Darcy’s Law for Groundwater Flow

Calculation of groundwater flow is very important for a different of applications, such as groundwater flow through from any aquifer to wells for water supply or irrigation purpose, and minimise of groundwater contamination. The basic actual tool for groundwater calculation is Darcy’s Law, the subject of this paragraph.

Introduction of Darcy’s Law

Darcy’s Law is expressed as an empirical relationship equation for liquid flow through a porous medium of soil strata. The most common example of application is groundwater flow through an confined or unconfined aquifer. Darcy’s Law provides the relationship equation between the flow rate of the groundwater seepage, the affected cross-sectional area of the aquifer perpendicular to the direction of flow, the value of  hydraulic gradient, and the data of  hydraulic conductivity system of the aquifer.

Equation of Darcy’s Law

The empirical equation for Darcy’s Law is fully based on the observations and experimental result that the rate of flow through a porous medium (i.e aquifer) is directly proportional to the cross-sectional area perpendicular to the flow and it is also strictly proportional to the loss of head per unit length to the direction of flow of liquid. Considering these two proportionalities together simultaneously comes up the following equation, and it may be written as:

Q = KA(hL/L), where all can be denoted as,

• Q = rate of flow of liquid through the porous medium of strata,
• A = cross-sectional area which totally perpendicular to flow,
• hL = head loss over the horizontal length portion, L, in the direction of flow

Now,

Darcy’s Law is scientifically valid only in case laminar flow condition, which actually happen whenever, Reynold’s number is less than 1.

• Reynold’s number (Re) for flow through a porous medium is scientifically defined as: Re = ρVL/μ, where,
•  ρ and μ are the density and viscosity of the liquid,
• V is the velocity of flow (Q/A), and
• L is a characteristic length,

Which is taken as the mean grain diameter of the medium strata. Most authentic way out of applications of groundwater flow have Re < 1, and then it can be calculated with Darcy’s Law of equation.

The hydraulic grade line, is a line whose height above any flowing fluid is equal to the height of water rises in a piezometer apparatus at that point. Now that measured height of the water in a piezometer above the flowing liquid level is equal to P/γ at any point in the flow field medium.

The actual calculated slope of the hydraulic grade line which is hL/L and is often called as the hydraulic gradient. As symbolic representation i is some time  used to represent the hydraulic gradient equation. Then i = hL/L ,

So, the Darcy’s Law can be given as Q = KAi.

Hydraulic Conductivity, K

The hydraulic conductivity, (K) is defined as a constant for a particular given porous medium of strata. The excessive porous medium with little bit resistance to flow will give a high value for K, on the other side a compactly packed medium with highly resistance or obstacles to flow of fluid which will give a low value result for K.

Resulting effect of flow of the medium and all the properties of the liquid flowing through inside it can be easily separated by the use of the specific permeability, k, as shown in the following below written equation:

K = kγ/μ, where

• k = specific permeability value, ft2 (a property of the porous medium only),
• γ = specific weight value of flowing liquid, lb/ft3,
• m =viscosity value of flowing liquid, lb-sec/ft2,
• K = value of hydraulic conductivity, ft/sec (ft3/sec/ft2).

Values of specific permeability are most of the times written  with the darcy as the unit, where 1 darcy = 1.062 x 10-11 ft2.