Review of te Fundamentals of Groundwater Flow Darc s Law 1 A 1 L A or L 1 datum L : volumetric flow rate [L 3 T -1 ] 1 : draulic ead upstream [L] A : draulic ead downstream [L] : draulic conductivit [L T -1 ] A: cross sectional area normal to flow direction [L ] L: distance between locations were 1 and are measured [L]
Darc s Law: 1 A L Alternativel: d A dl A d dl is called specific discarge also Darc flu or Darc velocit. However is not true flow velocit! Te true flow velocit v also referred to as seepage velocit or pore water velocit is defined as v na n d dl Bulk area is A Porosit is n Area for flow is (na)
Transmissivit (T) for confined auifer T b Confined auifer Confining bed b Potentiometric surface Confining bed Water table or free surface Unconfined auifer Transmissivit for unconfined auifer T : saturated tickness of auifer Confining bed
Darc s Law and Hdraulic Head 1. Hdraulic Head 1 L A 1 and are draulic eads associated wit points 1 and. Te draulic ead or total ead is a measure of te potential of te water fluid at te measurement point. datum Potential of a fluid at a specific point is te work reuired to transform a unit of mass of fluid from an arbitraril cosen state to te state under consideration. p1 1 p 1
Tree Tpes of Potentials A. Pressure potential work reuired to raise te water pressure W 1 1 1 m m P P VdP m dp ρ w P ρ w ρ w : densit of water assumed to be independent of pressure V: volume P v Reference state P P v v Current state
B. Elevation potential work reuired to raise te elevation 1 Z W mgd g m C. inetic potential work reuired to raise te velocit (d vdt) 1 Z 1 Z W mad m m m dv dt d v vdv v 3 Total potential: P Φ g ρ w v Unit [L T -1 ] Total [draulic] ead: Φ P g ρ g Unit [L] w v g
Total ead or draulic ead: Pieometer pressure ead [L] P ρ g w v g elevation [L] inetic term P1 ρg P ρg 1 1 datum A fluid moves from were te total ead is iger to were it is lower. For an ideal fluid (frictionless and incompressible) te total ead would sta constant.
For Groundwater Flow P v inetic term ρ g g w negligible draulic ead [L] P ρ w g pressure ead [L] elevation ead [L] Important: is relative to datum (reference state) pieometers 1 A B datum flow direction?
Hdraulic gradient 1 A L 1 A L ia d dl A L 1 datum Gradient of wit respect to at point A is: 1 d d A 1 1 lim A 1
θ dl d 4 3 1 Magnitude and direction of gradient vector in D dl d arctan θ
Etension of Darc s Law Darc s Law in 1D: dl d ( ) k j i k j i k j i grad dl d ; ; i j k: unit vectors NOTE: above euations are valid onl if is scalar i.e. auifer is isotropic. In oter words does not var wit direction at an location. Darc s Law in 3D:
In an anisotropic auifer is a second-order tensor. It as 9 components in te Cartesian coordinate sstem: Darc s Law becomes grad ( ) or in matri form
Darc s Law wit epanded terms: In tensor notation : i i ij j 13 j ( ) NOTE are principal components of te tensor; parallel or normal to te direction of te maimum or minimum. ij (were i j) are cross terms of te tensor representing contributions to te flow rate in direction i from te gradient in direction j.
General tensor: Auifer formation If te Cartesian coordinate aes are aligned wit te principal directions of te tensor te cross terms are all eual to ero:
Use onl one inde for te tensor: ; ; We ave components of specific discarge: or in tensor notation: ( ) i i i i 13
In isotropic media is scalar specific discarge is parallel to draulic gradient grad() In anisotropic media is tensor or vector specific discarge is NOT parallel to draulic gradient grad()
GROUNDWATER TORAGE a) torage Concept tead tate V w A V A w Non-tead tate if: 1 V w if: 1 V w torage tank A 1 torage Coefficient: Volume of water tat an auifer releases from or takes into storage per unit surface area of te auifer per unit cange in ead. Vw A 3 L LL
b) Unconfined Auifer (pecific Yield) n: Porosit; assume 1% drainable V w V w na na auifer torage from unconfined auifer involves psical dewatering V A w : specific ield n A: area Tpicall not 1% drainable V w V fna (f < 1) w fna V A w fn n r specific retention r
c) Confined Auifer V A V A w w Vw A water level pieometer pump V w : storage coefficient dimensionless s b [L -1 ] s : specific storage No psical dewatering involved. torage coefficient for confined auifer b water input 1 3 1 6 usuall << porosit A
I. Derivation of te Groundwater Flow Euation A. Continuit Principle (conservation of mass) inflow - outflow storage B. Darc s Law pecif ow inflow and outflow are calculated top back Control Volume left side () rigt side front bottom
t - Z Z Z Z Y Y Y Y X X X X top bottom front back rigt left ) ( storage sink top front rigt outflow source bottom back left inflow storage outflow inflow Z Y X Z Y X
t X X Y Y Z Z Divide bot sides b t t Z Y X Z Z Y Y X X were
t Heterogeneous/anisotropic wit sink/source transient 3D t General Flow Euation Tree dimensional transient eterogeneous anisotropic Laplace s Euation tead-state condition in omogeneous isotropic medium witout sink/source ( ) ( ) ( ) () Laplacian Operator
Two-dimensional t b b b b t T T ss * Homogeneous & Anisotropic (in terms of transmissivit) t T T ss * Homogeneous & Isotropic t T T ss * b: saturated tickness
II. Matematical model of groundwater flow A. Governing Euation (assumptions?) t (assumptions?) impler Eamples: t Assumptions: 3-D transient eterogeneous anisotropic wit sinks/sources (principal components aligned wit coordinate aes)
B. Initial Conditions For an transient problem te initial ead distribution must be known in order to solve for ead canges wit time i.e. ( ) f ( ) were f() is known For tis eample ( ) t t t
C. Boundar Conditions Heads flues or some combination of te two must be known at boundaries in order to solve for ead canges wit space in te interior of te flow field. (Wang and Anderson 198)
1. pecified ead (Diriclet condition) ( t) ( t) Γ1 were (t) is known For eample 1 on te left boundar on te rigt boundar 1
. pecified flu (Neumann condition) η o ( t) Γ were o (t) is a known flu across te boundar. Γ η A special case is ero flu boundar condition i.e. η Γ UETION: A river could be treated as eiter a specified-ead or specified flu boundar. Wat s te difference between te two treatments?
3. Combination of 1 & (Cauc condition) also referred to as ead-dependent boundar condition te flu into te auifer troug te auitard is dependent on te ead in te auifer i.e. η / / b b Γ 3 b / / b / UETION: Could a river also be treated as a ead-dependent boundar condition? If so wat s te difference wit te previous two treatments?
III. olutions of Matematical Model olutions to a matematical model of groundwater flow can be obtained analticall or numericall. Analtical olution: Head is epressed eplicitl as matematical formula (function) of t. For eample stead-state flow in a 1-D confined auifer: ( ) 1 L 1 1 L ceck d 1 d L constant slope!
Eample of Analtical olutions 1 b datum L 1. Governing euation:. Boundar conditions: T t ( ) ( ) X 1 X L
d d d T d d d d d d d a ( ) d d d ad ( ) a b wit boundar condition (1): () 1 we found: b 1 wit boundar condition (): L () 1 we found: a L 1 ( ) L 1
Numerical olution: Head is solved approimatel at predefined nodal points as illustrated below. Numerical solution is tpicall obtained troug a computer code. (Wang and Anderson 198)