1 PAPER 2008-110 Effect of Fracture Dip and Fracture Tortuosity on Petrophysical Evaluation of Naturally Fractured Reservoirs R. AGUILERA University of Calgary This paper is accepted for the Proceedings of the Canadian International Petroleum Conference/SPE Gas Technology Symposium 2008 Joint Conference (the Petroleum Society’s 59 th Annual Technical Meeting), Calgary, Alberta, Canada, 17-19 June 2008. This paper will be considered for publication in Petroleum Society journals. Publication rights are reserved. This is a pre-print and subject to correction. Abstract A model is developed for petrophysical evaluation of naturally fractured reservoirs where dip of fractures ranges between zero and 90 degrees, and where fracture tortuosity is greater than 1.0. This results in an intrinsic porosity exponent of the fractures (m f ) that is larger than 1.0. The finding has direct application in the evaluation of fractured reservoirs and tight gas sands, where fracture dip can be determined, for example, from image logs. In the past, a fracture-matrix system has been represented by a dual porosity model which can be simulated as a series-resistance network or with the use of effective medium theory. For many cases both approaches provide similar results. The model developed in this study leads to the observation that including fracture dip and tortuosity in the petrophysical analysis can generate significant changes in the dual porosity exponent (m) of the composite system of matrix and fractures. It is concluded that not taking fracture dip and tortuosity into consideration can lead to significant errors in the calculation of water saturation. The use of the model is illustrated with an example. Introduction The petrophysical analysis of fractured and vuggy reservoirs has been an area of interest in the oil and gas industry. In 1962, Towle 1 considered some assumed pore geometries as well as tortuosity, and noticed a variation in the porosity exponent m in Archie’s 2 equation ranging from 2.67 to 7.3+ for vuggy reservoirs and values much smaller than 2 for fractured reservoirs. Matrix porosity in Towle’s models was equal to zero. Aguilera 3 (1976) introduced a dual porosity model capable of handling matrix and fracture porosity. That research considered 3 different values of Archie’s 2 porosity exponent: One for the matrix (m b ), one for the fractures (m f =1), and one for the composite system of matrix and fractures (m). It was found that as the amount of fracturing increased, the value of m became smaller. Rasmus 4 (1983) and Draxler and Edwards 5 (1984) presented dual porosity models that included potential changes in fracture tortuosity and the porosity exponent of the fractures (m f ). The models are useful but must be used carefully as they result incorrectly in values of m > m b as the total porosity increases. PETROLEUM SOCIETY 2 Serra et al. 6 developed a graph of the porosity exponent m vs. total porosity for both fractured reservoirs and reservoirs with non-connected vugs. The graph is useful but must be employed carefully as it can lead to significant errors for certain combinations of matrix and non-connected vug porosities (Aguilera and Aguilera 7 ). The main problem with the graph is that Serra’s matrix porosity is attached to the bulk volume of the “composite system”. More appropriate equations should include matrix porosity (ø b ) that is attached to the bulk volume of the “matrix system” (Aguilera, 1995). Aguilera and Aguilera 7 published rigorous equations for dual porosity systems that were shown to be valid for all combinations of matrix and fractures or matrix and non- connected vugs. The non-connected vugs and matrix equations were validated using core data published by Lucia. 8 The fractures and matrix equations were validated originally with data from the Altamont trend in Utah and the Big Horn Basis in Wyoming (Aguilera 3 ). Subsequently, Aguilera 9 illustrated the use of these equations with core data from Abu Dhabi limestones and dolomites (Borai, 10 Aguilera 11 ), and carbonates from various locations in the USA and the Middle East (Ragland 12 ). The models can also be shown to be valid with published data from vuggy carbonates from the Lower Congo Basin of Angola 13 , vuggy dolomites and limestones from the Simonette area, Swann Hills formation of Alberta 14 . Aguilera and Aguilera 15 researched instances where the reservoir is composed mainly by matrix, fractures and non- connected vugs, which are sometimes observed in cores, or deduced from micro-resistivity and/or sonic images. In these cases a triple porosity model is more suitable for petrophysical evaluation of the reservoir. In the above cases, it has been assumed that the flow of current is parallel to the fractures. More recently Aguilera and Aguilera 16 investigated the effect on m of current flow that is not parallel to the fractures. This type of anisotropy, which can be correlated with fracture dip, is important to avoid potential errors in the calculation of water saturation. This model assumed a fracture tortuosity is equal to 1.0. A comparison of results with those obtained by Berg 17 using effective medium theory yields an excellent agreement for fracture angles of zero and ninety degrees. The comparison for other angles is reasonable but there are some differences that will be evaluated based on results from core laboratory work. The present paper extends the Aguilera and Aguilera 16 model to cases where tortuosity is larger than 1.0. THEORETICAL MODEL Figure 1 shows schematics of the fracture dip model considered in this study. Schematics 1-A through 1-D assume that fracture porosity is equal to 1% and that current flow direction is horizontal in all cases thus the angle corresponds to fracture dip. Schematic 1-A displays a horizontal fracture with tortuosity equal to 1.0. In this case the porosity exponent of the fractures (m f ) is also equal to 1.0 and fracture dip is equal to zero. Schematic 1-B presents a horizontal fracture with a fracture tortuosity greater than 1.0. In this case the tortuosity leads to porosity exponent of the fractures (m f ) equal to 1.3. It is important to note that although fracture dip is equal to zero, as in the case of schematic 3-A, the porosity exponent (m f ) is larger than 1.0 due to tortuosity. Schematic 1-C is for a fracture with a dip  equal to 50°. Tortuosity is equal to 1.0, and as a result the porosity exponent of the fractures (m f ) is equal to 1.0. However, the 50° angle leads to a pseudo fracture porosity exponent (m fp ) equal to 1.19. Schematic 1-D shows a non-horizontal fracture (dip  = 50°) with a certain amount of tortuosity that leads to m f = 1.3. The 50° angle leads to a pseudo fracture porosity exponent (m fp ) equal to 1.49. Aguilera and Aguilera 16 have presented results associated with schematics 1-A and 1-C. This paper presents research results for schematics 1-B and 1-D when tortuosity greater than 1.0 is taken into account. Permeability of idealized fracture rock, including fluid flow through anisotropic media, has been discussed in detail by Parsons 18 and need not be repeated here. Although Parson’s model is strictly for fluid flow, we have used it for current flow with reasonable results. 16 Parsons fluid flow anisotropy concepts can be combined with Equations A-4 and A-5 in Appendix A and the formation factor for calculating the porosity exponent m of the composite system at any angle of interest. Sihvola 19 considers the flow of fluids through a host medium, and how the addition of an inclusion would affect the flow. Figure 2 shows a mixture with aligned ellipsoidal inclusions. The host environment has a permittivity ε e and the ellipsoidal inclusion has a permittivity ε i . The mixture effective permittivity ε eff is anisotropic as on the different principal directions the mixture possesses different permittivity components. For these conditions the dual porosity exponent, m, is given by: 16 ( ) ( ) ( ) () [ ] φ θθ θθ log F/sinF/coslog m 90 2 0 2 == + = … …. (1) where, ( ) ( ) [ ] bff m b mm F '1/1 220 φφφ θ −+= = … ……… …. (2) ( ) ( ) [ ] bff m b mm F − = −+= '1 2290 φφφ θ …………… …. (3) f f b 2 2 1 ' φ φφ φ − − = ………………………………………… (4) m f is the porosity exponent of the fractures and, 2 ln ln )1( φ φ −−= ff mmf ………… …………………… (5) Equation 5 is valid for ø 2 >0; f has been found to range exponentially between 1.0 at ø = ø 2 , and m f at ø = 1.0, using numerical experimentation. 20 Development of the above equations is presented in Appendix A. The total porosity of the system is represented by ø. The angle between the fracture and the current flow direction is 3 equal to θ. If the flow of current is horizontal the angle corresponds to fracture dip. The formation factor F θ=0 applies to a systems in parallel (zero angle). The formation factor F θ=90 applies to systems in series (90-degree angle). This study also presents cases for various intermediate values of θ between 0 and 90 degrees. The equation for total porosity is: 7, 15 () 22b2m 1 φ φ φ φ φ φ + −=+= ……………………… …. (6) where ø m is matrix porosity attached to the bulk volume of the composite system, and ø b is matrix porosity attached to bulk volume of only the matrix block. RESULTS Figure 3 shows a crossplot of the porosity dual porosity exponent m vs. total porosity calculated from equations 1 to 6 for angles θ equal to 0 and 90 degrees. The graph is constructed for a constant value of m f equal to 1.3, a porosity exponent of the matrix m b equal to 2.0 and fracture porosity (PHI2 or ø 2 ) values of 0.001, 0.01, and 0.1. The same type of graph is presented in Figure 4 for a constant fracture porosity ø 2 equal to 0.01 and values of m f equal to 1.0, 1.3 and 1.5. The values of the dual porosity exponent m increase significantly for a given total porosity as the values of m f become larger. Not taking this into account can lead to significant errors in the calculation of water saturation. Figure 5 shows values of the dual porosity exponent, m, vs. total porosity calculated from equations 1 to 6 for different angles, for constant porosity exponent of the matrix m b equal to 2.0, and for a constant porosity exponent of the fractures m f equal to 1.3. Note that if the current flow is horizontal, the angle corresponds to the fracture dip. The larger the angle, the bigger is the value of m for a given total porosity. All curves eventually converge at a porosity exponent m b of the matrix equal to 2.0. EXAMPLE 1 Given an angle θ of 50 degrees between the direction of current flow and the fracture, what is the value of m for a dual-porosity system, if total porosity equals 0.05, fracture porosity is 0.01, the porosity exponent (m b ) of only the matrix is 2.0, and the porosity exponent of the fractures (m f ) affected by tortuosity is 1.3? The first step is calculating matrix porosity, ø m , which is equal to total porosity minus fracture porosity (ø m = 0.05 – 0.01 = 0.04); matrix porosity, ø b , which is equal to 0.040404 from equation 6 (ø b = 0.04/(1 – 0.01) = 0.040404); f that is equal to 1.104845 from equation 5, and ø’ b that is equal to 0.0441017 from equation 4. The inverse of the formation factor, 1/F θ=0 , is equal to 0.004452 from equation 2. The inverse of the formation factor, 1/F θ=90 , is equal to 0.00195 from equation3. Finally, the value of m for the composite system is calculated to be 1.941 from equation 1. EXAMPLE 2 What is the error in m and water saturation if θ is assumed to be equal to zero and m f is assumed to be equal to 1.0 in the previous example? What is the value of the pseudo porosity exponent of the fractures (m fp ) resulting from the 50-degree angle? If anisotropy and tortuosity are ignored leading to θ = 0 and m f = 1.0, the value of m is calculated to be 1.487 following the procedure explained in Example 1. This corresponds to an error of 23.4%. The error in the calculated water saturation is determined from: 7 ])(1.[100 /1 )( n mm w ic ErrorS − −= φ …… (8) If the water saturation exponent, n, is 2.0 the error in the calculated water saturation is 100[1-(0.05 1.721-1.487 ) 1/2 ] = 49.3%. Finally the pseudo porosity exponent of the fractures (m fp ) resulting from the 50-degree angle between the fracture orientation and direction of current flow, and the tortuous value of m f (1.3) is m fp = 1.49. This is calculated repeating the same steps shown above but assuming matrix porosity equal to zero (in reality use a very small of fracture porosity for the equations to work. For example, I have used ø b = 1E-12). In this case the inverse of the formation factor, 1/F θ=0 , is equal to 0.002512 from equation 2. The inverse of the formation factor, 1/F θ=90 , is essentially equal to 0.0 (in reality 1.69E-24) from equation 3. Finally, the value of m fp for the fractures is calculated to be 1.49 from equation 1. Conclusions 1) The effect of current flow that is not parallel to fractures has been investigated for cases where the porosity exponent of the fractures, m f , is greater than 1.0 due to fracture tortuosity. It has been found that the larger the amount of fracture tortuosity, the greater is the dual porosity exponent, m, of the composite system of matrix and fractures. 2) Not taking into account variations in fracture dip and fracture tortuosity can lead in some cases to significant errors in the calculations of the dual porosity exponent, m, of matrix and fractures; and water saturation. For the examples presented in this paper the water saturation error is 49.3%. Acknowledgements Parts of this work were funded by the Natural Sciences and Engineering Research Council of Canada (NSERC agreement 347825-06), ConocoPhillips (agreement 4204638) and the Alberta Energy Research Institute (AERI agreement 1711). Their contributions are gratefully acknowledged. NOMENCLATURE f - volume fraction which the inclusions occupy F - formation factor of the matrix system F t - formation factor of the composite system F θ =0 - formation factor of composite system at θ = 0° F θ =90 - formation factor of composite system at θ = 90° m – dual porosity exponent (cementation factor) of composite system of matrix and fractures m b - porosity exponent (cementation factor) of the matrix block m c – correct dual porosity exponent (cementation factor) of composite system m i – incorrect dual porosity exponent (cementation factor) of composite system m θ =0 – dual porosity exponent (cementation factor) of the composite system at θ = 0° m θ =90 – dual porosity exponent (cementation factor) of the composite system at θ = 90° 4 m f - porosity exponent (cementation factor) of the fracture system m fp - pseudo porosity exponent of the fractures (cementation factor) resulting from θ n - water saturation exponent N x - depolarization factor in x direction R o - matrix resistivity when it is 100% saturated with water (ohm-m) R o θ =0 - resistivity of the composite system (matrix plus fractures) at θ = 0 when it is 100% saturated with water (ohm-m) R o θ =90 - resistivity of the composite system (matrix plus fractures) at θ = 90 when it is 100% saturated with water (ohm-m) R w - water resistivity at formation temperature (ohm-m) S w – water saturation, fraction ε e - host environment permittivity ε i - inclusion permittivity ε eff - effective permittivity ε effx - effective permittivity in x direction ø - total porosity ø b - matrix block porosity attached to bulk volume of the matrix system ø m - matrix block porosity attached to bulk volume of the composite system ø 2 - porosity of natural fractures θ - angle between fracture and current flow direction REFERENCES 1. Towle, G., An analysis of the formation resistivity factor-porosity relationship of some assumed pore geometries; Paper C presented at Third Annual Meeting of SPWLA, Houston, 1962. 2. Archie, G. E., The electrical resistivity log as an aid in determining some reservoir characteristics;” Trans. AIME, vol. 146, p. 54-67, 1942. 3. Aguilera, R., Analysis of naturally fractured reservoirs from conventional well logs: Journal of Petroleum technology; v. XXVIII, no.7, p. 764-772, 1976. 4. Rasmus, J. C., A variable cementation exponent, m, for fractured carbonates; The Log Analyst, vol. 24, no. 6, p. 13-23, 1983. 5. Draxler, J. K. and Edwards, D. P., Evaluation procedures in the Carboniferous of Northern Europe; Ninth International Formation Evaluation Transactions, Paris, 1984. 6. Serra, O. et al, Formation Micro Scanner image interpretation; Schlumberger Educational Service, Houston, SMP-7028, 117 p, 1989. 7. Aguilera, R. and Aguilera, M.S., Improved models for petrophysical analysis of dual porosity reservoirs; Petrophysics, Vol. 44, No. 1, p. 21-35, January- February, 2003. 8. Lucia, F. J., Petrophysical parameters estimated from visual descriptions of carbonate rocks: A field classification of carbonate pore space; Journal of Petroleum Technology, v. 35, p. 629-637, 1983. 9. Aguilera, R., 2003, Discussion of trends in cementation exponents (m) for carbonate pore systems; Petrophysics, Vol. 44, No. 1, p. 301-305, September-October, 2003. 10. Borai, A. M., A new correlation for cementation factor in low-porosity carbonates; SPE Formation Evaluation, vol. 4, no. 4, p. 495-499, 1985. 11. Aguilera, R., Determination of matrix flow units in naturally fractured reservoirs; Journal of Canadian Petroleum Technology, vol. 12, pp. 9-12, December 2003. 12. Ragland, D. A., Trends in cementation exponents (m) for carbonate pore systems; Petrophysics, vol. 43, no. 5, p. 434-446, 2002. 13. Guillard, P. and Boigelot, J., Cementation factor analysis – a case study from Albo-Cenomanian dolomitic reservoir of the lower Congo basin in Angola; SPWLA, circa 1990. 14. Batem an, P. W., Low resistivity pay in carbonate rocks and variable “m”; The CWLS Journal, vol. 21, p. 13-22, 1988. 15. Aguilera, R. F. and Aguilera, R, A Triple Porosity Model for Petrophysical Analysis of Naturally Fractured Reservoirs; Petrophysics, vol. 45, No. 2, pp. 157-166, March-April 2004 . 16. Aguilera, C. G. and Aguilera, R.: “Effect of Fracture Dip on Petrophysical Evaluation of Naturally Fractured Reservoirs,” paper CICP 2006-132 presented at the Petroleum Society’s 7 th Canadian International Petroleum Conference (57 th Annual Technical Meeting), Calgary, Alberta, Canada, June 13 – 15, 2006. 17. Berg, C. R., Dual and Triple Porosity Models from Effective Medium Theory, SPE 101698-PP presented at the Annual Technical Conference and Exhibition held in San Antonio, Texas, Sept 24-27, 2006. 18. Parsons, R. W., Permeability of Idealized Fractured Rock; Society of Petroleum Engineers Journal, p. 126-136, June 1966. 19. Sihvola, A., Electromagnetic Mixing formula and Applications; The Institution of Electrical Engineers, London, United Kingdom, 1999. 20. Aguilera, R.: “Role of Natural Fractures and Slot Porosity on Tight Gas Sands,” SPE paper 114174 presented at at the 2008 SPE Unconventional Reservoirs Conference held in Keystone, Colorado, U.S.A., 10–12 February 2008. APPENDIX A The development presented here assumes that fluid flow equations though porous media have application in the flow of current through porous media. Equations published originally by Parsons 18 for fluid flow through anisotropic porous media are used as a base for developing the model presented in this paper that permits evaluating the effect of fracture dip and fracture tortuosity on the petrophysical evaluation of dual porosity naturally fractured reservoirs. Figure 2 shows a mixture with aligned ellipsoidal inclusions. The host environment has a permittivity  e and the ellipsoidal inclusion has a permittivity  i . The mixture effective permittivity  eff is anisotropic as on the different principal directions the mixture possesses different permittivity components. In this case, the Maxwell Garnett formula for the x-component is given by: 19 5 () eixe ei eex,eff N)f1( f εεε ε ε εεε −−+ − += …… (A-1) where f is the volume fraction which the inclusions occupy and N x is the depolarization factor in the x direction. In the case of naturally fractured reservoirs, f is the equivalent of fracture porosity (ø 2 ). The balance (1-f) is equivalent to the summation of matrix porosity and solid rock. Making the depolarization factor (N x ) in equation (A-1) equal to zero results in: () eieff ff ε ε ε −+= 1 max, ….…. (A-2) Making the depolarization factor (N x ) equal to one leads to: ie ei eff ff εε ε ε ε )1( min, −+ = ………. (A-3) For the case at hand, the permittivity concept is associated with the dielectric constant for mixtures of particles (rock crystals and grains) and water. Permittivity 19 has also been called dielectric permeability. Permittivity equals the conductivity of the composite system of matrix and fractures. Since resistivity is the inverse of conductivity, equations A-2 and A-3 can be re-written in more standard oil and gas notation as: () ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = o 2 w 2 o R 1 1 R 1 R 1 0 φφ θ ….…. (A-4) () ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = w 2 o 2 ow o R 1 1 R 1 R 1 R 1 R 1 90 φφ θ ….…. (A-5) Equations A-4 and A-5 are for a system consisting of matrix- fractures at zero and ninety degrees, respectively. The situation is presented schematically in Figure 6. 0 o R = θ represents the resistivity of the composite system at zero degrees when it is 100% saturated with water of resistivity R w . 90 o R = θ is the resistivity of the composite system at ninety degrees when it is 100% saturated with water of resistivity R w . ø 2 represents the porosity of fractures; this porosity is attached to the bulk volume of the composite system, i.e., it is equal to fracture void space divided by the bulk volume of the composite system. R w is water resistivity at reservoir temperature, and R o is the resistivity of the matrix (when S w =100%). The formation factor F =0 of a system in parallel is given by: ( ) wo m 0 R/RF 0 0 = = == − = θ θ φ θ ………. (A-6) The formation factor F =90 of a system in series is given by: ( ) wo m 90 R/RF 90 90 = = == − = θ θ φ θ ……. (A-7) The formation factor F of only the matrix is given by: ( ) wo m b R/RF b == − φ ………. (A-8) Combining equations (A-4), (A-6) and (A-8) leads to: ( ) ( ) [ ] b m b220 1/1F φφφ θ −+= = ………. (A-9) Combining equations (A-5), (A-7) and (A-8) leads to: ( ) ( ) [ ] b m b F − = −+= φφφ θ 2290 1 .……. (A-10) Equations A-9 and A-10 assume that the fracture porosity exponent, m f , is equal to 1.0. The equations can be extended to the case where m f is greater than 1.0 as follows: ( ) ( ) [ ] bff m b mm F '1/1 220 φφφ θ −+= = ……… (A-11) ( ) ( ) [ ] bff m b mm F − = −+= '1 2290 φφφ θ ………. (A-12) where a modification is entered from ø b to ø’ b for taking into account the possibility of an m f >1.0. The modification is: f f b 2 2 1 ' φ φφ φ − − = . ….…. (A-13) 2 ln ln )1( φ φ −−= ff mmf ……… (1-14) The equation is valid for ø 2 >0; f has been found to range exponentially between 1.0 at ø = ø 2 , and m f at ø = 1.0, using numerical experimentation. Equations (A-11) and (A-12) can be combined as follows for calculating the porosity exponent m for current flowing at any angle with respect to the fractures: θθ θθ 2 90 2 0t sin F 1 cos F 1 F 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ == … (A-15) Knowing that F t = ø -m leads to: θθ φ θθ 2 90 2 0 m sin F 1 cos F 11 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ == − … (A-16) Solving for m of the composite system at any angle, we obtain: ( ) ( ) ( ) () [ ] φ θθ θθ log F/sinF/coslog m 90 2 0 2 == + = …… (A-17) which is the same as equation (1) in the main body of the text. 6 Θ = 50° m f = 1.0 m fp = 1.19 Θ = 50° m f = 1.3 m fp = 1.49 Θ = 0° m f = 1.0 m fp = 1.0 Θ = 0° m f =1.3 m fp = 1.3 (A) (B) (C) (D) CURRENT DIRECTION IN ALL CASES DUAL POROSITY Ø 2 = 0.01 FIGURE 1. Schematics assume that current direction is horizontal in all cases, thus the angle θ in the schematic corresponds to fracture dip. Fracture porosity (Ø 2 ) = 0.01. (A) horizontal fracture with unity tortuosity (m f = 1.0), (B) horizontal fracture with tortuosity larger than 1.0 that leads to a porosity exponent of the fractures (m f ) equal to 1.3, (C) non-horizontal fracture (θ = 50°) with unity tortuosity (m f = 1.0); the 50° angle leads to a pseudo fracture porosity exponent (m fp ) equal to 1.19, (D) non-horizontal fracture (θ = 50°) with tortuosity (m f = 1.3). The 50° angle leads to a pseudo fracture porosity exponent (m fp ) equal to 1.49. If the flow of current is vertical, the angle corresponds to 90 minus fracture dip. This paper discusses research associated with cases (B) and (D). Research associated with cases (A) and (C) were discussed previously. 16 FIGURE 2. Schematic of mixture and aligned ellipsoidal inclusions. The host environment has a permittivity ε e and the ellipsoidal inclusion has a permittivity ε i . The mixture effective permittivity ε eff is anisotropic as on the different principal directions the mixture possesses different permittivity components. (Source: Sihvola 19 ) 7 0.001 0.010 0.100 1.000 123 DUAL-POROSITY EXPONENT, m (m f of only fractures = 1.3) TOTAL POROSITY PHI2 = 0.001 PHI2=0.01 PHI2=0.1 THETA = 0 O THETA = 90 O FIGURE 3. Total porosity versus dual porosity exponent (m) for different values of fracture porosity (PHI2). The matrix porosity exponent (m b = 2.0) and the fracture porosity exponent (m f = 1.3) are constant. 0.01 0.10 1.00 123 DUAL-POROSITY EXPONENT, m (fracture porosity, φ 2 = 0.01) TOTAL POROSITY mf = 1.0 mf = 1.3 mf = 1.5 THETA = 0 O THETA = 90 O FIGURE 4. Total porosity versus dual porosity exponent (m) for different values of the fracture porosity exponent (m f ). Fracture porosity (Ø 2 = 0.01) and the matrix porosity exponent (m b = 2.0) are constant. 8 0.01 0.10 1.00 123 DUAL-POROSITY EXPONENT, m (fractures exponent, m f = 1.3) TOTAL POROSITY 0 degrees 50 degrees 70 degrees 80 degrees 90 degrees FIGURE 5. Total porosity versus dual porosity exponent (m) for different fracture angles ( θ ). Fracture porosity (Ø 2 = 0.01), matrix porosity exponent (m b = 2.0) and fracture porosity exponent (m f = 1.3) are constant. A B C D FIGURE 6. Systems where host and inclusion run (A) parallel and (B) perpendicular to flow (Source: Sihvola 19 ). In these cases fracture tortuosity is equal to 1.0 and the fracture porosity exponent m f = 1.0. In cases C and D, object of this study, the values of m f are larger than 1.0 due to tortuous paths of the fractures. . to fracture dip. Schematic 1-A displays a horizontal fracture with tortuosity equal to 1.0. In this case the porosity exponent of the fractures (m f ) is also equal to 1.0 and fracture dip. naturally fractured reservoirs where dip of fractures ranges between zero and 90 degrees, and where fracture tortuosity is greater than 1.0. This results in an intrinsic porosity exponent of the fractures. θ in the schematic corresponds to fracture dip. Fracture porosity (Ø 2 ) = 0.01. (A) horizontal fracture with unity tortuosity (m f = 1.0), (B) horizontal fracture with tortuosity larger than