos ith the viscosity ratio closer to one or the smaller interfacial tension have straihter channels. Liquid superficial velocity (m/s).... S. Bubble flo Slu flo Unexplored Unstable channels T=4 o C Aperature=3µm : Transitional flo Channel flo... Gas superficial velocity (m/s) Fiure. Flo structure map for air-ater flo in the smooth-alled fracture. Fiure shos a flo structure map for the experiment. The correspondin S is also provided in this fiure. It is easy to see that the channel flo spans most of the S rane and major bubble and slu flos only exist in hih S situations. The shado areas in the flo maps indicate transitional flo hich means the flo structure is a combination of the to neihborin structures. The transition beteen the slu flo and channel flo is defined as unstable channel flo. In this reion, some lare finerin slus ere able to bride the to ends of the fracture. Hoever, the brided slu collapsed in a short time due to its unstable structure, ater intrusion and the insufficient as supply. The flo structure map is similar to the flo map (lass fracture) presented by Fourar and Bories (995). 4. Relative Permeability Calculations The absolute permeability of the smooth-alled fracture as first measured in sinle-phase ater flo. The absolute permeability of the fracture deduced from measurements by usin the Darcy equation is.5-9 m (5 darcies). The ap idth (aperture) of the fracture is 3 µm, the estimated permeability from Equation (4) is then.4-9 m (4 darcies). As can be seen in Fiure, measured values are in a ood areement ith the theoretical value of the absolute permeability. Air-ater relative permeabilities ere calculated by usin Equations () and (3) at one-second frequency in each run. The instantaneous flo rate as obtained from the FFRD. Comprehensive air-ater relative permeabilities obtained from thousands of data points are plotted in Fiure. Data points in the fiure sho acceptable correlation. Comprehensive ater 8

9 relative permeabilities are scattered at hih ater saturation oin to the slu and unstable-channel flos in the as-phase as shon in Fiures 8a and 8b. The vertically scattered effect observed ith the asphase relative permeability under extremely lo values of ater saturation may be associated ith either the pressure fluctuation due to the movin ater slus or the difficulty in sensin the instantaneous f from the FFRD at lo f. k (m ) k cubic Q (ml/min) Fiure. Absolute permeability of the fracture (aperture = 3 µm). kr kr kr. S Fiure. Comprehensive air-ater relative permeabilities in the smooth-alled fracture. 4.3 Relative Permeabilities Interpretation The averaed values of the relative permeabilities and the saturation for each pair of liquid and flo rates are presented in Fiure 3. Also plotted in this fiure are the curves of the X-model (Equations (5) and (6)), the Corey-model (Equations (7) and (8)) ith S rl = and S r =, and the viscous-couplin model (Equations () and ()). It appears that the X-model does not fit the experimental data. For a iven value of S, the k rl and k r are belo the straiht lines. Althouh as-phase experimental values are closer to the Corey curve, ater-phase values are much hiher than the ater-phase Corey curve. Therefore, neither the Corey curve nor the X- curve can fit the experimental result. On the other hand, the viscous-couplin model is in a ood areement ith experimental results, especially for the ater-phase. For the as phase, the viscouscouplin model shos acceptable fittin, but it seems to be less accurate at hih ater saturation. The photoraphs in Fiure 8 provide some clue about this less accurate result. At hih values of S, most of the ater channels extend from lonitudinal boundaries of the fracture space and flo straihter and more continuously. On the other hand, the active as flos as finerin slus and channels ith amorphous shapes. These chanin shapes enerate the correspondin pressure, saturation and flo rate chanes as illustrated in Fiure 9. The relative permeability may be affected by these chanes that are not considered by the previous models (X, Corey and viscous-couplin). The further study of flo structure effects on relative permeabilities ill be presented in the next section. kr. kr (averae) kr (averae) V-C model Corey curve X-curve. S Fiure 3. Comparison of averae experimental relative permeability ith the Coreycurve, X-curve and viscous-couplin models. 5 EFFECTS OF FLOW STRUCTURES ON RELATIVE PERMEABILITIES Fiure shos that the channel flo is the dominant flo structure in our experiments. We miht expect that these flos ould be described by the X-model. Instead, they are rather in accordance ith the viscous-couplin model. This can be explained by characterizin the morpholoy of phase channels and quantifyin the manitude of the tortuosity in this flo structure. A ne approach ill be discussed here to include the tortuosity created by the phases. 5. Tortuous-Channel Approach Considerin a simplified fracture space ith a dimension of L (lenth) W (idth) b (aperture) as shon in Fiure 4, the fluids, ater and as, form ideal and straiht channels throuh the fracture. From fundamentals of multiphase flo, the X-curve behavior is readily derived from Darcy s la and the cubic la by assumin neliible capillary pressure, 9

10 as slippae, inertia effect and ettin-phase stratified flo. W Water Top all b z y x L Gas Flo direction W Bottom all Ly CAAR proram (reconize and isolate channels) Lx Fiure 4. A simple model of a straiht as channel in a smooth-alled fracture. Consequently, the straiht phase-channel in the smooth-alled fracture ill yield the X-curve relative permeability as shon by Romm (966). Hoever, as shon in this study, the actual flo structures in fractures seldom reach the ideal straiht channel even thouh the fracture is smooth-alled. Most of the flo structures are either finerin or tortuous channels instead, and the relative permeability obtained from this experiment shos a deviation from X-curve as presented earlier in Fiure 3. As a result, a physical, tortuous-channel approach to modify X-curve and take into account the channel tortuosity is required. The concept of the tortuous-channel approach is based on the relationship beteen the fluid tortuosity and the relative permeability. As the channel flo observed in our study is tortuous (Fi. 9c), e define a tortuosity coefficient for the channel flo, c, to characterizin its morpholoy. The definition of this apparent parameter is based on the area of the channel and the smallest boundin rectanle that covers the hole channel for a specific phase. As shon in Fiure 5, the binary imaes processed from continuous true-color imaes ere input into another imae-processin proram, hich allos reconition and separation of the different flo structures. Then, the channel area, A c (unit: pixel ), lenth and idth of the smallest boundin box, L x and L y (unit: pixel), ere computed. The channel tortuosities for as and ater are then defined as: L L x y c, = ( ) and Ac L L x y c, = ( ) (4) Ac Lx L c = A Compute:. Channel area, A c.. Smallest boundin rectanle, L x L y. c y Fiure 5. Evolution of channel tortuosity alorithm. In this study, thousands of continuous imaes extracted from the video of the experiment ith a.33 second period ere analyzed, and the tortuosities of as and ater ere deduced for the channel flo confiuration. Illustrations of these calculations are presented in Fiure 6. It appears that ater channels are less tortuous than as channels. This may be due to the fact that the aterviscosity is hiher than the as-viscosity, and that ater channels tend to adhere to lonitudinal boundaries of the fracture, hile as channels flos in beteen. In order to characterize the deviation from the X- model, e propose the folloin relationships: S kr = (5) k c, S r = (6) c, This relationship assumes that the channel tortuosity is directly proportional to the deviation of X-curve behavior. In the case here each phase flos in the form of homoeneous strata (no dispersed phase), c = and c = and Equations (5) and (6) reduce to the X-model. As for the relative permeability, coefficient c for each phase varies beteen and. This coefficient is related to the interfacial area and, consequently, allos quantifyin the shear stress at the interface beteen the to fluids.

11 Oriinal imae Gas channel imae Water channel imae S = 46 c, =.63 c, =. S =.739 c, =.86 c,(averae) =.85 S = 5 c, =. c,(averae) =.5 S =.377 c,(averae) =.8 c, =.63 S =.7 c, =.34 c, = NA (no channels) Fiure 6. Representative imaes and correspondin processed as-channel and ater-channel imaes extracted from the CAAR imae processin. (Phase channels are in hite color in the channel imaes. Rectanles indicate the boundin boxes of the phase channels.) 5. Reproduction of Relative Permeabilities After deducin both ater and as tortuosities from the extracted imaes, the comprehensive ater-phase and as-phase relative permeabilities obtained from Equations (5) and (6) are shon in Fiure 7. The oriinal experimental results (Fiure ) are also provided in this fiure. The results from the tortuouschannel approach sho ood reproduction of the experimental results in both ater and as phases. Thus, the tortuous-channel approach leads to lo scatter in comparison ith the experimental relative permeabilities. This is because some unstable phenomena, bubbles and slus, ere excluded and there are almost no sources of measurement errors in the tortuous-channel approach, except the imae processin errors. 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 usin the averae tortuosity in Table and Equations (5) and (6) are presented in Fiure 9. They are in a ood areement ith averae experimental relative permeabilities for both as and ater phases. Aside from the enerally matchin performance, the tortuous-channel approach not only approximates the experimental result but closely reproduce some experimental points, especially the portion here the viscous-couplin model performs poorly. The mean tortuosity of as and ater channels as then computed by averain all tortuosities in each run. Table presents the mean channel tortuosity in each run for the experiments. The averae phase tortuosity (reciprocal) versus the phase saturation is shon in Fiure 8. Values of tortuosity reciprocal approach. hen the channels increase straihtness. A tortuosity value (or reciprocal) of. means perfect straiht phase channels. From these to plots, it is evident that the hiher the phase saturation, the

12 kr (a) kr (b) kr,experiment kr,t-c approach. S kr,experiment kr,t-c approach. S Fiure 7. Relative permeabilities from tortuouschannel approach usin phase tortuosities obtained from the processin of continuous imaes and its comparison ith the oriinal result: (a) as phase, (b) ater phase. Table. Averaes of tortuous-channel parameter obtained from CAAR imae processin proram and the relative permeavility values for the tortuous-channel approach and experiment. Saturation Averae channel Experiment Tortuous-channel tortuosities results approach S S c, c, kr, expe kr, expe kr, T-C kr, T-C NA * NA NA NA * : No channel detected. c.. c, =.5 S +.9 S R = (a) S c.. c, = 77S +.33S R = (b) S Fiure 8. Reciprocal of averae phase-channel tortuosity versus phase saturation in the air-ater experiment: (a) as phase, (b) ater phase kr. Kr(averae) Kr(averae) kr,t-c model kr,t-c model. S S,cri Fiure 9. Comparison of averae relative permeabilities from the tortuous-channel approach and from experiment. (The dash circles may contain more uncertainty due to fe samplin points.) 5.3 Comparison ith Models To compare the proposed tortuous-channel approach ith other theoretical and empirical models mentioned previously, simple curve fits to express the tortuosity of phase channels mathematically ere done in this preliminary stae. The optimal equations for fittin phase tortuosities as shon in Fiure 8 are:

13 c, = 77S +.33S c, =.5S +.9S (7) (8) This leads Equations (5) and (6) to a tortuouschannel model: k = 77S +.33S S (9) r r 3 3 k =.5S +.9S S () Fiure shos the comparison of the tortuouschannel model and viscous-couplin model. These to models perform equally accurate for the ater phase. For the as phase, the tortuous-channel model fits the experimental data better in hih S rane. On the other hand, the viscous-couplin model shos better performance in lo S rane. This model transition may indicate that the dominant force shifts from the horizontal (xz-plane) shear stress to vertical (xy-plane) shear stress. The referred coordinate system is illustrated in Fiure 4. Considerin the as phase, except in lo S rane, most of the force contributed by the flo as applied for explorin and sustainin the as reion horizontally. Therefore, the manitude of the vertical shear stress as relatively small and neliible. This leads to the feasibility of usin the tortuous-channel approach. Hoever, hen most of the fracture space as occupied by the as at lo S, less force as used to sustain the as reion horizontally since the as channels ere extended and straihter. Instead, most of the force as contributed by the vertical shear force because of the importance of the vertical stratified flo at this stae. This ill be close to the viscous-couplin behavior. kr. kr (averae) kr (averae) T-C model V-C model. S Fiure. Comparison of the experimental relative permeability ith the tortuous-channel model and viscous-couplin model for the air-ater experiment. To compare these models visually and quantitatively, the comparison beteen measured and predicative ater saturation and relative permeabilities as made usin Equations (5) to () and Equations (9) and (). The result is shon in Fiure. The statistical analysis of the absolute errors and standard deviations beteen experimental and modelin relative permeabilities and saturations is also presented in Table. Apparently, both the viscouscouplin model and tortuous-channel model can interpret and predict the relative permeability and ater saturation ith ood accuracy, hereas both the X-curve and Corey-curve models sho orse performance. From the statistical mean absolute error and standard deviation, the tortuous-channel model performs slihtly better than the viscous-couplin model in predictin both relative permeability and ater saturation. To sum up, from validations of both the comprehensive and averae results and model comparisons, the proposed tortuous-channel approach can interpret the relative permeabilities at the airater experiment successfully. This tells us that the relative permeability is flo structure dependent. At the moment, the flo structure prediction is a complicated and immature subject. Calculated S X-curve. Corey curve V-C model T-C model. (a) Measured S Calculated kr X-curve. Corey curve V-C model T-C model. (b) Measured kr Fiure. Comparison beteen measured and calculated (a) ater saturation, (b) relative permeavility. 3

14 Table. Statistical analysis descriptors (mean absolute error and standard deviation) of relative permeabilities and ater stauration beteen experiment and models. Models k r S MAE (k r) σ MAE(S ) σ X-curve Corey curve (Corey, 954) Viscous-couplin model* Tortuous-channel model (this ork) * : Fourar and Lenormand, (998) N : Mean absolute error; MAE( Y ) = Y e,i Y exp mod el,i N N σ = Y e,i Y el,i MAE Y mod ( N : standard deviation, ( exp )) 6 CONCLUSION Experimental results of air-ater flo throuh a smooth-alled fracture ere presented. Usin a hihspeed data acquisition system and diital imae processin technoloy, the experimental apparatus as able to characterize the unstable nature of the flo. Different flo confiurations ere visualized: bubbles, slus and channels. Experimental results of pressure drop, flo rates, and saturation ere interpreted by usin the relative permeability concept and compared to three models from the literature: the X-model, the Corey-model and the viscous-couplin model. The results ere ell described by the viscous-couplin model shoin more interactions beteen phases than the X-model, but less interference than predicted by the Corey-model. By introducin the tortuous-channel concept, e shoed that the deviation from the X-model is due to the flo confiuration. The proposed tortuouschannel approach has reproduced both oriinal and averae experimental relative permeabilities. After applyin mathematical expressions of phase tortuosity coefficients, the preliminary tortuouschannel model performed best amon other models. Therefore the concept of the tortuosity coefficient may allo modelin to-phase flo in a fracture hatever the flo confiuration. To this end, the coefficient c must be related physically to the shear stress at the surface beteen the to fluids and to the interface area. Unfortunately, up to no there is no model that allos us to predict flo structures in a fracture. It as also noticed that the values of the asphase relative permeability from the experiment tend to shift from the proposed tortuous-channel model to the viscous-couplin model in the lo ater saturation rane, hich may be because the vertical stratified flo is no loner neliible in this situation. Acknoledement Financial support for this research as provided by the US Department of Enery under contract DE- FG7-ID448. The authors ould like to thank Dr. Keen Li for helpful discussions. REFERENCES Brooks, R. H. and Corey, A. T., Properties of Porous Media Affectin Fluid Flo, J. Irri. Drain. Div., 6, 6, 996. Chen, C.-Y., Diomampo, G., Li, K. and Horne, R.N., Steam-Water Relative Permeability in Fractures, Geothermal Resources Council Transactions Vol.6, pp ,. Corey, A. T., The Interrelation beteen Gas and Oil Relative Permeabilities, Prod. Mon., 9, 38, 954. Diomampo, G., Relative Permeability throuh Fractures, MS thesis, Stanford University, Stanford, California,. Fourar, M. and Bories, S., Experimental Study of Air-Water To-Phase Flo Throuh A Fracture (Narro Channel), Int. J. Multiphase Flo Vol., No. 4, pp , 995. Fourar, M., and Lenormand, R., A Viscous Couplin Model for Relative Permeabilities in Fractures, Paper SPE 496, Presented at the SPE Annual Technical Conference and Exhibition, Ne Orleans, Louisiana, USA, September 7-3, 998. Mendoza, C.A. and Sudicky, E.A., Hierarchical scalin of constitutive relationships controlloin multi-phase flo in fractured eoloic media, paper presented at the Third Int. Conf. on Reservoir Characterization Requirements for Different Staes of Development, Tulsa, OK, 99. Murphy, J.R. and Thomson, N.R., To-Phase Flo in a Variable Aperture, Water Resources Research, vol. 9, No, pp , 993. Persoff, P. K., Pruess, K., and Myer, L., To-Phase Flo Visualization and Relative Permeability Measurement in Transparent Replicas of Rouh- Walled Rock Fractures, Proc. 6 th Workshop on Geothermal Reservoir Enineerin, Stanford University, Stanford, California, January 3-5, 99. Persoff, P., and Pruess, K., To-Phase Flo Visualization and Relative Permeability Measurement in Natural Rouh-Walled Rock Fractures, Water Resources Research Vol. 3, No. 5, May, pp ,

15 Pan, X., Won, R.C., and Maini, B.B., Steady State To-Phase Flo in a Smooth Parallel Fracture, presented at the 47 th Annual Technical Meetin of the Petroleum Society in Calary, Alberta, Canada, June -, 996. Pyrak-Nolte, L.J., Heleston, D., Haley, G.M., and Morris, J.W., Immiscible fluide flo in fracture, Proceedin of the 33 rd U.S. Rock Mech. Symp., edited by Tillersson and Waersik, pp , A. A. Balkema, Rotterdam, Netherlands, 99. Romm, E.S., Fluid Flo in Fractured Rocks, Nedra Publishin House, Mosco, 966 (Translated from the Russian). Rossen, W.R., and Kumar A.T.A., Sinle and Tophase flo in natural fractures, SPE 495, the 67 th SPE Annual Technical Conference and Exhibition, Washinton D.C., Oct. 4-7, 99. Scheideer, A.E., The Physics of Flo Throuh Porous Media, 3 rd ed., University of Toronto, Toronto Witherspoon, P.A., Wan, J.S.W., Iai, K. and Gale, J.E., Validity of Cubic La for Fluid Flo in a Deformable Rock Fracture, Water Resources Research, Vol. 6, No. 6, pp 6-4, 98. Yih, C.S., Fluid Mechanics, McGra-Hill, Ne York