CE 3354 Engineering Hydrology Lecture 22: Groundwater Hydrology Concepts – Part 2

Outline Direct Application of Darcy’s Law Steady flow solutions Rectilinear flow Flow to wells

Direct Application of Darcy’s Law A groundwater map is a topographic representation of the 3-D piezometric or water table surface. Darcy’s law implies that flow should be perpendicular to the lines of constant head (or potential) unless the medium is anisotropic (permeability is direction dependent).

Direct Application of Darcy’s Law Flowlines are thus usually plotted perpendicular to the water level distribution. Flowlines diverge away from a recharge area or source of water to the subsurface, Flowlines converge towards a discharge area. The concept is similar to water flowing downhill in a surface water system. A groundwater map is a topographic representation of the 3-D piezometric or water table surface. Darcy’s law states that flow should be perpendicular to the lines of constant head (or potential), unless the medium is anisotropic (permeability is direction dependent)1. Flowlines are thus usually plotted perpendicular to the water level distribution. Flowlines diverge away from a recharge area or source of water to the subsurface, and converge towards a discharge area. The concept is mapped similar to water flowing downhill in a surface water system. Figure 14 illustrates these concepts. Care must be used to be sure the maps are interpreted as water elevations and not depth to water. Depth to water is what is typically measured, while water surface elevation is what applies in analysis.

Direct Application of Darcy’s Law A groundwater map is a topographic representation of the 3-D piezometric or water table surface. Darcy’s law states that flow should be perpendicular to the lines of constant head (or potential), unless the medium is anisotropic (permeability is direction dependent)1. Flowlines are thus usually plotted perpendicular to the water level distribution. Flowlines diverge away from a recharge area or source of water to the subsurface, and converge towards a discharge area. The concept is mapped similar to water flowing downhill in a surface water system. Figure 14 illustrates these concepts. Care must be used to be sure the maps are interpreted as water elevations and not depth to water. Depth to water is what is typically measured, while water surface elevation is what applies in analysis.

Direct Application of Darcy’s Law The ideas of Darcys law extends to three-dimensional flow fields. In many practical cases 1-D concepts are valid and useful. For example, Figure 15 is a sketch of a section of aquifer. For the flow system in Figure 15 the flow is given by Equation 20 , a one-dimensional concept. Notice that the area incorporated the height of the water table directly - this is called the Dupuit assumption and is typically how flow in unconfined aquifers is modeled.

Direct Application of Darcy’s Law Another example using a plan-view groundwater map is that the flow between any two flow lines is determined by the head change along the path of flow, the width between the flow lines, the thickness of the aquifer and the hydraulic conductivity The ideas of Darcys law extends to three-dimensional flow fields. In many practical cases 1-D concepts are valid and useful. For example, Figure 15 is a sketch of a section of aquifer. For the flow system in Figure 15 the flow is given by Equation 20 , a one-dimensional concept. Notice that the area incorporated the height of the water table directly - this is called the Dupuit assumption and is typically how flow in unconfined aquifers is modeled.

Confined-Rectilinear Darcy’s Law Rearrange into differential equation Integrate to recover the equation of head Consider a confined aquifer of constant thickness, b that is fully penetrated by a stream and flow occurs from the stream to the aquifer, as depicted in the sketch in Figure 18. The water level in the stream is at elevation ho above the horizontal datum. The stream aquifer interface is the origin (x = 0); assume system is in equilibrium so that no changes occur with time. Also suppose that along a reach of stream of lengthw, the total rate of loss from the stream is 2Q (volume/time) and half this flow (Q) enters the part of the aquifer to the right of the origin (as in the sketch). This flow then moves away from the stream in a steady flow along the x – direction. The resulting head distribution is indicated by the piezometric surface. This surface actually traces the static water levels in wells or pipes tapping the aquifer at various points. Darcys law for this situation is given as Equation 23. The head distribution is described by one of the solutions to the differential equation. This solution must satisfy the differential equation and the boundary condition h = h0 at x = 0. Then verify that it actually satisfies the differential equation (i.e. substitute and differentiate). The boundary condition simply states that h must have a certain value along a boundary of the problem. The differential equation alone is insufficient to define h in terms of x. All it specifies is that the graph of h versus x will be a line with slope equal to Q/Kbw. There are an infinite number of such lines; however the information supplied by the boundary condition selects a single line from this infinitude by supplying the intercept. This information provides the reference value from which changes in head indicated by the differential equation may be measured. This process of (1) differentiation to establish that a given solution is indeed a solution and (2) application of a boundary condition to determine the particular solution is the essence of mathematical analysis of groundwater flow problems.

Unconfined – Rectilinear Darcy’s Law Rearrange into differential equation Recall some calculus The upper limit of flow is the water table itself the working definition of an unconfined aquifer. Again assume uniform flow away from the stream. The datum is the bottom of the aquifer and vertical components of flow are assumed to be negligible. This last assumption is never completely satisfied because movement cannot lateral near the free surface because of the slope of the free surface itself. Typically the vertical component is very small compared to the lateral component and can be neglected. In steep terrain neglecting vertical components is not justified near sources or sinks to the aquifer (streams, ponds, springs, etc.). This assumption (neglecting vertical components) is called the Dupuit hypothesis. An important difference in this situation as compared to the confined flow problem is that the cross sectional area of flow decreases along the flow path. As before, suppose that along a reach of stream of length w, the total rate of loss from the stream is 2Q (volume/time) and half this flow (Q) enters the part of the aquifer to the right of the origin (as in the sketch). This flow then moves away from the stream in a steady flow along the x-direction. The resulting head distribution is indicated by the water table (piezometric) surface. This surface actually traces the static water levels in wells or pipes tapping the aquifer at various points. Darcys law for this situation is Rearrangement into a typical differential equation is h @h @x = − Q K w (27) where K is the hydraulic conductivity. The head distribution is described by one of the solutions to the differential equation. This solution must satisfy the differential equation and the boundary condition h = h0 at x = 0. This differential equation is slightly different but recall from calculus @h2 = 2h (28) Thus multiplication of both sides by the constant 2 produces an equation that is linear in the square of head.

Confined – Cylindrical Now consider a circular, confined aquifer with a well in the center

Confined – Cylindrical Darcy’s Law Differential Equation Integrate Calculus

Unconfined – Cylindrical Unconfined aquifer on circular island

Unconfined – Cylindrical Unconfined aquifer on circular island

Unconfined – Cylindrical Unconfined aquifer on circular island

Transmissivity Transmissivity is the term that refers to the amount of water that will flow through an entire thickness of aquifer The term Kb is the transmissivity it represents the discharge per unit width under unit gradient through an aquifer The usual symbol is T

Flow Between Two Ditches

Flow Between Two Ditches

Flow Between Two Ditches

Flow Between Two Ditches

Different material properties Use hydraulics principles (head loss) to find average K (or T)

Different material properties Use hydraulics principles (head loss) to find average K (or T)

Summary Porosity, Specific Yield Storage Darcy’s Law – Hydraulic Conductivity Gradients and head Some steady flow solutions – use geometry, Darcy’s law, and calculus

Next Time Transient (time-varying) solutions Superposition of solutions Well hydraulics