A Comparative Examination of Seepage Analysis Techniques

(5.30) and to solve for h(x, y) in shallow, horizontal flow fields by analog or numerical simulation. It is possible to set up steady-state boundary-value problems based on Eq. In other words, h2 rather than h must satisfy Laplace’s equation. This paradoxical situation identifies Dupuit-Forchheimer theory for what it is, an empirical approximation to the actual flow field. Figure 5.14(c) shows the equipotential net for the same problem as in Figure 5.14(a) but with the Dupuit assumptions in effect. We will encounter seepage faces in a practical sense when we examine hillslope hydrology (Section 6.5) and when we consider seepage through earth dams (Section 10.2).

• However, it may be referred to as a potentiometric map, water table map or a flow net.
• If we know the hydraulic conductivity K for the material in a homogenous, isotropic region of flow, it is possible to calculate the discharge through the system from a flow net.
• For electrical flow through an individual resistor, the I in Eq.
• If discharge quantities or flow velocities are required, it is often easiest to make these calculations in the transformed section.
• We have therefore produced N linear, algebraic equations in N unknowns.
• Among the various methods of flow net construction, the most convenient method is the Graphical method.

Flow nets – Flow through sand dams – Non-restricted flow

It is also the least expensive method. Determination of the seepage force. Determination of the seepage discharge. Explanation of paradoxes in Dupuit-Forchheimer seepage theory.

3 Drawing a Flow Net for Flow Beneath an Impermeable Dam

• Figure 5.6 is a qualitatively sketched flow net for the dam seepage problem first introduced in Figure 5.3, but with a foundation rock that is now layered.
• Let us first consider a region of flow that is homogenous, isotropic, and fully saturated.
• If the complex flow net is desired, a transformed section is in order, but if flow directions at specific points are all that is required, there is a graphical construction that can be useful.
• Flow lines diverge on the upgradient side of the bedrock island in the middle of the aquifer and converge on the down gradient side.
• Boundary conditions are discussed in another Groundwater Project book (Woessner and Poeter, 2020).
• This equation is one of the most commonly occurring partial differential equations in mathematical physics.
• Figure 5.3 is a flow net that displays the seepage beneath a dam through a foundation rock bounded at depth by an impermeable boundary.

Potentiometric maps represent flow in the horizontal plain and they are often used to represent conditions in a single hydrogeologic unit or aquifer. Conceptually, the discharge through flow tubes is constant under steady state conditions. However, it may be referred to as a potentiometric map, water table map or a flow net. Hydrogeologic investigations have specific goals defined to answer questions related to applied groundwater projects or to meet research objectives. The problem of changing water conditions is omnipresent in geotechnical engineering.

In large regional studies, where head changes in the down gradient direction are large, transient head errors may not impact interpretations of flow directions and fluxes because changes are small (a few centimeters) compared to the spatial variation in heads (meters). Often, well hydrographs that reflect groundwater level changes over time are recorded during the data collection period and can be referred to for data adjustments or error analyses. The data sets are then represented as a snapshot in time when conditions are constant (e.g., March 15, 2020; or March 2019 to May 2019, average water levels). The exception to flow being parallel to the aquifer boundaries in confined aquifers occurs in recharge and discharge areas where vertical gradients are present as is the case in unconfined systems (Figure 79). Cross-sectional representations would use all of the data to show vertical components of flow (Figure 78). Flow within an aquifer or between aquifers can be represented by the construction of vertical cross sections and/or multiple potentiometric maps.

Then, the equipotential lines are drawn such that they cut or intersect the flow lines at right angles. The space enclosed by the adjacent equipotential lines and the flow lines must be curvilinear squares. It is curvilinear in nature and formed by the combination of the flow lines and the equipotential lines. Saturated-unsaturated flow nets are required to explain perched water tables (Figures 2.15 and Figure 6.11), and to understand the hydrogeological regime on a hillslope as it pertains to streamflow generation (Section 6.5). The flowlines and equipotential lines form a continuous net over the full saturated-unsaturated region. First there is the hydraulic-head pattern, h(x, z), that allows construction of the equipotential net (the dashed lines on Figure 5.13).

The problem in preparing a flow net for such cases lies in the fact that the position of the exit point that separates the two boundary conditions on the outflow boundary is not known a priori. Above E, along the line AE, the unsaturated pressure heads, ψ, are less than atmospheric, so outflow to the atmosphere is impossible. The water table EF intersects the outflow boundary AD at the exit point E. If there is no source of water at the surface, AB will also act like an impermeable boundary. The qualitative flow net in Figure 5.13 has been developed for a soil whose unsaturated characteristic curves are those shown on the inset graphs.

If cross sections are constructed at some angle other than parallel to groundwater flow, they will not represent vertical flow conditions along the flow path even though a vertical distribution of head can be plotted on the section (e.g., the W-NE section of Figure 80c and d). In confined systems, the aquifer is constrained by confining beds and has substantially higher hydraulic conductivity than confining beds, so near-horizontal flow occurs in much of the system (vertical equipotential lines) when the confined aquifer is a horizontal layer. Interpretation of the mapped gradients and behavior of groundwater flow can shed light on the aquifer conditions including changes in hydraulic conductivity, aquifer thickness, and can be used to identify the location of recharge and discharge areas. In flow nets summary, understanding and utilizing flow nets in geotechnical engineering enables engineers to effectively analyze groundwater movement through soils, assess potential issues related to seepage pressures, uplift forces, and design safer structures such as dams and retaining walls.

Accurate measurement of water flow has been for years difficult to perform, not to mention predictio… The difference in phreatic surface elevation is due to conservation of mass flow. For the saturated-unsaturated case, the phreatic surface intersects lower at the horizontal section of the drain.

9 Create and Investigate Topographically Driven Flow Systems

Flow lines illustrate the paths that water takes as it seeps through the soil, while equipotential lines connect points that have the same total hydraulic head or potential energy. Numerically generated flow nets are usually used to display flow patterns rather than to compute flow rates, because flow rates are calculated by numerically solving the groundwater flow equations. For example, in an aquifer with homogeneous and isotropic hydraulic conductivity, computer-generated equipotential lines cross flow lines at right angles. Groundwater professionals commonly use a groundwater model to compute hydraulic head, then later use a flow path tracking model (also known as a particle tracking model) to compute flow lines. Two types of boundary conditions are used in graphical construction of two-dimensional, steady-state flow nets. Graphical construction of a flow net solves the two-dimensional, steady-state groundwater equation in a homogeneous and isotropic material with defined boundary conditions.

Seepage Face, Exit Point, and Free Surface

Two common boundary conditions are (1) constant hydraulic head along the boundary and (2) no flow across the boundary. The change in width between the flow lines might reflect a change in hydraulic conductivity, cross-sectional area, and/or an increase or decrease of the discharge within the flow tube resulting from recharge or leakage to/from an adjacent formation (Figure 83). When parallel flow lines converge and diverge, again the relationships defined by Darcy’s law can be used to hypothesize possible subsurface conditions.

Figure 4 – A plan view of flow in a confined aquifer penetrated by a deep lake and pond and laterally constrained by bedrock. Figure 4 illustrates a plan view of a flow net between a lake and pond in an area constrained by bedrock. A flow net can also be constructed for two-dimensional flow in a plan view. A flow path tracking model enables one to draw a flow path starting from any location. An exception to these requirements may occur near the edge of the domain where a partial (or fractional) flow tube may be drawn.

The method we have called relaxation (after Shaw and Southwell, 1941) has several aliases. These are the same two steps that led to the development of Laplace’s equation in Section 2.11. In Appendix VI, we present a brief development along these lines. It is possible to begin with Laplace’s equation and proceed more mathematically toward the same result. The development of the finite-equations presented in this section was rather informal. Numerical simulation is almost always programmed for the digital computer, and computer programs are usually written in a generalized form so that only new data cards are required to handle vastly differing flow problems.

As an example of the application of this construction, one might compare the results of Figure 5.10 with the flowline/equipotential line intersections in the right-central portion of Figure 5.9(c). If the complex flow net is desired, a transformed section is in order, but if flow directions at specific points are all that is required, there is a graphical construction that can be useful. At the surface, this condition implies that the soil is just saturated. It will also expand the hydraulic conductivity ellipse into a circle of radius (the outer circle in Figure 5.7); and the fictitious, expanded region of flow will then act as if it were homogenous with conductivity Kx. The last two rules make it extremely difficult to draw accurate quantitative flow nets in complex heterogeneous systems. In aquifer-aquitard systems with permeability contrasts of 2 orders of magnitude or more, flowlines tend to become almost horizontal in the aquifers and almost vertical in the aquitards.

It is the simplest and the quickest method of flow net construction. The graphical method is the method in which the flow net is constructed by an intensive trial and error procedure. Among the various methods of flow net construction, the most convenient method is the Graphical method.

The most widespread use of electrical analog methods in groundwater hydrology is in the form of resistance-capacitance networks for the analysis of transient flow in aquifers. The electric analog Figure 5.11 (b) consists of a sheet of conductive paper cut in the same geometrical shape as the groundwater flow field. (5.18) and (5.19) reveals a mathematical and physical analogy between electrical flow and groundwater flow. If one attempts to draw the equipotential lines to complete the flow systems on the diagrams of Figure 5.5, it will soon become clear that it is not possible to construct squares in all formations. In homogeneous, isotropic media, the distribution of hydraulic head depends only on the configuration of the boundary conditions. Flowlines must meet the boundary at right angles, and adjacent equipotential lines must be parallel to the boundary.