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Transient Ground Water Flow Transient Equation The Transient equation is an extension of the Poisson and Laplace equations we considered before and allows accumulation A in IPOLA terms Not surprisingly the equation is a great deal like the heat diffusion equation we worked with previously and the solution methods are exactly the same We ll focus on an explicit solution Here we ll consider only 2 D flow but the approach for 3 D flows would be the same Derivation of Transient Ground Water Flow Equation The fundamental mass balance equation is where I inputs P production O outputs L losses A accumulation I P O L A 0 A non steady state or transient ground water system has changes in water storage with time water accumulation or removal hence A 0 Let s ignore Production Loss So 0 simplifies to I O A 0 Recall that accumulation of chemical mass in the control volume we used to study diffusion was very easy chemical mass was simply the concentration times the volume The volume was assumed constant So any change in concentration was immediately reflected as a change in stored chemical mass In a similar way the storage of water in a confined aquifer depends on the head Vw x y S h where S the Storage Coefficient is the proportionality factor When the head changes from 10 to 5 h ht ht 1 5 10 5 and the volume Vw is released from storage As an instantaneous rate of accumulation we have Vw h x yS t t 0 Considering a 3 D aquifer cell of thickness b in map view we have qy y y qx x y qx x x Vw x b qy y where qx x indicates the flux L3L 2T 1 or LT 1 in the x direction at the point x In order to satisfy 0 we must have qx x x qx x yb qy y y qy y xb S x y h t that is the flux into each wall times the area over which it occurs is the flow into or out of that wall and all of these flows must sum to S x y h t Ground water flows in response to head gradients e g h x 0 001 in accordance with Darcy s law q K h x 0 where q is the flux and K is the hydraulic conductivity LT 1 Incorporating Darcy



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