Darcys law Groundwater Hydraulics Daene C Mc Kinney

Darcy’s law Groundwater Hydraulics Daene C. Mc. Kinney 1

Outline • • • Darcy’s Law Hydraulic Conductivity Heterogeneity and Anisotropy Refraction of Streamlines Generalized Darcy’s Law 2

Darcy http: //biosystems. okstate. edu/Darcy/English/index. htm 3

Darcy’s Experiments • Discharge is Proportional to – Area – Head difference Inversely proportional to – Length • Coefficient of proportionality is K = hydraulic conductivity 4

Darcy’s Data 5

Hydraulic Conductivity • Has dimensions of velocity [L/T] • A combined property of the medium and the fluid • Ease with which fluid moves through the medium k ρ µ g = cd 2 = = = intrinsic permeability density dynamic viscosity specific weight Porous medium property Fluid properties 6

Hydraulic Conductivity 7

Groundwater Velocity • q - Specific discharge Discharge from a unit crosssection area of aquifer formation normal to the direction of flow. • v - Average velocity of fluid flowing per unit crosssectional area where flow is ONLY in pores. 8

Example h 1 = 12 m K = 1 x 10 -5 m/s f = 0. 3 Find q, Q, and v h 2 = 12 m /” 10 m Flow Porous medium 5 m L = 100 m dh = (h 2 - h 1) = (10 m – 12 m) = -2 m J = dh/dx = (-2 m)/100 m = -0. 02 m/m q = -KJ = -(1 x 10 -5 m/s) x (-0. 02 m/m) = 2 x 10 -7 m/s Q = q. A = (2 x 10 -7 m/s) x 50 m 2 = 1 x 10 -5 m 3/s v = q/f = 2 x 10 -7 m/s / 0. 3 = 6. 6 x 10 -7 m/s 9

Hydraulic Gradient vector points in the direction of greatest rate of increase of h Specific discharge vector points in the opposite direction of h 10

Well Pumping in an Aquifer Hydraulic gradient y Circular hydraulic head contours Dh K, conductivity, Is constant q Specific discharge x Well, Q h 1 h 2 h 3 h 1 < h 2 < h 3 Aquifer (plan view) 11

Validity of Darcy’s Law • We ignored kinetic energy (low velocity) • We assumed laminar flow • We can calculate a Reynolds Number for the flow q = Specific discharge d 10 = effective grain size diameter • Darcy’s Law is valid for NR < 1 (maybe up to 10) 12

Specific Discharge vs Head Gradient Re = 10 Re = 1 Experiment shows this Darcy Law predicts this a tan-1(a)= (1/K) q 13

Estimating Conductivity Kozeny – Carman Equation • Kozeny used bundle of capillary tubes model to derive an expression for permeability in terms of a constant (c) and the grain size (d) Kozeny – Carman eq. • So how do we get the parameters we need for this equation? 14

Measuring Conductivity Permeameter Lab Measurements • Darcy’s Law is useless unless we can measure the parameters • Set up a flow pattern such that – We can derive a solution – We can produce the flow pattern experimentally • Hydraulic Conductivity is measured in the lab with a permeameter – Steady or unsteady 1 -D flow – Small cylindrical sample of medium 15

Measuring Conductivity Constant Head Permeameter • Flow is steady • Sample: Right circular cylinder – Length, L – Area, A Continuous Flow head difference • Constant head difference (h) is applied across the sample producing a flow rate Q flow • Darcy’s Law Overflow A Outflow Q Sample 16

Measuring Conductivity Falling Head Permeameter • Flow rate in the tube must equal that in the column Initial head Final head flow Outflow Q Sample 17

Heterogeneity and Anisotropy • Homogeneous – Properties same at every point • Heterogeneous – Properties different at every point • Isotropic – Properties same in every direction • Anisotropic – Properties different in different directions • Often results from stratification during sedimentation www. usgs. gov 18

Example • • a = ? ? ? , b = 4. 673 x 10 -10 m 2/N, g = 9798 N/m 3, S = 6. 8 x 10 -4, b = 50 m, f = 0. 25, Saquifer = gabb = ? ? ? Swater = gbfb • % storage attributable to water expansion • • %storage attributable to aquifer expansion • 19

Layered Porous Media (Flow Parallel to Layers) Piezometric surface Dh h 1 h 2 datum Q b W 20

Layered Porous Media (Flow Perpendicular to Layers) Piezometric surface Dh 1 Dh 2 Dh Dh 3 b Q Q L 1 L 2 L L 3 21

Example Flow Q • Find average K 22

Flow Q Example • Find average K 23

Anisotrpoic Porous Media • General relationship between specific discharge and hydraulic gradient 24

Principal Directions • Often we can align the coordinate axes in the principal directions of layering • Horizontal conductivity often order of magnitude larger than vertical conductivity 25

Flow between 2 adjacent flow lines For the squares of the flow net so For entire flow net, total head loss h is divided into n squares If flow is divided into m channels by flow lines 26

KU/KL = 1/50 Flow lines are perpendicular to water table contours Flow lines are parallel to impermeable boundaries KU/KL = 50 27

Contour Map of Groundwater Levels • Contours of groundwater level (equipotential lines) and Flowlines (perpendicular to equipotiential lines) indicate areas of recharge and discharge 28

Groundwater Flow Direction • Water level measurements from three wells can be used to determine groundwater flow Head Gradient, J direction Groundwater Contours hi h 1(x 1, y 1) hi > h j > h k hj hk h 3(x 3, y 3) z y Groundwater Flow, Q h 2(x 2, y 2) x 29

Groundwater Flow Direction Head gradient = Magnitude of head gradient = Angle of head gradient = 30

Groundwater Flow Direction Head Gradient, J h 1(x 1, y 1) Equation of a plane in 2 D 3 points can be used to define a plane h 3(x 3, y 3) z y Groundwater Flow, Q h 2(x 2, y 2) x Set of linear equations can be solved for a, b and c given (xi, hi, i=1, 2, 3) 31

Groundwater Flow Direction Negative of head gradient in x direction Negative of head gradient in y direction Magnitude of head gradient Direction of flow 32

Example Find: y Well 2 (200 m, 340 m) 55. 11 m Well 1 (0 m, 0 m) 57. 79 m Magnitude of head gradient Direction of flow x Well 3 (190 m, -150 m) 52. 80 m 33

Example Well 2 (200, 340) 55. 11 m x Well 1 (0, 0) 57. 79 m q = -5. 3 deg Well 3 (190, -150) 52. 80 m 34

Refraction of Streamlines • Vertical component of velocity must be the same on both sides of interface y Upper Formation x • Head continuity along interface Lower Formation • So 35

Consider a leaky confined aquifer with 4. 5 m/d horizontal hydraulic conductivity is overlain by an aquitard with 0. 052 m/d vertical hydraulic conductivity. If the flow in the aquitard is in the downward direction and makes an angle of 5 o with the vertical, determine q 2. 36

Summary • Properties – Aquifer Storage • Darcy’s Law – – Darcy’s Experiment Specific Discharge Average Velocity Validity of Darcy’s Law • Hydraulic Conductivity – – Permeability Kozeny-Carman Equation Constant Head Permeameter Falling Head Permeameter • Heterogeneity and Anisotropy – Layered Porous Media • Refraction of Streamlines • Generalized Darcy’s Law 37

Example Flow Q 38

Flow Q Example 39