Who proposed this formula, k = 200D2ee2 where, k = coefficient of permeability, De = Effective Grain size?

Question

Prepp Team · Accepted Answer

Understanding Soil Permeability and Terzaghi's Formula

Soil permeability is a fundamental property in geotechnical engineering that describes how easily water can flow through a soil mass. It is quantified by the coefficient of permeability, denoted by \(k\). This coefficient is influenced by various factors, primarily the size and shape of the soil particles, the void ratio (the volume of voids to the volume of solids), the structure and arrangement of particles, the degree of saturation, and the properties of the fluid (like viscosity and density).

Determining the coefficient of permeability is crucial for analyzing seepage through earth dams, groundwater flow, drainage problems, consolidation settlement, and many other geotechnical applications.

The Proposed Permeability Formula: \(k = 200D^2ee^2\)

The question provides a specific formula relating the coefficient of permeability, \(k\), to the effective grain size, \(D\), and void ratio, \(e\):

\displaystyle k = 200D^2ee^2

Where:

\(k\) is the coefficient of permeability.
\(D\) is the Effective Grain size.
\(e\) is the void ratio of the soil.

This formula suggests that the coefficient of permeability is directly proportional to the square of the effective grain size and also related to the void ratio.

Attribution of the Formula: Who Proposed It?

Various researchers have proposed empirical and theoretical formulas to estimate the coefficient of permeability based on soil properties like grain size distribution and void ratio. Some of the notable contributors include Hazen, Kozeny, Carman, and Terzaghi.

Based on the options provided and the specific formula format, the question asks to identify the proposer of the formula \(k = 200D^2ee^2\).

Let's consider the options:

Allen Hazen: Hazen proposed a widely used empirical formula \(k = C D_{10}^2\) for filter sands, where \(D_{10}\) is the effective grain size (the diameter such that 10% of the soil by weight is finer) and \(C\) is a constant (typically around 100). Hazen's formula primarily depends on \(D_{10}\) and does not explicitly include the void ratio \(e\) in its standard form.
Terzaghi: Karl Terzaghi is considered the father of modern soil mechanics. He made significant contributions to understanding soil behavior, including permeability and consolidation. Terzaghi proposed formulas relating permeability to both effective grain size and void ratio. A common form attributed to Terzaghi is \(k = C D_{10}^2 \frac{e^2}{1+e}\), where \(C\) is a constant. While the given formula \(k = 200D^2ee^2\) differs slightly in the void ratio term compared to some standard forms attributed to Terzaghi, the inclusion of both effective grain size squared and void ratio in a formula for permeability aligns more closely with permeability relationships proposed by Terzaghi or derived from principles like the Kozeny-Carman equation (which Terzaghi utilized and expanded upon) than with Hazen's simpler formula or Jaky's work.
Jaky: Jozsef Jaky is known for his formula for the coefficient of earth pressure at rest, \(K_0 = 1 - \sin \phi'\). His primary contributions are in the field of lateral earth pressure, not directly permeability formulas involving grain size and void ratio.
Kozney: Josef Kozeny, and later modified by Philip Carman (Kozeny-Carman equation), developed a theoretical approach based on flow through porous media treated as bundles of capillary tubes. The Kozeny-Carman equation is often written as \(k = \frac{\gamma_w}{\mu} \frac{e^3}{1+e} \frac{1}{(S_s v)^2}\) or \(k = \frac{\gamma_w}{\mu} \frac{e^3}{1+e} \frac{D_{eff}^2}{C}\), where \(\gamma_w\) is unit weight of water, \(\mu\) is dynamic viscosity, \(S_s\) is specific surface area, \(v\) is specific volume, and \(D_{eff}\) is an effective diameter. This formula involves the term \(\frac{e^3}{1+e}\) and a \(D^2\) term (implicitly or explicitly). So, Kozeny-Carman also relates permeability to both \(D^2\) and \(e\).

Comparing the given formula \(k = 200D^2ee^2\) with the typical forms proposed by these scientists, it most closely aligns with the structure involving \(D^2\) and terms related to \(e\) found in formulas attributed to Terzaghi and Kozeny-Carman. However, given the options, and common attributions in geotechnical literature for formulas involving \(D^2\) and \(e\) terms, Terzaghi is a strong candidate for proposing such a relationship, even if the exact form \(ee^2\) is specific to this question's context.

Therefore, based on the question and the provided options, the formula \(k = 200D^2ee^2\) is attributed to Terzaghi.

Summary of Formula Components

The formula \(k = 200D^2ee^2\) highlights the key factors influencing permeability:

Effective Grain Size (\(D\)): Permeability is highly sensitive to grain size. Larger particles generally lead to larger pore spaces and higher permeability. The dependence on \(D^2\) indicates this strong relationship.
Void Ratio (\(e\)): The volume of voids affects the flow path. A higher void ratio means more space for water to flow, generally leading to higher permeability. The term \(ee^2\) (or potentially \(e^3\)) reflects this dependence on the pore volume.
Constant (200): This is an empirical constant that would depend on the units used and the specific characteristics of the soil for which the formula was derived or calibrated.

Revision Table: Key Figures in Soil Permeability

Scientist/Engineer	Key Contribution Related to Permeability	Associated Formula (Typical Form)
Allen Hazen	Empirical formula for sands	\(k = C D_{10}^2\)
Karl Terzaghi	Developed fundamental principles of soil mechanics, including consolidation and permeability concepts. Proposed formulas relating k to \(D\) and \(e\).	\(k = C D_{10}^2 \frac{e^2}{1+e}\) (or similar forms)
Josef Kozeny & Philip Carman	Theoretical formula based on capillary flow model	\(k = \frac{\gamma_w}{\mu} \frac{e^3}{1+e} \frac{1}{S_s^2}\) (or similar forms involving \(D_{eff}^2\))
Jozsef Jaky	Known for Earth Pressure at Rest	\(K_0 = 1 - \sin \phi'\)

Additional Information: Factors Affecting Soil Permeability

Beyond effective grain size and void ratio, several other factors influence soil permeability:

Particle Shape: Angular particles tend to pack less efficiently than rounded particles, potentially creating larger void spaces and higher permeability for a similar void ratio.
Soil Structure: The arrangement of particles (e.g., flocculated vs. dispersed clay structure) significantly impacts permeability, especially in fine-grained soils. Fissures or cracks in clay can drastically increase mass permeability.
Degree of Saturation: The presence of trapped air bubbles in partially saturated soil can block pore spaces and reduce permeability compared to fully saturated soil.
Fluid Properties: Permeability is defined with respect to a specific fluid (usually water). The viscosity and density of the fluid affect the flow rate through the pores. The coefficient of permeability \(k\) is typically reported for water at a standard temperature (e.g., 20°C). The coefficient of intrinsic permeability (\(\kappa\)) is a property of the soil medium alone, independent of the fluid, and is related to \(k\) by \(k = \kappa \frac{\gamma_w}{\mu}\).