Relative Permeability Measurement and Numerical Modeling of Two-Phase Flow Through Variable Aperture Fracture in Granite Under Confining Pressure

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1 GRC Transactions, Vol. 36, 2012 Relative Permeability Measurement and Numerical Modeling of Two-Phase Flow Through Variable Aperture Fracture in Granite Under Confining Pressure Noriaki Watanabe, Keisuke Sakurai, Takuya Ishibashi, and Noriyoshi Tsuchiya Graduate School of Environmental Studies, Tohoku University, Sendai, Miyagi, Japan Keywords Relative permeability, numerical modeling, two-phase flow, fracture, granite, confining pressure, geothermal reservoir Abstract Relative permeability measurement of decane-water twophase flow was conducted in a fracture, created in granite, at confining pressures of 5 MPa and 10 MPa. Non-wetting phase (decane) relative permeability decreased with decreasing capillary pressure, indicating significant phase interference due to capillarity in two-phase flow through a rock fracture under confining pressure. Moreover, the relative permeability was much smaller at the same capillary pressure for the higher confining pressure, indicating that the phase interference became much more significant at a higher confining pressure. Consequently, the X model and the viscous coupling model, ignoring any phase interference and phase interference due to capillarity, respectively, were expected to be inappropriate for a fracture under confining pressure, emphasizing importance of evaluating two-phase fracture flow characteristics under confining pressure. However, it is difficult to measure phase saturation within a fracture under confining pressure. Consequently, a numerical non-steady-state two-phase fracture flow model, which considers influences of both viscosity and capillarity, was developed to determine a relative permeability-saturation-capillary pressure relation for the fracture aperture distribution under confining pressure. The model, which could be verified by experimental results, provided a non-wetting relative permeability curve, which was entirely different from the X model, the viscous coupling model, and even the Corey model. Moreover, an application of the model for air-water two-phase fracture flow provided a similar conclusion. 1. Introduction Two-phase flows through fractures in subsurface rocks are of great importance in several domains, such as geothermal energy, petroleum recovery, and environmental engineering. Despite this importance few studies have been conducted, and the results presented in the literature seem to be contradictory. So far, two-phase fracture flows are not well understood. 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 permeabilies are presented in the literature: the X model with no phase interference, the viscous coupling model with phase interference due to viscosity, and the Corey model with phase interference due to capillarity. The experimental results in the literature show different behavior for the relative permeabilities. Some results are in accordance with the X model [Romm, 1966], whereas other results are in accordance with the viscous coupling model [Fourar and Bories, 1995] or the Corey model [Diomampo, 2001]. It is obvious that previous studies show a diversity of behavior for relative permeabilities in fractures. Moreover, the fractures in these previous studies are not real variable aperture fractures in rocks under confining pressure [Watanabe et al., 2008; Nemoto et al., 2009; Watanabe et al., 2009]. In order to better understand two-phase flows through fractures in subsurface rocks, relative permeability measurements were made on a fracture, created in granite, under confining pressure. Relative fracture permeability was measured with varying capillary pressure. However, phase saturation was not measured experimentally, because it would be difficult to determine saturation within a fracture in a rock under confining pressure, even with X-ray computed tomography [Watanabe et al., 2011a and 2011b]. Instead, a numerical non-steady-state two-phase fracture flow model was developed on the basis of experimental results, and utilized to determine a relative permeability-saturation-capillary pressure relation for the aperture distribution created from digital data of the fracture surface topography. 2. Experimental and Numerical Methods 2.1. Relative Permeability Measurements A cylindrical granite sample (50 mm in diameter, 50 mm in length) containing a single fracture (50 mm 50 mm) was prepared using Inada granite from Ibaraki, Japan. The fracture was 583

2 an induced tensile fracture having two rough surfaces, and the intrinsic fracture permeabilities by the cubic law assumption with negligible matrix permeability were m 2 and m 2 at 5 MPa and 10 MPa, respectively The surface topography for the two fracture surfaces were measured in a 1.7-mm square grid system by a laser scanning equipment. The aperture distribution within the fracture under confining pressure, which was used in the numerical model simulation described later, was determined using the digital data of the fracture surface topography. The Viton sleeved sample, which was placed in a core holder made of stainless steel, was subjected to a prescribed confining pressure (5 MPa or 10 MPa). Water (wetting phase) was first injected into the sample to saturate the fracture, and then n-decane (non-wetting phase) was injected into the sample to measure non-wetting phase relative permeability at room temperature. The viscosities were 1.0 mpa s and 0.9 mpa s for water and decane, respectively, and the interfacial tension at the fluid-fluid interface was 50 mn/m. In the present relative permeability measurement, the semi-dynamic method was used [Lenormand and Eisenzimmer, 1993]. Decane flowed through the sample at a constant flow rate, Q nw, while water at the outlet face of the sample was maintained at a constant pressure of 100 kpa, P w (Figure 1). At the steady state condition, non-flowing water had a constant pressure, P w, within the sample, and flowing decane had a pressure difference between the inlet and the Figure 1. Semi-dynamic method for the relative permeability measurement of the fracture in the sample under confining pressure. outlet pressures, P in,nw and P out,nw (= P w ), respectively. Non-wetting phase effective hydraulic aperture, a eff,nw, at a known inlet capillary pressure, P in,nw - P w, was determined using the cubic law assumption as follows: 12 µ L Q a eff, nw = nw, (1) ( P in,new P out,nw ) W where µ is the viscosity of decane, and L and W are the length and the width of the fracture. The corresponding non-wetting phase effective permeability, k eff,nw, was then determined as follows: k eff, nw = a 2 eff, nw 12. (2) Non-wetting phase relative permeability, k r,nw, was finally determined as follows: k r, nw = k eff,nw. (3) k where k is the intrinsic fracture permeability Numerical Modeling Since significant influence of capillarity was observed in the relative permeability measurement as described herein, a numerical non-steady-state two-phase fracture flow model, which considers influences by both viscosity and capillarity (the Young-Laplace equation), was developed to determine a relative permeability-saturation-capillary pressure relation for the aperture distribution within the fracture under confining pressure. In case of a single-phase flow through a variable aperture fracture in two dimensions (x-y coordinates), the steady-state laminar flow of a viscous and incompressible fluid may be modeled on the basis of the local cubic law [Watanabe et al., 2008; Nemoto et al., 2009; Watanabe et al., 2009]. For the single-phase fracture flow model with the present aperture distribution, which is created from the digital data of the fracture surface topography measured in the square grid system, the equation of continuity for the Darcy flow may be written as follows: p x e3 x + p y e3 y = 0, (4) where e is the local fracture aperture, and p is the local pressure of the fluid. Solving a finite difference form of Eq. 4 with boundary conditions provided a numerical intrinsic permeability, which may be compared to the experimental value. Consequently, the aperture distribution within the fracture at 5 MPa was determined by matching the numerical intrinsic permeability with the experimental value at that confining pressure (Figure 2). In this permeability matching, the apertures created by the two fracture surfaces were uniformly increased or decreased to vary the intrinsic permeability. Note that the aperture distribution at 10 MPa has not been addressed yet. On the other hand, in case of an immiscible two-component twophase flow through the determined aperture distribution, the steadystate laminar flow of two viscous and incompressible fluids with known phase distribution in a very short time may be described as follows: ( ) x Figure 2. Aperture distribution within the fracture at 5 MPa, determined for the numerical modeling of two-phase flow. e 3 p + 2γ cosθ / e x µ + e 3 ( p + 2γ cosθ / e = 0, (5) y µ y where µ is the viscosity of either wetting or non-wetting fluid, and γ and θ are the interfacial (or surface) tension and the contact angle at the fluid-fluid interface, respectively. In the present study, 584

3 a repeated solution of a finite difference form of Eq. (5) with changing phase distribution under the same boundary conditions as the relative permeability measurement was utilized to simulate the decane-water two-phase flow through the fracture at 5 MPa. Note that zero contact angle (perfect water wet) was assumed. In the numerical model simulation, the fracture was initially saturated by water, and then decane was injected at a constant flow rate. Inlet pressure of decane, phase distribution within the fracture, and the outlet flow rates of the two fluids changed with time, and became finally constant, providing a relative permeability-saturation-capillary pressure relation. The main steps in the numerical model simulation are listed below. 1. Steady state flow at the initial phase distribution (100% wetting phase saturation) is determined. 2. Time step is determined so that the local saturation of nonwetting phase does not exceed 100% at the next time step. 3. Changes in phase distribution during the time step are calculated. 4. Viscosity at grid-grid interface is changed depending on the flowing fluid. 5. Steady state flow at a known phase distribution is determined. 6. Steps 2-5 are repeated. Since the numerical model could be verified by experimental results as described later, the numerical model was also utilized to simulate air-water two-phase flow in the same aperture distribution under the same boundary conditions, by ignoring compressibility of non-wetting gas phase. In this application, the viscosity of air and the surface tension were 0.02 mpa s and 70 mn/m, respectively. 3. Results and Discussion 3.1. Experimental Relation Between Relative Permeability and Capillary Pressure both confining pressures of 5 MPa and 10 MPa (Figure 3). The relative permeability increased (or decreased) with increasing (or decreasing) inlet capillary pressure. Particularly, at lower inlet capillary pressures of <10 kpa at 5 MPa and <150 kpa at 10 MPa, the relative permeability drastically changed with inlet capillary pressure. Moreover, the relative permeability at the same inlet capillary pressure was much smaller for the higher confining pressure. These results indicated that capillarity had significant influence on two-phase flow through a fracture under confining pressure, where the influence became much more significant with increasing confining pressure due to smaller apertures. Consequently, the relative permeability curves by X model and viscous coupling model are expected to be inappropriate for two-phase flow through a fracture under confining pressure Validity of Numerical Non-Steady-State Two- Phase Fracture Flow Model The relative permeability measurement in the decane-water two-phase flow through the fracture at 5 MPa was simulated by the numerical model, and a numerical relation between the non-wetting phase relative permeability and the inlet capillary pressure was compared with the experimental relation (Figure 4). The numerical relative permeability changed with inlet capillary pressure as observed in the relative permeability measurement. Particularly, the drastic decrease in the relative permeability at smaller capillary pressures of <10 kpa was also observed in the numerical model simulation. Although the numerical results did not perfectly match the experimental results, the experimental relation between the relative permeability and the inlet capillary pressure could be reasonably reproduced by the numerical model. Consequently, it could be concluded that the numerical model was valid in simulating two-phase fracture flow under the conditions in the present relative permeability measurement. In the relative permeability measurement for the decane-water two-phase flow, the non-wetting phase relative permeability of the fracture significantly changed with inlet capillary pressure at Figure 4. Numerically and experimentally determined relations between the non-wetting phase relative permeability and the inlet capillary pressure for the decane-water two-phase flow through the fracture at 5 MPa. Figure 3. Experimentally determined relations between the non-wetting phase relative permeability and the inlet capillary pressure for the decanewater two-phase flow through the fracture at 5 MPa and 10 MPa Numerical Relative Permeability-Saturation- Capillary Pressure Relation Since the numerical model could be verified by the experimental results, the numerical model simulation has provided insights into relative permeability-saturation-capillary pressure relations in decane-water and air-water two phase flows in a fracture under confining pressure. In general, for both decane- 585

4 water and air-water two-phase flows, the non-wetting relative permeability decreased drastically with increasing wetting phase saturation (Figure 5). As a result, the relative permeability curves in the present study showed stronger saturation dependency (phase interference), compared with relative permeability curves by the X model, viscous coupling model, and the Corey model. Note that zero residual saturation was assumed for both phases in the Corey model. Since X model and viscous coupling model ignore any phase interference and phase interference due to capillarity, respectively, these models could not predict the relative permeability curves in the present study. However, it was surprising that the Corey model also could not predict the relative permeability curves. This may be due to a difficulty for one fluid to bypass the other fluid in a two dimensional porous media such as a fracture. It was predicted that the relative permeability became quite small even at intermediate wetting phase saturations. The difference in the relative permeability curves between decane-water and air-water two-phase flows may have been caused by the large difference in viscosities between decane and air since the interfacial and surface tensions were not significantly different. Figure 5. Numerically determined relations between the non-wetting phase relative permeability and wetting phase saturation for the decanewater and the air-water two-phase flows through the fracture at 5 MPa, compared with relative permeability curves by X, viscous coupling (VC), and Corey models. Due to the strong saturation dependency of the relative permeability, it is important to know a relation between the capillary pressure and the saturation when one needs to control or predict fracture flow. Since capillary pressure in the present study was not constant within the fracture as shown in Figure 1, average capillary pressure was calculated and related to the wetting phase saturation (Figure 6). The wetting phase saturation increased drastically with decreasing capillary pressure, particularly at lower capillary pressures of <10 kpa, which corresponded to available apertures of >13 mm for decane and >18 mm for air, based on the Young- Laplace equation. At such lower capillary pressures, majority of apertures of the fracture having the mean aperture of 12.6 mm (Figure 2), could not be available for the non-wetting phases, resulting in drastic increase in water saturation with decreasing capillary pressure. Conversely, it may be possible to determine a Figure 6. Numerically determined relations between the wetting phase saturation and the average capillary pressure for the decane-water and the air-water two-phase flows through the fracture at 5 MPa. capillary pressure to maintain low water saturation using the mean aperture, the interfacial (or surface) tension, and the contact angle at the fluid-fluid interface. 4. Conclusions Two-phase flows through fractures in subsurface rocks are of great importance in several domains. Despite this importance few studies have been conducted, and the results presented in the literature seem to be contradictory. The experimental results presented in the literature show different behavior for the relative permeabilities. Some results are in accordance with the X model, whereas other results are in accordance with the viscous coupling model or the Corey model. Moreover, the fractures in these previous studies are not real variable aperture fractures in rocks under confining pressure. So far, two-phase fracture flows are not well understood. In order to better understand two-phase flows through fractures in subsurface rocks, relative permeability measurements and numerical modeling were conducted on a variable aperture fracture in granite under confining pressure. It can be concluded that there is significant phase interference due to capillarity in twophase flow through a rock fracture under confining pressure, where the phase interference becomes much more significant at a higher confining pressure. Consequently, the relative permeability curves by the X model with no phase interference and viscous coupling model with phase interference due to viscosity are inappropriate for a fracture under confining pressure. Moreover, even the relative permeability curves by the Corey model with phase interference due to capillarity may be difficult to use for relative permeability curves of a fracture under confining pressure. It is important to evaluate two-phase flows through real variable aperture fractures in rocks under confining pressure. Acknowledgements The present study was supported in part by the Japan Society for the Promotion of Science (JSPS) through a Grant-in-Aid for Young Scientists (A), No The authors would like to thank Dr. Kimio Watanabe at Reneries, Ltd. for coding the numeri- 586

5 cal model simulation program. A part of the numerical results was obtained using supercomputing resources at Cyberscience Center, Tohoku University. References Diomampo, G., Relative permeability through fractures. M.S. thesis, Stanford Univ., Stanford, Calif. Fourar, M., and S. Bories, Experimental study of air-water two-phase flow through a fracture (narrow channel), International Journal of Multiphase flow, 21, Lenormand, R., and A. Eisenzimmer, A novel method for the determination of water/oil capillary pressures of mixed wettability samples. SCA Conference Paper Number Nemoto, K., N. Watanabe, N. Hirano, and N. Tsuchiya, Direct measurement of contact area and stress dependence of anisotropic flow through rock fracture with heterogeneous aperture distribution. Earth and Planetary Science Letters, 281(1-2), Romm, E. S., Fluid Flow in Fractured Rocks. translated from Russian, Nedra, Moscow. Watanabe, N., N. Hirano, and N. Tsuchiya, Determination of aperture structure and fluid flow in a rock fracture by a high-resolution numerical modeling on the basis of a flow-through experiment under confining pressure. Water Resources Research, 44, W Watanabe, N., N. Hirano, and N. Tsuchiya, Diversity of channeling flow in heterogeneous aperture distribution inferred from integrated experimental-numerical analysis on flow through shear fracture in granite. Journal of Geophysical Research, 114, B Watanabe, N., T. Ishibashi, N. Hirano, N. Tsuchiya, Y. Ohsaki, T. Tamagawa, Y. Tsuchiya, and H. Okabe, 2011a. Precise 3D numerical modeling of fracture flow coupled with X-ray computed tomography for reservoir core samples. SPE Journal, SPE PA. Watanabe, N., T. Ishibashi, Y. Ohsaki, Y. Tsuchiya, T. Tamagawa, N. Hirano, H. Okabe, and N. Tsuchiya, 2011b. X-ray CT based numerical analysis of fracture flow for core samples under various confining pressures. Engineering Geology, 123,

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