MIT 1 020 - Derived Supply, Equilibrium, Groundwater Pumping

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MIT OpenCourseWare http://ocw.mit.edu 1.020 Ecology II: Engineering for Sustainability Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.1.020 Ecology II: Engineering for Sustainability Lectures 08_19 Derived Supply, Equilibrium, Groundwater Pumping Motivation/Objective Extend the derived demand model of Lectures 08_16 & 08_17 to include the cost of supplying water from a single groundwater well. Combine derived supply and demand to obtain an equilibrium solution. Approach 1. Derive an expression for the marginal cost of supplying water from a single groundwater well. Identify factors affecting pumping cost. 2. Add the supply curve calculation to the MATLAB program of Lectures 08_16 & 08_17 3. Identify the equilibrium solution (intersection of supply and demand curves) 4. Consider how problem inputs (aquifer transmissivity, energy cost, etc.) affect solution. Concepts and Definitions Needed: Pumping cost at supply well depends on flow rate and depth to water: QhztgptEpFgcost)( −Δ=Δ=ρ& ($ season-1) p = energy price ($ joule-1) E& = power required to pump water (watts), tΔ= length of irrigation season (sec season-1) =iQ Pumping rate at i (m3 sec-1) , =gzground surface elev. (m) =hgroundwater elev. (m) Pumping at the well affects . Derive using rate form of mass balance eq in radial coords: h)(QhMass rate pumped from well = mass rate flowing through soil toward well Mass balance: rqQπρρ2= (kg sec-1) drrdhTq)(== radial flow per unit width (m3 m-1sec-1) T =Soil transmissivity (m2 sec-1) Substitute q in mass balance to get differential eq. for h: rTQdrQdhπ2)(= Solve for h(Q): ⎥⎦⎤⎢⎣⎡+=00ln2)(rrTQhQhwπ wrr >>0=wr well radius (m) distance where =0r0)( hQh= = water level at well when Q = 0 Marginal cost of supplying water: []⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−−Δ=−Δ∂∂=∂∂=00ln)]([)(rrTQhztgpQQhztgpQQFQwggcostπρρμ ($ m-3) Include this expression in crop allocation optimization code from Lectures 08_16 & 08_17. Plot of )(Qμ vs Q gives derived supply (a curve). Intersection of supply )(Qμ and demand )(Qλdefines equilibrium solution [].,eqeqQλ. Crop Allocation Results with Groundwater Supply Cost Included Supply curve is linear in Q. Note dependence of equilibrium solution on problem inputs (price, transmissivity, unpumped depths to groundwater, etc.)


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MIT 1 020 - Derived Supply, Equilibrium, Groundwater Pumping

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