By the end of this module, you will be able to:

• Understand porosity: Calculate and interpret porosity in different geological materials
• Apply Darcy’s Law: Use the fundamental equation to analyze groundwater flow
• Measure hydraulic properties: Distinguish between hydraulic head, conductivity, and permeability
• Analyze flow systems: Interpret flow patterns in heterogeneous and anisotropic media
• Use field methods: Design monitoring networks using wells and piezometers
• Create flow maps: Construct potentiometric surfaces and interpret hydrographs
• Solve practical problems: Apply groundwater principles to real-world scenarios

• This module introduces the principal laws that govern groundwater movement through porous media.
• We will explore:
  • Definitions and significance of porosity
  • Darcy’s Law: the foundational equation for flow in saturated zones.
  • Concepts of hydraulic head, conductivity, and velocity
• These principles form the basis for:
  • Understanding aquifer behavior
  • Designing wells and evaluating sustainability
  • Developing numerical models

• Porosity describes how much void space exists in a rock or soil—essentially, how much room is available to hold water.
• Groundwater resides in the pores of rocks and sediments.
• Total porosity n_T is defined as:n_T = \frac{V_v}{V_T} = \frac{V_T - V_s}{V_T}where:
  • V_v = volume of voids
  • V_s = volume of solids
  • V_T = total volume
• Typical values of n_T:
  • Sand and gravel: 0.25 – 0.35
  • Cemented sandstones: 0.05 – 0.15

• Porosity can also be calculated using bulk density:n_T = 1 - \frac{\rho_b}{\rho_s}where:
  • \rho_b = bulk density (mass/volume of dry sample)
  • \rho_s = particle density (mass/volume of solids)
• This approach is common in lab settings where volume and mass can be measured directly.

It is also useful to distinguish:

• Primary porosity: original voids formed during rock/soil formation (e.g., in sediments, vesicles in lava).
• Secondary porosity: formed later, e.g., via fractures, faults, or dissolution in carbonates.

• Primary porosity depends on:
  • Compaction: tighter packing = lower porosity
  • Grain shape and arrangement: more spherical = less compact = higher porosity
  • Sorting:
    • Well-sorted sediments (uniform size) → higher porosity
    • Poorly sorted (mixed sizes) → smaller grains fill the voids
• Porosity ranges:
  • 0 to >50% depending on material
  • Unlithified materials (e.g. sand) >> lithified ones (e.g. sandstone)

Good sorting and less compaction = more room for groundwater.

• For unlithified materials:
  • Smaller grain size → higher porosity (e.g. clays > sands)
  • Glacial till is variable (mixtures of sand/silt/clay + varying compaction)
• Sandstone:
  • Porosity reduced by cementation
• Carbonates (e.g., limestone, dolomite):
  • Diagenesis and reactivity make porosity highly variable
• Igneous rocks: low primary porosity
• Fractured rocks:
  • Have dual porosity: matrix + fractures
  • Fractures provide secondary porosity essential for flow

• Effective porosity: fraction of pore space that actually contributes to flow
  • Often < total porosity (Not all the pores are connected)
  • Especially important in fractured or poorly connected media
• Fractured media (e.g. shales, granite):
  • Matrix may store water, but fractures control flow
  • Fracture porosity $n_f$ for shale: ~0.05 or less
• Dual porosity systems:
  • Primary: matrix porosity
  • Secondary: fractures, dissolution features (e.g. caverns in limestone)
• Porosity must be paired with permeability (up next!) to understand flow.

• Water in the subsurface occupies pores in soil and rock.
• Unsaturated zone: pores contain both air and water
  • Water content fluctuates with infiltration and evapotranspiration
  • Plants use this water
• Saturated zone: pores are fully water-filled — this is groundwater
• The water table separates the two zones
• A small portion of rain infiltrates deep enough to reach the water table → recharge

Water in the ground isn’t just “there”—it’s dynamic and layered.

• After recharge, groundwater flows:
  • Downward from water table due to gravity
  • Sideways following pressure gradients, often toward streams
• This flow forms discharge zones (e.g. rivers, springs)
• Example: Rain → infiltration → percolation → river discharge
• Timescales vary:
  • Surface runoff = hours
  • Groundwater discharge = months to years
• Flow influenced by:
  • Water table shape
  • Conductivity of subsurface materials

• Henry Darcy (1800s) studied water filtration and developed an empirical law for groundwater flow.
• Darcy’s experiments showed that flow depends on energy differences:
  • Water moves from high to low gravitational energy
  • Like a drop on a slide — higher up means more potential energy
• Energy loss over distance gives rise to the hydraulic gradient:

• This principle forms the core of Darcy’s Law.

• Gradient = energy change per unit length of flow path
• In groundwater terms:
  • Energy is hydraulic head (gravitational + pressure)
  • Steeper gradient → faster potential flow
• Flow rate increases with:
  • Greater difference in head
  • Shorter flow distance
• Key analogy: sliding down a playground slide
  • Steeper slide → faster slide

The gradient controls the force pushing water through pores.

• Not all materials transmit water equally well.
• Hydraulic conductivity (K) quantifies this:
  • High K → gravel, coarse sand
  • Low K → clay, unfractured rock
• Units: length per time (e.g. m/day)
• K depends on:
  • Pore size and connectivity
  • Fluid properties (viscosity, density)

Even with the same gradient, some materials flow better than others.

• Darcy combined gradient (i) and material property (K) to describe flow:

  q = -K i
  • q: Darcy velocity (aka specific discharge), volumetric flow per area

• Alternate form using cross-sectional area A:

  Q = -K i A

• Negative sign: flow goes from high head to low head

• Valid for laminar flow in porous media

Darcy’s Law is foundational to all groundwater flow modeling.

• Darcy velocity (q) assumes flow spreads over the whole cross section.

• But in reality, water only flows through connected pores

• True or pore velocity (v) is:

  v = \frac{q}{n_e}

  where:

  • n_e: effective porosity (only connected pore space)
  • q: Darcy velocity

• Since n_e < 1, pore velocity is greater than Darcy velocity

• Rearranged Darcy’s Law:

  v = -\frac{K}{n_e} \cdot \frac{h_1 - h_2}{\Delta l}

This gives the actual speed water travels through aquifers.

• Hydraulic head is a measure of energy available to move groundwater

• It is expressed in units of length (e.g., m above sea level)

• In the field, head is measured with a standpipe or piezometer.

• Key components:

  • Elevation of top of casing (EL_{TOC})
  • Depth to water (DTW)

• The head is calculated by:

  h = EL_{TOC} - DTW

Measuring head tells us the direction and gradient of groundwater flow.

• Hydraulic head consists of:

  • Elevation head (z): height above datum
  • Pressure head (P/\rho_w g): height of water column exerting pressure
  • Velocity head: often negligible in groundwater

• General form (Bernoulli):

  h = z + \frac{P}{\rho_w g} + \frac{v^2}{2g}

• Simplified form (no velocity term):

  h = z + \frac{P}{\rho_w g}

• Units: meters (m), Pressure (Pa) can be converted from depth of water:

  P = \rho_w g (h - z)

• Hydraulic conductivity is a measure of how easily water flows through porous material

• Denoted by K, it appears in Darcy’s Law:

  q = -K \frac{dh}{dl}

• Units of K are velocity (e.g. m/day)

• ↑ K: sand, gravel, ↓ K: clay, shale

• K depends on:

  • Properties of the medium
  • Properties of the fluid (density, viscosity)

Hydraulic conductivity is widely used in field hydrogeology.

• Intrinsic permeability k characterizes the medium only:

  • Independent of fluid type
  • Units: \mathrm{m}^2, also expressed in darcy or cm²

• Darcy’s Law rewritten:

  q = -\frac{k \, \rho_w g}{\mu} \, \frac{dh}{dl}

• Conversion to hydraulic conductivity:

  K = \frac{k \, \rho_w g}{\mu}

• At 20°C and 1 atm: $ K = 9.77 \times 10^6 \cdot k$

  k = (1.023 ) \times 10^{-7} \, \mathrm{m \cdot sec} \cdot K

Use k for comparisons across different fluids.

• Used for coarse-grained materials (e.g. sands, gravels)
• A steady flow is established by maintaining a constant head difference across the sample
• Flow rate Q is measured at the outlet
• Hydraulic conductivity K is calculated as:K = \frac{Q L}{A h}where:
  • L = sample length
  • A = cross-sectional area
  • h = constant head difference

Works best when Q is large and easily measured.

• Preferred for fine-grained or less permeable materials (e.g. clays, silts)
• No continuous flow: water level in standpipe drops over time
• Measure time t_1 - t_0 for water head to fall from h_0 to h_1
• Hydraulic conductivity:K = 2.3 \, \frac{a L}{A (t_1 - t_0)} \log_{10} \left( \frac{h_0}{h_1} \right)where:
  • a = standpipe cross-section
  • A = sample cross-section

Falling-head is more accurate for slow flows and low K.

• In anisotropic materials, hydraulic conductivity depends on direction.

• Darcy’s Law becomes a vector-tensor equation:

  \mathbf{q} = - \mathbf{K} \nabla h

• In Cartesian form:

  \mathbf{q} = q_x \mathbf{i} + q_y \mathbf{j} + q_z \mathbf{k}\nabla h = \frac{\partial h}{\partial x} \mathbf{i} + \frac{\partial h}{\partial y} \mathbf{j} + \frac{\partial h}{\partial z} \mathbf{k}

• The hydraulic conductivity tensor is:

  \mathbf{K} = \begin{bmatrix} K_{xx} & K_{xy} & K_{xz} \\ K_{yx} & K_{yy} & K_{yz} \\ K_{zx} & K_{zy} & K_{zz} \end{bmatrix}

These terms describe how flow in one direction may be influenced by gradients in others.

• For anisotropic flow, each component of flow is influenced by multiple gradients:

• If the medium is aligned with coordinate axes, then off-diagonal terms vanish:

• This simplifies to:

This is common in layered sedimentary rocks where horizontal and vertical conductivities differ.

• Field and lab studies confirm many rocks are anisotropic
• Horizontal conductivity is often orders of magnitude greater than vertical
• This affects:
  • Recharge and discharge zones
  • Aquifer drawdown patterns
  • Contaminant transport direction

Below are typical values from core samples:

Table – Anisotropic Conductivity of Sedimentary Rocks

• A medium is homogeneous if permeability is constant within it.

• If permeability varies across space, it is heterogeneous.

• A common model: layered medium with different K_{hi}, K_{vi}, and thicknesses b_i

• For M horizontal layers, the equivalent horizontal conductivity is:

  K_h = \frac{ \sum_{i=1}^M b_i K_{hi} }{ \sum_{i=1}^M b_i }

• This is a thickness-weighted average.

• Horizontal conductivity dominates in flow parallel to bedding.

• In vertical direction, equivalent conductivity is given by:

  K_v = \frac{ \sum_{i=1}^M b_i }{ \sum_{i=1}^M \frac{b_i}{K_{vi}} }

• This is a harmonic average — low K_{vi} dominates vertical resistance.

• When groundwater crosses a boundary between layers, it refracts.

• Refraction law:

  \frac{K_1}{K_2} = \frac{\tan(\alpha_1)}{\tan(\alpha_2)}

• The angle of flow changes based on contrast in hydraulic conductivity across boundary.

• Hydraulic head can be measured with:
  1. Water wells
  2. Piezometers
  3. Water table observation wells
• These are often installed together at study sites
• Example system: deep aquifer beneath shallow, low-permeability unit
• Flow is:
  • Vertical in shallow layer
  • Horizontal in deeper aquifer

• Water wells:

  • Large-diameter casings
  • Long screens → integrate over large depth
  • Used in well-defined aquifers

• Measure average head where equipotential lines are vertical

• In horizontal aquifer flow, screen depth doesn’t affect head reading

• Piezometers:

  • Narrow diameter
  • Short screen → point-specific measurements
  • Ideal for contamination, dewatering, or vertical gradient studies
  • Often installed as nests at multiple depths

• Water well: h = 73 m (deep)
• Piezometers: show vertical gradient (e.g., 81 → 78 m)

Wells, piezometers, and equipotential lines

• Mapping the spatial distribution of hydraulic heads in an aquifer is a key method for evaluating aquifer conditions.
• A potentiometric surface map shows groundwater levels (hydraulic head) across an area.
• Important uses:
  • Identifying recharge and discharge areas
  • Detecting pumping impacts (cones of depression)
  • Predicting contaminant migration paths
• Meinzer (1923): defined potentiometric surface as the imaginary surface representing groundwater head in an aquifer.

• Three main steps:

  1. Plot points: map well/piezometer locations with measured head values
  2. Contour: draw equipotential lines connecting equal head values
  3. Flow arrows: add arrows perpendicular to equipotential lines (showing decreasing head)

• Key notes:

  • Map must be for one aquifer only
  • Flow in aquifers is assumed horizontal
  • Equipotential lines → vertical in cross-section, but projected to 2D on the map

• Example: Memphis Sand Aquifer (1995)
• Heavy pumping caused cones of depression
• Equipotential contours + flow arrows show water moving toward pumping centers
• Applications:
  • Tracking regional groundwater flow
  • Managing water supply stress
  • Identifying recovery trends after reduced pumping

• A hydrograph shows groundwater levels over time at a monitoring well
• Traditionally: manual measurements (monthly, seasonal)
• Now: automated, remote sensors (satellite uplinks, wireless)
• Hydrographs complement potentiometric maps:
  • Maps = snapshot at one time
  • Hydrographs = long-term temporal trends
• Useful for:
  • Detecting pumping impacts
  • Seasonal recharge/discharge cycles
  • Long-term aquifer depletion or recovery

• Vertical sections through aquifers showing:
  • Stratigraphy (layering, K contrasts)
  • Hydraulic head distributions
  • Flow lines and equipotentials
• Often constructed along mean flow direction
• Show how layering causes refraction of flow paths
• Vertical exaggeration helps visualize layering and flow

• We learned about the principles of groundwater flow, covering:
• Porosity and Physical Properties: void space in rocks and how it controls water storage
• Groundwater Occurrence: unsaturated vs saturated zones and the water table
• Darcy’s Law: the fundamental equation governing groundwater flow rates
• Hydraulic Properties: head, conductivity, and permeability measurements
• Heterogeneous Media: flow behavior in layered systems and flow refraction
• Field Investigation Methods: wells, piezometers, and mapping techniques
• Practical Applications: aquifer management and contaminant transport