7.4  Steady State Equations Describing Confined and Unconfined Flow

William W. Woessner; Eileen P. Poeter

7.4 Steady State Equations Describing Confined and Unconfined Flow

Up to this point, the development of the flow equations has represented transient conditions. For steady-state flow, the change in storage with time is equal to zero because the head values and gradients are constant. Consequently, the left-hand side in any of Equations 67 through 76 is set to zero. For example, the steady state equation representing confined, three-dimensional, steady-state, anisotropic, heterogeneous conditions is represented by Equation 77.

$\displaystyle 0=\frac{\partial}{\partial x}\left(K_x\frac{\partial h}{\partial x}\right)+\frac{\partial}{\partial y}\left(K_y\frac{\partial h}{\partial y}\right)+\frac{\partial}{\partial z}\left(K_z\frac{\partial h}{\partial z}\right)$

(77)

Whereas the equation for confined three dimensional isotropic and homogeneous conditions, referred to as the Laplace equation is represented by Equation 78.

$\displaystyle 0=\frac{\partial^2h}{\partial x}+\frac{\partial^2h}{\partial y}+\frac{\partial^2h}{\partial z}$

(78)

The equation for two-dimensional flow describing steady state unconfined anisotropic, heterogeneous conditions is represented by Equation 79 and unconfined steady-state two-dimensional flow under isotropic and homogeneous conditions is presented in Equation 80. Equation 80 is the Laplace equation.

$\displaystyle 0=\frac{\partial}{\partial x}K_x\left(h\frac{\partial h}{\partial x}\right)+\frac{\partial}{\partial y}K_y\left(h\frac{\partial h}{\partial y}\right)$

(79)

$\displaystyle 0=\frac{\partial^2h^2}{\partial x^2}+\frac{\partial^2h^2}{\partial y^2}$

(80)

The solution to any of the Equations 67 through 80 would produce head values at any location in the groundwater flow system: for the three-dimensional equations h(x,y,z); two-dimensional, h(x,y) or h(x,z); and h(x), h(y) or h(z) for one-dimensional representations. Regardless of the equation form, or dimensionality, given information about the boundary conditions of the problem domain, the resulting head distributions can be used to determine hydraulic gradients, flow directions and fluxes, and with effective porosity values, groundwater velocities.

These governing equations can be modified to account for many other hydrologic conditions that would impact the mass flux of water and the resulting head terms. For example, the effect of adding a constant or variable source of water (e.g., surficial recharge or an injection well) or extracting water (pumping well) can be represented by adding additional terms to the right-hand side of the governing equation because they are an additional change in mass in the REV. Governing equations are used to represent the general conditions of a groundwater system under investigation. They are combined with values of hydraulic conductivities, aquifer geometries, boundary conditions, and source/sink terms to generate head distributions and water balances for the specific system under investigation. Section 7.5 provides a brief explanation of boundary conditions and presents an example of application of the groundwater flow equations to field settings.

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