EM 1110-2-1901 30 Sep 86 CHAPTER 6 UNCONFINED FLOW PROBLEMS 6-1. Introduction. This chapter will consider unconfined flow problems for cases involving earth dams. Because of their ability to give a strong visual sense of flow and pressure distribution, flow nets will be used to define seepage. Other methods, such as transformations (Harr 1962), electrical analogy models, and numerical methods,can provide pressures and flows and be used to develop flow nets. Unconfined flow problems require the solution of flow and pressure distribution within the porous media and definition of the line of seepage boundary (phreatic surface within the dam). 6-2. Homogeneous Earth Dam on Impervious Foundation. The simplest earth dam configuration consists of a homogeneous, pervious embankment on animpervious foundation. Though rarely encountered in engineered embankments, this case will introduce general methods of defining flow in embankments. a. Definition of Unknown Seepage Boundaries and Calculation of Flow per Unit Length of Embankment, q . It is desired to define the flow and pressure distribution within the embankment and total flow through the embankment. The first step is determination ofthe upper flow line (which is the line of seepage boundary) and the length of the seepage exit face on the downstream slope of the earth dam. This provides all necessary boundary conditions for flow net construction and complete seepage definition. The two unknown boundaries, BC and CD, figure 6-1, are a combination of an entrance condition, figure 4-7(c), BBl; part of a parabola, B1B2; a smoothtransition between points of tangency, B2C, and a straight line discharge face along the downstream slope, CD. A parabola, shown by the dashed line, is the basic geometric member used to define the location and extent of the two boundaries. Casagrande (1937) provided the standard reference for flow through embankments while others (Harr 1962, Cedergren 1977, and others) added to and refined thebasic methods. Figure 6-2 provides the nomenclature and formulas for drawing the line of seepage and exit face and determining the quantity of seepage per unit length of embankment, q . In a given problem, embankment geometry and head water elevation provide values for h , m and which allow location of points A and B and determination of distance, d , as shown in figure 6-2.
Figure 6-1. Line ofseepage, BC, and seepage exit face, CD, for a homogeneous earth dam on an impermeable foundation (prepared by WES) 6-1
EM 1110-2-1901 30 Sep 86
Figure 6-2. Determination of line of seepage and seepage exit face for embankments on impervious foundations (adapted from 151 New England Waterworks Association )
EM 1110-2-1901 30 Sep 86 b. After this is done one of the four methodsshown in figure 6-2 and explained below can be used to determine the location of the exit face CD and the line of seepage BC. < 30º Schaffernak-Van Iterson. The two formulas for this method (1) given in figure 6-2 assume gradient equals dy/dx and allow direct determination of a and q . Construction of basic parabola shown in figure 6-3 is the first step in determining the upper line of seepage(Casagrande 1937). From embankment geometry and headwater height, point A is located. d and y are determined by scribing an arc, with radius DA through point E. Then the point of vertical tangency of the basic parabola, F, is determined. Line AG, parallel to the embankment base and horizontal axis of the parabola, is drawn and divided into an equal number of segments (6 in the case in figure 6-3). LineGF, the vertical tangent to the parabola, located at yo/2 from the downstream toe of the embankment is divided into the same number of equal segments as line AG. The points dividing line AG into segments are connected with point F. The intersection of these lines with their counterpart lines drawn from the points on line GF define the parabola. Thus the basic parabola, dashed line A-F, is...