Hydrology BC ENV 3025

# Darcy's law

## Introduction

• groundwater is the water in the saturated zone (Fig)
• recharge is the water entering the saturated zone
• 30% of freshwater on Earth trapped below the surface
• in many parts of the world, groundwater is the only source of fresh water
• in the US about 10% of the rainfall becomes groundwater eventually. This amount equals the annual use of water in the US, about 3 inch per year
• residence time = reservoir/flux = ~1000 m / 3 inch/year = 10,000 y! This is a very rough estimate.
• water may stay in the groundwater reservoir between several days and thousands of years. We will discuss tracer techniques that may be used to derive residence times later in the class
• management of catchment areas  requires understanding of groundwater flow
• many environmental issues involve groundwater
• WIPP site case study (A Tour of the WIPP Site)
• underground repository built for certain radioactive wastes
• not high-level, but they contain isotopes that remain radioactive for very long periods of time (tens of thousands of years)
• storage of waste in salt formations
• quantitative description of groundwater flow necessary to evaluate risk

## Conceptual model of groundwater flow

• the flow of water through a porous medium (Fig 6.1)
• water flows tortuous paths
• geometry of channels is very complex
• frictionles flow is totally meaningless!
• conceptual model of flow through a porous medium is flow through a bundle of very small (capillary) tubes of different diameters (Fig 6.2)
• the flow (Q) through a horizontal tube can be described as: Q = - π *D4/(128* µ )*dp/dx (Poiseuille's law)

• => size of the capillary tubes is important!

## Darcy's law

• what drives groundwater flow?
• water flows from high elevation to low elevation and from high pressure to low pressure, gradients in potential energy drive groundwater flow
• in groundwater flow, we cannot make the assumption that there is no friction, therefore the  head is not constant
• also u is so small that that term can be typically neglected  (example!)
• groundwater flows from high to low  head
• how do you measure the head or potential? => drill an observation well, the elevation of the water level in the well is a measure of the potential energy at the opening of the well
• in 1856, a French hydraulic engineer named Henry Darcy published an equation for flow through a porous medium that today bears his name (Fig. 6.3)
• Q = KA (h1-h2)/L or q = Q/A = -K dh/dl, h: hydraulic head, h = p/(ρg) + z
• thought experiment: hydraulic head distribution in a lake
• q = Q/A is the specific discharge [L/T], dh/dl is the hydraulic gradient
• K is the hydraulic conductivity [L/T]
• the law is very similar to Ohm's law for electrical curcuits I = 1/R * U (current = voltage divided by resistance)
• the original Darcy experiment yielded these data (Fig 6.4)
• the analogy between Darcy's law and Poiseulle's law suggests that K = (const*d2)*ρg/m
• the first term (const*d2) is k, the intrinsic permeability [L2], summarized the properties of the porous medium, while rg/m describe the fluid
• hydraulic conductivities and permeabilities vary over many orders of magnitude (Fig 6.5)

• Example: calculation of a typical hydraulic gradient of 1/100 in a salt formation with a hydraulic conductivity of 10-10 m s-1 will produce a specific discharge of 10-12 m s-1, or less than 1 mm per 30 years!
• specific discharge has the dimension of a velocity, but it is not the velocity at which the water flows in the porous medium, the water has to squeeze through the pores
•  tagged parcels that are averaged together, will appear to move through a porous medium at a rate that is faster than the specific discharge
• porosity n is the fraction of a porous material which is void space n = Vvoid/Vtotal
• the mean pore water velocity is then: v = q/n (Fig) (experiment)
• Darcy's law has been found to be invalid for high values of Reynolds number and at very low values of hydraulic gradient in some very low-permeability materials, such as clays.
• example :
• K= 10-5 m/s, h2-h1 = 100m, L = 10km, A = 1m2 > Q = 3.15 m3/y; the K value above is typical for a sandstone aquifer
• the actual flow velocity v may be calculated with the following formula: v=Q/(A*f)=q/n, n is the porosity, and q the specific discharge
• if the porosity n is 30%, the flow velocity in the example above is 10.5 m/y

## Water in natural formations

• an aquifer is a saturated geological formation that contains and transmits "significant" quantities of water under normal field conditions (=> gravel, sand, volcanic and igneous rocks, limestone) (Fig 6.6)
• an aquitard is a formation with relatively low permeability
• an aquiclude is a formation that may contain water but does not transmit significant quantities (clays and shales)
• an aquifuge is a foamtion that does not contain or transmit significant amounts of water
 formation contains water permeability aquifer Y high aquitard Y low aquiclude Y very low aquifuge N negligible
• confined and unconfined (water-table) aquifers
• an unconfined aquifer has a water table (water table aquifer)
• a confined aquifer does not have a water table. If you drill a well, water will rise (in the well) above the top of the aquifer
• perched groundwater is groundwater sitting on top of a poorly permeable layer with an unconfined aquifer underneath
• the height to which water rises in a well defines the piezometric or potentiometric surface
• geology of aquifers (show examples)
• unconsolidated sediments: loose granular deposit, particles are not cemented together (e.g.: Long Island)
• consolidated sediments, most important: sandstone, porosity varies depending on the degree of compaction (e.g. Zion, Bryce, and Grand Canyon National Parks)
• limestone: composed mainly of calcium carbonate, CO2 rich water dissolves limestone, e.g.: limestone caves, karst (e.g. Floridan aquifer)
• volcanic rock
• basalt lava, fractures (e.g.: Hawaii, Palisades)
• crystalline rocks: igneous and metamorphic rocks, e.g. Granite, have often very low porosity, flow through fractures
• porosities and hydraulic conductivities of different aquifer rocks (Fig 6.5)

• flow in a horizontal confined aquifer  (Fig 6.7)
• lines of equal hydraulic head are called equipotentials
• flow occurs perpendicularly to those, lines indicating those are called flowlines
• together, the equipotentials and the streamlines constitute a flow net (Fig 6.8)
• generally, groundwater flow follows topography, in detail the situation can be more complicated though
• groundwater flow not only occurs near the water table, but does penetrate deep into the aquifer (Fig 6.9)
• flownets provide a lot of information about groundwater flow, they are generated by computer models these days

## Quantifying groundwater flow using flownets

•  T = Kb [L2 T-1] is called the transmissivity of the aquifer, this term is often the more useful parameter for estimating the yield of an aquifer, it is relevant when we want to estimate the discharge per unit length of stream, for example
• the area between a pair of streamlines is referred to as a streamtube
• in more complicated flow nets, these squares might become "curvilinear squares," as can be seen in (Fig 6.9)
• if we isolate one of these squares (ds=dm)  (Fig 6.10) and make use of Darcy's law, we can calculate the discharge through the streamtube: Q = K*b*dh
• you can imagine each streamtube as a "pipe," because water cannot cross a streamline
• the specific discharge will be greatest where the streamtube is narrowest, analogue to the laminar flow table
• the total discharge through the streamtube must be the same at any cross section
• by counting the number of stream tubes, we can determine the total flow
• another example is steady flow under a dam (Fig 6.11)
• the dam is 100m wide (direction into the page), and the hydraulic conductivity beneath the dam is 10-10 m s-1
• we use the length of the dam (100 m) in place of the aquifer thickness (b)

## Heterogeneity and anisotropy

• so far we have considered only homogeneous aquifer (the same K everywhere)
• virtually all natural materials through which groundwater flows display variations in intrinsic permeability from point to point, this is referred to as heterogeneity (Example: Fig)
• permeable zones tend to focus groundwater flow, while, conversely, flow tends to avoid less permeable zones
• in anisotropic media the permeability depends on the direction of measurement, in isotropic media, it does not

### Resources

• Manning, J.C. (1997) Applied Principles of Hydrology. Prentice Hall, third edition, 276p.
• Freeze, R.A. and Cherry, J.A. (1979) Groundwater. Prentice Hall, 604p.
• WIPP
• WIPP Information Site - EHC