Lecture9

Lecture9 - SIMPLIFIED SOIL MOISTURE DYNAMICS Andres...

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SIMPLIFIED SOIL MOISTURE DYNAMICS Andres Tremante Florida International University Mechanical Engineering Department
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Project Goals Soil Dynamics Florida International University 1/26 The goal of this project is to study soil moisture dynamics by means of a simplified version of the Richards equation The project will consist in the following cases: method of columns which breaks the Richards equation into a set of ordinary differential equations Finite difference approach in both explicit and implicit forms. Solve the problem using a computational fluid dynamics package.
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Florida International University 2/26 Soil Dynamics Governing Equations = 1 Z H k Z t S b h h e S S S H s H = 1 ) ( Simplified Richards Equation The pressure head and saturation Nomenclature H = Pressure head in water H e = Entry pressure head Z = Vertical height S = Local Saturation K = Unsaturated Hydrologic conductivity K sat = Saturated hydrologic conductivity S h = Hygroscopic saturation Q = Volumetric flow rate d = Saturation numerical decay Ke = Conductivity decay b = Retention curve parameter T = Time The partial derivative of S with respect to Z was descritized in time using a forward difference scheme. Eq (1) Eq (2) t S S t S t i t i = + 1 Eq (3)
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Florida International University 3/26 Kez sat e K K Z K = min ) ( Hydraulic Conductivity Local Saturation dz ae c z S = ) ( ) 0 ( 1 , 1 S a c = = Local saturation as a function of vertical depth at different values of numerical decay Simplified Parameters Local saturation as a function of vertical depth at different values of numerical decay Soil Dynamics
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Florida International University 4/26 Soil Dynamics Numerical Model To solve the mathematical model 4 cases were considered: Case 1: Assumes that K and the descritized form of Eq.(3) are constant. These simplifications make it possible to solve Eq. (1) by means of a set of ordinary differential equations. Case 2: is the same as case 1, but this time K is considered dependent on Z. Case 3 : solves the Richards equation using a finite volume scheme. To solve cases 1-2, a single step first degree Runge-Kutta ODE solver was used. A constant time step was used for each time iteration. Case 4: consists in solving the problem with a commercial C.F.D software
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Numerical Model: Case 1 Soil Dynamics Florida International University 5/26 Constant value of hydraulic conductivity 2 2 1 dZ H d K t S S t i t i = + ( ) ( ) + = = t i h h e S S H S H tK dZ dY Y dZ dH 1 1 + = 1 dz dH k Q Set of 1 st order ODEs Were Y would represent the first derivative of H with respect to Z. The initial condition for H was found using Eq.(2), and for the initial condition for Y Eq.(5) Eq. (5) Table 1: Case 1 parameter values Parameter Value Unit H e 3.0 cm S h 0.02 S(0) 0.2 K 50 cm/day ∆t 0.02 days b 1.0 d 1.2 The numerical values for the model parameters used for solving this case are giving bellow in Table 1 Eq. (4)
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Lecture9 - SIMPLIFIED SOIL MOISTURE DYNAMICS Andres...

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