" = ˆ i # #x + ˆ j # #y + ˆ k # #z. "t = D #2 h

Transcription

1 del operator " = ˆ i # #x + ˆ j # #y + ˆ k # #z Hydrology Gradient: "h = ˆ i #h #x + ˆ j #h #y + k ˆ #h #z q = - K"h Darcy s Law Divergence: " q = #q 1 #x + #q 2 #y + #q 3 #z Laplacian: " 2 h = #2 h #x + #2 h 2 #y + #2 h 2 #z 2 Diffusion Equation: Hydrologic "h "t = D #2 h D = hydraulic diffusivity = K/Ss = T/S

2 Analogy: Gravitational Potential Φg " g = GM r #" g = $ GM r 2 ˆ r = Force # 2 " g = 4%G& = 0 in free space

3 If fdx +gdy+hdz is an exact differential (= du), then it is easy to integrate, and the line integral is independent of the path: Q " P fdx + gdy + hdz = Q " P du = u(q)# u(p) Exact differential: du = "u "x dx + "u "y dy + "u "z dz If true: f = "u "x g = "u "y h = "u "z Condition for exactness: => Curl u = 0 "h "y = "g "z "f "z = "h "x "g "x = "f "y

4 If the function is a vector (v) rather than a scalar, there are two different types of differential operations, somewhat analogous to the two ways of multiplying two vectors together {i.e. the cross (vector) and dot (scalar) products}: For v( x, y, z) = v 1ˆ i + v 2ˆ j + v 3ˆ k Type 1: the Curl of v is a vector: % ˆ i ˆ j k ˆ ( ' $ $ $ * Curl v = " # v = ' * ' $x $y $z* & v 1 v 2 v 3 ), = $v 3 $y + $v / $z ˆ, i + $v 1 0 $z + $v /, 3. 1 ˆ j + $v 2 - $x 0 $x + $v 1. - $y / 1 k ˆ 0 Type 2: the Divergence of v is a scalar: $ Div v = " v = & ˆ i # #x + ˆ j # #y + k ˆ # % #z ' ) v 1ˆ i + v 2ˆ j + v 3ˆ k ( ( ) So: " v = #v 1 #x + #v 2 #y + #v 3 #z Great utility for fluxes & material balance

5 Conservative Forces Suppose that force F = fi +gj + hk acts on a line segment dl = idx+jdy+kdz : Q Q Work = # F " dl = # ( f ˆ i + g ˆ j + h k ˆ )" ˆ i dx + ˆ j dy + k ˆ dz P P ( ) = Q # P fdx + gdy + hdz Q =(if exact) = $!"# dl = $ d" = "(Q)% "(P) P If fdx + gdy + hdz is exact, then the work integral is independent of the path, and F represents a conservative force field that is given by the gradient of a scalar function Φ (= potential function). Q P In general: 1. Conservative forces are the gradients of some potential function Φ. 2. The curl of a gradient field is zero i.e., Curl (grad Φ) = 0 " # F = " # "$ = 0

6 Equipotential Lines Lines of constant hydraulic head Gradient defines direction of groundwater flow Contours on potentiometric surface or on water table map! => Equipotential Surfaces in 3D!

7 Cones of depression, Sacramento Valley Criss & Davisson 1996, after DWR 1986

8 Black Hills Potentiometric Surface, Dakota Aquifer after Darton 1909

9 Hubbert (1940)

10 Equipotential Lines Contour lines of constant hydraulic head on a map Vertical equipotential planes are typically assumed, in which case, an individual equipotential line (contour line) represents the intersection of a given equipotential plane with the Earth s surface Gradient defines direction of groundwater flow

11 Potentiometric Surface Equipotential Line Equipotential Planes (assumed)

12 Potentiometric Surface Equipotential Line q Equipotential Planes (assumed)

13 Potentiometric? Surface? Equipotential Line q Equipotential Planes (actual)

14 Potentiometric Surface q Confined Aquifer q Equipotential Planes

15 Potentiometric Surface: Map of the hydraulic head on a plane of interest, projected vertically upward to Earth s surface Map of level to which water would rise in a piezometer A piezometer is a small dia, non-pumping well with a short screen that is cased to the aquifer * (Fetter, p. 134). A piezometer measures the hydraulic head at a point!contours on the potentiometric sfc. are equipotential lines! Direction of GW flow is down gradient Potentiometric surface is 2D representation of 3D phenomenon Vertical planes assumed- no vertical flow Concept is rigorously valid only for horizontal flow within a horizontal aquifer. For an Unconfined Aquifer, Potentiometric Sfc = Water Table For a Confined Aquifer; there is no Water Table

16 Flow Nets:!!Set of intersecting Equipotential lines and Flowlines! Flowlines!= Streamlines!!!!!= Instantaneous flow directions!! Pathlines!= Actual particle path!!pathlines Flowlines for transient flow!!!flowlines to Equipotential planes if K is isotropic!!!can be conceptualized in 3D!

17 No Flow No Flow No Flow Fetter

18 Flow Net Rules: Flowlines are perpendicular to equipotential lines (isotropic case) No Flow Boundaries Equipotential lines meet No Flow boundaries at right angles Flowlines are tangent to such boundaries (// flow) Constant Head Boundaries Equipotential lines are parallel to constant head boundaries Flow is perpendicular to constant head boundary Spacing between equipotential lines L: If spacing between lines is constant, then K is constant K 1 /L 1 ~ K 2 /L 2 for aquifer of constant thickness

19 Flow Net Rules: Flowlines are perpendicular to equipotential lines (isotropic case) No Flow Boundaries Equipotential lines meet No Flow boundaries at right angles Flowlines are tangent to such boundaries (// flow) Constant Head Boundaries Equipotential lines are parallel to constant head boundaries Flow is perpendicular to constant head boundary Spacing between equipotential lines L: Better rule: K 1 m 1 /L 1 ~ K 2 m 2 /L 2

20 FLOW NETS Impermeable Boundary Constant Head Boundary Water Table Boundary after Freeze & Cherry