LAPLACE'S EQUATION OF CONTINUITY

Transcription

1 LAPLACE'S EQUATION OF CONTINUITY y z Steady-State Flow around an impervious Sheet Pile Wall Consider water flow at Point A: v = Discharge Velocity in Direction Figure Das FGE (2005). v z = Discharge Velocity in z Direction Y Direction Out Of Plane Slide 1 of 23

2 LAPLACE'S EQUATION OF CONTINUITY Consider water flow at Point A (Soil Block at Pt A shown left) Rate of water flow into soil block in direction: v dzdy Figure Das FGE (2005). Rate of water flow into soil block in z direction: v v v z ddy Rate of water flow out of soil block in,z directions: z v vz z ddzdy dz ddy Slide 2 of 23

3 LAPLACE'S EQUATION OF CONTINUITY Consider water flow at Point A (Soil Block at Pt A shown left) Total Inflow = Total Outflow Figure Das FGE (2005). v v vz ddzdy vz dz ddy z v dzdy v ddy 0 v z or vz 0 z Slide 3 of 23

4 Slide 4 of GEOTECHNICAL LABORATORY z h k h k z h k i k v h k i k v z z z z z Consider water flow at Point A (Soil Block at Pt A shown left) Figure Das FGE (2005). Using Darcy s Law (v=ki) LAPLACE'S EQUATION OF CONTINUITY

5 FLOW NETS: DEFINITION OF TERMS Flow Net: Graphical Construction used to calculate groundwater flow through soil. Comprised of Flow Lines and Equipotential Lines. Flow Line: A line along which a water particle moves through a permeable soil medium. (a.k.a. streamline). Flow Channel: Strip between any two adjacent Flow Lines. Equipotential Lines: A line along which the potential head at all points is equal. NOTE: Flow Lines and Equipotential Lines must meet at right angles! Slide 5 of 23

6 FLOW NETS FLOW AROUND SHEET PILE WALL Figure 5.12a. Das FGE (2005). Slide 6 of 23

7 FLOW NETS FLOW AROUND SHEET PILE WALL Figure 5.12b. Das FGE (2005). Slide 7 of 23

8 FLOW NETS: BOUNDARY CONDITIONS 1. The upstream and downstream surfaces of the permeable layer (i.e. lines ab and de in Figure 12b Das FGE (2005)) are equipotential lines. 2. Because ab and de are equipotential lines, all the flow lines intersect them at right angles. 3. The boundary of the impervious layer (i.e. line fg in Figure 12b Das FGE (2005)) is a flow line, as is the surface of the impervious sheet pile (i.e. line acd in Figure 12b Das FGE (2005)). 4. The equipontential lines intersect acd and fg (Figure 12b Das FGE (2005)) at right angles. Slide 8 of 23

9 FLOW NETS FLOW UNDER AN IMPERMEABLE DAM Figure Das FGE (2005). Slide 9 of 23

10 q q q GEOTECHNICAL LABORATORY FLOW NETS: DEFINITION OF TERMS Rate of Seepage Through Flow Channel (per unit length): Using Darcy s Law (q=va=kia) 3 q n q k h h 1 2 l 1 k h h 2 3 l 2 k h h 3 4 l 1 l 2 Potential Drop h 1 h 2 h 2 h 3 h 3 h 4... H N d l 3 l 3... Figure Das FGE (2005). Where: H = Head Difference N d = Number of Potential Drops Slide 10 of 23

11 FLOW NETS: RULES FOR CREATING FLOW NETS (FROM UTEXAS) 1. Head drops between adjacent equipotential lines must be constant (or, in those rare cases where this is not desirable, clearly stated, just as in topographic contour maps)! 2. Equipotential lines must match known boundary conditions. 3. Flow lines can never cross. Equi. Flow Line Slide 11 of 23

12 FLOW NETS: RULES FOR CREATING FLOW NETS (FROM UTEXAS) 4. Refraction of flow lines must account for differences in hydraulic conductivity. 5. For isotropic media (what you have). Equi. a) Flow lines must intersect equipotential lines at right angles. b) The flow line-equipotential polygons should approach curvilinear squares, as shown in the Figure to the right. Flow Line Slide 12 of 23

13 FLOW NETS: RULES FOR CREATING FLOW NETS (FROM UTEXAS) 6. The quantity of flow between any two adjacent flow lines must be equal. Equi. 7. The quantity of flow between any two stream lines is always constant. Flow Line Slide 13 of 23

14 FLOW NETS: DRAWING PROCEDURE (AFTER HARR (1962, P. 23) 1. Draw the boundaries of the flow region to scale so that all equipotential lines and flow lines that are drawn can be terminated on these boundaries. 2. Sketch lightly three or four flow lines, keeping in mind that they are only a few of the infinite number of curves that must provide a smooth transition between the boundary flow lines. As an aid in spacing of these lines, it should be noted that the distance between adjacent flow lines increases in the direction of the larger radius of curvature. 3. Sketch the equipotential lines, bearing in mind that they must intersect all flow lines, including the boundary streamlines, at right angles and that the enclosed figures must be (curvilinear) squares. Slide 14 of 23

15 FLOW NETS: DRAWING PROCEDURE (FROM HARR (1962, P. 23) 4. Adjust the locations of the flow lines and the equipotential lines to satisfy the requirements of step 3. This is a trail-and-error process with the amount of correction being dependent upon the position of the initial flow lines. The speed with which a successful flow net can be drawn is highly contingent on the eperience and judgment of the individual. A beginner will find the suggestions in Casagrande (1940) to be of assistance. 5. As a final check on the accuracy of the flow net, draw the diagonals of the squares. These should also form smooth curves that intersect each other at right angles. Slide 15 of 23

16 FLOW NETS: EXAMPLES Wrong Wrong Correct! Unconfined groundwater flow nets on a slope Slide 16 of 23

17 FLOW NETS: EXAMPLES Cross-sectional flow net of a homogeneous and isotropic aquifer (Hubbert, 1940). Slide 17 of 23

18 FLOW NETS: EXAMPLES Contour map of the piezometric surface near Savannah, Georgia, 1957, showing closed contours resulting from heavy local groundwater pumping (from Bedient, after USGS Water-Supply Paper 1611). Slide 18 of 23

19 FLOW NETS: DAM EXAMPLES Slide 19 of 23

20 Therefore, flow through one channel is: q k q k N GEOTECHNICAL LABORATORY H N d If Number of Flow Channels = N f, then the total flow for all channels per unit length is: HN d f FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE Figure 5.12b. Das FGE (2005). Slide 20 of 23

21 GIVEN: Flow Net in Figure N f = 3 N d = 6 k =k z =510-3 cm/sec DETERMINE: a. How high water will rise in piezometers at points a, b, c, and d. b. Rate of seepage through flow channel II. c. Total rate of seepage GEOTECHNICAL LABORATORY FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE Figure Das FGE (2005). Slide 21 of 23

22 SOLUTION: Potential Drop = (5m 1.67m) GEOTECHNICAL LABORATORY H N d 0.56m At Pt a: Water in standpipe = (5m 10.56m) = 4.44m At Pt b: Water in standpipe = (5m 20.56m) = 3.88m At Pts c and d: Water in standpipe = (5m 50.56m) = 2.20m FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE Figure Das FGE (2005). Slide 22 of 23

23 FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE SOLUTION: q k k = cm/sec k = m/sec H N d q = (510-5 m/sec)(0.56m) q = m 3 /sec/m q k HN N d f qn q = ( m 3 /sec/m) * 3 q = m3/sec/m f Figure Das FGE (2005). Slide 23 of 23