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FLOW THROUGH POROUS MEDIA

z. D z. x. D y. D x. y. FLOW THROUGH POROUS MEDIA. DERIVATION OF RICHARD’S EQUATION IN RECTANGULAR COORDINATES. The general continuity equation is: q = a v where q is the flow rate, volume/time (L 3 /T) a is the cross-section area perpendicular to the flow, (L 2 )

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FLOW THROUGH POROUS MEDIA

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  1. z Dz x Dy Dx y FLOW THROUGH POROUS MEDIA Porous Media Flow

  2. DERIVATION OF RICHARD’S EQUATION IN RECTANGULAR COORDINATES The general continuity equation is: q = a v where q is the flow rate, volume/time (L3/T) a is the cross-section area perpendicular to the flow, (L2) v is the flow velocity, length/time (L/T) Porous Media Flow

  3. Flow in the x-direction Porous Media Flow

  4. Flow in the y-direction Porous Media Flow

  5. Flow in the z-direction Porous Media Flow

  6. From Continuity of mass Where Q is the volumetric water content and t is time. Porous Media Flow

  7. From Continuity of mass Porous Media Flow

  8. By canceling out terms Porous Media Flow

  9. Applying the Darcy Law to each velocity term: Porous Media Flow

  10. FLOW THROUGH POROUS MEDIA Porous Media Flow

  11. FLOW THROUGH POROUS MEDIA In unsaturated soil the total potential can be estimated as the sum of the matric potential and the gravity potential: Since the gravity potential only acts in the vertical, or z-direction, the total potential, h, can be replaced by the matric potential, Y, in all terms except the one involving z: Porous Media Flow

  12. FLOW THROUGH POROUS MEDIA Porous Media Flow

  13. FLOW THROUGH POROUS MEDIA This equation is known as the Richard's Equation. When only the terms involving z on the left are used, this equation can be used to simulate the vertical infiltration of water into the soil profile. Porous Media Flow

  14. FLOW THROUGH POROUS MEDIA For saturated flow h will be the total head, F and there can be no change in moisture content with time. Porous Media Flow

  15. FLOW THROUGH POROUS MEDIA For a homogeneous, isotropic soil Kx = Ky = Kz = K Porous Media Flow

  16. FLOW THROUGH POROUS MEDIA Since K is not 0, the term inside the bracket must be 0 Porous Media Flow

  17. FLOW THROUGH POROUS MEDIA This is called the LaPlace Equation for Saturated Flow in a homogeneous soil. Porous Media Flow

  18. Porous Media Transport of Chemicals • DIFFUSION: transport from points or higher concentration to points of lower concentration. • Molecular Diffusion is due to the random movement of molecules • Turbulent Diffusion is due to the random movement of the fluid carrier. Also called dispersion. Porous Media Flow

  19. Molecular Diffusion • Many molecules move from High to Low • Few molecules move from Low to High • Result is decrease in high concentration and increase in low concentration High Concentration Low Concentration Porous Media Flow

  20. HYDRODYNAMIC DISPERSION DUE TO UNEVEN FLOW VELOCITY WITHIN A PORE Porous Media Flow

  21. HYDRODYNAMIC DISPERSION DUE TO UNEVEN VELOCITIES BETWEEN PORES Porous Media Flow

  22. HYDRODYNAMIC DISPERSION DUE TO VARYING FLOW PATHS Porous Media Flow

  23. FICKS LAW OF DIFFUSION • D = Diffusion Coefficient, L2 / T • C = Chemical Concentration , M / L3 • qx = Rate of mass transport in the x-direction. M/ L2T Porous Media Flow

  24. FICKS LAW OF DIFFUSIONCOMBINED WITH THE CONTINUITY Dx A Volume = ADx Porous Media Flow

  25. FICKS LAW OF DIFFUSION • Considering one-dimensional flow in the x-direction, for continuity: Porous Media Flow

  26. FICKS LAW OF DIFFUSIONThe Combined Diffusion Equation Porous Media Flow

  27. ADVECTION • Chemical Transport due to bulk movement of the fluid. • The fastest form of chemical transport in porous media. • Concentration decreases in the direction of fluid movement. Porous Media Flow

  28. The Combined Advection/ Dispersion Equation where s represents all the source and sink terms that occur in the real environment. Porous Media Flow

  29. The Combined Advection/ Dispersion EquationAssumptions • one dimensional flow • uniform flow velocity in the column • constant moisture content • linear, instantaneous, reversible adsorption Porous Media Flow

  30. Chemical Breakthrough Co A Qs L C/Co = relative concentration C Porous Media Flow

  31. Plug Flow Q Porous Media Flow

  32. Porous Media Flow

  33. Flow with Advection Q Porous Media Flow

  34. Porous Media Flow

  35. Porous Media Flow

  36. Sorption Retardation Coefficient R = number of pore volumes @ C/C0=0.5 Porous Media Flow

  37. The Combined Advection/ Dispersion Equation considering adsorption: Porous Media Flow

  38. Porous Media Flow

  39. Porous Media Flow

  40. Porous Media Flow

  41. Combining Adsorption Advection-Dispersion Degradation and Source-Sinks Porous Media Flow

  42. Travel Time of a Chemical Through Porous Media • Continuity Equation: Q = AV • Q is the flow rate (Volume / Time) • A is the total media flow cross-section area. • V is the average flow velocity through the media. • Q is the water fraction by volume in the media. Porous Media Flow

  43. Travel Time of a Chemical Through Porous Media Q=AV V=Q/A V A Porous Media Flow

  44. Travel Time of a Chemical Through Porous Media Q=AaVa Va=Q/Aa Aa=AQ Va = V/Q Va Aa Porous Media Flow

  45. Velocity Through Porous Media Flow Rate = 1 CMS Total Area = 2 M2 V=Q/A=1/2= 0.5MPS Q = 1/2 = 0.5 Open Area = 1 M2 Va=V/Q = 0.5/0.5 = 1 MPS Porous Media Flow

  46. Travel Time of a Chemical Through Porous Media • Va is the actual flow velocity in the pores of the media. • Va = V / Q • Mw = CwQ = Mass of chemical in the flowing water • MT = Cw(Q+KdBd) = The total mass of chemical in the media. Porous Media Flow

  47. Travel Time of a Chemical Through Porous Media • Z = the distance over which travel occurs • T = Travel time of the chemical over distance Z. • V = Z / T • Therefore T = Z / V • Actual Ta = Z / Va = ZQ / V Porous Media Flow

  48. Travel Time of a Chemical Through Porous Media • When adsorption of the chemical occurs within the media, the travel time must account for this by: • Tr = Z ( Q + Kd Bd ) / V • V may be the infiltration rate Porous Media Flow

  49. Factors Influencing Chemical Leaching (Rate of Water Movement) • Soil Properties • Infiltration Rate • Porosity • Soil Moisture • Physical Properties • Surface Roughness • Slope Porous Media Flow

  50. Factors Influencing Chemical Leaching (Rate of Water Movement) • Physical Properties • Rainfall amount and intensity • crop species and stage of growth • weather • temperature • solar radiation • wind velocity Porous Media Flow

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