PU

2D SEEPAGE AND FLOW NETS

  • Introduction to Multi-Dimensional Seepage

    • Previously, seepage analysis focused on 1D space, meaning water flow was either solely horizontal or vertical.

    • In real-world geotechnical problems, water often flows in two or three dimensions, moving at an angle, necessitating 2D analysis.

    • The aim is to characterize this multi-directional seepage and apply it to geotechnical design problems.

  • Concept of Flow Net Construction

    • Definition: A graphical representation used to solve 2D seepage problems.

    • Applications of Flow Nets:

      • Calculating flow rate (Q).

      • Determining stresses at any point within the soil domain.

      • Calculating water pressures acting on structures (e.g., uplift pressure under a dam, pressures on a sheet pile) for structural design.

      • Assessing the risk of piping erosion and developing mitigation strategies.

  • Learning Objectives

    • Explain the effects of seepage through soil deposits in a 2D space.

    • Use 2D seepage concepts to describe stresses within the soil.

  • Recommended Resource

    • Textbook: "Courageous Soil Mechanics: An Introduction to Geotechnical Engineering" by Holsenko Bags.

  • Understanding 2D Seepage Scenarios

    • Practical Examples: Water flow around dams, sheet piles, or temporary structures like cofferdams (e.g., for bridge pier construction).

    • Water always moves from an area of higher total head to an area of lower total head.

    • In 2D, velocity vectors are angled, unlike the purely horizontal or vertical vectors in 1D problems.

    • Simplification: Complex 3D problems are often simplified to 2D by assuming plane strain/strength conditions, where the problem extends infinitely in the third dimension, allowing analysis of a single 2D slice.

  • Derivation of Laplace's Equation for 2D Seepage

    • Approach: Analyze the flow through an infinitesimal soil element with dimensions dx (horizontal), dy (vertical), and a unit thickness (e.g., 1 ext{ m}) into the page.

    • Velocity Components: The discharge velocity vector (V) entering the element is split into horizontal (vx) and vertical (vy) components.

    • Continuity Equation (Conservation of Mass):

      • Assumptions: Water is incompressible, no volume change in the soil, and the soil is fully saturated.

      • The total flow rate entering the element must equal the total flow rate exiting the element (Q{in} = Q{out}).

      • Flow rate entering in x-direction: Q{x,in} = vx imes (dy imes 1).

      • Flow rate entering in y-direction: Q{y,in} = vy imes (dx imes 1).

      • As water flows through the element, velocities change. The outflow velocities include these changes:

        • vx + rac{ ext{d}vx}{ ext{d}x}dx in the x-direction.

        • vy + rac{ ext{d}vy}{ ext{d}y}dy in the y-direction.

      • Equating inflow and outflow rates and simplifying (assuming dx imes dy
        eq 0) leads to:
        rac{ ext{d}vx}{ ext{d}x} + rac{ ext{d}vy}{ ext{d}y} = 0

    • Incorporating Darcy's Law (v = k imes i):

      • The hydraulic gradient (i) is the change in total head (H) over a distance.

      • vx = kx imes rac{ ext{d}H}{ ext{d}x}.

      • vy = ky imes rac{ ext{d}H}{ ext{d}y}.

      • Substituting these into the continuity equation results in:
        rac{ ext{d}}{ ext{d}x} (kx rac{ ext{d}H}{ ext{d}x}) + rac{ ext{d}}{ ext{d}y} (ky rac{ ext{d}H}{ ext{d}y}) = 0

    • Assumption of Isotropic Conditions:

      • Assuming the coefficient of permeability is the same in all directions: kx = ky = k.

      • Since k
        eq 0, it can be factored out, leading to Laplace's Equation:
        rac{ ext{d}^2H}{ ext{d}x^2} + rac{ ext{d}^2H}{ ext{d}y^2} = 0

      • This equation is fundamental in various fields, including heat transfer.

  • Solving Laplace's Equation

    • Purpose: The primary goal is to determine the total head (H) at any point (x, y) within the seepage domain. Once H is known, all other seepage parameters can be derived.

    • Complementary Functions: Laplace's equation is satisfied by two sets of functions that describe seepage:

      • Potential Functions (Equipotential Lines): Curves along which the total head (H) is constant.

      • Flow Functions (Flow Lines): Curves representing the actual paths of water flow.

      • These two families of curves are orthogonal (intersect at right angles) at any point.

  • Methods for Solving Seepage Problems (Satisfying Laplace's Equation)

    • 1. Mathematical Closed-Form Solutions: Requires defining boundary and initial conditions to derive an exact mathematical expression. (Generally beyond the scope of this course).

    • 2. Physical Models: Using scaled models to simulate the problem and measure parameters. (Can be impractical).

    • 3. Numerical Methods: Techniques like Finite Element Method (FEM) or Finite Difference Method (FDM), which are computational approaches. (Introduced in later courses).

    • 4. Graphical Methods (Hand Sketching): Drawing a network of lines (flow net) based on specific rules. This is the primary method taught for this topic.

      • While hand sketching can be an