FLOW IN POROUS MEDIA LUÍS RIBEIRO INSTITUTO SUPERIOR TECNICO UNIVERSITY OF LISBON

Transcription

1 FLOW IN POROUS MEDIA LUÍS RIBEIRO INSTITUTO SUPERIOR TECNICO UNIVERSITY OF LISBON

2 Where does groundwater flow?

3 How water flows? Groundwater moves from areas of high hydraulic head to areas of low hydraulic head United States Environmental Protection Agency

4 Type of hydrogeological formations Aquifer A formation, group of formations, or part of a formation that contains sufficient saturated permeable material to yield significant quantities of water to wells and springs (after Lohman and others, 1972). Aquitard - A confining bed that retards but does not prevent the flow of water to or from an adjacent aquifer; a leaky confining bed. It does not readily yield water to wells or springs, but may serve as a storage unit for ground water (AGI, 1980) Aquiclude - A hydrogeologic unit which, although porous and capable of storing water, does not transmit it at rates sufficient to furnish an appreciable supply for a well or spring (after WMO, 1974). Aquifuge A hydrogeologic unit which has no interconnected openings and, hence cannot store or transmit water.

5 Groundwater residence times

6 Water in the ground Source: United States Geological Survey

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8 Ψ < 0 K e θ are function of ψ

9 Type of aquifers Unconfined aquifer Ground surface Limit of the saturated zone = Phreatic level

10 Type of aquifers Confined aquifer What does it happen when the saturated zone reaches the top of the aquifer? Piezometric level Limit of the saturated zone Note that: The pressure that water places on top of the aquifer can be measured by measuring the height h that water rises in a piezometric tube that taps the confined aquifer

11 Type of aquifers Perched aquifer PERCHED AQUIFERS are aquifers that have a confining layer below the groundwater, and sits above the main water table.

12 Effects of groundwater pumping Source: United States Geological Survey

13 Artesian wells

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15 DARCY LAW The French municipal hydraulic engineer Henry Darcy ( ) studied the movement of water through sand and from empirical observations defined the basic equation, universally known as Darcy s law, that governs groundwater flow in most alluvial and sedimentary formations. Darcy s law is the foundation of the theoretical aspects of groundwater flow

16 DARCY LAW Q/A = - K Δh / Δ l q = - K i

17 Darcy velocity The term q, referred to as the specific discharge, has the dimensions of velocity and is also known as the darcy velocity or darcy flux. It is important to remember that the darcy velocity is not the true, microscopic velocity of the water moving along winding flow paths within the soil or rock. Instead, by dividing the specific discharge by the fraction of open space (in other words, effective porosity) through which groundwater flows across a given sectional area, this provides an average measure of groundwater velocity.

18 Macroscopic Law

19 Validity of Darcy law Re number of Reynolds

20 q = K im

21 Hydraulic conductivity and permeability K Hydraulic Conductivity [ LT -1 ] K = k ( ρ g / μ ) k - Permeability [L 2 ] k = C d 2

22 DENSITY AND VISCOSITY The density and viscosity of water are functions of temperature and pressure but these effects are not great for the ranges of temperature and pressure encountered in most groundwater situations

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24 POROSITY

25 θ

26 Heterogeneity

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28 K is lognormal distributed 16 Histogram: ce K-S d=.18063, p<.20 ; Lilliefors p<.01 Expected Normal No. of obs X <= Category Boundary

29 K is a tensor 9 components 3 components, Main directions

30 Heterogeneity vs Anisotropy Kz Kx K x K m i m i i K z m i m / i K i mi thickness of layer i

31 Head as Energy

32 Kinetic Energy. Energy required to accelerate fluid packet from velocity v1 to velocity v2. Eq 1 Gravitational work. Energy required to raise fluid packet from elevation z1 to elevation z2. Eq 2

33 Pressure work. Energy required to raise fluid packet pressure from P1 to P2. Eq 3 assuming a unit mass of incompressible fluid

34 the sum of eq1, eq2 and eq3 is the total mechanical energy for the unit mass (i.e. m = 1) Assuming that v is small (true for flow in porous media).

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37 Hydraulic head datum h = ψ + z [ L ]

38 HYDRAULIC HEAD Equation confirms that the hydraulic head at a point within a saturated porous material is the sum of the elevation head, z, and pressure head, ψ, thus providing a relationship that is basic to an understanding of groundwater flow.

39 EQUIPOTENTIAL LINES Observation boreholes and piezometers located within a district provide a picture of the threedimensional distribution of hydraulic head throughout an aquifer system. Lines drawn joining points of equal groundwater head, or groundwater potential, are termed equipotential lines. Lines perpendicular to the equipotential lines are flow lines and can be used in the construction of a flow net

40 EQUIPOTENTIAL CONTOURS In plan view, the construction of equipotential contours results in a map of the potentiometric surface. In an unconfined aquifer, the potentiometric surface is a map of the water table, where the groundwater is by definition at atmospheric pressure. In a confined aquifer the potentiometric surface predicts the position that the water level would rise to in a borehole that penetrates the buried aquifer.

41 The areas of high hydraulic head may be interpreted as groundwater recharge zones while areas of low hydraulic head are typically in groundwater discharge zones.

42 GROUNDWATER FLOW THEORY

43 At the beginning of the last century, Meinzer and Hard (1925) observed in a study of the Dakota sandstone that more water was pumped from the region than could be accounted for (as water was pumped, a cone of depression developed and the rate of abstraction decreased, but with no apparent effect on groundwater levels in the recharge zone), such that the water-bearing formation was demonstrating elastic behaviour in releasing water from storage. Later, in deriving the general partial differential equation describing transient groundwater flow, Jacob (1940) formally described the elastic behaviour of porous rocks. There are two mechanisms that explain how water is produced by confined aquifers: the porosity of the aquifer is reduced by compaction and groundwater is released; and the water itself expands since water is slightly compressible

44 The total downward stress, σt, applied at the top of a confined aquifer is supported by an upward effective stress, σe, on the aquifer material, and the water pressure contained in the pore space Pw. eq1

45 If the pore water pressure is decreased by groundwater pumping or by natural groundwater outflow, the stress on the aquifer material will increase causing it to undergo compression.

46 Specific storage Specific storage represents the volume of water that an aquifer releases from storage per unit surface area of aquifer per unit decline in the component of hydraulic head normal to that surface

47 Compressibility of water The compressibility of water β is defined as: eq2

48 Compressibility of aquifer material The compressibility of aquifer materiasl α is defined as: eq3

49 eq4

50 dσe = 0 ρgdψ = ρgdh eq5 For a unit decline in head, dh = 1, and if unit volume is assumed (VT = 1), then eq4 becomes: dvw = α(1)( ρg)( 1) = αρg eq6 The water produced by the expansion of water is found from eq2 thus: dvw = βvwdpw eq7

51 Recognizing that the volume of water, Vw, in the total unit volume of aquifer material, VT, is nvt where n is porosity, and that dp = ρgdψ or ρg for a unit decline in hydraulic head (where ψ = h z with z remaining constant), then for unit volume, VT = 1 : dvw = βn(1)( ρg) = βnρg eq8

52 eq9

53 In other words, groundwater pumped from a confined aquifer does not represent a dewatering of the physical pore space in the aquifer but, instead, results from the secondary effects of aquifer compaction and water expansion. As a consequence, for an equivalent unit decline in hydraulic head, yields from confined aquifers are much less than from unconfined aquifers. Hence, storage coefficient values of confined aquifers are much smaller than for unconfined aquifers.

54 Transmissivity T = K x b [L 2 T -1 ] It represents the rate at which water of a given density and viscosity is transmitted through a unit width of aquifer or aquitard under a unit hydraulic gradient.

55 VALUES OF GOOD AQUIFER PRODUCTIVITY T > m 2 /s S entre e

56 Transmissivity and specific yield of unconfined aquifers For an unconfined aquifer, the transmissivity is not as well defined as in a confined aquifer, but the equation can be applied with b now representing the saturated thickness of the aquifer or the height of the water table above the top of a lower aquitard boundary. The transmissivity will, therefore, vary if there are large seasonal fluctuations in the elevation of the water table or if the saturated thickness of the aquifer shows lateral variation as a result of an irregular lower aquitard boundary or differences between recharge and discharge areas in the same aquifer. The storage term for an unconfined aquifer is known as the specific yield, Sy, (or the unconfined storativity)

57 Equations of groundwater flow Equations of groundwater flow are derived from a consideration of the basic flow law, Darcy s law, and an equation of continuity that describes the conservation of fluid mass during flow through a porous material Under steady-state conditions, the magnitude and direction of the flow velocity at any point are constant with time. For transient conditions, either the magnitude or direction of the flow velocity at any point may change with time, or the potentiometric conditions may change as groundwater either enters into or is released from storage.

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59 Steady-state saturated flow

60 Equation of continuity If the fluid is incompressible, then density, ρ(x, y, z), is constant and previous equation becomes :

61 From Darcy s law, each of the specific discharge terms can be expressed as:

62 Transient saturated flow The law of conservation of mass for transient flow in a saturated porous material requires that the net rate of fluid mass flow into the control volume is equal to the time rate of change of fluid mass storage within the control volume. The equation of continuity is now:

63 The first term on the right-hand side of equation describes the mass rate of water produced by expansion of the water under a change in its density, ρ, and is controlled by the compressibility of the fluid, β. The second term is the mass rate of water produced by the compaction of the porous material as influenced by the change in its porosity, n, and is determined by the compressibility of the aquifer, α.

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65 By expanding the terms on the left-hand side of equation using the chain rule (eliminating the smaller density gradient terms compared with the larger specific discharge gradient terms) and, at the same time, inserting Darcy s law to define the specific discharge terms, then:

66 LEAKAGE

67 LEAKAGE MEASURES

68 LEAKAGE RATE 1 unconfined aquifer ; 2 confined aquifer 3 aquitard h2 h1 h2 1 h h1 > h2 h1 < h2

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71 AQUIFER SYSTEM OF QUERENÇA- SILVES Influent and Efluent river sector streams 0m 5000m 10000m Arade Alcantarilha Quarteira