Basics of permeability and groundwater flow

1. Permeability Permeability – Part A Dr O.Hamza Part A Prepared by Dr O. Hamza o_hamza at hotmail dot com Lecture reference: OH_GA03_A

4. Soil as a porous media Nature of soil Soil is a natural particulate earth material which has three intermixed phases: (1) Solid (mineral particles) (2) Gas (air, co2), (3) Liquid (usually water, but also possibly oil, chemical solutions) So soil can be considered as a porous media. Permeability – Part A Dr O.Hamza

5. Soil as a porous media Nature of soil Can any porous material be permeable? Permeability – Part A Dr O.Hamza A material is said to be permeable if it contains continues voids .

6. Soil as a porous media Nature of soil Void ratio, e = Vv / Vs Degree of saturation, S r =Vw / Vv Water content, w = Mw/Ms Porosity, n = Vv/V Permeability – Part A Dr O.Hamza

7. Soil as a porous media Seepage velocity The volume flow rate q is calculated as the product of flow velocity v and total cross sectional area A : q = v. A Assume a column of soil and water flowing through. q q Permeability – Part A Dr O.Hamza

8. Soil as a porous media Seepage velocity The ratio of volume flow rate q to the average area of voids A v on a cross section normal to the macroscopic direction of flow is called the seepage velocity v s At the particulate level the water follows a tortuous path through the pores. Permeability – Part A Dr O.Hamza Flow line

9. Soil as a porous media Seepage velocity In general, the velocity of water through soil has been found to be relatively small (e.g. less than 0.61 m/min). Permeability – Part A Dr O.Hamza Flow line

11. Hydraulic head and hydraulic gradient Pressure, Elevation and total head Pore pressure at a given point (e.g. point A in the diagram) can be measured by the height of water in a standpipe located at that point. Pore pressures are often indicated in this way on diagrams. The height of the water column is the pressure head (h w ) Pressure head Permeability – Part A Dr O.Hamza

12. Hydraulic head and hydraulic gradient Pressure, Elevation and total head Elevation head The elevation head ( h z ) of a point is its height above the datum line. To identify significant differences in pore pressure at different points, we need to eliminate the effect of the points' position. A height datum is required from which locations are measured. h w Permeability – Part A Dr O.Hamza Total head The height above the datum of the water level in the standpipe is the total head ( h ). h = h z + h w

13. Hydraulic head and hydraulic gradient Pressure, Elevation and total head Total head, h = Elevation head, h z + Pressure head, h w + Velocity head, h v h w The total head h includes also velocity head h v ; this head in soils is usually neglected . However, the velocity head must be considered when dealing with flow through pipe and open channels. Permeability – Part A Dr O.Hamza

14. Hydraulic head and hydraulic gradient Hydraulic gradient The hydraulic gradient is the rate of change of total head along the direction of flow It is the difference in total head that are important. The direction of flow depends on these differences. Flow of pore water in soils is driven from positions of higher total head towards positions of lower total head. Datum Permeability – Part A Dr O.Hamza

15. Hydraulic head and hydraulic gradient Hydraulic gradient Example. In each diagram there are two points, a small distance  s apart, h z1 and h z2 above datum. Permeability – Part A Dr O.Hamza Datum

16. Hydraulic head and hydraulic gradient Hydraulic gradient The hydraulic gradient In the first diagram, the total heads are equal . The difference in pore pressure is entirely due to the difference in altitude of the two points and the pore water has no tendency to flow  h=0 thus i =0  No flow Permeability – Part A Dr O.Hamza Datum

17. Hydraulic head and hydraulic gradient Hydraulic gradient The hydraulic gradient In the second diagram, the total heads are different . The hydraulic gradient is i ≠ 0 and the pore water tends to flow.  h=0 thus i =0  No flow  h ≠ 0 thus i = (h 2 - h 1 ) /  s  flow Permeability – Part A Dr O.Hamza Datum

19. Darcy’s law and permeability Darcy’s law in 1856, a French hydraulic engineer named Henry Darcy published an equation for flow through a porous medium. v ~ i or q ~ i In a saturated porous media, the flow velocity v or the rate of flow of water q (volume/time) through cross-sectional area A is found to be proportional to hydraulic gradient i Permeability – Part A Dr O.Hamza

20. Darcy’s law and permeability Darcy’s law Permeability – Part A Dr O.Hamza where v is flow velocity k is coefficient of permeability with dimensions of velocity (length/time) q is flow rate = = ----------------------- Quantity of water Time Higher hydraulic gradient i  faster flow ? agree disagree Higher coefficient of permeability k  faster flow ? agree disagree

21. Darcy’s law and permeability Darcy’s law The coefficient of permeability of a soil is a measure of the water flow conductivity. Permeability – Part A Dr O.Hamza Higher hydraulic gradient i  faster flow ? agree disagree Higher coefficient of permeability k  faster flow ? agree disagree

22. Darcy’s law and permeability Soil permeability Coefficient of permeability may vary from as large as 1 m/s for clean gravels to a small value of the order of 10 -10 m/s for very fine soil such as clay-silt (BS 8004, 1986). Coefficient of permeability k Permeability – Part A Dr O.Hamza clay Gravel

23. Darcy’s law and permeability Temperature and permeability The flow of water through confined spaces is controlled by its viscosity  and the viscosity is controlled by temperature. The values of k at 0°C and 10°C are 56% and 77% respectively of the value measured at 20°C. An alternative permeability K (dimensions: length²) is sometimes used as a more absolute coefficient depending only on the characteristics of the soil skeleton. Permeability – Part A Dr O.Hamza (Ref. Geotechnical on the Web)

24. Darcy’s law and permeability Stratified soil and permeability Representative graphic lithology log and core photos of an insitu In nature, soil is stratified

25. Darcy’s law and permeability Stratified soil and permeability Consider a stratified soil having horizontal layers of thickness t 1 , t 2 , t 3 , etc. with coefficients of permeability k 1 , k 2 k 3 , etc How can we apply the concept of hydraulic gradient and coefficient of permeability? Permeability – Part A Dr O.Hamza For a single layer, Darcy's law indicates:

26. Darcy’s law and permeability Stratified soil and permeability (1) total flow rate q = q1= q2= q3..  head drop  h=  h1 +  h2+  h3 (2) total flow rate q = q1+ q2+ q3..  head drop  h=  h1 =  h2 =  h3 Which condition of these two is valid for vertical flow? Permeability – Part A Dr O.Hamza Vertical flow For vertical flow , the flow rate q through area A of each layer is the same. Hence the head drop across a series of layers can be given.

27. Darcy’s law and permeability Stratified soil and permeability For vertical flow , the flow rate q through area A of each layer is the same. Hence the head drop across a series of layers can be given. Head drop across a series of layers is The average coefficient of permeability is q = q1= q2= q3..  h=  h1 +  h2+  h3 Permeability – Part A Dr O.Hamza Vertical flow

28. Darcy’s law and permeability Stratified soil and permeability For horizontal flow , the head drop  h over the same flow path length  s will be the same for each layer. So i 1 = i 2 = i 3 etc. Horizontal flow The flow rate through a layered block of soil of breadth B The average coefficient of permeability q = q1+ q2+ q3..  h=  h1 =  h2 =  h3 Permeability – Part A Dr O.Hamza

32. Determination of coefficient of permeability Laboratory measurements of permeability Permeameter 2. Falling head Permeability test Recommended for fine-grained soils Total head h in standpipe of area ‘ a ’ is allowed to fall; heads h 1 and h 2 are measured at times t 1 and t 2 . Hydraulic gradient  h/L varies with time. Permeability – Part A Dr O.Hamza

33. Determination of coefficient of permeability Laboratory measurements of permeability Permeameter 2. Falling head Permeability test The quantity of flow through the sample in time dt: By integrating between two test limits and re-arranging the equation: dQ = -adh a A 1 h 2 Permeability – Part A Dr O.Hamza

34. Determination of coefficient of permeability Laboratory measurements of permeability Coefficient of permeability k can be indirectly estimated from Oedometer test k = c v. m v .  w c v coefficient of consolidation m v one-dimensional compliance Oedometer Permeability – Part A Dr O.Hamza

38. A horizontal cylinder of homogonous sand is tested as shown in the figure. Imaginary points A, B, C, D and E are located at the centreline of the cylinder. What is the hydraulic gradient expected along the soil. Determine the pressure, elevation and total head at different points to be able to plot them versus horizontal distance. Quiz 3 (10 minutes) Working on Quizzes and Example problems Permeability – Part A Dr O.Hamza 4 cm 5 5 5 cm 5 cm 3 A B C D E Datum

39. Solution Quiz 3 Working on Quizzes and Example problems The hydraulic gradient along the soil is i BC =  h / L = -5/10= -0.5 Since this gradient is constant along the soil: i BC = i CD = i BD = - 0.5 i BC =  h BC /L BC =(h C -h B )/ L BC - 0.5 = (h C – 5)/ 5 h C = 2.5 L=10 cm  h= -5 cm 4 cm 5 5 5 cm 5 cm 3 A B C D E Datum

40. Problem 1 . Hydraulic head and hydraulic gradient For the soil shown in the figure calculate the total head and pore water pressure at point P and Q . Then determine the direction of the flow . What is the hydraulic gradient between these two points? Working on Quizzes and Example problems Permeability – Part A Dr O.Hamza 5 m Pressure head Pressure head Recall: h (total head) = h z (elevation head) + h w (pressure head) pressure head h w = height of water above the point

41. Pore water pressure : Pressure head thus, pore water pressure at P: u P = 1.7 x 9.81 ≈ 17 kPa; pore water pressure at Q: u Q = 2.2 x 9.81 ≈ 22 kPa Hydraulic gradient i PQ =  h /  S = (0 -1.2)/ 5 = - 0.24 Direction of flow: Water moves from P to Q i.e. from high head to low head Problem 1 . Hydraulic head and hydraulic gradient Total head h = h z (elevation head) + h w (pressure head) h P = -0.5 + 1.7 = 1.2 m h Q = -2.2 + 2.2 = 0 m Working on Quizzes and Example problems Solution Permeability – Part A Dr O.Hamza

42. Problem 2 . Laboratory measurement of the coefficient of permeability The results of a constant head permeability test for a cylindrical fine sand sample having a diameter of 150mm and a length of 300mm are as follows: Constant head difference = 500 mm Time of collection of water = 5 min Volume of water collected = 350 cc Find the coefficient of permeability for the soil in m/s. Note. cc is cubic centimetre, cm 3 Working on Quizzes and Example problems Permeability – Part A Dr O.Hamza  h

43. Problem 2 . Laboratory measurement of the coefficient of permeability Solution: Using the equation of constant head permeability test where: -  h is the c onstant head difference = 500 mm - The cross-section area of the sample, A=  D 2 /4 = 17662 mm 2 - The flow rate, i.e. the quantity of water that flows in a unit of time, q = volume of water collected (mm) / time of this collection (sec) = 350x10 -6 / (5x60) = - Thus k = …………. Working on Quizzes and Example problems Permeability – Part A Dr O.Hamza