Groundwater Flow Equations and Flow Nets

Objective Need values of head, flow, and groundwater velocity Resource use (well development) environmental impact contaminant transport remediation In many cases we need to make predictions Approach: use models Take into account aquifer and fluid properties (pore size, K, porosity, etc) Account for boundary conditions: head, flow

Types of Models Physical models (scaled) Analog models Electrical Viscous flow Mathematical models Material, energy and momentum balance Constitutive “laws” – e.g., Darcy’s Law Input account aquifer and fluid properties (pore size, K, porosity, etc Input boundary and initial conditions Output head, flow, velocity

Groundwater Flow Equation Confined Flow

Homogeneous Isotropic Hydraulic conductivity does not depend on position or direction

Steady State Conditions Mass accumulation is equal to zero THE LaPLACE EQUATION

Unconfined Flow General Equation Steady State Chain rule

Unconfined Flow Assume changes in h are small Steady State LaPlace Equation!

Model Solution We have the model: head (h) is described in the equations in terms of x, y, z, and t Need to solve the differential equation! Solve model using integral calculus Conditions of steady state (no change over time) Simple boundary conditions Solve model using techniques for PDEs Conditions change with time Laplace Transforms, combination of variables, perturbation Solve model numerically Write differential equations as algebraic equations Solve model graphically

Flow Nets Graphical solution to LaPlace’s Equation Graphical depiction of equipotential lines and flow lines Valid for 2-D, incompressible media and fluid, steady state conditions

FLOW LINE - the average path a particle of water follows through the porous media (the vector mean path) Here shown as a dye trace in flow through a dam Rule: flow lines can never intersect or meet

EQUIPOTENTIAL LINE - line of equal potential (f) or head (f/g). FLOW LINE Ho EQUIPOTENTIAL LINE EQUIPOTENTIAL LINE - line of equal potential (f) or head (f/g). The same energy level exists everywhere along a given equipotential line Equipotential lines can never intersect or meet

Flow Lines Groundwater flows from high head (potential) to low head (potential) More specifically, it follows the maximum gradient in head. dh/dl is maximum for the shortest dl (l1) Therefore, flow lines cross equipotential lines at right angles (isotropic medium) If we know the distribution of equipotential lines in a homogeneous and isotropic aquifer, flow lines can easily be drawn. l4 l3 h1 l2 l1 h2 h1 h2 h3 h4

Flow and Equipotential Lines

Simple Flow Net ho – 1DhT /4 ho – 3DhT /4 ho – 1DhT /2

Boundary Conditions Impermeable Boundary (aka no-flow boundary) (for example, an ‘impermeable’ rock, concrete) Constant Head Boundary (for example, a standing body of water) Variable Head Boundary (for example, a water table)

Impermeable Boundary “no-flow boundary” no-flow boundaries Adjacent flowlines must parallel a no-flow boundary Equipotential lines must be perpendicular to no-flow boundaries

Constant Head Boundary “equipotential surface” boundaries Flow lines must intersect a constant head boundary at right angles Adjacent equipotential lines must ‘parallel’ the constant head boundary

What Type of Boundary ?

Variable Head Boundary (variable but known) Neither a flowline nor an equipotential line A line (surface) with known heads provides known intersection points for equipotential lines WT h = 93 92 91 90 92 91 90

Groundwater Mapping Hydrologic map 2-D representation of 3-D flow system Measure water levels in piezometers at same point in time Construct contours of equal head Use interpolation Equipotential lines are intersections of equipotential surfaces with the plane of the diagram Isotropic aquifer: direction of flow at every point is perpendicular to contours Hydraulic gradient computed from contour spacing

Contoured Equipotential Data

Groundwater Flow Draw flow lines perpendicular to equipotential lines Identify sources and sinks Identify impervious zones Identify surface/gw interactions Identify GW divides Assess environmental impact Evaluate contaminant transport

Intuitions from Flownets Converging flowlines Equipotential lines are more closely spaced Groundwater velocities increase Discharge areas (e.g. - springs, effluent streams, pumping wells, etc.) Radiating (diverging) flowlines Associated with recharge areas Water is entering the ground - e.g. influent streams, some lakes, hilltops, injection wells, etc.

Map Features Recharge Discharge

Map Features Surface water interactions Changes in T Impervious zones

Map Features Impermeable zones Faults GW divides

Effect of Symmetry Flow does not cross AB and DC No-flow boundaries Can use symmetry to isolate individual flow systems

Flow Net Construction General Rules For homogeneous material, flow net construction is independent of K!! Impermeable (no flow) boundaries Flow lines parallel Equipotential lines perpendicular Constant head boundaries Flow lines perpendicular Equipotential lines parallel

Flow Net Construction Steps Identify boundary conditions No flow, constant head, variable but known head Sketch boundaries to scale Choose equipotential line contour interval and flowline spacing such that the grid constructed approximates squares an enclosed circle is tangential to all 4 sides ERASE AND REDRAW until all flow lines are perpendicular to equipotential lines, boundary conditions are satisfied, and grid is made up of squares can have partial streamtube at edges KEEP A SENSE OF HUMOR!

Simple Flow Net ho – 1DhT /4 ho – 3DhT /4 ho – 1DhT /2

Simple Flow Net ho – 1DhT /5 ho – 4DhT /5 ho – 1DhT /2

Constant head boundary No flow boundary Squares?

Curvilinear squares An enclosed circle is tangential to all 4 sides

Example Identify boundary conditions No flow, constant head, variable but known Choose equipotential line contour interval and flowline spacing to produce squares ERASE AND REDRAW all flow lines are perpendicular to equipotential lines boundary conditions are satisfied grid is made up of squares can have partial streamtube at edges

Streamtubes The region between adjacent flow lines is called a STREAMTUBE Discharge through each stream tube is constant flowlines cannot intersect As flowlines converge, the cross-sectional area perpendicular to flow gets smaller Flow stays constant within a streamtube, Hydraulic gradient must increase Equipotential lines get more closely spaced low gradient high gradient

Flow in Streamtubes is Equal dQ1 dQ2 dQ3 dQ1= dQ2= dQ3 dQ1+ dQ2+ dQ3 = QTOTAL Total width, W Flow lines are evenly spaced to make squares Total flow is divided into equal increments Therefore, every streamtube has an equal discharge

Darcy’s Law dq’1 dq’2 dq’3 dq’ = flow per streamtube per unit width h - Dh dq’1 Dl dq’2 Dz dq’3 dq’ = flow per streamtube per unit width Dh = head drop btw equipotential lines Dl = flow length between equipotential lines Dz = height of streamtube Unit width dQ1 = – KDzDw(Dh/Dl) dq’ = –KDz(1)(Dh/Dl)

Darcy’s Law dq’ = –KDz(1)(Dh/Dl) h - Dh dq’1 Dl dq’2 Dz dq’3 Flownet is made of squares so Dl = Dz dq’ = –K(Dh) q’ = –pK(Dh) where p = number of streamtubes Dh = DhT/f where f = number of head drops Unit width q’ = –pKDhT/f Q = q’W

Simple Flow Net Q = –KA(DhT/L) = –K(10)(5)(3/50) = –3K DhT = 3 cm W = 5 cm B = 10 cm L = 50 cm Q = q’W = –pK(DhT/f)W = –2K(3/10)(5) = –3K

Total width = 80 m K = 10-3 cm/sec z = 60 m Data: p = 2; f = 7; DhT = 60 m q’ = –pK(DhT/f) = –2(10–5 m/s)(–60 m/7) = 1.714x10–4 m3/sec/m Q = q’*W = (1.714x10–4 m3/sec/m) x 80 m = 1.371x10-2 m3/sec Q = 1185 m3/day; 313055 gal/d, 217.4 gpm; 0.48 cfs

Pipe Flow What size pipe would be required to carry the quantity of seepage flow through the dam? Mannings Eq: Pipes flowing full: A = pD2/4 R = A/Wp = pD2/4/pD = D/4 6” concrete pipe laid at 1% grade (small pipe, flat grade) n = 0.013 A = 3.14 x 0.52/4 = 0.1963 R = 0.5/4 = 0.125 Q = 1.486/0.013 x 0.1963 x 0.125(2/3) x 0.01(1/2) = 0.56 cfs

p/f = Shape factor There are an infinite number of possible flownets that can be constructed for any set of boundary conditions. However, the shape factor p/f will be constant (if the flownet is constructed with squares) p = 3, f = 6, p/f = 0.5 p = 6, f = 12, p/f = 0.5

Refraction of Flow Lines Heterogeneous aquifers K1 q1 K2 q2 Where q1 and q2 are the angles of the flowlines with respect to the perpendicular to the conductivity boundary

Equipotential Lines K1 K2 Draw equipotential lines as squares in material 1 Equipotential lines must be continuous across boundary Isotropic – flow lines and equipotential lines form right angles in material 2 Rectangles are formed in material 2 K1 q1 K2 q2

Relate K1 and K2 by Continuity & Geometry Flow in streamtube is constant Head drop between equipotential lines is same in both materials Flow area Flow length Q1 l1 K1 a q1 b c K2 l2 Q2 q2

Tangent Law Eliminate Dh Substitute Cancel b, w Q1 l1 K1 a q1 b c K2

Flow from High K to Low K K1 K1 > K2 K2 Flow lines refracted towards the perpendicular Flow net squares shorten to rectangles Flow path btw equipotential lines, l, becomes smaller, increasing Dh/l Steeper gradient and greater areas needed to produce the same Q

Flow from Low K to High K K1 K1 < K2 K2 Flow lines refracted away from the perpendicular Flow net squares elongate to rectangles Flow path btw equipotential lines, l, becomes greater, decreasing Dh/l Smaller gradient and flow area needed to produce the same Q

Flow Net Shape Write continuity Eq. Substitute Recognize K1 K2 l1 l2 Q1 Q2 q1 q2 a b c Flow net squares elongate (or shorten) by the ratio of hydraulic conductivities

Path of Least Resistance GW flows obliquely across heterogeneous systems Flow lines ‘prefer’ to use high K formations as conduits; flow lines ‘prefer’ to cut across low K units by the shortest route In flow systems with aquifer/aquitard conductivity contrasts of 2 orders of magnitude or more (K1 > 100K2) Flow lines tend to become ‘parallel’ to high K units, and ‘perpendicular’ to low K units

Effect of Lenses – High K Low K FLOW LINE REFRACTION NEAR HIGH K LENS

Effects of Lenses – Low K High K FLOW LINE REFRACTION NEAR LOW K LENS

Flow Nets in Anisotropic Systems Directional variation in K Flow lines no longer meet equipotential lines at right angles We can still construct the flow net by transforming the coordinates no longer 90o

Coordinate Transformation Anisotropic System Flownet construction For example, due to sedimentary layering Generally Expand vertical coordinate by the ratio Construct flow net as usual using square elements Collapse back to original dimensions KZ KX KX > KZ

Coordinate Transformation Example Transformed section