# 4.4 Hydraulic Conductivity

The *hydraulic conductivity* proportionality constant, *K*, can be conceptualized as the relative ease of fluid passage through a porous material. It has direction and magnitude and is represented as a vector; however, the first part of this discussion presents it as a scalar value. Rearranging Darcy’s law to solve for hydraulic conductivity generates Equation 25.

(25) |

In this configuration, it becomes clear that the units of *K* are L/T because *Q* units are (L^{3}/T), *A* units (L^{2}), *h* units are (L), and *L* units are (L). The units are presented as Equation 26.

(26) |

Thus, the constant of proportionality, *K*, has units of velocity (e.g., meters/seconds, meters/day). However, *K* is not a velocity, rather it represents the transmission properties of the porous material. If water easily passes through a porous material it is described as having a high hydraulic conductivity; if water is poorly transmitted through a material it has a low hydraulic conductivity. These conditions are also referred to as permeable or of low permeability, respectively.

# Intrinsic Permeability

Freeze and Cherry (1979) describe the results of hydraulic conductivity experiments used to explore the relationship between physical properties of the porous media and the fluid. A number of columns filled with different sizes of glass beads were set up as Darcy columns and changes in specific discharge were observed. Based on these observations, the relationship of the specific discharge to measurable characteristics of the porous media and were noted as shown in Equations 27, 28, and 29.

q ∝ d^{2} |
(27) |

where:

q |
= | specific discharge (L/T) |

d |
= | diameter of uniform glass beads comprising the porous medium (L) |

q ∝ ρ g |
(28) |

where:

ρ |
= | fluid density (M/L^{3}) |

g |
= | gravitational constant (acceleration of gravity) (L/T^{2}) |

(29) |

where:

μ |
= | dynamic viscosity (M/(LT)) |

When considered together with Darcy’s original observation that *q* ∝ −*dh*/*dl*, these three relationships lead to a definition of hydraulic conductivity that includes physical characteristics of the porous media and the influence of fluid properties (Equation 30).

(30) |

where:

Cd^{2} |
= | intrinsic permeability (k) of the porous medium (L^{2}) |

C |
= | a constant of proportionality representing effects of particle shape, tortuosity, and pore size distribution of the porous medium independent of fluid properties (dimensionless) |

Once the properties of the fluid are known, the hydraulic conductivity can be calculated from the intrinsic permeability as shown in Equation 31.

(31) |

For sediments and rocks,* intrinsic permeability* (*k*) incorporates the influence of all the media properties that affect flow, not only the mean grain diameter as was the case for the uniform glass spheres. It has units of L^{2}. The intrinsic permeability represents the magnitude of variation in the diameters of the interconnected pores as well as the amount of branching and reconnecting of the pore pathways over a linear travel path, referred as the degree of tortuosity. *Tortuosity* is a measure of actual distance traveled divided by the shortest distance between two locations. In general, the larger the diameter of the pores and the more efficiently they are interconnected (less tortuosity), the larger the intrinsic permeability. In contrast, a porous material with small diameter pores and many circuitous interconnected pathways (high tortuosity) would have a lower intrinsic permeability. Intrinsic permeability can also be computed if the hydraulic conductivity and fluid properties are known by rearranging Equation 31.

# Fluid Properties

The specific weight and dynamic viscosity of the fluid also influence the hydraulic conductivity. The larger the specific weight of a fluid ( *γ* = *gρ*) and the lower the dynamic viscosity (*μ*), the higher the hydraulic conductivity. The *specific weight*, *γ*, of a fluid is its density times the gravitational constant, *ρg*, and the *dynamic viscosity*, *μ*, is the ratio of the shearing stress on a plane to the rate at which fluid velocity changes across the plane (internal resistance to flow). The influence of fluid properties on the value of hydraulic conductivity can be illustrated by visualizing how the flow rate would be affected if two Darcy sand-filled columns under the same hydraulic gradients were set up such that water flowed through one and molasses flowed through the other (Freeze and Cherry, 1979). It is assumed that the sand in each column has the same the structure and number of interconnected pores. Clearly the flow rate of the molasses would be slower than that of water. This is because the viscosity of molasses is typically more than a thousand times higher than that of water, while the specific weight of molasses is only about 1.5 times higher than water.

For groundwater systems, changes in density and viscosity caused by temperature need to be considered when computing hydraulic conductivities. Dynamic viscosity and density of water as a function of the water temperature is shown in Figure 28. Temperature has a more significant impact on viscosity than density.

For example: If a sand has an intrinsic permeability, *k*, of 1 × 10^{-7} cm^{2}, and the water moving through the sand has a temperature of 10 °C, then (from Figure 28):

*μ*= 1.3 milliPascal-second, which is 0.013 gram/(centimeter-second)*ρ*= 0.9997 g/cm^{3}, and with*g*= 980.67 cm/s^{2}(constant independent of temperature), then*K*would be 7.54 × 10^{-3}cm/s ~ 8 × 10^{-5}m/s

Repeating the calculation assuming the water temperature is 20 °C:

*μ*= 1.0 milliPascal-second, which is 0.01 gram/(centimeter-second)*ρ*= 0.998 gm/cm^{3}, and with*g*= 980.67 cm/s^{2}, then- K would be 9.787 × 10
^{-3}cm/s ~ 1 × 10^{-4}m/s

These calculations illustrate that although the changes in *K* are rather small, *K* increases as water temperature increases, because the magnitude of the decrease in dynamic viscosity (in the denominator) with temperature is larger than the corresponding decrease in density (in the numerator).

As fluids of different composition pass through saturated porous medium the properties of the composition of the fluid will influence hydraulic conductivity characterization (remember the molasses example). For example, if pure benzene was spilled during a train accident and entered the top of the 10 °C groundwater system as liquid benzene, the saturated *K* value of the sand for benzene could be computed. In this simplified example, at a temperature of 10°C, the *K* of sand with an intrinsic permeability, *k*, of 1 × 10^{-7} cm^{2}, for benzene would be governed by the properties of benzene, not properties of water:

*μ*_{benzene}= 1.25 milliPascal-second, which is 0.0125 gram/(centimeter-second)*ρ*_{benzene}= 0.727 gm/cm^{3}, and with*g*= 980.67 cm/s^{2}, then*K*_{benzene}would be 5.7 × 10^{-3}cm/s ~ 6 × 10^{-5}m/s

This indicates the *K* for benzene flow (~ 6 × 10^{-5} m/s) is lower than the *K* for water flow (~ 8 × 10^{-5} m/s) at the same temperature. This is because, although benzene is less viscous than water at 10 °C, the density of benzene is considerably less than that of water. If the benzene dissolved in the water such that benzene molecules were being carried as a dissolved substance (solute) within the water, the flow rate would be controlled by the properties of the water because no liquid benzene would be present.

To reiterate, hydraulic conductivity is symbolized by a capital *K* with units of length over time L/T. *K* reflects the impact of the properties of both the fluid and the medium on the ease with which fluid passes through a medium. Intrinsic permeability is usually represented by a lowercase *k* with units of length squared (L^{2}). *k* reflects the impact of only the properties of the medium on the ease with which fluid passes through it. Occasionally *k*_{i} is used to represent intrinsic permeability. In older literature, and a few modern writings, hydraulic conductivity is referred to as *permeability* or the *coefficient of permeability*. In some literature (e.g., Freeze and Cherry, 1979) the term “permeability” is used to represent *k* without the “intrinsic” modifier. Consequently, care needs to be exercised when reading a report or textbook to clearly understand and apply the information. It is useful to check the units associated with each term to decipher how the terms are being used. In the petroleum industry, the term “permeability” usually refers to *k* (L^{2}). Petroleum engineers often work with materials of low *k* and if the values are given in units of m^{2} or cm^{2} the number is very small (e.g., 1 × 10^{-8} to 1 × 10^{-16} cm^{2}). As a result, they introduced a unit called a “darcy” that is equivalent to 9.87 x 10^{-9} cm^{2} or, approximately 1 × 10^{-8} cm^{2} to describe the capacity of petroleum reservoir rocks to transmit fluids. A sandstone with an intrinsic permeability of 1 × 10^{-9} cm^{2} would have a permeability of 0.1 darcy.