Darcy's law

Take away ideas and understandings

• Definitions of saturated/unsaturated zone, groundwater.
• Groundwater flows from high to low elevations, or more precise from high potential energy (=hydraulic head) to low potential energy.
• The hydraulic head is measured by determination of the vertical position of the water table in a well relative to  a reference surface.
• Darcy's law says that the discharge rate q is proportional to the gradient in hydrauolic head and the hydraulic conductivity (q = Q/A = -K*dh/dl).
• Definitions of aquifers, aquitards, and aquicludes and how hydraulic conductivity relates to geology.

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

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 = -p*D⁴/(128*m)*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
• 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/rg + 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 orginal Darcy experiment yielded these data (Fig 6.4)
• the analogy between Darcy's law and Poiseulle's law suggests that K = (const*d²)*rg/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!
• 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
• 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 is the fraction of a porous material which is void space f = V_void/V_total
• the mean pore water velocity is then: v = q/f (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/f, f 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 aquiclude is a formation that may contain water but does not transmit significant quantities (clays and shales)
• an aquitard is a formation with relatively low permeability
• 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)

Steady groundwater flow

• 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

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.