Brief Review of Vector Calculus

Transcription

1 Darc s Law in 3D Toda Vector Calculus Darc s Law in 3D q " A scalar as onl a magnitude A vector is caracteried b bot direction and magnitude. e.g, g, q, v,"," Vectors are represented b : boldface in boos, papers & reports, or Caracters topped b a bar or arrow, or b an underline eg, andwritten on paper or te board: e.g., seepage velocit r v, v, v or v. & v # v, v, v v % v " v v v e.g., µ, ",, n,, S, S s Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall 006 1

2 We ll top a caracter wit a bar, or use boldface, to indicate a vector. f a b c were a, b, c are all scalars unit vector in te Z direction unit vector in te Y direction unit vector in te X direction f a b c c a f b Project vector f onto eac ais Ortogonal projection onto eac aes. Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall 006

3 To find te magnitude lengt of a vector: Te magnitude of vector f written f is f a b c were f a b c To multipl a vector b a scalar: & ma# m f ma mb mc mb % mc " A vector multiplied b a scalar results in a vector. Multiplication b a scalar onl affects te vector lengt. Multipling vectors b eac oter: For groundwater flow, te dot or inner product is te most commonl used product. Given two vectors: g g g g g Te dot product g is a scalar: g g dot is te product of te component of in te direction of wit magnitude Te result is a scalar. Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall 006 3

4 g g g dot is te product of te component of in te direction of Te result is a scalar. wit magnitude Tis is te component of in te direction of. It equals g cos. g g g cos" g g g sum products of components Reminder from fresman Psics: pusing a car Force in te -direction moves te car. Find and sum te -direction forces: Force contributed is F cos 45 F 45 F3 No contribution F1 100% of force contributed Direction of movement Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall 006 4

5 Special Cases: Ortogonal Vectors: dot product is ero g g g cos90 o g 0 0 Parallel Vectors: dot product is te product of vector magnitudes g g g cos 0 o g Gradient operator: Te gradient operator is a wa of doing differentiation wit vectors. It is a vector operator. Gives te rate of cange of a scalar field in te direction of te greatest rate of cange. 1 : pronounced del Te indicates te vector del is operating on a scalar 1 1 In tis case it is because del is operating on elevation. Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall 006 5

6 Gradient operator: Eample: ",, Hdraulic gradient:,, Te driving force for groundwater flow is -, were is ead, because points in te increasing direction. Define te draulic gradient vector J " For a one-dimensional sstem a Darc column varies onl in -direction, ie Gradient of : " d d If conductivit and area is uniform witin te column and no sources/sins, ten d d is linear, so d d l Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall 006 6

7 Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall f " Divergence: Te divergence is obtained b taing te dot product of te gradient operator and anoter vector, eg., specific discarge. Te result is a scalar. Divergence operator: It s basicall taing te partial derivative of eac component of a vector and ten summing te result. is te divergence of vector f " c b a c b a f # " % & " [ ] " is te divergence of te gradient. [ ] " " # It is basicall taing te second partial derivative. Eample. Given ten Written as and called te Laplacian Operator.,,,,,,,, "

8 If 0 ten Laplace s equation If " a constant or a function of space ten Poisson s equation Te divergence sows te presence of sources/sins and oter internal forcings. More on Tensors & % # " ij is te entr from te i t row, j t column ij gives flu in i t -direction for a unit gradient applied in te j t -direction is a smmetric tensor ij ji Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall 006 8

9 Etrapolation of Darc s Law to 3D for anisotropic, eterogeneous porous media q " or q " # q i " ij j Einsteinian notaton Etrapolation of Darc s Law to 3D " " q % " % q q # q & # & # " % % & # & q Z i.e., q " Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall 006 9

10 Etrapolation of Darc s Law to 3D q q q " " " Darc s Law in 3D ma q " min If our coordinate aes for our problem are aligned differentl tan te principal directions of, we must use a full tensor for Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall

11 Instead, we can rotate coordinate aes so te are parallel to te aes of te conductivit ellipsoid Ten & % # & 0 " % # 0 " all off-diagonal components equal ero q " # " Darc s Law in 3D Suppose is isotropic: tis implies off-diagonal terms in and tat are ero, i.e., flow is strictl in te direction of gradient # # # q " " # # # If we consider flow in onl one direction, sa : q " Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall

12 Darc s Law in 3D Review Vector Calculus Darc s Law in 3D Net time Conservation of Mass Continuit Field Equations # S s " q # t Aquifer wide: Groundwater Balance Cange in storage Recarge Pumping GW Discarge ETGW ± Underflow Forcings Wat about te water balance witin an individual Control Volume CV Pumping flow Pumping somewere an arbitrar CV witin te aquifer? Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall 006 1

13 Continuit Equation Assumptions 1. Medium is not deformable but undergoes elastic compression in te vertical direction Teragi assumption. Fluid is onl sligtl compressible densit is nearl constant From 1 and all compressibilit effects are described b specific storage, S s, wic is te onl form of storage 3. Darc s law applies Statement of Mass Conservation or Continuit Rate of cange of storage Volumetric flu in volumetric flu out No oter sources/sins or forcings Continuit Equation Volumetric flu into te CV: Volumetric flu into te left face Volumetric flu into te bac face Volumetric flu into te top face constant face q, in A q, in A q A, in qdd qdd q dd Total volumetric flu in q dd qdd q dd Note positive down in tis derivation Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall

14 Continuit Equation Volumetric flu out of te CV: Volumetric flu out of te rigt face At d constant face q q q,out A X q A "q " da q dd "q " ddd dd Were does tis come from? If q,out q,in, ten q must ave canged over d. Te "q / " grasps tis cange. Matematical justification: Talor Series Approimation Continuit Equation Volumetric flu out of te CV: Volumetric flu out of te rigt face Volumetric flu out of te front face Volumetric flu out of te bottom face At d constant face q q d dd q q d dd qz qz d dd Total volumetric flu out q q q q d dd q d dd q d dd Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall

15 Continuit Equation q dd qdd qdd q q q " [ q d dd q d dd q d dd] # " q Net volumetric flu into te CV: Net volumetric flu into te CV volumetric flu in volumetric flu out q q q " [ ddd ddd ddd] q q q " ddd ddd Continuit Equation Rate of cange of storage in te CV: Rate of cange in storage t V w But Vw Ss VT were V T d d d Rate of cange in storage S s V T t S s d d d t Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall

16 Continuit Equation Statement of Mass Conservation or Continuit Rate of cange of storage Volumetric flu in volumetric flu out S d d d s t q q q " ddd # " q ddd q Ss " q q S s # # t " q # or Ss " q 0 # t Continuit Equation Statement of Mass Conservation or Continuit Rate of cange of storage Volumetric flu in volumetric flu out S s " "t # % q q "# "# q "# "# # If draulic conductivit is isotropic: " S s # #t " S s # #t Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall

17 Verticall Integrated Aquifer Equation Aquifer equation, for omogeneous : Multipl b ticness, b to upscale Define T and S " S S t b T " " SSb t S t,,,t,,t Rewrite as " S T t a diffusion equation Anoter omolog to eat conduction or solute diffusion S T an upscaled draulic diffusivit Darc s Law in 3D Review Conservation of Mass Continuit Field Equations Net time Boundar Conditions Initial Conditions Proper Matematical Statement ` # S s " q # t Hdrolog Program, New Meico Tec, Prof. J. Wilson, Fall