FLOW THROUGH EARTH DAMS
Flow through earth dams is an important design consideration. We need to ensure that the porewater pressure at the downstream end of the dam will not lead to instability, and the exit hydraulic gradient does not lead to piping. The major exercise is to find the top fl ow line called the phreatic surface (Figure ). The pressure head on the phreatic surface is zero.
Phreatic surface within an earth dam
Casagrande (1937) showed that the phreatic surface can be approximated by a parabola with corrections at the points of entry and exit. The focus of the parabola is at the toe of the dam, point F (Figure). The assumed parabola representing the uncorrected phreatic surface is called the basic parabola. Recall from your geometry course that the basic property of a parabola is that every point on it is equidistant from its focus and a line called the directrix. To draw the basic parabola, we must know point A, the focus F, and f (one-half the distance from the focus to the directrix). Casagrande recommended that point C be located at a distance 0.3AB, where AB is the horizontal projection of the upstream slope at the water surface. From the basic property of a parabola, we get
The equation to construct the basic parabola is
Solving for z, we obtain
Since H and b are known from the geometry of the dam, the basic parabola can be constructed. We now have to make some corrections at the upstream entry point and the downstream exit point.
The upstream end is corrected by sketching a transition curve (BE) that blends smoothly with the basic parabola. The correction for the downstream end depends on the angle b and the type of discharge face. Casagrande (1937) determined the length of the discharge face, a, for a homogeneous earth dam with no drainage blanket at the discharge face and b # 308, as follows. He assumed that Dupuit’s assumption, which states that the hydraulic gradient is equal to the slope, dz/dx, of the phreatic surface is valid. If we consider two vertical sections—one is KM of height z, and the other is GN of height a sin b—then the flow rate across KM is
and across GN is
From the continuity condition at sections KM and GN, qKM 5 qGN; that is,
which simplifi es to
We now integrate Equation within the limits x1 5 a cos b and x2 =b, z1 = a sin b and z2 = H.
Simplifi cation leads to