Journal of Petroleum Science and Engineering 48 (2005) 185 – 198 www.elsevier.com/locate/petrol Effects of petrophysical rock properties on tortuosity factor Attia M. Attia 1 Faculty of Petroleum Engineering-Suez Canal University-Suez, 43721, Egypt Received 30 April 2004; received in revised form 26 April 2005; accepted 25 June 2005 Abstract Tortuosity factor is an important parameter of formation resistivity factor calculations in the Archie formula, which is used to predict water saturation. The objective of this work was to study the effects of petrophysical rock properties, i.e., amount of finegrains, porosity, cementation factor, formation resistivity factor,electrolyte concentrations and degree of brine saturation, on the tortuosity factor using Berea and synthetic sandstones cores. This study also trying to formulate empirical correlations between tortuosity factor and these studied petrophysical rock properties. The obtained results showed that the tortuosity factor is not a constant value, but it varies largely according to many parameters such as were studied in the present article. It was found that it increases as a result of decreasing the amount of fine grains, increasing formation resistivity factor, and cementation factor, and decreasing both porosity and degree of brine saturation. Tortuosity obtained from electrical resistivity measurements is very close to the tortuosity obtained from capillary pressure data. The analysis demonstrated that the correlations between the tortuosity factor and the petrophysical rock properties would yield a strong relationship with most accurate coefficients. D 2005 Elsevier B.V. All rights reserved. Keywords: Tortuosity; Fine grains; Formation resistivity factor; Degree of brine saturation 1. Introduction Carman (1939) as referenced by Adisoemarta et al. (2000) shows that the microscopic flow path through a porous media is approximately at 458 with respect to the direction of the bulk ionic current through saturated unconsolidated material as: Xa ¼ X =cos 45 ¼ 1:414 X : ð1Þ E-mail address: aattia2@Lsu.edu. Academic Visitor at Petroleum Engineering Department, Louisiana State University, USA. 1 0920-4105/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2005.06.012 Where X a is the actual flow path length and X is the thickness of porous media. The higher value reflects a microscopic flow path with a larger component perpendicular to bulk ionic flow compared to the component parallel to flow. They concluded that the minimum value of tortuosity factor equal to one and the maximum value is 1.4, leasing the tortuosity factor equal to one can be unfavorable to the oil in place calculation. Also they found that for recalculation of water saturation by setting tortuosity factor to a unity is between 5% less to 30% more than the real assumed water saturation. In addition, Carman defined the tortuosity factor is equal to the effective length of 186 A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 the fluid flow path within the porous media X a over the apparent length X. Tortuosity is the ratio between the actual length of true flow paths to the theoretical length of the sample or porous material. Adisoemarta et al. (2000) reported that the tortuosity factor, which is equal to the square root of tortuosity, is a function of the average angle of electrical movement with respect to the bulk fluid flow and cementation factor (m) and is related to the flow area difference between pore throat and pore body. Hypothetically it is unfeasible to have tortuosity factor less than one and must only advance to one for a nearly linear ionic flow path through the porous media. Then the minimum values of the tortuosity factor are equivalent to the theoretical length of porous media. Koponen et al. (1996) defined tortuosity as a specific transportation mechanism, and it is evident that it is a physical quantity not uniquely defined. They constructed an empirical equation between porosity and tortuosity as: s ¼ 0:8ð1 UÞ þ 1: ð2Þ They reported that the idea of tortuosity is often introduced in the context of solving the closure problem for moving in porous media by driving the macroscopic transport equations in terms of average quantities alone. A usual method of deducing the appropriate form of the drag force between fluid and the solid matrix is to use some simplified models of the porous material, such as the capillary model and to generalize the results for more realistic material. This generalization may be attempted by introducing an additional parameter that is supposed to take care of the more complicated transportation path neglected in the model. Fellah et al. (2003) studied tortuosity and porosity measurements in the pore space for packing of glass beads. This technique for measuring porosity and tortuosity is based on a time model of the direct and inverse scattering problem for the propagation of transient ultrasonic waves in a homogeneous isotropic slab of porous material with a rigid frame. Olny et al. (2001) reported an analytical solution for describing tortuosity and typical length of permeable material by acoustical measurements under the homogenization theory; wave propagation in permeable materials is dependant upon the numerical geometrical parameters. Normally, five parameters are used: por- osity, static airflow resistance, tortuosity, and viscous and thermal characteristic lengths. The three previous parameters are normally complex to be measured with present direct methods, for a wide range of materials. The estimated process is based on the measurement of dynamic density and compressibility, in order to separate viscous and thermal property. Perkins et al. (1956) concluded that for a predetermination of the tortuosity, of fully brine saturation sandstones, has shown that the formation resistivity factor, F r, the tortuosity, t, and porosity are related by Fr ¼ s2 =U: ð3Þ For partial saturation in the presence of more than one phase in the sample, the resistivity factor is related to the porosity, tortuosity and saturation as: Fr ðSr Þ ¼ s2 =UTSr : ð4Þ In addition, concluded saturation exponent dnT may be articulated by the tortuosity and apparent cross section area of the electrolyte during electric current stream, the Brine saturation exponent is a function of resistivity ratio as follows: Fr =Fr ðSr Þ ¼ Srn ¼ Ro =Rt : ð5Þ Wildenschild and Jensen (1999), Knight (1991) concluded the relationships between tortuosity and saturation of formation. Five homogeneous and three heterogeneous sands are used in this study. Although the tortuosity flow patterns, they found that the effective unsaturated hydraulic conductivity as well as the retention curves for the three heterogeneous sands were quite related, thus signifying that this type of heterogeneous flow system can be treated as a corresponding homogeneous medium characterized by effective parameters. Revil et al. (1998) built up a new electrical conductivity equation based on Bussian’s model and accounting for the special performance of ions in the pore space. The tortuosity of the transport of anions is self-sufficient of the salinity and corresponds to the bulk tortuosity of the pore space, which is obtained by the result of the formation resistivity factor and the porosity. For the cations, the situation is dissimilar. At elevated salinities, the main paths for the electro passage of the cations are situated in the interconnected pore space, and the tortuosity for the transport of cations is therefore the bulk tortuosity. As the A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 Salem (1993) concluded that tortuosity is a key parameter controlling the modification in jointly cementation factor and Kozeny–Carman constant designated the difficult electric and hydraulic tortuous passageway in the sediments. In addition, tortuosity shows specific control on the variation of formation resistivity factor ( F r), where porous media show additional resistance to electric current. The correlation between cementation factor and tortuosity maybe suggested that the existence of very small pores force the fluid to stream through a longer path (higher tortuosity) than when the fluid flows in larger pores or fractures. The empirical equation of tortuosity with cementation factor is: m ¼ 0:91365 ln s þ 1:55514: ð6Þ One potential challenge is the difficulty associated with attempting to extract a relationship among porosity, permeability, and particles diameter in this equation. Using a simple plot, Carman–Kozeny, estimated permeability over different data ranges to study the sensitivities associated with tortuosity, particle size, and porosity. Their study indicates that the most significant parameter is porosity. Tortuosity is found to have little effect. Permeability increases significantly at the same time that porosity and particle size increases. Porosity is the most sensitive to permeability; the large uncertainty in particle diameter creates the parameter that controls the accuracy of permeability estimates from the Carman–Kozeny relationships. Particle size also controls the permeability, while porosity exhibiting slighter significance. Kewen and Roland (2002) obtained the pore size distribution index (k) where Purcell (1949) intro- 10 100% fines Tortuosity Factor, t salinity decreases, the main paths for transport of the cations transfer from the pore space to the mineral water boundary and subsequently are focus to different tortuosities. This transfer is function of salinity, ratio between the surface conductivity of the grains and the electrolyte conductivity. The electrical conductivity of grainy porous media is resolute as a function of pore fluid salinity, temperature, water and gas saturations, shale content, and porosity. They found that the model provides an excellent detail for the variation of electrical conductivity with last parameters. Surface conduction at the mineral water interface is described with the Stern theory of the electrical double layer and is shown to be independent of the salinity in shaly sands above 10 3 mol L 1. Stephanie and Richard (1998) concluded that the electrical resistivity of siliciclastic rock depends on the pore fluid resistivity and formation resistivity factor. They studied for low and high porosity. They found that for low porosity sediments the formation resistivity factor depends on clay conduction and porosity. For high-porosity sediments (0.3–0.6), the clay conduction is a minor effect, but controlled primarily by porosity and pore geometry. Porosity versus formation resistivity factors for Amazon Fan well logs show two separate trends that function on the amount of shale present. Muds, with more than about 0.4 shale fractions, follow a trend that increases too much higher formation resistivity factor while porosity decreases, compared to sands and a little muddy sands. They concluded that in high-porosity muds, existence of clays decreases the formation conductivity by increasing the tortuosity of pores. Christian David (1993) concluded that from numerical simulations of two-dimensional networks used as analogs to pore space in porous rocks are presented to emphasize the existence of preferential paths for transport processes in heterogeneous media. Results showed that hydraulic flow and electrical current are generally determined in the so-called bcritical pathsQ when the pore size distribution has a declining exponential-like shape in opposition to nearly homogeneous or uniform like distributions. Shogenova et al. (2001) studied the electrical properties of 273 Cambrian quartz sandstones and concluded that the lithology and cementation are the main factors controlling properties of Cambrian quartz sandstones rock. 187 80% fines 60% fines 40% fines 1 0 0.2 0.4 0.6 0.8 Brine Saturation, Sr 1 1.2 Fig. 1. Tortuosity factor versus brine saturation with different percentage of fine grains at consolidation pressure 20.7 Mpa. 188 A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 2.5 10 At 5 % NaCL Resistivity,Ro, OHm Tortuosity Factor, t 100% fines "-80% fines" 60% fines 40% fines 1 0 0.2 0.4 0.6 0.8 Brine Saturation, Sr 1 At 5 % NaCL( 2 ) 1.5 Ro (5%) = -0.02(FG )+ 3.19 2 R = 0.96 1 0.5 1.2 0 Fig. 2. Tortuosity factor versus brine saturation with different percentage of fine grains at consolidated pressure 27.6 MPa. duced lithology factor index, is equivalent to inverse of tortuositys. It is obtained from fitting the experimental data of the normalizing water saturation versus capillary pressure in order to determine relative permeability. El-Khatib (1995) suggested that the J-function, as intended by Levertt, is not only one of its kinds for all porous media. The integral of the reciprocal of the square of the function over the saturation range is dependent on the tortuosity that will be diverse for the different formations. He concluded that the modified averaged petrophysical rock parameter, J-Function, depends on the tortuosity in term of porosity, irreducible water saturation, and permeability for a given formation. This modified relation, J-function, can be used to estimate permeability from logs resulting in values of brine saturation and porosity. Most of the previous works, concluded that tortuosity,is an important parameter for the recalculation of water saturation, lithology factor index, relative permeability and J-function calculations. They also mentioned that the tortuosity is a key parameter 0 20 40 60 80 Amount of fine grains,FG, % 100 120 Fig. 4. Effect of percentage fine grains in the samples on the resistivity measurements at fully saturation with 5% NaCL. controlling the modification in jointly cementation factor. However, They did not consider the effects of amount of fine grains, pressure consolidation and electrolytes concentrations on the tortuosity factor. The objective of the present work is to study the effects of the amount of fine-grains, pressure consolidation, porosity, cementation factor, formation resistivity factor,electrolyte concentrations and degree of brine saturation on the tortuosity factor using Berea and synthetic sandstones cores. Moreover, in this study it was attempted to formulate empirical correlations between tortuosity factor and the petrophysical rock properties. The output of these correlations would help to improve well log interpretation. 2. Results and discussion The results of the present study are demonstrated graphically in Figs. 1–29. These figures show the effects of the studied parameters on the tortuosity fac2 10 Tortuosity Factor, t 100% fines Tortuosity factor, t 80% fines 60% fines 40% fines 1 0 0.2 0.4 0.6 0.8 1 1.2 Brine Saturation, Sr Fig. 3. Tortuosity factor versus brine saturation with different percentage of fine grains at consolidated pressure 34.5 MPa. 1.6 t = -0.01(FG) + 2.02 2 R = 0.94 1.2 0.8 t-fines t-fines( 0.4 0 0 20 ) 40 60 80 Amount of fine grains,FG,% 100 120 Fig. 5. Effect of percentage fine grains in the sample on the tortuosity factor at 5% NaCL. A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 100 10 20.7 MPa Tortuosity Factor, t Tortuosity Factor, t 20 MPa 27.6 MPa 34.5 MPa 10 1 0.01 189 0.1 27.6 MPa 34.5 MPa 1 0.1 1 1 Brine Saturation, Sr Brine Saturation, Sr Fig. 6. Effect of consolidation pressure on the tortuosity factor at 40% fine grains. Fig. 8. Effect of consolidation pressure on the tortuosity factor at 100% fine grains. tor: these studied influencing parameters are the amount of the fine grained sands in the core (by %),the cementation factor,the consolidation pressure, the degree of saturation, the formation resistivity factors and the salinity of brine water. Tables 1 and 2 shows the summarized results of the petrophysical rock properties for Berea and synthetic sandstone cores, respectively. Presentation and discussion of the obtained results are given in the following separate heading. MPa. Various brine water salinity, which varies from 0.2% to 5% NaCL. Inspection of these figures shows that the same trend is given in all these figures,i.e., decreasing the values of tortousity factor with increasing the percentage of fine materials in the cores. Fig. 1. which shows the effect of the percentage of fine grains at consolidation pressure of 20.7 MPa, and various brine saturation values. It indicates that the amount of fines in core has a great influence on the tortuosity factor, however,changing the concentration of the brine has to a little extent,some influence at the 20.7 MPa consolidation pressure. Figs. 2 and 3 shows that increasing the pressure to 27.6 or to 34.5 MPa, the tortuosity factor will decrease significantly by increasing the brine saturation, however, the effect of the percent of fine grains of sands in the n core still as it stated above, i.e., tortuosity increases with decreases of the percent of fines in the core. It is clear that with decreasing percentage of fines, the tortuosity factor increases for all synthetic cores. In addition, the increase is also owing to the surface conduction area, 2.1. Effect of amount of fine grains on tortuosity factor The results are shown in Figs. 1–3 show the effects of the amount of fines in the core specimens on tortuosity factor at various consolidation pressures. The percentage of fine sand in the core were changed to give different cores having 40%, 60%, 80% and 100% fines. These cores were consolidated under three different consolidation pressure: 20.7, 27.6 and 34.5 10 2.5 20.7 MPa Tortuosity Factor, t Tortuosity Factor, t 27.6 MPa 34.5 MPa t = 0.36Fr0.56 R2 = 0.92 2 1.5 1 1 0.1 1 Brine Saturation, Sr Fig. 7. Effect of consolidation pressure on the tortuosity factor at 60% fine grains. 5 10 15 20 25 30 Formation Resistivity Factor, Fr Fig. 9. Empirical correlation between tortuosity factor and formation resistivity factor for Berea sandstones cores at 5% NaCL. 190 A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 2.5 Tortuosity Factor, t Tortuosity factor, t 2 1.5 t = 0.87Fr0.16 R2 = 0.97 1 0.5 5.5 10.5 15.5 20.5 2 25.5 Formation Resistivity Factor, Fr Fig. 10. Empirical correlation between formation resistivity factor and tortuosity factor for synthetic cores at 5% NaCL. because the conductivity of the matrix is equal to the conductivity of brine plus conductivity of the surface conduction area. At this point, the conductivity of brine is constant; however, the percentage of fines varies from 40% to 100 %. The conductivity is affected by the percentage of fine grains; Fig. 4. shows that the resistivity increases as the percentage of fines decrease. This is due to a decrease in surface transmission; therefore, the tortuosity increases as shown in Fig. 5. In addition, Figs. 6–8 the tortuosity increases as compaction increases, owing to porosity decreases and cementation factor increases. This change in the porosity and cementation factor is due to the rearrangement of grains as pressure increases. Fines in the sample, with pressure help to increase the cementation factor and create a longer path (higher tortuosity), with narrow pores. This behavior is very clear at lower brine saturation (close to irreducible water saturation zone). A best fit is created for the results between percentage of fine grains in samples and true resistivity, formation resistivity factor and tortuosity factor. The 14 Fr = -0.15(FG) + 19.79 2 R = 0.96 12 10 8 6 4 1.8 1.82 1.84 1.86 1.88 1.9 Cementation Factor,m 1.92 1.94 Fig. 12. Relation between tortuosity factor and cementation factor at 5% NaCl for Berea sandstone cores. empirical equations for each one relation were obtained as follow: For synthetic sandstones cores: Rt ¼ 0:02FG þ 3:19 ð7Þ Fr ¼ 0:15FG þ 19:79 ð8Þ s ¼ 0:01FG þ 2:02: ð9Þ Where F G represents the percentage of fine grains in samples. 2.2. Tortuosity and Formation resistivity factor relationship The effects of formation resistivity factor on tortuosity for Berea and synthetic cores were studied. Fig. 9. for Berea sandstones and Fig. 10. for synthetic cores represent the different percentages of fines as grain size increases. These figures show the influence of the formation resistivity factor in fully saturated brine samples on tortuosity. It is clear that with an increasing formation resistivity factor the rock becomes more Tortuosity Factor, t 16 1.33 2 R = 0.9619 1.5 1.78 0 0.5 Formation resistivity factor,Fr t = 0.90 m 2 1.5 0.99 t = 0.85 m 2 R = 0.91 1 2 0 0 20 40 60 80 100 120 Amount of Fines grains,FG,% Fig. 11. Effect of percentage of fine grains on the formation resistivity factor. 0.5 0.5 1 1.5 Cementation Factor,m 2 Fig. 13. Relation between tortuosity factor and cementation factor at 5% NaCl for synthethic cores. A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 6 Tortuosity , T Tortuosity Factor, t 5 5 T = 1.94Sr -0.43 R2 = 0.92 4 3 2 T-Sr T-Sr( 1 0.3 % 1%NaCL 5%NaCL 4 3 2 1 0 ) 0 0 0 0.2 0.4 0.6 0.8 Fig. 14. Effect of degree of brine saturation on the tortuosity for Berea sandstone specimens at 5% NaCl concentration. resistance to electric current. Tortuosity therefore increases with more tortuous passages. For synthetic cores, Fig. 11 shows the correlation between formation resistivity factors and percentage of fines in the samples. It is clear from this figure that the formation resistivity factor increases when the percentage of fines is reduced; therefore, irregularity strongly affects the formation resistivity factor and tortuosity (Fricke, 1931). In addition, a best fit is created between tortuosity and formation resistivity factor. Thereby, one can find the empirical equations for relationships such as: For Berea sandstones cores: s ¼ 0:36Fr0:56 : ð10Þ For synthetic sandstones cores s ¼ 0:87Fr0:16 : ð11Þ 2.3. Tortuosity and porosity exponent (cementation factor) relationship T Vs Sr 10 ) -0.55 T = 1.54 Sr 2 R = 0.80 1 1.2 Figs. 12 and 13 show that the tortuosity factor increases with increasing cementation factor (porosity exponent) for Berea and synthetic cores, respectively. Cementation may have an influence; therefore, the electric flow expands and the effectiveness of electric current path decreases. Tortuosity, therefore, increases as cementation increases. For synthetic cores the cementation factor can indicate the type of rock according to grain size. For studying each group, experiments show that the porosity and cementation factor increase as grain size decreases, as in Table 3. As the percentage of fine grains decrease, from 100% to 40% the degree of sorting decrease close to poor sorting and very close to irregularity grains which means the grains become close to each other and consequently the pores become slighter or even bunged. This improves the direct interaction between cementation factor and tortuosity. The cementation factor increases as pores decrease. Subsequently, the forces push the fluid, thus increasing flow. Consequently, tortuosity is 5 Tortuosity Factor, t 20 0.4 0.6 0.8 Brine Saturation, Sr Fig. 16. Effect of salinity on the tortuosity–saturation curve at 100% fine grains and consolidation pressure 20.7 Mpa. The relation between tortuosity factor and porosity exponent for Berea and synthetic cores was studied. T Vs Sr( 0.2 1 Brine Saturation, Sr Tortuosity , T 191 0.30% 4 1%NaCL 3 5%NaCL 2 1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Brine Saturation , Sr 0.8 0.9 1 Fig. 15. Effect of degree of saturation on the tortuosity using synthetic cores. 0 0.2 0.4 0.6 0.8 1 1.2 Brine Saturation, Sr Fig. 17. Effect salinity on the tortuosity–saturation curve at 100% fine grains and consolidation pressure 27.6 MPa. 192 A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 8 10 8 1%NaCL 6 5%NaCL Tortuosity Factor, t Tortuosity Factor, t 0.30% 4 2 0 0 0.2 0.4 0.6 0.8 Brine Saturation, Sr 1 0.3 % NaCL 1%NaCL 5%NaCL 6 4 2 0 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Brine Saturation, Sr Fig. 18. Effect salinity on the tortuosity–saturation curve at 100% fine grains and consolidation pressure 34.5 MPa. enhancing, as fluid flows through a longer tortuous path, compared to the path that fluid follows in a more porous rock. Because of the increased force, the fluid flows through a longer path, smaller pores (higher tortuosity) than when the fluid flows in larger pores. This phenomenon is related to pore size and pore area in porous media. In addition best fit is created for the results between the tortuosity and porosity exponent or shape factor (cementation factor). The best empirical equations obtained are as follows: For Berea sandstones cores: s ¼ 0:90m1:33 : ð12Þ For synthetic sandstones cores: s ¼ 0:93m 0:13: ð13Þ Moreover, the relation between tortuosity factor and porosity for twelve synthetic cores is: s ¼ 2:03U þ 1:78: ð14Þ Fig. 20. Effect of salinity on the tortuosity–saturation curve at 80% fine grains and consolidation pressure 27.6 Mpa. 2.4. Degree of saturation and Tortuosity relationship It is necessary to study the relation between tortousity factors and fluid saturation. With reduced fluid saturation, the saturation of both wetting and nonwetting phase’s changes, as does the water saturation exponent. There is an independent relation between tortuosity and the degree of fluid saturation. Nevertheless, tortuosity is very important for explaining the change in wettability using a water saturation exponent and is useful for log interpretation. Figs. 14 and 15 for Berea and synthetic cores respectively show that as soon as the degree of brine saturation reduces, there is an increase in tortuosity which means an increase in electrical resistance of the core samples. Due to a decrease in cross-sectional area of the electrolyte, there is an increase in electrical path length. Therefore, at surface water saturation (irreducible water saturation) the cross-sectional area is reduced to the value of a fully saturated zone, although the tortuosity increases. At high brine saturation, the water in the pore space becomes highly conductive and the surface conduction 8 8 1%NaCL 6 Tortuosity Factor, t Tortuosity Factor, t 0.3 NaCL 5%NaCL 4 2 0 0.3 % NaCL 1%NaCL 5%NaCL 6 4 2 0 0 0.2 0.4 0.6 0.8 1 1.2 Brine Saturation, Sr Fig. 19. Effect of salinity on the tortuosity–saturation at 80% fine grains and consolidation pressure 20.7 MPa. 0 0.2 0.4 0.6 0.8 Brine Saturation, Sr 1 1.2 Fig. 21. Effect of salinity on the tortuosity–saturation curve at 80% fine grains and consolidation pressure at 34.5 Mpa. A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 100 0.3 % NaCL 1%NaCL 5%NaCL 4 Tortuosity Factor, t Tortuosity Factor, t 5 193 3 2 1 0.3 % NaCL 1%NaCL 5%NaCL 10 1 0 0 0.2 0.4 0.6 0.8 1 0.1 1.2 0 Brine Saturation, Sr Fig. 22. Effect of salinity on the tortuosity–saturation curve at 60% fine grains and consolidation pressure at 20.7 Mpa. of brine/air interface is very small, relative to the total conductivity of the rock. Hence, tortuosity is small when brine saturation is high, due to the higher efficiency of the current flow. This is confirmed by Kozeny –Carman equation with decreasing the specific surface area led to increasing the tortuosity. With decreasing the brine saturation the surface area of the air/water interface increases; however, this increases the amount of insulator in the porous media, consequently decreasing the conductivity of the rock and also the efficiency of electric current flow through the porous media; hence, the tortuosity increases. These processes continue with the decrease of saturation (gas extended), thus, increasing air/brine interface and decreasing conductivity; hence, tortuosity increases as the efficiency of electric flow path decreases, Knight (1991). This process continue until no decrease in brine saturation while the cores having a small amount of brine saturation, this brine exists in the pore space as thin film of brine covered the surface of the rock (surface water) defined as connate water saturation. At this stage, the nonwetting phase saturation (insulator) is very high 0.2 0.4 0.6 0.8 Brine Saturation, Sr consequently smallest conductivity; hence, the maximum value of tortuosity factor at the smallest conductivity (highest resistivity value) is obtained. In addition, a best fit is created for the results between tortuosity and degree of saturation. Moreover, empirical equation is given as follows: For Berea sandstones cores: s ¼ 1:94Sr0:43 : ð15Þ For synthetic sandstones cores: s ¼ 1:54Sr0:55 : ð16Þ 2.5. Tortuosity and electrolyte concentrations relationship The relation between tortuosity factor and electrolyte concentration is an autonomous relationship as shown in Figs. 16–27. It’s clear that tortuosity factor increases with salinity increase especially for all cores having amount of fines less than 100%. Notice that 8 Tortuosity Factor, t Tortuosity Factor, t 10 0.3 % NaCL 1%NaCL 5%NaCL 6 4 2 0 1 0 0.2 0.4 0.6 0.8 Brine Saturation, Sr 1 1.2 Fig. 23. Effect of salinity on the tortuosity–saturation curve at 60% fine grains and consolidation pressure at 27.6 Mpa. 1.2 Fig. 24. Effect of salinity on the tortuosity–saturation curve at 60% fine grains and consolidation pressure at 34.5 Mpa. 100 0.3 % NaCL 1%NaCL 5%NaCL 1 0 0.2 0.4 0.6 0.8 1 1.2 Brine Saturation, Sr Fig. 25. Effect of salinity on the tortuosity–saturation curve at 40% fine grains and consolidation pressure at 20.7 Mpa. A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 16 0.3 % NaCL 1%NaCL 5%NaCL 6 4 2 0 0 0.2 0.4 0.6 0.8 Brine Saturation, Sr 1 1.2 Formation resistivity Factor,Fr Tortuosity Factor, t 8 10 12 8 10 8 6 2 6 Ro = -10.39Rw +2 20.986Rw - 0.8425 R =1 Salinity-resistivity 4 F-salinity Salinity-resistivity. ( 2 F-salinity. ( ) 4 ) 0 0 Fig. 26. Effect of salinity on the tortuosity–saturation curve at 40% fine grains and consolidation pressure at 27.6 Mpa. 12 Fr = -6.3024Rw2 + 2.9813Rw + 13.764 14 R2 = 1 2 Core Resistivity,Ro,OHm 194 0 1.5 0.5 1 Brine resistivity,Rw,OHm Fig. 28. Effect of salinity on the resistivity and formation resistivity factor. the tortuosity factor increases in conjunction with the electrolyte concentration, until the level of concentration becomes less effective with increasing salinity concentrations. In fact, the conductivity of the rock is equal to the conductivity of electrolyte concentration, plus the conductivity of the surface area in question, which is constant while the conductivity of brine changes; thus, the conductivity of the rock depends on the salinity concentration. As the conductivity of electrolyte increases the formation resistivity factor decreases; thus, resistivity decrease. Fig. 28 shows the relation between formation resistivity factors and electrolyte resistivity. It is clear that the formation resistivity factor increases with increasing resistivity (decreasing electrolyte concentrations). 2.6. Tortuosity from capillary pressure and from electrical resistivity methods such as, capillary pressure, acoustic properties and electrical properties of core samples. Fig. 29. Show the tortuosity from capillary pressure and tortuosity obtained from electrical measurements. It is clear from this figure that similarity for results but the accuracy is not high. The results from electrical properties are more accurate because these are measured data. However, the tortuosity from capillary pressure depends on the slope of the capillary pressure versus normalized water saturation. 2.7. Comparison between the correlations in the literature and correlations obtained in this article Figs. 30–34 show that the comparison between the correlations which were mentioned in the introduction in the present study and those obtained from resistiv- Tortuosity is very important in fluid flow through porous media. Tortuosity can be obtains by different 6 5 0.3 % NaCL 1%NaCL 5%NaCL Tortuosity Tortuosity Factor, t 100 10 4 3 2 electrical 1 Capillary pressure 0 1 0 0.2 0.4 0.6 0.8 Brine Saturation, Sr 1 1.2 Fig. 27. Effect of salinity on the tortuosity–saturation curve at 40% fine grains and consolidation pressure at 34.5 Mpa. 21 23 24 30 31 32 33 34 3.01 4.35 4.26 3.78 4.7 4.04 3.82 4.48 electrical Capillary pressure 3.84 4.75 3.87 3.61 3.6 4.02 4.95 4.33 Core # Fig. 29. Tortuosity obtained from capillary pressure and obtained from electrical properties. A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 195 Table 1 Summarized results of porosity, formation resistivity factor, tortuosity, brine saturation exponent for Berea sandstone cores Core no. 21 23 24 30 31 32 33 34 Porosity Tortuosity, T 0.19 0.19 0.186 0.187 0.181 0.182 0.192 3.84 4.35 4.26 3.78 4.71 4.04 3.82 4.48 Tortuosity factor, t Cementation factor, m Formation resistivity factor, F r Water saturation exponent, n 2.09 2.06 1.94 2.17 2.01 1.95 2.12 1.89 1.88 1.79 1.92 1.82 1.79 1.91 22.78 22.21 20.34 25.2 22.34 21 23.34 2.12 1.8 2.42 1.93 1.98 1.62 2.02 1.96 Table 2 Physical properties of the synthetic cores Specimen name Porosity, / 2 6 14 18 22 28 32 38 42 48 54 58 0.26 0.29 0.29 0.22 0.27 0.25 0.19 0.24 0.22 0.20 0.20 0.19 Permeability, K air (mD) Consolidation pressure (MPa) Grain Size (%) Quartz flour From 0.126 to 0.149 mm From 0.149 to 0.177 mm From 0.177 to 0.210 mm 48 37 32 68 49 40 70 55 38 333 132 118 20.7 27.6 34.5 20.7 27.6 34.5 20.7 27.6 34.5 20.7 27.6 34.5 100 100 100 80 80 80 60 60 60 40 40 40 0 0 0 6 6 6 12 12 12 18 18 18 0 0 0 6 6 6 12 12 12 18 18 18 0 0 0 8 8 8 16 16 16 24 24 24 Table 3 Petrophysical rock properties for synthetic cores Core no. Fine grains, (%) Porosity Formation resistivity factor at 5% NaCl, F r Water saturation exponent at 5% NaCl, n Cementation factor at 5% NaCl, m Tortuosity factor at 5% NaCl 2 6 14 18 22 28 32 38 42 48 54 58 100 100 100 80 80 80 60 60 60 40 40 40 0.26 0.29 0.29 0.22 0.27 0.25 0.19 0.24 0.22 0.20 0.20 0.19 5.6 7.1 7.5 6.8 9.8 10.2 11.1 13.8 15.2 14.1 21.1 18.6 1.7 1.8 2.2 1.7 2 1.7 1.7 2 1.6 2.1 2.2 1.8 1.3 1.5 1.6 1.3 1.6 1.6 1.5 1.6 1.5 1.6 1.7 1.7 1.31 1.44 1.48 1.38 1.61 1.61 1.64 1.81 1.83 1.71 2.06 1.86 A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 2.2 Tortuosity factor Tortuosity predicted 196 tortuosity presented in this work Tortuosity by Koppen (1996) Maximum value by Carman 1.4 Minimum value by Carman 1 0.6 0.26 0.29 0.29 0.27 0.25 0.24 0.22 0.2 Porosity 0 0.19 Fig. 30. Comparison between tortuosity calculation by Koponen and tortuosity presented in this article. ity measurements data. Fig. 30 shows that the comparison between tortuosity factor that was predicted by Koponen et al. (1996) and tortuosity factor that was obtained in the present study. It is clear from this figure that the values of tortuosity factor for both correlations were very close to each other. Figs. 31 and 32. show that, the difference between tortuosity factor according to Carman ( which ranges from 1 and 1.4 ) and tortuosity factor obtained in the present study. It is clear that from this comparison that, for Berea sandstones cores, these values of tortuosity factor were more than the maximum value (1.4) that predicted by Carman. Fig. 32. shows that some of these values of tortuosity factor for synthetic cores (compacted cores) are in the interval of Carman range and others values found were above the maximum level (1.4) of Carman. These results showed that tortuosity factor is not a constant value, but it varies largely according to type of rocks and petrophysical rock properties. 2.6 20 40 Cores # 60 80 Fig. 32. Comparison between tortuosity obtained from Carman and obtained from presented in this article using synthetic sandstones cores. Fig. 33 Shows a good agreement between tortuosity factor obtained according to Eq. (11) and tortuosity factor obtained from resistivity measurements. Fig. 34 Represent the relations, between tortuosity factor obtained by Salem (1993) and those obtained in the present article, using Berea and synthetic cores. It is clear from the figure, that a variation between the results from these correlations. This may be due to; the difference of core preparations, amount of fine and consolidation pressure in synthetic cores. 3. Conclusions 1. For all cores tested, the results showed that tortuosity factor is not a constant value, but it varies largely according to many parameters such as studied in the present article. 2. Tortuosity factor increases as a result of decreasing the amount of fine grains, increasing formation resistivity factor, consolidation pressure and cementation factor. 2.2 5 Tortuosity factor predicted,tp Tortuosity factor Tortuosity factor of Synthetic cores 1.8 1.8 Maximum value by Carman 1.4 Minimum value by Carman 1 4 1:1 tp vs. tm 3 2 1 0 0.6 0 10 20 30 40 0 1 2 4 3 5 Tortuosity Factor from electrical measurements ,tm Cores # Fig. 31. Comparison between tortuosity obtained from Carman and obtained from presented in this article using Berea sandstones cores. Fig. 33. Validation of tortuosity factor from formation resistivity factor according to the empirical equation (t = 0.87F0.2 r ) for synthetic cores at 5% NaCL. A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198 Tortuosity factor 2.5 k U 2 197 Lithology factor index Porosity 1.5 1 Acknowledgments " Berea" " synthetic cores" 0.5 0 "Tortuosity predicted by Salem(1993) 1.21 1.41 1.43 1.21 1.54 1.45 1.35 1.46 1.42 1.49 1.55 1.50 Cementation factor Fig. 34. Comparison between tortuosity factor obtained by Salem (1993) and presented in this article as a function of cementation factor. 3. Tortuosity factor increases with decreasing both porosity and degree of brine saturation. 4. Tortuosity factor increases with electrolyte concentration increase for all cores having different amount of fine-grains. 5. Tortuosity factor obtained from electrical resistivity measurements very close to the tortuosity factor obtained from capillary pressure data. 6. New correlations for the tortuosity factor and the petrophysical rock properties such as porosity, cementation factor and formation resistivity factor have been developed to cover a wide range of the amount of fine grains in the samples. 7. The correlations between the tortuosity factor and petrophysical rock properties yielded the strongest relationship with most accurate coefficients. Nomenclature FG Percentage of fine grains in samples Fr Formation resistivity factor J Average rock properties ( J-Function) m Cementation factor n Brine saturation exponent Rw Water resistivity, V.m Rt True rock resistivity, V.m Ro Formation resistivity when 100% saturated with brine, V.m Sr Brine saturation T Tortuosity s Tortuosity factor Xa Actual flow path length X Thickness of porous media Sincere appreciation to Dr. Zaki Bassiuoni for his valuable advice and helpful discussions. I would like to thanks Dr. Hamid Khattab and Dr. Amen AbdElraheem for useful discussions and review this paper. 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