- Slides: 14
Topic 3: Darcy`s Law Application and Aquifer Characterstics
Darcy’s (1856) law forms the foundation of quantitative groundwater hydrology Control valve l. A L h. A z. A B Porous mediu m Datum Along flow direction A l To tank B h. B z. B
v = - k dh /dl k is the constant of proportionality known as the hydraulic conductivity (m/d) v = Q/A Q = v A= - k A dh / d. L flow quantity can be estimated In Cartesian coordinate system, velocity in x, y and z direction can be written as vx= - kx h / x, vy = - ky h / y, vz= - kz h / z
Darcy`s law is valid only for laminar flow, Reynold’s number Re is < 10, Re = vd 10/ , where v is Darcy velocity d 10 is the average mean grain diameter of the porous media (90% retained on the sieve) and is the kinematic viscosity of water (= 0. 01 cm 2/s at 20 c).
In groundwater flow, the groundwater head h is taken as the summation of the pressure head + velocity head + elevation head. Since the velocity head is negligibly small compared to the other two values, h = p/ g + z is the head normally used in groundwater flow problems. When the flow is not laminar, the hydraulic gradient is not proportional to the velocity, but involves terms containing higher powers of velocity vn, where n is generally ≤ 2. Point of measurement is the bottom of the piezometer.
Vertical variation k =10 m/d fine sand k =50 coarse sand Z k =100 m/d fine gravel X Y X k =100 k =50 Lateral variation k =10 Schematic representation of the vertical and lateral variation of hydraulic conductivity
Homogeneous isotropic Homogeneous anisotropic ky K Y ky =10 m/d k. Kx. X K 4 X k=7 Heterogeneous isotropic kx =20 m/d Heterogeneous anisotropic (after Freeze and Cherry 1979) Aquifer classification based upon the hydraulic conductivity of the media
Homogeneous Heterogeneous source. Singhal (1985)
Isotropic Anisotropic source. Singhal (1985)
Aquifer formation Hydraulic conductivity (m/d) Clay soil (surface) 0. 01 – 0. 2 Deep clay bed 10 -8 – 0. 01 Loam soil 0. 1 - 1 Fine sand 1 -5 Medium sand 5 - 10 Coarse sand 20 - 100 Gravel 100 - 1000 Sand+gravel mix 5 - 100 Clay+sand+gravel mix 0. 001 -1 Sand stone 0. 01 - 1
Average groundwater flow velocity is always higher than the Darcy velocity
Numerical Problem: Assume that three piezometers are installed very close to each other but penetrate up to different depths as given below: Piezometer Elevation at the surface (m) amsl Depth of piezometer (m) Depth of water (m) a 450 b 450 c 450 100 50 27 47 36 (modified from Freeze and Cherry 1979) Let A, B, and C refer to the points of measurement of piezometers a, b, and c, respectively. Calculate
1. Groundwater head at a, b, and c in m. 2. The pressure head at A, B, and C in m. 3. The elevation head at A, B, and C in m. 4. The fluid pressure at B(N/m 2). 5. Can you conceive of a hydrogeological situation that would lead to the directions of flow indicated by these data? 6. If the formation also has two aquitards with a thickness of 15 m (lower) and 20 m (upper) respectively, and hydraulic conductivity 0. 01 m/d, compute the rate of fluid movement across the two aquitards for an aquifer area of 4 km 2
Area = 4 km 2 a b c Ground surface 36 m Aquifer 1 47 m 50 m C 150 m 100 m k` =. 01 m/d 27 m 20 m 15 m B A Q 2 Aquifer 2 k` =. 01 m/d Aquifer 3 350 m Q 1 400 m 300 m Datum 450 m