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Calculus and Groundwater Basics: Understanding Derivatives and Porosity Fundamentals

Explore derivative relationships in calculus, including numerical approximations and partial derivatives. Dive into the basics of porosity, pore volume calculations, and its relevance in ground water flow. Understand concepts such as total head, Darcy's Law, and Kozeny-Carman Equation in hydrogeology.

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Calculus and Groundwater Basics: Understanding Derivatives and Porosity Fundamentals

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  1. Midterm Review

  2. Calculus • Derivative relationships • d(sin x)/dx = cos x • d(cos x)/dx = -sin x

  3. Calculus • Approximate numerical derivatives • d(sin)/dx ~ [sin (x + Dx) – sin (x)]/ Dx

  4. Calculus • Partial derivatives • h(x,y) = x4 + y3 + xy • The partial derivative of h with respect to x at a y location y0 (i.e., ∂h/∂x|y=y0), • Treat any terms containing y only as constants • If these constants stand alone they drop out of the result • If the constants are in multiplicative terms involving x, they are retained as constants • Thus ∂h/ ∂x|y=y0 = 4x3 + y0

  5. Ground Water Basics • Porosity • Head • Hydraulic Conductivity • Transmissivity

  6. Porosity Basics • Porosity n (or f) • Volume of pores is also the total volume – the solids volume

  7. Porosity Basics • Can re-write that as: • Then incorporate: • Solid density: rs = Msolids/Vsolids • Bulk density: rb = Msolids/Vtotal • rb/rs = Vsolids/Vtotal

  8. Cubic Packings and Porosity Simple Cubic Body-Centered Cubic Face-Centered Cubic n = 0.48 n = 0. 26 n = 0.26 http://members.tripod.com/~EppE/images.htm

  9. FCC and BCC have same porosity • Bottom line for randomly packed beads: n ≈ 0.4 http://uwp.edu/~li/geol200-01/cryschem/ Smith et al. 1929, PR 34:1271-1274

  10. EffectivePorosity

  11. EffectivePorosity

  12. Porosity Basics • Volumetric water content (q) • Equals porosity for saturated system

  13. Ground Water Flow • Pressure and pressure head • Elevation head • Total head • Head gradient • Discharge • Darcy’s Law (hydraulic conductivity) • Kozeny-Carman Equation

  14. Multiple Choice:Water flows…? • Uphill • Downhill • Something else

  15. Pressure • Pressure is force per unit area • Newton: F = ma • Fforce (‘Newtons’ N or kg m s-2) • m mass (kg) • a acceleration (m s-2) • P = F/Area (Nm-2 or kg m s-2m-2 = kg s-2m-1 = Pa)

  16. Pressure and Pressure Head • Pressure relative to atmospheric, so P = 0 at water table • P = rghp • r density • g gravity • hpdepth

  17. P = 0 (= Patm) Pressure Head Pressure Head (increases with depth below surface) Elevation Head

  18. Elevation Head • Water wants to fall • Potential energy

  19. Elevation Head (increases with height above datum) Elevation Elevation Head Elevation datum Head

  20. Total Head • For our purposes: • Total head = Pressure head + Elevation head • Water flows down a total head gradient

  21. P = 0 (= Patm) Pressure Head Total Head (constant: hydrostatic equilibrium) Elevation Elevation Head Elevation datum Head

  22. Head Gradient • Change in head divided by distance in porous medium over which head change occurs • dh/dx [unitless]

  23. Discharge • Q (volume per time) Specific Discharge/Flux/Darcy Velocity • q (volume per time per unit area) • L3 T-1 L-2→ L T-1

  24. Darcy’s Law • Q = -K dh/dx A where K is the hydraulic conductivity and A is the cross-sectional flow area 1803 - 1858 www.ngwa.org/ ngwef/darcy.html

  25. Darcy’s Law • Q = K dh/dl A • Specific discharge or Darcy ‘velocity’: qx = -Kx∂h/∂x … q = -K gradh • Mean pore water velocity: v = q/ne

  26. Intrinsic Permeability L2 L T-1

  27. Kozeny-Carman Equation

  28. Transmissivity • T = Kb

  29. Potential/Potential Diagrams • Total potential = elevation potential + pressure potential • Pressure potential depends on depth below a free surface • Elevation potential depends on height relative to a reference (slope is 1)

  30. Darcy’s Law • Q = -K dh/dl A • Q, q • K, T

  31. Mass Balance/Conservation Equation • I = inputs • P = production • O = outputs • L = losses • A = accumulation

  32. qx|x Dz qx|x+Dx Dx Dy Derivation of 1-D Laplace Equation • Inflows - Outflows = 0 • (q|x - q|x+Dx)DyDz = 0 • q|x – (q|x +Dx dq/dx) = 0 • dq/dx = 0 (Continuity Equation) (Constitutive equation)

  33. General Analytical Solution of 1-D Laplace Equation

  34. Particular Analytical Solution of 1-D Laplace Equation (BVP) BCs: - Derivative (constant flux): e.g., dh/dx|0 = 0.01 - Constant head: e.g., h|100 = 10 m After 1st integration of Laplace Equation we have: After 2nd integration of Laplace Equation we have: Incorporate derivative, gives A. Incorporate constant head, gives B.

  35. Finite Difference Solution of 1-D Laplace Equation Need finite difference approximation for 2nd order derivative. Start with 1st order. Look the other direction and estimate at x – Dx/2:

  36. Finite Difference Solution of 1-D Laplace Equation (ctd) Combine 1st order derivative approximations to get 2nd order derivative approximation. Set equal to zero and solve for h:

  37. 2-D Finite Difference Approximation

  38. Matrix Notation/Solutions • Ax=b • A-1b=x

  39. Toth Problems • Governing Equation • Boundary Conditions

  40. Recognizing Boundary Conditions • Parallel: • Constant Head • Constant (non-zero) Flux • Perpendicular: No flow • Other: • Sloping constant head • Constant (non-zero) Flux

  41. Internal ‘Boundary’ Conditions • Constant head • Wells • Streams • Lakes • No flow • Flow barriers • Other

  42. Poisson Equation • Add/remove water from system so that inflow and outflow are different • R can be recharge, ET, well pumping, etc. • R can be a function of space and time • Units of R: L T-1

  43. Poisson Equation (qx|x+Dx - qx|x)Dyb -RDxDy = 0

  44. Dupuit Assumption • Flow is horizontal • Gradient = slope of water table • Equipotentials are vertical

  45. Dupuit Assumption (qx|x+Dx hx|x+Dx- qx|x hx|x)Dy - RDxDy = 0

  46. Capture Zones

  47. Water Balance and Model Types

  48. 2Dy Y 1Dy 0 0 1Dx 2Dx Effective outflow boundary Block-centered model Only the area inside the boundary (i.e. [(imax -1)Dx] [(jmax -1)Dy] in general) contributes water to what is measured at the effective outflow boundary. In our case this was 23000  11000, as we observed. For large imax and jmax, subtracting 1 makes little difference. X

  49. Effective outflow boundary Mesh-centered model 2Dy An alternative is to use a mesh-centered model. This will require an extra row and column of nodes and the constant heads will not be exactly on the boundary. Y 1Dy 0 0 1Dx 2Dx X

  50. Summary • In summary, there are two possibilities: • Block-centered and • Mesh-centered. • Block-centered makes good sense for constant head boundaries because they fall right on the nodes, but the water balance will miss part of the domain. • Mesh-centered seems right for constant flux boundaries and gives a more intuitive water balance, but requires an extra row and column of nodes. • The difference between these models becomes negligible as the number of nodes becomes large.

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