## Step-by-Step Solutions and Explanations ### Part A (i): Derivation of the Kozeny-Carman Equation Start with the **Hagen-Poiseuille equation** for laminar flow through a single capillary tube: \[ Q = \frac{\pi r^4}{8\mu} \frac{\Delta P}{L} \] where \( Q \) is volumetric flow rate, \( r \) is tube radius, \( \mu \) is dynamic viscosity, \( \Delta P \) is pressure drop, and \( L \) is length. For a porous medium modeled as a bundle of tubes: - Permeability (\( k \)) relates to Darcy’s Law: \( Q = -kA(\Delta P/\mu L) \) - Porosity (\( \phi \)) is the fraction of pore volume to total volume. - The **specific surface area** (\( S_v \)) is the surface area of grains per unit volume. Expressing permeability, use the geometry of packed spheres and relate pore size to grain size: \[ k = C \frac{\phi^3}{S_v^2 (1-\phi)^2} \] where \( C \) is a dimensionless constant (Kozeny constant) encapsulating shape and tortuosity. Including **tortuosity** (\( \tau \)), which accounts for the actual length of flow paths being longer than the straight-line distance, the Kozeny-Carman equation is: \[ k = \frac{1}{K_c} \frac{\phi^3}{S_v^2 (1-\phi)^2} \] where \( K_c \) (Kozeny constant) incorporates both the pore shape factor and tortuosity. --- ### Part A (ii): Deduction of the Kozeny Constant (\( K_c \)) The **Kozeny constant** (\( K_c \)) arises from the geometric complexity and tortuosity of the flow path. If \( \tau \) is the tortuosity (actual path length/straight length) and \( \alpha \) is a shape factor, then: \[ K_c = \alpha \tau^2 \] Here, \( \tau^2 \) represents the increase in flow path length squared, and \( \alpha \) accounts for non-cylindrical pore shapes. For a bundle of straight capillaries, \( \tau = 1 \) and \( \alpha = 5 \), so \( K_c = 5 \). For real rocks, \( \tau > 1 \), so \( K_c > 5 \). --- ### Part B (i): Mathematical Relation for Sorting—Trask Sorting Coefficient **Sorting** refers to the distribution of grain sizes. Trask's sorting coefficient (\( S_o \)) is defined as: \[ S_o = \sqrt{\frac{d_{75}}{d_{25}}} \] where \( d_{75} \) and \( d_{25} \) are grain diameters at 75% and 25% cumulative frequency, respectively. This coefficient can be correlated with permeability at constant porosity: higher sorting (lower \( S_o \)) usually means higher permeability for a given porosity. A typical empirical relation is: \[ k \propto d^2 S_o^n \] where \( d \) is mean grain size, and \( n \) is an exponent (often negative, showing that poor sorting reduces permeability). --- ### Part B (ii): Specific Surface Area Expression For a pack of spherical grains with dominant diameter \( d \): \[ S_v = \frac{\text{total grain surface area}}{\text{total grain volume}} \] For spheres, surface area = \( 4\pi r^2 \), volume = \( \frac{4}{3}\pi r^3 \), so for one sphere: \[ \frac{4\pi r^2}{\frac{4}{3}\pi r^3} = \frac{3}{r} \] For a collection of spheres with diameter \( d \): \[ S_v = \frac{6}{d} \] To include sorting, use a weighted average or more complex function involving the sorting coefficient. For a log-normal distribution of grain sizes (mean \( d_m \), variance \( \sigma^2 \)), a simplified expression is: \[ S_v \approx \frac{6}{d_m} (1 + \sigma^2) \] where \( \sigma \) is the standard deviation of the grain size distribution. --- ### **Final Answers** #### Kozeny-Carman Equation: \[ k = \frac{1}{K_c} \frac{\phi^3}{S_v^2 (1-\phi)^2} \] #### Kozeny Constant: \[ K_c = \alpha \tau^2 \] #### Trask Sorting Coefficient: \[ S_o = \sqrt{\frac{d_{75}}{d_{25}}} \] #### Specific Surface Area for Spheres: \[ S_v = \frac{6}{d} \] or, with sorting, \[ S_v \approx \frac{6}{d_m}(1+\sigma^2) \]