Darcy’s law

In document Groundwater Lowering in Construction (Page 50-54)

Groundwater and permeability

3.2 Permeability and groundwater flow

3.2.1 Darcy’s law

The modern understanding of flow of groundwater through permeable ground originates with the researches of the French hydraulics engineer, Henri Darcy (1856), see Section 2.3. He investigated the purification of water by filtration through sand and developed an equation of flow through a granular medium based on the earlier work of Pouseuille con-cerning flow in capillary tubes. His conclusions can be expressed alge-braically as equation (3.1), universally referred to as Darcy’s law, which forms the basis for most methods of analysis of groundwater flow.

Q:kA

冢 冣

(3.1)

where (see Fig. 3.2 which schematically shows Darcy’s experiment) Q is the volumetric flow of water per unit time (the ‘flow rate’); A is the cross sec-tional area through which the water flows; l is the length of the flow path between the upstream and downstream ends; dh is the difference in total hydraulic head between the upstream and the downstream ends; and k is the permeability of the porous media through which the water flows.

The total hydraulic head at a given point is the sum of the ‘pressure head’

and the ‘elevation head’ at that point, as shown in Fig. 3.3. The elevation head is the height of the measuring point above an arbitrary datum, and the pressure head is the pore water pressure u, expressed as metres head of water.

Total hydraulic head is important because it controls groundwater flow.

Water will flow from high total hydraulic head to low total hydraulic head.

It follows that water does not necessarily flow from high pressure to low pressure or from high elevations to low elevations – it will only flow in

dh l

Figure 3.2 Darcy’s experiment.

response to total hydraulic head, not pressure or elevation considered in isolation.

A term which is used frequently in any discussion of groundwater lower-ing is ‘drawdown’. Drawdown is the amount of lowerlower-ing (in response to pumping) of total hydraulic head, and is a key, measurable, performance target for any dewatering operation. Drawdown is equivalent to:

i The amount of lowering of the ‘water table’ in an unconfined aquifer.

ii The amount of lowering of the ‘piezometric level’ in a confined aquifer.

iii The reduction (expressed as metres head of water) of pore water pressure observed in a piezometer (see Section 6.5). In this case the drawdown of total hydraulic head can be estimated directly from the change in pressure head, since the level of the piezometer tip does not change, so the change in elevation head is zero. If the reduction in pore water pressure is ⌬p, the drawdown is equal to ⌬p/␥w, where ␥wis the unit weight of water.

Darcy’s law is often written in terms of the hydraulic gradient i which is the change in hydraulic head divided by the length of the flow path (i: dh/l).

Equation (3.1) then becomes

Q:kiA (3.2)

In this form the key factors affecting groundwater flow are obvious.

i If other factors are equal, an increase in permeability will increase the flow rate.

ii If other factors are equal, an increase in the cross-sectional area of flow will increase the flow rate.

iii If other factors are equal, an increase in hydraulic gradient will increase the flow rate.

Figure 3.3 Definition of hydraulic head.

These points are vital in beginning to understand how groundwater can be manipulated by groundwater lowering systems.

In the presentation of his equation Darcy left no doubt of its origin being empirical. His important contributions to scientific knowledge were based on careful observation in the field and in the laboratory, and on the conclusions that he drew from these. Permeability is in fact only a theoretical concept, but one vital to realistic assessments of groundwater pumping requirements and so an understanding of it is most desirable. In theory, permeability is the notional (or ‘Darcy’) velocity of flow of pore water through unit cross sec-tional area. In fact, the majority of the cross secsec-tional area of a soil mass actu-ally consists of soil particles, through which pore water cannot flow. The actual pore water flow velocity is greater than the ‘Darcy velocity’, and is related to it by the soil porosity n (porosity is the ratio of voids, or pore space, to total volume).

The main condition for Darcy’s law to be valid is that groundwater flow should be ‘laminar’, a technical term meaning the flow is smooth. Flow will be laminar at low velocities but will become turbulent above some velocity, dependent on the porous media and the permeating fluid. Darcy’s law is not valid for turbulent flow. In most groundwater lowering applications flow can safely be assumed to be laminar. The only locations where turbulent flow is likely to be generated is close to high flow rate wells pumping from coarse gravel aquifers. The implications of this flow to wells are discussed in Section 3.4.

For idealized, homogeneous, soils permeability depends primarily on the properties of the soil including the size and arrangement of the soil particles, and the resulting pore spaces formed when the particles are in contact. For example, consider an assemblage of billiard balls of similar size (Fig. 3.4(a)).

This is analogous to the structure of a high permeability soil (such as a coarse gravel) where the voids (or pore water passages) are large, and the pore water can flow freely. Next, consider an assemblage of billiard balls with marbles placed in the spaces between the billiard balls (Fig. 3.4(b)); this is analogous to a soil of moderate permeability because the effective size of the pore water passages are reduced. Finally, consider a structure with lead shot particles placed in the voids between the billiard balls and the marbles – the passages for the flow of pore water are further reduced; this simulates a low perme-ability soil (Fig. 3.4(c)). It is logical to infer from this analogy that the ‘finer’

portion of a sample dominates permeability. The coarser particles are just the skeleton of the soil and may have little bearing on the permeability of a sam-ple or of a soil mass in situ.

The proportion of different particle sizes in a sample of soil (as might be recovered from a borehole) can be determined by carrying out a particle size distribution (PSD) test, see Head (1982). The results of the test are normally presented as PSD curves like in Fig. 3.5, which shows typical PSD curves for a range of soils. PSD curves are sometimes known as grading curves or sieve analyses, after the sieving methods used to categorize the coarser particle

(a)

(b)

(c)

Figure 3.4 Soil structure and permeability. (a) High permeability, (b) medium permeability, (c) low permeability.

Figure 3.5 Particle size distribution.

sizes. Since there is a fairly intuitive link between particle size and perme-ability, over the last 100 years or so many researchers and practitioners have developed empirical correlations between permeability and certain particle size characteristics. Some of these methods are still in use today, and are described in Section 6.6, along with the drawbacks and limitations of such correlations.

As discussed earlier, the permeability is also dependent on the properties of the permeating fluid. Groundwater lowering is concerned exclusively with the flow of water, so this issue is not a major concern. It is worth not-ing, however, that the viscosity of water varies with temperature (and varies by a factor of two for temperatures between 20⬚C and 60⬚C), so in theory permeability will change with temperature. In practice, groundwa-ter temperatures in temperate climates vary little and errors in permeabil-ity due to temperature effects tend to be small in comparison with other uncertainties.

In document Groundwater Lowering in Construction (Page 50-54)