10. Steady two-dimensional flow in the vertical plane

So far in this book, solutions were derived based on the Dupuit approximation. This means that the resistance to flow in the vertical direction is neglected in an aquifer. As a result, the head does not vary in the vertical direction. The vertical component of flow inside an aquifer is computed from continuity of flow. This is a very good approximation for many groundwater problems where the flow is predominantly horizontal. But there are flow situations, of course, where vertical flow in an aquifer does play an important role. Such problems may include flow to partially penetrating streams or wells, flow in aquifers with an irregular top or base, and flow toward the bottom of lakes or oceans. For such problems, the question arises whether the Dupuit approximation is still a good approximation, especially when the vertical hydraulic conductivity is (much) smaller than the horizontal hydraulic conductivity. In this chapter, solutions are presented for two-dimensional flow in a vertical cross section where the Dupuit approximation is not adopted. The range of applicability of Dupuit solutions is assessed by comparing several two-dimensional solutions to equivalent Dupuit solutions.
      Solutions for steady two-dimensional flow in the vertical plane tend to be derived with advanced mathematical techniques such as conformal mapping, the hodograph method, or separation of variables. The solutions presented in this chapter are all written in terms of a complex specific discharge potential, which is introduced in the following. All solutions are presented rather than derived.
      Two dimensional flow in the vertical plane is formulated in terms of the specific discharge vector with components \(q_x\) in the horizontal direction and \(q_z\) in the vertical direction. Continuity for steady flow in the vertical \(x\), \(z\) plane is (see Eq. (1.28))

(10.1)\[ \frac{\partial q_x}{\partial x} + \frac{\partial q_z}{\partial z} = 0 \]

where the \(z\)-axis points vertically upward. The components of the specific discharge vector for an aquifer with isotropic and homogeneous hydraulic conductivity \(k\) may be written as

(10.2)\[ q_x = -k\frac{\partial h}{\partial x} = -\frac{\partial \phi}{\partial x} {\hskip 2em} q_z = -k\frac{\partial h}{\partial z} = -\frac{\partial \phi}{\partial z} \]

where

(10.3)\[ \phi=kh \]

is the specific discharge potential. The dimensions of the specific discharge potential \(\phi\) are \(L^2/T\), which is in contrast to the discharge potential \(\Phi\) used in previous chapters, which has dimensions \(L^3/T\). The negative gradient of the specific discharge potential \(\phi\) results in the specific discharge vector \(\vec{q}\), while the negative gradient of the discharge potential \(\Phi\) results in the discharge vector \(\vec{Q}\). Substitution of the components of the specific discharge vector (Eq. (10.2)) into the continuity equation (Eq. (10.1)) gives Laplace’s equation.

(10.4)\[ \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial z^2} = 0 \]

For two-dimensional steady flow in the vertical plane governed by Laplace’s equation, there is also a stream function, which is written as \(\psi\) [L\(^2\)/T], as was done for flow in a vertical cross-section in Chapters 1-4. The specific discharge potential and stream function are combined in a complex potential

(10.5)\[ \omega = \phi + i\psi \]

The complex specific discharge may be obtained from \(\omega\) as

(10.6)\[ q_x - iq_z = -\frac{\text{d}\omega}{\text{d}\zeta} \]

where \(\zeta=x + iz\) is the complex coordinate in a vertical cross-section.