Transient Ground Water FlowTransient EquationDerivation of Transient Ground Water Flow EquationFinite Difference Expression of Poisson EquationAnalytical solution of 1-D Transient Ground Water Flow EquationAssignmentReferencesTransient Ground Water FlowTransient EquationThe 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 EquationThe fundamental mass balance equation is      ALOPI( 0 )where:- I = inputs- P = production- O = outputs- L = losses- A = accumulationA 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   AOI( 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 thySxtVw( 0 )Considering a 3-D aquifer ‘cell’ of thickness b in map view, we havewhere 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:xhKq( 0 )where q is the flux and K is the hydraulic conductivity (LT-1). Incorporating Darcy’s law into our flow balance equation, we havethyxSxbyhKyhKybxhKxhKyyyxxx( 0 )-x-yqx|xqx|x+-xqy|yqy|y+-y-VwbThe product of the hydraulic conductivity K and the aquifer thickness b is known as the transmissivity T and has units of L2T-1. Both K and b are considered constant here. Incorporating T, dividing both sides by T-x-y and by -1thTSyyhyhxxhxhyyyxxx( 0 )We are taking the gradients of the gradients, and if we shrink -x and -y to differential size we havethTSyyhyhxxhxhyyyxxx( 0 )or equivalently,thTSyhxh2222( 0 )This is the 2-D version of the transient flow equation.Finite Difference Expression of Poisson EquationFrom our previous work we know that2222xhhhxhxxxxx( 0 )and2222yhhhyhyyyyy( 0 )We need a finite difference estimate for ∂h/∂t too. This is a little more complicated because now we have to include time.The simplest approach is to use the ‘known’ head from the previous time step. At the startof any numerical solution, the Initial Condition will specify the time 0 heads. So, we can approximate the derivative ∂h/∂t by -h/-t or thhththttyxtyx,,,,( 0 )Therefore, the 2-D transient flow equation ( 0 ) will have the following finite difference representation:thhTSyhhhxhhhttyxtyxttyyxttyxttyyxttyxxttyxttyxx ,,,,2,,,,,,2,,,,,,22( 0 )Note that we have chosen to compute the spatial gradients at the t – -t time level.We assume that -x = -y (i.e., we work on a simple isotropic grid). Then h|x,y,t--t appears twice of the left side of the equation and can be combined so that we can simplify tothhTSxhhhhhttyxtyxttyyxttyyxttyxxttyxttyxx ,,,,2,,,,,,,,,,4( 0 )Finally, we can multiply both sides by –(T-t)/S and rearrange to solve for the head at x,y.2,,,,,,,,,,,,,,4xhhhhhStThhttyyxttyyxttyxxttyxttyxxttyxtyx( 0 )The finite difference expression for the 1-D equation can be obtained by eliminating all terms at y +/- -y, and 2 of the h|x,y,t--t terms. We end up with 2,,,,,,,,,,2xhhhStThhttyxxttyxttyxxttyxtyx( 0 )C|x, txx +-x-C/-t|t--t/2Estimate heret--ttx --xC|x, t-1Notice that this is exactly the solution we had for the heat/diffusion equation in 1-D with diffusion coefficient D = T/S. Hence the same stability criterion applies: T-t/(S-x2) - ½.Analytical solution of 1-D Transient Ground Water Flow EquationCarslaw and Jaeger (1959) offer an analytical solution to the 1-D transient equation for uniform fixed initial temperature (initial head) and an instantaneous change in temperature (head) at distance L:    )/(1,00,100,00,100,0,222sin)cos(2SLtTnnttttttxeLxnnnhhLxhhhh( 0 )Sum the series until the change in the sum becomes insignificant.AssignmentModify your heat/diffusion code to solve the 1-D transient ground water flow equation. We wish to simulate temporal changes in the confined aquifer heads between two reservoirs (shown below). Assume the aquifer is 100 m long, has a T of 0.02 m2 min-1, and a storage coefficient of 0.002. The head is initially uniform at 16 m (i.e., the initial condition is h|x,0 = 16 m) and drops to 11 m at x = 100 at time 0 (i.e., the boundary conditions are h|0,t = 16 m and h|100,t = 11 m).Figure 1. Wang and Anderson, 1982. Introduction to Groundwater Modeling. W. H. Freeman and Company, San Francisco. 237 pp.Plot the