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- 1. Basics of groundwater hydrology in geotechnical engineering Part B Prepared by Dr O. Hamza o_hamza at hotmail dot com Lecture reference: OH GA03 B OH_GA03_B Permeability – Part B Dr O.Hamza
- 2. Content • Introduction • Quasi-one-dimensional Quasi one dimensional and radial flow • Field determination of coefficient of permeability • Summary S mmar • Example problems Permeability – Part B Dr O.Hamza
- 3. Introduction Darcy’s law q = Aki where k i coefficient of permeability with is ffi i t f bilit ith (Ref. Geotechnical on the Web) dimensions of velocity (length/time) Q Quantity of water q is flow rate = = ----------------------- t Time In a saturated porous media, the rate of flow of water q (volume/time) through cross-sectional area ‘A’ is found to be proportional to hydraulic th h ti l i f dt b ti lt h d li gradient ‘ i ’ Permeability – Part B Dr O.Hamza
- 4. Introduction Aquifer and Darcy’s law Aquifer is a term used to designate a porous geological formation that: - contains water at full saturation - permits water to move through it under ordinary field conditions Permeability – Part B Dr O.Hamza
- 5. Introduction Aquifer and Darcy’s law The horizontal flow rate q is constant. For an aquifer of width B and varying thickness tt, Darcy's Law indicates that q=Aki =Btki or Hydraulic gradient varies inversely with aquifer thickness Where flow occu s in a co e e o occurs confined aqu e whose t c ess varies ge t y with ed aquifer ose thickness a es gently t position the flow can be treated as being essentially one-dimensional. Permeability – Part B Dr O.Hamza
- 6. Quasi-one-dimensional Quasi one dimensional and radial flow • Cylindrical flow: confined aquifer • Cylindrical flow: groundwater lowering • Spherical flow Permeability – Part B Dr O.Hamza
- 7. Quasi-one-dimensional and radial flow Cylindrical flow: confined aquifer Pumping aquifer Confined aquifer Steady-state pumping from a well which extends the full thickness of a confined aquifer is one of the one dimensional problem which can be one-dimensional analysed in cylindrical coordinates. Permeability – Part B Dr O.Hamza
- 8. Quasi-one-dimensional and radial flow Cylindrical flow: confined aquifer Darcy's Law still applies, with hydraulic gradient dh/dr and area A varying with radius: A = 2πr.t 2 rt In this case pore pressure or head varies only with radius r r. Permeability – Part B Dr O.Hamza
- 9. Quasi-one-dimensional and radial flow Cylindrical flow: confined aquifer Integrating between the borehole and at variable distance r: di t where ro is the radius of the borehole and h0 the constant head in the borehole. Permeability – Part B Dr O.Hamza
- 10. Quasi-one-dimensional and radial flow Cylindrical flow: groundwater lowering Pumping from a borehole can be used for deliberate groundwater lowering in order to facilitate excavation. Permeability – Part B Dr O.Hamza
- 11. Quasi-one-dimensional and radial flow Cylindrical flow: confined aquifer Permeability – Part B Dr O.Hamza
- 12. Quasi-one-dimensional and radial flow Cylindrical flow: groundwater lowering This is an example of quasi-one- dimensional radial flow with flow thickness t=h Then A=2πr h and t=h. A=2πr.h Groundwater lowering Original level of water table Integrating between the borehole and at variable distance r: Drawdown The radius of influence Permeability – Part B Dr O.Hamza
- 13. Quasi-one-dimensional and radial flow Spherical flow Darcy's Law still applies, with hydraulic gradient dh/dr and area A varying with radius: A=4πr² where r0 is the radius of the piezometer and h0 the constant head in the piezometer head varies only with radius r r. Variation of pore pressure around a point source or side (for example, a piezometer being used for in situ determination of permeability) is a one in-situ one- dimensional problem which can be analysed in spherical coordinates. Permeability – Part B Dr O.Hamza
- 14. Determination of coefficient of permeability • Laboratory measurement of the coefficient of permeability L b t t f th ffi i t f bilit • Field measurement of the permeability • Empirical relations for the coefficient of permeability Permeability – Part B Dr O.Hamza
- 15. Determination of coefficient of permeability Field measurement of the permeability Field measurement Laboratory measurement • Field or in-situ measurement of permeability avoids the difficulties involved in obtaining and setting up undisturbed samples • Field or in-situ measurement of permeability provides information about bulk permeability, rather than merely the permeability of a small and possibly unrepresentative sample. Is field measurement of permeability better than the lab y measurement? Permeability – Part B Dr O.Hamza
- 16. Determination of coefficient of permeability Field measurement of the permeability Well-Pumping test Observational boreholes In a well-pumping test, a number of observation boreholes at radii r1 and r2 are monitored to measure the pressure heads. Permeability – Part B Dr O.Hamza
- 17. Determination of coefficient of permeability Field measurement of the permeability If the pumping causes a drawdown in an unconfined (i.e. open surface) soil stratum Well-Pumping test then the quasi one dimensional flow equation quasi-one is applied. Integrating between the two test limits and rearranging the equation: Impermeable (Assuming the pumping causes a drawdown in an unconfined (i.e. open surface) soil stratum then) Observational b h l Ob ti l boreholes Permeability – Part B Dr O.Hamza
- 18. Determination of coefficient of permeability Field measurement of the permeability If the soil stratum is confined and of thickness t Well-Pumping test and remains saturated th d i t t d then Confined stratum Permeability – Part B Dr O.Hamza
- 19. Determination of coefficient of permeability Empirical relations for the coefficient of permeability Empirical relations p k = function of (other parameters) Permeability of all soils is strongly influenced by the density of packing of the soil particles which can be simply described through void ratio e. Several empirical equations for estimation of the coefficient of permeability have been proposed in the past. Permeability – Part B Dr O.Hamza
- 20. Determination of coefficient of permeability Empirical relations for the coefficient of permeability Permeability of granular soils P bilit f l il For fairly uniform sand Hazen (1930) proposed the following relation between the coefficient of permeability k (m/s) and the effective particle size D10 (in mm) (the particle size than which 10% soil is finer): k = C D10 C.D 2 where C is a constant approximately equal to pp y q 0.01 (see the figure beside) Hazzan equation and data relating coefficient of permeability and effective grain size of granular soils Permeability – Part B Dr O.Hamza
- 21. Determination of coefficient of permeability Empirical relations for the coefficient of permeability Permeability of soft clays P bilit f ft l Samarasinghe, H S i h Huang and D d Drnevich (1982) h i h have suggested th t th t d that the coefficient of permeability of clays can be given by the equation: en k=C 1+ e where h e is void ratio C and n are constant to be determined experimentally Consolidation of soft clay may involve a significant decrease in void ratio and therefore of permeability. Permeability – Part B Dr O.Hamza
- 22. Summary • All soils are permeable materials, water being free to flow through the interconnected pores between the solid particles. • Water in saturated soil will flow in response to hydraulic g p y gradient and occurs towards the lower total head. • Flow rate is proportional to the hydraulic gradient and can be affected by the geometry of the pores. pores • The hydraulic gradient may be associated with natural flow or induced by loading the soil (i.e. due to excavation or construction). • Coefficient of permeability may be determined from laboratory experiments or from in situe measurements • Pore water pressure u at any point of the soil is computed from the definition of the hydraulic head, u = γw(h-hz) (where h is total head and hz is elevation head). Permeability – Part B Dr O.Hamza
- 23. Quizzes and example problems Work on: • Quizzes: quiz 3 to 6 * • Example problems: * problem 3 and problem 4 * Note. quiz 1 and problem 1 and 2 are covered in Part A of Permeability lecture Permeability – Part B Dr O.Hamza
- 24. Working on Quizzes and Example problems Quiz 3 The sets of nested piezometers shown below penetrate a layered aquifer. •For one of the piezometers, indicate graphically the elevation head, pressure head, and total head. • For both cases, indicate the direction of the vertical flow between the layers. • F case 2, what is a realistic situation th t might result i a set of h d For 2 h ti li ti it ti that i ht lt in t f heads such as this? Note: The wells are drawn with some separation between them to allow you room to label the heads. Assume, however, that they are truly nested, i.e., that they penetrate the surface of the aquifer at the same location. datum Case 1 Case 2 Permeability – Part B Dr O.Hamza
- 25. Working on Quizzes and Example problems Quiz 3 Solution: S l ti hw h Flow Flow hz datum Case 1 Case 2 The situation in Case 2 might happen if the middle layer is being pumped OR if the middle layer is a zone of incredibly high conductivity. Permeability – Part B Dr O.Hamza
- 26. Working on Quizzes and Example problems Quiz 4 An inclined permeameter tube is filled with three layer of soil of different permeabilities as shown in the figure figure. (i) Formulate q in terms of the different dimensions and permeabilities for each soil element (ii) D t Determine th h d l i the head loss (Δh) b t between each soil element assuming h il l t i k1=k2=k3 (iii) Re-determine the head loss (Δh) between each soil element assuming 3k1=k2=2k3 (iv) Express the head at points A, B, C, and D A B C (with respect to the datum) (v) Plot the various heads versus horizontal distance. Permeability – Part B Dr O.Hamza
- 27. Working on Quizzes and Example problems Quiz 4 (i) Flow rate q in each soil element is equal: Δh q = Aki = Ak L Δh Δh 2 q1 = Ak1 1 q 2 = Ak 2 q 3 = ... L1 L2 q = q1 = q 2 = q 3 Δh = Δh1 + Δh 2 + Δh 3 Permeability – Part B Dr O.Hamza
- 28. Working on Quizzes and Example problems Quiz 4 (ii) Flow rate q in each soil element is equal: q = q1 = q 2 = q 3
- 29. Working on Quizzes and Example problems Quiz 4 (iii) Flow rate q in each soil element is equal: q = q1 = q 2 = q 3
- 30. Working on Quizzes and Example problems Quiz 4 (iv) Heads
- 31. Working on Quizzes and Example problems Quiz 4 (v) Plotting NOTE: It is coincident that ll heads th t all h d appears in a straight line.
- 32. Working on Quizzes and Example problems Quiz 5 The site consists of an unconfined aquifer and a confined aquifer separated by a 5-m thick 5 thi k confining layer. Water in the unconfined aquifer i f h and water fi i l W t i th fi d if is fresh, d t in the confined aquifer is saline. Two nested piezometers have been drilled, one penetrating the unconfined aquifer (P1), and one penetrating the confined aquifer (P2) ). Land surface elevation: 68.1 m Temperature of water in P1 and P2: 16° C Depth to P1: 21.2 m Depth to P2: 38.6 m Depth to water in the well at P1: 4.3 m Depth to water in the well at P2: 4.9 m Unit weight of fresh water at 16° C: 9.99 kN/m3 Unit weight of water in P2: 10.21 kN/m3 • Sketch a diagram (doesn’t have to be to scale) showing the information described above. • What is the total head (h1) for P1? • Determine the pressure head for P2 (hw2-saline), and the equivalent fresh-water pressure head for P2 (hw2-frish) w2 frish • What is the total fresh-water head (h2-fresh) for P2? • Will you issue a permit to inject hazardous waste into the deep aquifer ? Why or why not? Permeability – Part B Dr O.Hamza
- 33. Working on Quizzes and Example problems Quiz 5 4.3t 4.9 21.2 38.6 m 68.1 m Datum Permeability – Part B Dr O.Hamza
- 34. Working on Quizzes and Example problems Quiz 5 Fresh water total head for P1 is 68.1 – 4.3 = 63.8 m Saline pressure head for P2 is 38.6 – 4.9 = 33.7 m For the equivalent fresh-water pressure head, pressure must be equal: fresh water head uSaline = ufirsh So γSaline x 33.7 = γfrish x hw2-frish solve for hw2-frish: = γSaline x 33.7 / γfrish = 10.21 x 33.7 /9.99 = 34.4 m so, so h2-fresh = hz2 + hw2-frish = (68 1 – 38 6 ) + 34.4 = 63 9 m f f (68.1 38.6 34 4 63.9 Thus flow is in an UPWARD direction from the lower aquifer, and you should not issue the permit (In addition if you inject waste into the lower aquifer permit. addition, it will further increase the pressure head and increase the upward gradient.) Permeability – Part B Dr O.Hamza
- 35. Working on Quizzes and Example problems Quiz 6 A soil profile consists of th il fil i t f three l layers with properties shown i th t bl b l ith ti h in the table below. Calculate the equivalent coefficients of permeability parallel and normal to the stratum. Layer Thickness (m) kx (parallel, m/s) kz (normal, m/s) 1 3 2x10-6 6 1.0x10 6 1 0x10-6 2 4 5x10-8 2.5x10-8 3 3 3x10-5 1.5x10-5 Answers: For the flow parallel to the layers: kx= 9.6x10^-6 m/s For the flow normal to the layers: kz=6.1x10^-8 m/s Permeability – Part B Dr O.Hamza
- 36. Working on Quizzes and Example problem Problem 3 Field measurement of the coefficient of permeability 3. A stratum of sandy soil overlies a horizontal bed of impermeable material; the surface of which is also horizontal. In order to determine the in situ permeability of the soil, a test well was driven to the bottom of the stratum. Two observation boreholes were made at distances of 12.5m and 25m respectively from the test well. Water was pumped from the test well at the rate of 3x10-3 m3/s until the water level became steady. The heights of the water in the two observation boreholes were then found to be 4.25m and 6.5m above the impermeable bed. Find the value, expressed in m3/day, of the Impermeable coefficient of permeability of the sandy soil ffi i t f bilit f th d il Permeability – Part B Dr O.Hamza
- 37. Working on Quizzes and Example problem Problem 3 Field measurement of the coefficient of permeability 3. Key solution This is a quasi-one dimensional flow, from which we found that: where: q (rate of flow) = 3x10-3 m3/s = 3x10-3 x 60 x 60 x 24 = 259 2 m3/day 259.2 Impermeable r1= 12.5m and r2 = 25m h1= 4.25m and h2= 6.5m ln(r2/r1) = 0.693 Note ‘ln’ is the logarithm to base e, also called the natural logarithm. Permeability – Part B Dr O.Hamza
- 38. Working on Quizzes and Example problems Problem 4 E i i l relations of th coefficient of permeability 4. Empirical l ti f the ffi i t f bilit For a clay soil, the following are given: soil Void ratio 1.1 0.9 k (cm/s) ( /) 0 302 x 10-7 0.302 7 0 12 x 10-7 0.12 7 en Use the following empirical relation: k=C 1+ e proposed by Samarasinghe, Huang and Drnevich (1982) to estimate the coefficient of permeability of the clay at a void ratio of 1 2 1.2. Hint: form two equations with two unknowns C and n by substituting the experimental values given in the table in the equation. Permeability – Part B Dr O.Hamza

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