Net flow

Flow nets consist of two types of lines streamlines and isopotentials. Streamlines are lines that follow the path of representative parcels of water water always flows parallel to streamlines. Isopotentials, drawn perpendicular to streamlines, are lines along which the hydraulic head, h, is constant. Therefore, water always flows perpendicular to isopotentials. Flow nets are often drawn to represent the horizontal movement of groundwater and associated chemicals in an aquifer the plane of the flow net then represents the horizontal aquifer surface, and it is assumed that underneath each point on the surface, flow is essentially the same at all depths in the aquifer. An example of such a flow net is shown in Fig. 3-8. 

Although many groundwater hydrologists customarily employ computerized techniques for analyzing groundwater flow problems, it is valuable for the student of groundwater transport to be able to draw and interpret flow nets. To draw a flow net, the following rules must be observed  

Streamlines are always perpendicular to isopotentials in an isotropic aquifer. 

Flow boundaries—impermeable surfaces across which flow cannot occur, such as the surfaces of clay lenses, concrete, or buried tanks—should be thought of as streamlines because water flows parallel to them. Like streamlines, flow boundaries are perpendicular to isopotentials. 

Boundaries that can act as either sources or sinks for water (such as rivers and lakes) sometimes can be treated as lines of known hydraulic head. 

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To form a heat balance for the flows. Net sensible heat flow from the space = Sensible heat absorbed by the ventilating air. 

As a more complete picture of the project emerges, the cash flows through the project life can be projected. This allows more detailed evaluation of project profitability on the basis of cash flows. Net present value can be used to measure the profit taking into account the time value of money. Discounted cash flow rate of return measures how efficiently the capital is being used. 

The oldest and most widely used method of estimating water age is the calculation of travel times using Darcy s law combined with an expression of continuity. If a field of steady-state, groundwater flow is subdivided into a two-dimensional flow net (figure 1), then Darcy s law can be written as ... 

Figure 2 was constructed by contouring of values calculated from the flow net of figure 1. Much more complex diagrams are possible provided sufficient information concerning the aquifer and fluid-flow conditions can be obtained. For simple boundary conditions and homogeneous aquifers, a direct analytical solution for isochronal surfaces is available [2]. 

Figure 1. Flow net representing groundwater circulation near a river which intercepts homogeneous and isotropic aquifer of infinite thickness.

The central recursion traverses the product flow net. At arbitrary points of the explosion additional search trees to examine alternative explosions can be built up... 

Physical Approaches. Groundwater-exchange rates with lakes are traditionally estimated by careful measurements of hydraulic potentials within the groundwater system, followed by application of Darcy s law in the form of flow-net analysis or numerical modeling. However, these measurements can be time-consuming and costly, and can require monthly to weekly measurements at many piezometers to examine the three-dimensional nature of the hydraulic-potential field. In addition, characterization of the hydraulic conductivity of the aquifer is critical to physical approaches and typically leads to results with large uncertainties (I, 2). 

The equation of continuity and the Bernoulli theorem together show, for a stream of incompressible fluid, that (a) where the cross-sectional area is large and the streamlines are widely spaced, the velocity is low and the pressure is high, and (6) where the cross-sectional area is small and the streamlines are crowded together, the velocity is high and the pressure is low Hence a flow net gives a picture not only of the velocity field but also of the pressure field. 

It is not easy to compute contraction coefficients in other cases, but they can be determined reasonably well by means of flow nets. However, without taking this much trouble, a little judgment will enable one to make a fair estimate of the contraction. 

The net present value is always less than the total future worth of the project because of the discounting of future cash flows. Net present value is easily calculated using spreadsheets, and most spreadsheet programs have an NPV function. 

FIGURE 3-8 A flow net for a hypothetical aquifer formed in a drowned river valley, which is assumed to be of uniform thickness over its area and located between two water-impermeable walls of bedrock. The hydraulic head varies between 2 m and 4.9 m. Squares A and B and points x and y are used in Example 3-2. 

Assume the flow net shown in Fig. 3-8 was drawn for an aquifer 3 m thick with a hydraulic conductivity of 10 cm/sec and a porosity of 0.24. Calculate specific discharge in square B, discharge in each stream tube, and total aquifer discharge. What is the travel time for a dissolved chemical across square B From point x to point yl... 

According to the rules of flow nets, discharge is equal in each stream tube. Thus, total aquifer discharge is equal to the number of stream tubes multiplied by the discharge in each tube ... 

The groundwater flow into a well also can be analyzed by using a flow net. Because the well is symmetrical, lines of constant drawdown (isopotentials) are circles centered at the center of the well (Fig. 3-11). Streamlines are arranged radially around the well if there is no regional flow in the aquifer. 

The flow net is a helpful tool for depicting the area of an aquifer from which a well captures water. It is essential to be able to delineate this area to avoid pulling contaminated groundwater into a well used for drinking water production areas of known contamination must lie outside the well s capture zone. Alternatively, if a well is to be used for groundwater remediation (removal of contaminated groundwater), the capture zone must enclose the contaminated areas. Capture curves, which are the boundaries of capture zones, may be drawn readily by inspection if a flow net is available the capture zone includes all stream tubes that terminate in the well. The following are two examples of capture curves. 

A small shop has been dumping used solvents into an abandoned well, shown as follows. Sketch a flow net indicating where the solvents are likely to appear at the lake bed. If dissolved solvent moves with... 

Permeability Constant head Falling head Flow problems, such as flow nets and drainage... 

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