A flownet is a graphical representation of two-dimensional steady-state groundwater flow through aquifers. Construction of a flownet 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 flownet. The flownet 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 which are at constant values of potential or hydraulic head, and the other surfaces are no-flow boundaries, which define the sides of the outermost streamtubes. Mathematically, the process of constructing a flownet 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 and lines tangent to flowpaths. 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 flownet 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 and a little practice:
streamlines and equipotentials meet at right angles,
diagonals drawn between the cornerpoints of a flownet will meet each other at right angles,
streamtubes and drops in equipotential can be halved and should still make squares,
flownets 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 ; only a section of the flownet 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 flownets
The first flownet pictured here illustrates and quantifies the flow which occurs under the dam ; from the pool behind the dam to the tailwater downstream from the dam. There are 16 green equipotential lines between the 5 m upstream head to the 1m downstream head. The blue streamlines show the flowpath taken by water as it moves through the system; the streamlines are everywhere tangent to the flow velocity. The second flownet pictured here shows a flownet being used to analyze map-view flow, 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 flownet to a point sink, there must be a recharge boundary nearby to provide water and allow a steady-state flowfield to develop.
Flownet results
describes the flow of water through the flownet. 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. An equivalent amount of flow is passing through each streamtube, therefore narrow streamtubes are located where there is more flow. The smallest squares in a flownet are located at points where the flow is concentrated, and high flow at the land surface is often what the civil engineer is trying to avoid, being concerned about or dam failure.
Singularities
Irregular points in the flow field occur when streamlines have kinks in them. This can happen where the bend is outward, and there is infinite flux at a point, or where the bend is inward where the flux is zero. The second flownet illustrates a well, which is typically represented mathematically as a point source ; this is a singularity because the flow is converging to a point, at that point the Laplace equation is not satisfied. These points are mathematical artifacts of the equation used to solve the real-world problem, and do not actually mean that there is infinite or no flux at points in the subsurface. These types of points often do make other types of solutions to these problems difficult, while the simple graphical technique handles them nicely.
Extensions to standard flownets
Typically flownets are constructed for, isotropicporous media experiencing saturated flow to known boundaries. There are extensions to the basic method to allow some of these other cases to be solved:
inhomogeneous aquifer: matching conditions at boundaries between properties
anisotropic aquifer: drawing the flownet in a transformed domain, then scaling the results differently in the principle hydraulic conductivity directions, to return the solution
one boundary is a seepage face: iteratively solving for both the boundary condition and the solution throughout the domain
Although the method is commonly used for these types of groundwater flow problems, it can be used for any problem which is described by the Laplace equation, for example electric current flow through the earth.
