Surface Area, Lecture notes of Law

Download Surface Area and more Lecture notes Law in PDF only on Docsity! Surface Area The specific surface area is a dominant parameter in models for permeability and in the transport of a species that can adsorb on the mineral surfaces. The specific surface area is usually expressed as square meters of surface per gram of solid. Here we will factor out the grain density and express the specific surface area as square meters per cubic centimeter of solid. (Later we will express the specific surface as a ratio of pore surface/pore volume.) The solid will be modeled as an oblate spheroid. This is a solid of revolution of an ellipse about its minor axis. The minor radius is b and the major radius is a. The ratio, Sb/V, is given by the following formula. (Mensuration formulas) 23 3 1 1log 2 4 1e S b V ε ε ε ε ⎛ ⎞− +⎛ ⎞ ⎛= + ⎜ ⎟⎜ ⎟ ⎜ −⎝ ⎠ ⎝⎝ ⎠ ⎞ ⎟ ⎠ where the eccentricity is 2 2a b a ε − = The group, (Sb/V), will have consistent units if S is in square meters, V is in cubic centimeters, and b is in microns. Figure 3.32 plots the specific surface as a function of the minor radius, b. The grain density (2.65 gm/cm3 for quartz) has been factored out and the specific surface area is expressed as per unit cm3 rather than gram. A upper coarse sand grain has a radius of about 103 microns Fig. 3.31 Parameters of an ellipse (CRC Standard Mathematical Tables, 1987) Fig 3.32 Surface to volume ratio of oblate spheroid 3- 26 (one millimeter) and it has a surface area of about 10-3 m2 /cm3. A silt or clay particle with a minor radius of about 1.0 micron has a surface area of about 1.0 m2/cm3. A smectite sheet with a thickness of about 10-3 micron (1.0 nm) will have a surface area of about 103 m2/cm3. (Note: Is something is wrong here? The sphere appears to have a greater specific area than an oblate spheroid. A sphere should be a body of minimum area for a given volume. Answer: For the same volume, an oblate spheroid will have its surface to volume ratio increasing in proportion to the 2/3 power of the aspect ratio. The specific surface is plotted as a function of the radius of the minor axis. The major axis is greater than the minor axis ratio by the aspect ratio.) Clay type Area, m2/gram kaolinite 45 illite 175 montmorillonite 800 When evaluating adsorption, the specific surface area of sand grains usually is not of much interest compared to the clays contained in the rock. For example the following table illustrates the range of specific areas that can be expected from clays (Corey 1990) In addition to the importance of the surface/volume ratio to adsorption on porous media, the ratio of surface area to pore volume will be shown later to be an important parameter in models of permeability and NMR relaxation of fluids in the pore space. The expression for the specific surface shows the surface to pore volume ratio to be inversely proportional to the length of the minor axis, b, for a given eccentricity. The constant of proportionality is 3 for a sphere and is equal to 3/2 for a thin disk. (Note: I think it should be 2 for a thin disk.) Porosity Porosity is the fraction (or percent) of the rock bulk volume occupied by pore space. The porosity may be divided into macro porosity and micro porosity in rocks that have a bimodal pore size distribution. Some examples include: (1) sandstones with a significant amount of clays, (2) sandstones with microporous chert grains, i.e., interparticle and intraparticle porosity, (3) carbonate rocks with vuggy porosity (caverns are an extreme case) and matrix porosity, (4) carbonate rocks with moldic porosity and matrix porosity, (5) carbonate rocks with interparticle porosity and intercrystalline porosity, (6) fracture porosity and matrix porosity. The total porosity can also be divided into effective porosity and ineffective porosity. Ineffective pores are pores with no openings or zero coordination number. Effective porosity can be divided into Cul-de-sac or dead-end pores with a coordination number of one and catenary pores with coordination number of two or more. These types of porosity are illustrated in Fig. 3.33. 3- 27 Permeability The mobility, denoted by λ, is a transport coefficient of the porous medium for the volumetric flux of a fluid just as electrical conductivity and thermal conductivity are transport coefficients for the flow of electrical current and heat, respectively. This transport coefficient was divided by Darcy into two factors (λ = k/µ), the permeability, k, which is a property of the porous medium and the viscosity, µ, which is a property of the fluid. The permeability was originally conceived as a constant of a particular medium. However, in reality the permeability is generally not spatially uniform, i.e., porous media are usually heterogeneous, depends on direction, i.e., is not isotropic, depends on the current stress conditions and past stress history, is a function of the electrolyte composition of the fluids, and depends on the amount and distribution of the fluid phases, i.e., depends on relativity permeability. It is because of this highly variable nature of permeability that we need to know the factors that govern the value of permeability. We will describe two models of the permeability. They are both based on a bundle of capillary tubes model. However, one is based on a packed bed of spherical particles and the other is based on a pore size distribution model. Packed Bed of Spherical Particles One model for relating the flow resistance of porous media to the dimensions of the pores or particles is the Blake-Kozeny model (Bird, Stewart, and Lightfoot 1960). This model represents the pore network of the porous medium as a bundle of capillary tubes with an average or equivalent radius, R, and an average length, L', that is somewhat longer than the system length. The effective radius is related to a particle diameter, Dp, by applying the hydraulic radius concept and assuming that the porous medium is a bed of uniform particles. The resulting expression is then compared with Darcy's law to determine an expression for the permeability of the medium in terms of the particle diameter and porosity. Darcy's law is an empirical relationship between the flux and the driving force for laminar, single phase flow through porous media. /u q A P L k P L λ µ = ∆ = − ∆ = − where P p g zρ= − 3- 30 The constant of proportionality between the flux and the driving force, commonly known as the mobility, λ, is directly proportional to the permeability, k, which is a property of the porous medium, and inversely proportional to the viscosity, µ, which is a property of the fluid. The porous medium is modeled as a bundle of capillary tubes with a length L', that is greater than the system length, L, due to the tortuosity of the pore network. It has been empirically determined that this tortuosity factor can be approximated by the factor 25/12. ( 2'/L Lτ = ) ( )2'/ 25 12L Lτ = = The average velocity in a capillary tube is given by the Hagen-Poiseuille law. ( )2 8 ' o LR P P v Lµ − < >= The average velocity in the bundle of tubes is greater than the average velocity in the pore space of the medium because of the greater length traversed in the tortuous capillary. Alternatively, it can be argued that the fluid it the porous medium must also traverse a greater length but the transverse components of velocity cancel in averaging over the porous medium and thus the average velocity in the pores of the medium is less than the average velocity in a tortuous capillary. ( )/ 'pore capillary v v L L= The average velocity of the fluid in the pores (v, the interstitial velocity) is related to the flux (u, superficial velocity, filtration velocity, or Darcy velocity) by the porosity of the porous medium (φ, pore volume/bulk volume). If the porous medium is random, then the fraction of the cross-sectional area open to pores is equal to the porosity. Thus the flux is 3- 31 ( ) ( ) ( ) ( ) ( ) 2 2 2 / ' 8 ' '/ 8 3 25, 50 12 o L o L o L qu A v v L L R P P L L L R P P L R P P L φ φ φ µ φ µ τ φ τ µ = = = < > − = − − = = By comparing the above equation with Darcy's law we have, 23 50 50 3 Rk kR φ φ = = The above equation is an expression for the equivalent pore radius of the porous medium assuming a bundle of capillary tubes model with a tortuosity of 25/12. The wetted surface of a porous medium can be related to the permeability and porosity by introducing the concept of the hydraulic radius. For flow in a capillary, the hydraulic radius is related to the radius as follows. 2 2 2 h RR R R π π = = In porous media, the hydraulic radius can be determined as follows: ( ) ( ) ( ) ( ) cross section available for flow wetted primeter volume available for flow total wetted surface volume of voids volume of bed wetted surface volume of bed hR a φ = = = = 3- 32 Permeability (darcies) of artificially mixed and wet-packed sand [Jorden and Campbell 1984 (Beard and Weyl 1973)] Size Coarse Medium Fine Very Fine Sorting Upper Lower Upper Lower Upper Lower Upper Lower Extremely well sorted 475. 238. 119. 59. 30. 15. 7.4 3.7 Very well sorted 458. 239. 115. 57. 29. 14. 7.2 3.6 Well sorted 302. 151. 76. 38. 19. 9.4 4.7 2.4 Moderately sorted 110. 55. 28. 14. 7. 3.5 Poorly sorted 45. 23. 12. 6. Very poorly sorted 14. 7. 3.5 Porosity of artificially mixed and wet-packed sand [Jorden and Campbell 1984 (Beard and Weyl 1973)] Size Coarse Medium Fine Very Fine Sorting Upper Lower Upper Lower Upper Lower Upper Lower Extremely well sorted 0.431 0.428 0.417 0.413 0.413 0.435 0.423 0.430 Very well sorted 0.408 0.415 0.402 0.402 0.398 0.408 0.412 0.418 Well sorted 0.380 0.384 0.381 0.388 0.391 0.397 0.402 0.398 Moderately sorted 0.324 0.333 0.342 0.349 0.339 0.343 0.356 0.331 Poorly sorted 0.271 0.298 0.315 0.313 0.304 0.310 0.305 0.342 Very poorly sorted 0.286 0.252 0.258 0.234 0.285 0.290 0.301 0.326 Assignment 3.4 Calculation of Permeability as a Function of Grain Size Calculate and plot the permeability (darcy) as a function grain size (mm) and porosity for grain size in the range (10-4 mm to 10 mm) and porosity of (0.2, 0.3. 0.4, 0.5). Also plot the measured values for the extremely well sorted sand packs listed above. Post the average value of the sand pack porosity. Use Fig. 2.8 to determine the grain size. For a porosity of 0.4 tabulate the approximate grain size (in descriptive scale, mm, and µm) that will result in a permeability of 100 darcy, 1 darcy, and 1 md. 3- 35 Estimation of Permeability from Pore Size Distribution Rapid methods to estimate rock permeability has always been a high priority in the petroleum industry. Mercury porosimetry for measuring capillary pressure and calculation of permeability therefrom was introduced by Bob Purcell of Shell Oil Co. in 1949. The method treats the porous medium as a bundle of capillary tubes with the pore size distribution quantified by the mercury-air capillary pressure curve. The tortuosity is an empirical factor that brings the calculation into correspondence with measured permeability. The average velocity in a capillary tube of radius Ri is described by the Hagen-Poiseuille law. 2 8 i i R Pv Lµ ∆ = The capillary radius can be determined for the relation of the capillary pressure to an equivalent pore radius. ( ) ( ) 2 cos 2 cos,c ii i c i P R R P σ θ σ = = θ Thus the average velocity in a capillary tube can be expressed in terms of the capillary pressure at which that capillary is being entered be a nonwetting fluid. ( ) ( ) 2 2 cos 1 2i c i P v L P σ θ µ ∆ = Let S(Pc) denote the fraction of the pore space that is occupied by the wetting phase when the capillary pressure is equal to Pc. Then dS is the incremental fraction of the pore space corresponding to Pc and Pc-dPc. The interstitial velocity is the integral over all pores. 1 0 v v d= ∫ S The superficial velocity (q/A) is then as follows. ( ) ( ) 2 1 20 cos 2 c P dSu v L P σ θ φ φ µ ∆ = = ∫ S This equation can be compared with Darcy’s law. k Pu Lµ ∆ = 3- 36 By comparing the last two equation, an expression can be derived for the permeability. ( ) ( ) 2 1 20 cos 2 c dSk P S σ θ φ = ∫ Tortuosity has not yet been considered to this point. Purcell introduced a factor, called the “lithology factor” to bring the calculated permeability into correspondence with the measured air permeability. We will use the tortuosity factor here to parallel the nomenclature for the packed bed. ( ) ( ) 2 1 20 cos 2 c dSk P S σ θ φ τ = ∫ Purcell observed that τ ranged from 2.8 for 1500 md sandstone to 12 for 1 md sandstone. This may be compared with the value of 25/12 ≈ 2 for a packed bed of spheres. Thomeer (1960) refined the method by introducing a model for fitting the measured capillary pressure data. Mercury capillary pressure curves can be measured from drill cuttings when cored samples are not available. Swanson (1981) observed that the low pressure portion of the capillary pressure curve was often different between measurements with small samples (e.g. drill cuttings) and larger core samples. This difference is thought to be due to the sample surface roughness and/or the accessibility of pores to the external surfaces. The low pressure portion corresponds the larger pores which contribute the most to permeability. Thus he suggested using a point on the capillary pressure curve that is independent of sample size. This point is the point of tangency of the curve of Pc versus mercury volume as a percent of bulk Fig. 3.36 Comparison of capillary pressure measured on plugs and cuttings (Swanson, 1981) 3- 37 The transformation from the qualitative sorting, to the sorting coefficient (Beard and Weyl, 1973), and to the standard deviation of the log normal distribution is summarized in following table. Sorting So σ Extremely well sorted 1.05 0.072 Very well sorted 1.15 0.207 Well sorted 1.3 0.389 Moderately sorted 1.7 0.787 Poorly sorted 2.35 1.267 Very poorly sorted 4.2 2.128 Correlation of porosity with sorting. The porosity data of Beard and Weyl was correlated with the standard deviation (of the logarithm grain size distribution), Fig. 3.38. Correlation of Porosity with Sorting 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.000 0.500 1.000 1.500 2.000 2.500 Standard Deviation Po ro si ty UC LC UM LM UF LF UVF LVF FIT Fig. 3.38 Correlation of porosity with sorting (R2 =0.93, excluding very poorly sorted data) The regression of porosity with standard deviation excluding the Very poorly sorted data gives the following linear relationship. 0.428 0.0998φ σ= − 3- 40 Tortuosity. The Blake-Kozeny model determined a value of 25/12 for the tortuosity of a bed of uniform spherical particles. We will let the tortuosity, τ, be a function of the sorting. The Carman-Kozeny model is as follows. ( ) ( ) 3 2 2 3 2 2 72 1 or 72 1 p p D k D k φ τ φ φ τ φ = − = − The tortuosity required to fit the Carman- Kozeny equation to the measured permeability of Beard and Weyl was calculated from the above equation. The grain size was estimated by transforming from the qualitative grain size to diameter in mm. The calculated tortuosity and the regression excluding the coarse sand data are illustrated in Fig. 3.39. Grain size Dp Upper coarse 1.30 Lower coarse 0.70 Upper medium 0.40 Lower medium 0.30 Upper fine 0.20 Lower fine 0.13 Upper very fine 0.10 Lower very fine 0.07 Tortuosity Required to Fit Permeability Model 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 0.000 0.500 1.000 1.500 Standard Deviation To rt uo si ty UC LC UM LM UF LF UVF LVF Fit Fig. 3.39 Tortuosity required to fit Kozeny model for permeability (R2=0.93, excluding coarse sand data) Note that the tortuosity extrapolates to 2.5 for zero standard deviation, a value very close to the 25/12 determined by Blake for a spherical bead back. The linear regression, excluding the coarse sand data, give the following result. 2.46 7.72τ σ= + 3- 41 Since both the porosity and tortuosity are a function of the sorting, one would expect a cross-correlation between tortuosity and porosity. The cross-correlation of the linear correlations for porosity and tortuosity and of the porosity and tortuosity of the individual sands are shown in Fig. 3.40. The equation for the cross- correlation of the porosity and tortuosity correlations with sorting is as follows. Cross-Correlation of Tortuosity and Porosity 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 0.250 0.300 0.350 0.400 0.450 Porosity To rt uo si ty Fig. 3.40 Cross-correlation of porosity and tortuosity (upper coarse sand data omitted) 35.6 77.3τ φ= − Permeability predicted from porosity, grain diameter, and sorting. The permeability predicted from the Carman-Kozeny model using the correlation for tortuosity given above is compared with the measured permeability of Beard and Weyl in Fig. 3.41. The predicted values for the upper coarse sand were much larger than the measured values and some are off the figure (i.e., >1000 darcy). Permeability Predicted from Porosity, Grain Diameter, and Sorting 1 10 100 1000 1 10 100 1000 Measured Permeability, darcy Pr ed ic te d Pe rm ea bi lit y, d ar cy Fig. 3.41 Permeability predicted from porosity, grain size, and sorting (R2=0.87) 3- 42