A flow net is a graphical representation of two-dimensional steady-state groundwater flow through aquifers.
Construction of a flow net is often used for solving groundwater flow problems where the geometry makes analytical solutions impractical. The method is often used in civil engineering, hydrogeology or soil mechanics as a first check for problems of flow under hydraulic structures like dams or sheet pile walls. As such, a grid obtained by drawing a series of equipotential lines is called a flow net. The flow net is an important tool in analysing two-dimensional irrotational flow problems. Flow net technique is a graphical representation method.
Basic method
The method consists of filling the flow area with stream and equipotential lines, which are everywhere perpendicular to each other, making a curvilinear grid. Typically there are two surfaces (boundaries) which are at constant values of potential or hydraulic head (upstream and downstream ends), and the other surfaces are no-flow boundaries (i.e., impermeable; for example the bottom of the dam and the top of an impermeable bedrock layer), which define the sides of the outermost streamtubes (see figure 1 for a stereotypical flow net example).
Mathematically, the process of constructing a flow net consists of contouring the two harmonic or analytic functions of potential and stream function. These functions both satisfy the Laplace equation and the contour lines represent lines of constant head (equipotentials) and lines tangent to flowpaths (streamlines). Together, the potential function and the stream function form the complex potential, where the potential is the real part, and the stream function is the imaginary part.
The construction of a flow net provides an approximate solution to the flow problem, but it can be quite good even for problems with complex geometries by following a few simple rules (initially developed by Philipp Forchheimer around 1900, and later formalized by Arthur Casagrande in 1937) and a little practice: streamlines and equipotentials meet at right angles (including the boundaries), diagonals drawn between the cornerpoints of a flow net will meet each other at right angles (useful when near singularities), streamtubes and drops in equipotential can be halved and should still make squares (useful when squares get very large at the ends), flow nets often have areas which consist of nearly parallel lines, which produce true squares; start in these areas — working towards areas with complex geometry, many problems have some symmetry (e.g., radial flow to a well); only a section of the flow net needs to be constructed, the sizes of the squares should change gradually; transitions are smooth and the curved paths should be roughly elliptical or parabolic in shape.
Example flow nets
The first flow net pictured here (modified from Craig, 1997) illustrates and quantifies the flow which occurs under the dam (flow is assumed to be invariant along the axis of the dam — valid near the middle of the dam); from the pool behind the dam (on the right) to the tailwater downstream from the dam (on the left).
There are 16 green equipotential lines (15 equal drops in hydraulic head) between the 5 m upstream head to the 1m downstream head (4 m / 15 head drops = 0.267 m head drop between each green line). The blue streamlines (equal changes in the streamfunction between the two no-flow boundaries) show the flowpath taken by water as it moves through the system; the streamlines are everywhere tangent to the flow velocity.
The second flow net pictured here (modified from Ferris, et al., 1962) shows a flow net being used to analyze map-view flow (invariant in the vertical direction), rather than a cross-section. Note that this problem has symmetry, and only the left or right portions of it needed to have been done. To create a flow net to a point sink (a singularity), there must be a recharge boundary nearby to provide water and allow a steady-state flowfield to develop.
Flow net results
Darcy's law describes the flow of water through the flow net. Since the head drops are uniform by construction, the gradient is inversely proportional to the size of the blocks. Big blocks mean there is a low gradient, and therefore low discharge (hydraulic conductivity is assumed constant here).