GG655/CEE63 Groundwater Modeling Model Theory Water Flow Aly I. El-Kadi Hydrogeology 1
Saline water in oceans = 97.% Ice caps and glaciers =.14% Groundwater = 0.61% Surface water = 0.009% Soil moisture = 0.005% Atmosphere = 0.001% Aquifers Store reasonable amount and transmit water fast enough Permeability k > 10-1 cm Aquitard : leaky aquifer of k less than that Aquifer types Confined Unconfined
Changing from unconfined to confined conditions Perched aquifer Water table For unconfined aquifer At water table: Pore water pressure = atmospheric pressure Capillary fringe 3
Aquifer parameters Porosity Due to cracks or voids created by physical and chemical weathering processes Vv n V V v = volume of void space V = total volume ρ 1 b n 100 ρd e.g., n = 0.3 or 30% b = bulk density d = particle density (~.65 gm/cm3) Effective porosity: portion available for water flow Specific Yield Volume of water drained by gravity S y Total volume of sample n S y S r S r = Specific retention 4
Hydraulic (total) head h = z + H z = elevation H = pressure head [L] [L] [L] H = P/ w g P = Pressure [M/L/T ] w = Density [M/L 3 ] g = Gravity acceleration [L /T] Darcy s eperiment Hydraulic conductivity Ability of aquifer (geologic formation) to conduct (transmit) water Darcy s Law Discharge Q: h a L h b Q ~ h b h a 1 Q ~ L Q ~ A head difference length -sec area h h b a Q ~ A L h h b a L Q KA ; K = hydraulic conductivity L T dh Q KA dl dh/dl = hydraulic gradient Negative sign because water moves in direction of decreasing head 5
K is related to permeability by fluid properties: K kγ μ kρg μ k = permeability [L ] and = specific weight, density, and viscosity of fluid, respectively g = acceleration of gravity Transmissivity T = Kb [L /T] K = conductivity [L/T] b = aquifer saturated thickness Assumes horizontal flow with uniform vertical head conditions dh Q Kb 1.0 d T Q for dh 1.0 d Amount of water transmitted horizontally through a unit Width under unit gradient Storage Specific storage (S s ) Volume of water absorbed or epelled per unit aquifer volume per unit change in head [1/L] S s = w g(+n) w = water density [M/L 3 ] g = acceleration of gravity [L/T ] = = matri compressibility [LT /M] n = porosity [dimensionless] = water compressibility [LT /M] Storage coefficient (Storativity) (S) Volume of water absorbed or epelled per unit aquifer area per unit change in head [dimensionless] 6
Confined aquifer: Water is released due to compressibility: S = S s b Unconfined aquifers: Water is released by draining the pores (voids) and compressibility; the latter can be ignored: S = S y +S s h S ~ S y Homogeneity Homogenous aquifers: same properties at all locations Opposite is called heterogeneous K 1 K K K 1 K K 3 T is heterogeneous K is heterogeneous Isotropy Isotropic aquifer: same conductivity in all directions Opposite is called anisotropic K K K 1 K 1 Anisotropic Isotropic 7
Math Refresher Derivatives A variable h that is a function of space and time t can be written as h=h(,t) and t: : independent variables h: : dependent variable. Derivatives Partial derivatives h ( a, b ) h (, b ) h ( a, b ) lim a a h ( a, b ) h ( a, t ) lim t t b t h ( a, b ) b slopes of the function h(,t) and at point =a and t=b 8
Partial derivative wrt,b) h( h=h(,t=b) Tangentatpoint(a,b) X (a,b) Mathematical Notations Quantity Definition Eample Scalar Characterized by magnitude Head only Vector Characterized by magnitude Velocity and direction Second- Characterized by magnitude, Hydraulic order direction, and magnitude conductivity Tensor changes with direction Mathematical Notations Gradient operator h h h h g rad ( h ) i j k y z Divergence operator v v v div ( v ) y v z y z 9
Mathematical Notations Laplacian operator h h h h div grad( h) y z Eample: Dispersive flu F D C D g ra d ( C ) D Groundwater Flow Equation Darcy's law describes a linear relationship between the rate of flu and head gradient in one dimension: v K dh d in three dimensions: h h h v K K y K z y z in which h = hydraulic head; K is hydraulic conductivity; and, y, and z are coordinate systems 10
This equation can be written as: v K ij h i, j =1,, and 3. If the principal components of K (i.e., K ii coincide with the coordinates i : h v K ii i and K ij = 0 for i = j j ii ) Derivation of Equation = fluid density n = porosity R = direct injection/discharge Q = flu rate R (Q) z+z (Q) + (Q) y (Q) y+y (Q) (Q) z z y 11
Q Q Q y y z z t Q Q Q Ryz y z n n yz 0 t t t v Q y z v v v lim 0 v v v n y z R y z t v v y vz h R Ss y z t K h y K h y z K h h S yy zz s z t S g n s Saturated 3-D equation Condensed form: Kh R S h s t Steady State: K h R 0 Steady state homogenous: h 0 Integrating the flow over the vertical: h h h T T S yy y y t T = Kb S = S s b 1
Boundary and Initial Conditions Third type q n =q(,y,h;t) First type h=h(,y;t) Initial Condition h i =h(,y;t=t 0 ) Second type q n =q(,y;t) 13