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 ($v<em>x$) and vertical ($v</em>y$) 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 ($Q<em>{in} = Q</em>{out}$).

  • Flow rate entering in x-direction: $Q<em>{x,in} = v</em>x imes (dy imes 1)$.

  • Flow rate entering in y-direction: $Q<em>{y,in} = v</em>y imes (dx imes 1)$.

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

    • $v<em>x + rac{ ext{d}v</em>x}{ ext{d}x}dx$ in the x-direction.

    • $v<em>y + rac{ ext{d}v</em>y}{ ext{d}y}dy$ in the y-direction.

  • Equating inflow and outflow rates and simplifying (assuming $dx imes dy <br>eq 0$) leads to:
    $rac{ ext{d}v<em>x}{ ext{d}x} + rac{ ext{d}v</em>y}{ ext{d}y} = 0$

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

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

  • $v<em>x = k</em>x imes rac{ ext{d}H}{ ext{d}x}$.

  • $v<em>y = k</em>y imes rac{ ext{d}H}{ ext{d}y}$.

  • Substituting these into the continuity equation results in:
    $rac{ ext{d}}{ ext{d}x} (k<em>x rac{ ext{d}H}{ ext{d}x}) + rac{ ext{d}}{ ext{d}y} (k</em>y rac{ ext{d}H}{ ext{d}y}) = 0$

• Assumption of Isotropic Conditions:

  • Assuming the coefficient of permeability is the same in all directions: $k<em>x = k</em>y = k$.

  • Since $k <br>eq 0$, 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