9. Transient two-dimensional flow

All two-dimensional solutions presented so far were for steady flow: The head and flow are a function of \(x\) and \(y\), but not of time \(t\). In this chapter, wells are considered with a discharge that varies through time. Both the head and flow are a function of \(x\), \(y\), and \(t\). The changes in head and flow caused by transient wells are discussed for a variety of settings.
      Continuity of flow for two-dimensional transient flow in the horizontal \(x\), \(y\) plane may be written as

(9.1)\[ \frac{\partial Q_x}{\partial x} + \frac{\partial Q_y}{\partial y} = - S\frac{\partial h}{\partial t} + N \]

where \(N\) is the areal recharge and \(S\) is the storage coefficient, which is equal to the phreatic storage \(S = S_p\) for unconfined flow and the product of the specific storage \(S_s\) and aquifer thickness \(H\) (\(S = S_sH\)) for confined flow (see Chapter 5). Note that this equation differs from the continuity of flow equation for one-dimensional transient flow (Eq. (5.5)) only by the additional term \(\partial Q_y / \partial y\).
      The components of the discharge vector are written in terms of the derivatives of the head as

(9.2)\[ Q_x = -T\frac{\partial h}{\partial x} \quad\quad Q_y = -T\frac{\partial h}{\partial y} \]

where the transmissivity is approximated as constant, also for unconfined flow (see Chapter 5). Substitution of Eq. ((9.2)) for \(Q_x\) and \(Q_y\) in Eq. ((9.1)) gives the differential equation for two-dimensional transient flow

(9.3)\[ \nabla^2 h = \frac{S}{T}\frac{\partial h}{\partial t} -\frac{N}{T} \]

The discharge potential is not very useful for transient flow (except for the alternative formulation for unconfined flow discussed in Section 5.5), and the stream function and complex potential are only valid for flow governed by the Laplace equation, so they are not used here. For the case that areal recharge is negligible (\(N = 0\)), Eq. ((9.3)) reduces to the two-dimensional diffusion equation

(9.4)\[ \nabla^2 h = \frac{S}{T}\frac{\partial h}{\partial t} \]

If, in addition, the flow is radially symmetric, the differential equation becomes (compare Eq. ((6.8)))

(9.5)\[ \frac{\partial^2 h}{\partial r^2}+\frac{1}{r}\frac{\partial h}{\partial r}= \frac{S}{T}\frac{\partial h}{\partial t} \]

      Solutions to the two-dimensional diffusion equation are difficult to derive mathematically, even for many cases of radially symmetric flow. Mathematical solution techniques to obtain exact solutions include similarity solutions, separation of variables, and a variety of transform techniques (Laplace, Hankel, and Fourier). In the following, several solutions for transient flow to wells are presented rather than derived. The solutions are discussed, programmed, and visualized as in the other chapters. The mathematical correctness of the presented solutions may be verified through differentiation of the solution and substitution into the differential equation (either analytically or numerically), and by verifying that the boundary conditions and initial conditions are met.
      All solutions presented in this chapter are for the head change caused by a transient well, which allows for the superposition of transient solutions in time, and for the superposition of steady solutions and transient solutions. Transient flow solutions are presented for wells in confined, unconfined, semi-confined, and two-aquifer systems, including the estimation of aquifer parameters from pumping tests. The last two sections concern the effect of wellbore storage and skin effect, and transient pumping in a two-aquifer system. These problems are solved in the Laplace domain, and the solution in the physical domain is obtained through a numerical back-transformation.