Fundamentals of Fluid flow, Poiseulles law, Laplace’s equation, Darcy’s law in saturated and unsaturated flows; differential equations, capillary conductivity and diffusivity; numerical solution for one dimensional water flow

Fluid flow fundamentals encompass the principles governing the movement of liquids and gases through various mediums. Poiseuille's law, for instance, illuminates the flow of incompressible fluids in cylindrical pipes, where the flow rate (Q) is intricately linked to pressure difference (ΔP), viscosity (μ), and pipe dimensions (length, L, and radius, r). Laplace's equation, on the other hand, guides the steady-state potential distribution, such as pressure, in a fluid. In hydrology, Darcy's law plays a pivotal role, particularly in saturated flow, where it connects specific discharge (q), hydraulic conductivity (K), and hydraulic gradient (i). This law also extends its utility to unsaturated flow, albeit with varying hydraulic conductivity based on soil moisture content. Developing differential equations, often rooted in mass conservation, underpin modeling porous media flow. These equations are coupled with Darcy's law, adapting to the specific conditions of saturated and unsaturated flow. Within this context, capillary conductivity and diffusivity become vital parameters governing water movement in porous materials. However, it's essential to acknowledge Darcy's law's limitations, as it assumes steady-state, laminar flow, and homogeneity, which may not hold in highly heterogeneous or fractured media. Finally, numerical techniques, like finite difference or finite element modeling, are indispensable tools for solving these complex differential equations, often used for one-dimensional water flow simulations. These concepts and methodologies are fundamental in various fields, including hydrogeology, civil engineering, and environmental science, where understanding and modeling fluid flow through porous media are of paramount importance.

1. WELCOME

2. PRESENTATION TITLE ©

Fundamentals of fluid flow, Poiseulles law, Laplace’s
equation, Darcy’s law in saturated and unsaturated flows;
development of differential equations in saturated and
unsaturated water flow, capillary conductivity and
diffusivity; limitations of Darcy’s law; numerical solution for
one dimensional water flow

Presented and Submitted by
Mr. ANIKET SUNIL GAIKWAD
(Reg. No.: Ph.D. 2021/22)

3. ©

COURSE TITLE
ADVANCES IN SOIL PHYSICS

Presented and Submitted by
Mr. ANIKET SUNIL GAIKWAD
(Reg. No.: Ph.D. 2021/22)

Submitted to
Department of Soil Science and Agricultural Chemistry,
Post Graduate Institute, M. P. K. V., Rahuri

4. 6 Fundamentals of fluid flow 2
7 Poiseuilles law, Laplaces equation 3
8and 9 Darcy’s law in saturated and unsaturated flows, 3
development of different equations in saturated and
unsaturated water flow
10 and 11 |Capillary conductivity and diffusivity; Limitations of 5

Darcy’s law; Numerical solution one dimension water flow

5. [ MAJOR CONCEPT OF TRANSPORT | ©

Solute molecules, dissolved in gas or liquid, move from high to low concentration.
Driving force for movement is difference in concentration with distance Transport is
formulated by Fick’s law of diffusion

Heat moves from high to low temperature. Driving force for heat transport is
difference in temperature with distance. Heat transport is formulated by Fourier’s
law of heat conduction

Current (movement of electrons) is caused by electrical potential differences.
Electrons move from high to low potential. Current is computed from Ohm’s law.
Water moves from high to low pressure or potential. Driving force is difference in its
pressure with distance. In addition water will move by gravitational forces. In soils,
water moves by pressure (or water potential) and gravitational forces. Water flow is

described by DARCY’s law

6. = Saturated flow of water : Condition of soil, when all large and small pores are filled
with water is called saturated flow. The direction of flow is from a zone of higher
moisture potential to a lower moisture potential.

= Unsaturated flow of water : Soil pores contain some air as well as water is called

unsaturated soil.

1. Fluids flow in soils only when there is a gradient in piezometric head (h). Lack of
a gradient in piezometric head implies that fluid is not flowing.

2. Whenever there is fluid flow in soils, there is energy dissipation.

3. In soils, fluids always flow down the gradient in piezometric head. That is, fluids

flow from high energy regions to low energy regions.

7. = WHAT IS FLUID..?
A substance which is capable of flowing and deform continuously
under acting shear and stress.
" Fluids:
1. Liquid - Incompressible, close molecules.
2. Vapors - compressible.

3. Gas - compressible.

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8. Ideal fluid : The fluid which are incompressible and has no viscosity (not exist)
Real fluid : The fluid which is compressible and has viscosity
Eg. kerosene, petrol, castor oil)

Two types of real fluid :

a) Newtonian fluid : Follow newton's law of viscosity.

Eg. water, air, alcohol, glycerol)
b) Non-Newtonian fluid : Not Follow newton's law of viscosity.

Eg. soap solution, oobleck (starch mixed with water, blood, gel)

9. WHAT IS NEWTON'S LAW OF VISCOSITY..? ©

= Shear stress (t) is directly proportional to the rate of shear strength
or velocity gradient (du/dy).
ie. toc du/dy

Where,
t= Shear stress
u = Coefficient of dynamic viscosity
du/dy = Velocity gradient

10. FORMS OF FLUID FLOW IN SOIL ©

> Gaseous Flow -
1. Diffusion - Due to partial pressure differences of water vapour.
2. Mass Flow - Due to the differences in total air pressure.

> Solid Flow - only at glacial movement.

> Liquid Flow - two types
1. Saturated flow

2. Unsaturated flow

11. THERE ARE TWO TYPES OF FLOW

laminar flow

= Laminar flow - —

= Turbulent flow - i ee A

©

12. REYNOLDS NUMBER (Osborne Reynolds, 1883) ©

The ratio of inertial forces of flow to frictional forces is called reynolds
number.

P, dv P,,= Density of water
Re = ---------- d = Diameter of effective pores
n v = Flow velocity
1 = Viscosity of water

Reynolds number for laminar flow (Re) :
1. Flow through pipes Re < 2000

2. Flow through parallel plates Re < 1000
3. Flow through open channel Re < 500
4. Flow through soil Re < 1

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13. POISEUILLE’S LAW ©

+ The discharge or volume flow of water (q in cm*/hr) through a narrow tube is
directly proportional to fourth power of the radius (r* in cm) and pressure
difference (AP) and inversely proportional to the viscosity (1) and length of tube

(inc), Viscosity 7 =e

gs Po

Where, 1 ———+l
AP = Pressure difference (Dynes /cm?).
r?= Radius of tube (cm)
L= Length of tube (cm)
1 = Coefficient of Viscosity of liquid (dynes-sec em? (poise))

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14. NN BB Y D =

ASSUMPTIONS OF POISEUILLE’S LAW ©

. Flow is laminar and steady state.

. Liquid in contact with wall have speed zero.

. Pressure at any point of certain cross section is constant.
. Fluid is incompressible and Newtonians (water)

. The tube is stiff , straight and uniform.

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15. KEY POINTS ©

Soil have different types of pore sizes and channels through which
water may flow.

Frequently fluid velocity changes from point to point even along
same passage.

It is very complicated to describe the actual geometry and flow
pattern of a typical soil specimen under microscopic details.

Flow through complex porous media is generally described in terms

of a macroscopic flow velocity vector.

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16. &

= Hydraulic head [total hydraulic head (Ht) = gravitational head (Hg) + pressure head (Hp)]:

It is the elevation with respect to a standard datum at which water stands in a rise pipe or
manometer connected to a point in question in the soil. Hydraulic head (H) is the common
name of total potential head i.e. the potential energy per unit weight of soil water and is also
known as hydraulic potential. In saturated soil, the hydraulic head is measured with
piezometer which is simply the open pipe installed in the soil to measure the ground water
table. In unsaturated soil, the tensiometer is used to measure the hydraulic head.

= Hydraulic gradient: It is the total head drop per unit distance in the direction of flow. OR
it is the rate of change of hydraulic head with distance. It records the head consumed by
friction in the flow in unit distance. It has no dimension. It is the driving force of water flow.
It is the difference in hydraulic head (H) between two points divided by the distance

between the points. i= AH + L

17. Tortuosity : It is the average ratio of the actual length of flow path of the water (Le) to the straight
length of the flow path (L). Tortuosity is always > land may exceed 2. Tortuosity = Le + L

Flux and Flow Velocity : Flux is the rate of flow i. e. the quantity of material or energy transferred
through a system or a portion of a system in unit time. Generally, the flux of a matter is measured as
mass moving through system in unit time is called mass flux. If the moving matter is fluid (soil water),
the flux may be measured as volume of fluid moving through a system in a unit time and is called
volume flux. The flow of heat through a system in a unit time is called heat flux.
The flux per unit area is called the flux density. So, the flux density of water transported through a unit
cross-sectional area of a system in a unit time. Sometimes flux density is also called simply flux.
Therefore flux (q) of water is the volume of water (V in cm?) flowing through a unit cross-sectional area
(A in cm?) per unit time (t in second).
Q=V+(Ax0)
Flux is same as the flow of water, when water is flowing through a pipe. In soil, flux differs from the flow
velocity because flow does not take place through the entire cross-sectional area of the soil and soil
particles reduces this area; thus the real area through which flow takes place is smaller than the cross-
sectional area of the soil column. So the flow velocity of water in soil is greater than flux.

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18. = Permeability: It is the ease with which a porous medium like soil transmits water or air and is
usually being used in a qualitative sense. Quantitatively it is the specific property of governing
the rate or readiness with which a porous medium like soil transmits fluids down ward under
standard (usually saturated) conditions. Its dimension is L/T and c.g.s unit is cm/sec.

Intrinsic Permeability (K”): The hydraulic conductivity of soil is dependent upon properties of
both soil and water that saturates the soil. The soil characteristics which affect the rate of flow
are pore geometry (total porosity, the distribution of pore size and tortuosity) of the soil. The

rate of flow that flow which is dependent on soil characteristics only is termed as “intrinsic

permeability”.

Fluidity (f): The fluid properties that control the rate of flow in the soil are viscosity and
density of the fluid. It is directly proportional to the density (p) of fluid (water) and inversely
proportional to the viscosity (n) of fluid (water).

f = pg*n where g= acceleration due to gravity

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19. In soil, the change in fluidity is caused by the change in temperature and electrolytic
concentration which in turn change the density and viscosity of water. The change in viscosity
is associated with the change in hydraulic conductivity (K) of soil.

Hydraulic Conductivity (KX): It the ease with which a soil conducts or transmits water through
its pores. It is the proportionality constant in Darcy’s law indicating the ability of the soil to
transmit flowing liquid (water). Thus hydraulic conductivity is the flux or effective flow
velocity (q in cm/sec.) at unit hydraulic gradient (i is unit less).

Thus hydraulic conductivity may be defined as the rate of flow of liquid through a porous
medium under unit hydraulic gradient. It has dimension of flux or velocity i.e. length per time
(L/T) and c.g.s. unit of a cm/sec.

The hydraulic conductivity of of a saturated soil is sometimes termed as permeability. When
basically it is dependent on properties of the soil termed as intrinsic permeability (K’) and

properties of the fluid termed as fluidity (f).

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20. Sand 210.0
Loamy Sand 61.0
Fine Sandy Loam 26.0
Loam 13.0
Sandy Clay Loam 4.3
Clay Loam 2.3
Clay 0.6

Q Factors affecting permeability or hydraulic conductivity of a saturated soil

Soil texture: Saturated hydraulic conductivity is maximum in sandy soils and it is decreases

with decrease in coarseness. Texture finer than sandy clay loam may not follow this, because

in those fine textured soils, aggregation may over ride the effect of soil texture.

21. 2. Soil structure : It is maximum in spheroidal type and minimum in plate like structure. Block like
and prism like structure have moderate permeability. Hydraulic conductivity of a massive
structure is very slow. More aggregation with greater amount of non capillary pores, greater is the
hydraulic conductivity.

3. Compaction : Compaction of a soil decreases the non capillary pores as well as total pore space of
the soil resulting reduction of hydraulic conductivity of a soil.

4. Exchangeable cations : Presence of higher amount of exchangeable sodium cause a heavy reduction
in hydraulic conductivity due to dispersion of soil colloids.

5. Organic matter : Organic matter improves the hydraulic conductivity of those soils which have slow
permeability like fine textured soils due to formation of large amount of water stable aggregates.

6. Fluidity : Fluidity id depends on concentration and composition of salts dissolved in water and
viscosity of a water, which is dependent on temperature and electrolyte concentration. If the salt is
sodium hydroxide or sodium carbonate it would cause dispersion of soil colloids and reduce

hydraulic conductivi

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22. E

FORMULAS
Change in head - AH = H,- H,
Flow volume or Flux - q = V/At
Where,

V = Volume of water flowing through column
t= Time (hours)
A= Cross sectional area (cm?)

Hydraulic gradient - i= A H/L

Hydraulic conductivity or Permeability - K = q/i

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23. DARCY’S LAW &

French Engineer Henry Darcy (1856) studied the flow of water through a
column of sand filtered beds and found that the flow or discharge (Q in
cm*/hr), being the volume (V in cm?) of water flowing through the column per
unit time (t in hrs.) is directly proportional to the cross sectional area ( A in
cm?) of the column and to the hydraulic head drop (AH=h,-h,) and inversely

proportional to the length (L in cm) of the soil column. i.e. (q « i)

Vv AAH Here, q= Q/A, i= AH /L

qui

Where,
V = Volume of water flowing (cm°), t = Time of flow (hrs), A = Area of cross section (cm?),

AH = Hydraulic head drop, L = Length of column (cm)

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24. u =

p— de eg

25. LIMITATION OF DARCY’S LAW ©

Valid if the flow through soil is laminar.

Law valid only limited conditions of liquid flow in saturated porous medium
till a constant ratio of flux hydraulic gradient is obtained.

The law is not valid at high flow velocities from which the linearity of flux
verses hydraulic gradient relationship fails.

Law valid in lower ranges of flow velocities.

It is valid for flow in a course, silts and fine sand.

Flow of water through porous medium in soil is laminar only as long as

Reynolds number (Re) is smaller than 1.

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26. DARCY’S LAW IN SATURATED FLOW FAN

Darcy law equation for saturated water flow under steady state condition in

homogeneous and isotropic porous medium (soil) in to a 3 dimensional
differential equation form is
= Liquid flow in homogeneous medium «© Driving force (Hydraulic gradient)

and property of conducting medium. q = -K VH

Where,
q = Flux density (q= V/At), K = Saturated hydraulic conductivity,
H = Hydraulic head , VH = Gradient of hydraulic head in 3D space
= Direction of flow is determined by AH
= Negative sign indicate water flowing in a direction opposite to that of

hydraulic head gradient. RES

27. IN ONE DIMENSIONAL SYSTEM THE DARCY’S EQUATION
BECOMES

Where,
qx = flux in x direction.

28. FACTORS AFFECTING THE SATURATED FLOW ©

. Texture - Sand > Loam > Clay

. Structure - Micro pores < Macro pores

. Amount of organic matter - Maintain high proportion of macropores
. Temperature

. Pressure - Entrapped air pressure reduce flow

. Amount of water in the soil

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29. LAPLACE EQUATION ©

= This equation is derived from Darcy’s law.

= The steady state flow of groundwater is described by a form of the Laplace
equation.

= This equation is used to study hydrogeology.

= Laplace equation in the three space variables and is one of the studied equation
of mathematical physics.

= Used in solving equation of saturated flow.

= It can be solved to obtain a quantitative description of water flow in various
systems such as impervious boundaries, free water surfaces, known inflow

outflow rates etc.

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30. LAPLACE EQUATION ©

= One of the most useful field equations employed in hydrogeology.
= The solution to this equation describes the value of the hydraulic

head at any point in a 3-dimensional flow field.

Q How this equation is obtained..... ?

31. . For homogeneous but

. For homogeneous and

. For horizontal flow

. Steady state flow

SIMPLIFICATIONS OF THE EQUATION

anisotropic aquifer

isotropic

©

32. ASSUMPTION OF LAPLACE EQUATION ©

. As compared to the viscous flow the internal forces are negligible.

2. The water is continuously flowing through the system.

3. The flow is in isothermal condition.

4. Darcy’s law is valid i.e. flow is laminar.

5. The soil is completely saturated i.e. degree of saturation is 100%.

6. The soil is homogeneous i.e. coefficient of permeability is constant everywhere in
soil medium.

7. The soil is isotropic, i.e. coefficient of permeability is same at all direction.

8. During flow, volume of soil and water remains constant i.e. no expansion and
contraction.

9. The soil and water are incompressible i.e. no volume change occurs

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33. UNSATURATED FLOW ©

Macro pores full of air.

Micro pores = water + air

Moisture tension gradient creates unsaturated flow.

Movement is from a zone of low suction to high suction.

With increasing degree of unsaturation more soil pores become devoid
of water

Contribution of hydraulic head or the gravitational components to
total potential becomes progressively less.

Driving force for water flow through the matric potential gradient (i.e.

Wim) ES

34. &

= The modified form of Darcy’s law for unsaturated soil as given by
Richards 1931.
a= -K(y) VH

(Hydraulic conductivity is the function of matric potential) i.e. -K(y)

Where,

q= Flux or flow
VH= Potential or hydraulic head gradient

(i.e. Suction head (y) + Gravitational head (z))

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35. FACTORS AFFECTING THE UNSATURATED FLOW

Nature of soil —
1. Distribution of pores
2. Size of pores.
Soil moisture content - Higher percentage of water in moist soil

more rapid is the delivery.

©

36. Flow of water occurs when essentially all the|Flow of water occurs when macroposre are filled

pores are filled with water and water is not under|with air and the microporse are filled with water

The moving force is the gradient of positive] The moving force is the gradient of negative
pressure potential and water flows from a zone of] pressure potential termed as suction or tension and
higher hydrostatic pressure to a zone of lower|water flows spontaneously from a zone of lower!

hydrostatic pressure. matric suction to hig

Sandy soil (more macro pores) conducts water|Clayey soil (more micro pores) conducts water
more rapidly than a clayey soil. more rapidly than a sandy soil.

The hydraulic conductivity value is generally|The hydraulic conductivity values varies with

independent of the magnitude of water potential|suction that is hydraulic conductivity value
decreases with increase in suction values.
The movement of water more but, the actual flow
ath is less than that in unsaturated soil. ath is more than that in saturated soil.

37. CAPILLARY CONDUCTIVITY ©

The property of an unsaturated porous medium to transmit liquid.
OR

Coefficient which express the extent to which an unsaturated
permeable medium allows flow of water through its interstices, under a

unit gradient of capillary potential.

rd movement by

38. DIFFUSIVITY ©

Term diffusivity is introduced to express the water flow process in
unsaturated soil along the water content gradient.

The soil water diffusivity D(® which is the simpler mathematical
form of water flow and easy to estimate experimentally in all range
of interest is less affected by hysteresis.

To express the unsaturated flow of water in term of diffusivity it is
assumed that flux of water is proportional to the water content

gradient.

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39. NUMERICAL SOLUTION &
ONE DIMENSIONAL WATER FLOW
= From one porous medium some amount of water is draining. In such condition cross
sectional area is 40 m?, hydraulic head drop is 105 cm, hydraulic conductivity is 10°
cm/sec? and length of the given column of medium is 35 cm. find out the amount of
discharged water from the medium?
Solution :
Given : Cross sectional area (A)= 40 m?,
Hydraulic head drop (AH)= 105 cm,
Hydraulic conductivity (K)= 10° cm/sec,
Length of column of medium (L)= 35 cm
Discharged water (Q)= ?

Answer : Q = K.A.AH /L, Q = 40 x 105 x 103/35, Q = 12.00 cm*/sec

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40. &
THANK YOU