PET524-6-steadystate

Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids

7.1

7.1 Horizontal, 1D, Steady-State Flow

Steady state condition

The steady state flow regime can be characterized by pressure and fluid velocity

distributions that are independent of time; therefore, the continuity equation can be

simplified to:

0v (7.1) An application of Eq. (7.1) is the measurement of permeability in laboratory samples.

For linear, horizontal flow of an incompressible fluid through an isotropic porous media

the diffusivity equation becomes;

0k

(7.2)

Consider the example shown in figure 7.1. A long cylindrical core of length, L, and

cross-sectional area, A, has applied pressure of )()0( Lppp . The flow rate, q, is

constant and follows Darcys Law. Also allow permeability to be a function of position

as well as pressure.

X=0

P0

X=L

q

PL

Figure 7.1 Schematic of flow through a horizontal core

The solution is:

L

pAkq

(7.3)

where

L

0)x(k

dx

k

L (7.4)

where k is the harmonic mean permeability of the sample.


7.2

Ideal Gas Flow

As a second example, lets use gas as the flowing fluid; therefore, we must

account for both compressibility and gas-slip effects (Collins,1961). The diffusivity

equation becomes

0dx

dp

p

b1pk

dx

d

(7.5)

where the Klinkenberg Effect is given by

p

b1

)x(kk (7.6)

Note the equivalent liquid permeability is a function of distance, x. Integrating twice

results in

cL

p

p

bpk

1 (7.7)

where c is a constant, 2

)()0( Lppp

and average permeability is given by Eq. (7.4)

with k replacing k . Comparing with Darcys Law yields,A

pqc

, where the flow

rate is evaluated at the mean pressure.

Radial Geometry

The equivalent form of the continuity equation in radial geometry for steady state

condition is,

01

vr

rr (7.8)

Assuming an incompressible fluid, isotropic media, and constant fluid properties results

in the following continuity equation.

01

rr

rr

(7.9)

The general solution is

2)ln(

1)( CrCr (7.10)


7.3

where C1 and C2 are constants to be determined from the boundary conditions.

if assume Pwf at rw and Pe at re then obtain Darcys steady state flow equation.

wr

er

rwrekh

q

ln

)(2

if source located at (a,b), in Cartesian coordinates,

Cbyaxkh

qyx

2)(2ln

4),(

(7.11)

if anisotropic,

Cbyy

k

xk

axh

yk

xk

qyx

2)(2ln

4),(

(7.12)

7.2 Streamlines, Isopotentials, source/sink

A streamline indicates the instantaneous direction of flow for a fluid particle

throughout the system. For steady state, this direction of flow is constant for all time,

however, for transient conditions the streamline is valid only for an instant in time.

Streamlines must be orthogonal to the equipotential lines throughout the region of flow.

An exception to this rule will be shown later for anisotropic media. Isopotential or

equipotential lines indicate lines of constant potential. A streamtube defines the area

between two adjacent streamlines. A Flow net is a set of equipotential lines and

streamlines. A simple example is shown in Figure 7.2.

Figure 7.2 Flow net for a simple flow system (Freeze and Cherry, 1979)

Constant hydraulic head, h, of 100 m and 40 m, exists at the left and right boundaries,

respectively. No flow boundaries occur at the top and bottom. As can be seen, the


7.4

flowlines indicate the direction of flow from higher potential to lower, i.e., to the right.

The equipotential lines (dashed lines) are orthogonal to these flowlines. For

homogeneous, isotropic media, flow nets must obey the following rules.

a. flowlines and equipotential lines are normal to each other

b. equipotential lines must intersect impermeable boundaries at right angles

c. equipotential lines must parallel constant potential boundaries

Notice the boundary conditions are special forms of flowlines or equipotential lines. For

a closed boundary the fluid velocity normal to the boundary is zero (See figure 7.3).

Since there is no flow across the boundary, the flowlines adjacent to the boundary must

be parallel to it, and therefore the equipotential lines must meet the boundary at right

angles (See fig. 7.2 as an example).

ln

Flux normal to boundary is zero

Figure 7.3 Schematic of a closed boundary

Mathematically, this boundary can be expressed as:

0

0

nl

nv

(7.13)

where n is the unit vector normal to the boundary and ln is the distance measured parallel

to n.

In the case of an open boundary, flux is continuous, therefore, the boundary is an

isopotential line, (= constant) and no flux exists tangential to the boundary. Figure 7.4

represents this condition.


7.5

Flux normal to boundary is

constant

Figure 7.4 Schematic of an open boundary

Effect of anisotropy on flow net

An example of an anisotropic system is illustrated in Figure 7.5. The vertical

section represents flow from a surface pond at h = 100 toward a drain at h = 0. The drain

is considered to be one of many parallel drains set at a similar depth oriented

perpendicular to the page.

h=100

h=0

Figure 7.5 Flow problem in a homogeneous, anisotropic system.

The vertical impermeable boundaries are "imaginary"; they are created by the

symmetry of the overall flow system. The lower boundary is a real boundary; it

represents the base of the surficial soil, which is underlain by a soil or rock formation

with a conductivity several orders of magnitude lower. If the vertical axis is arbitrarily set

with z = 0 at the drain and z = 100 at the surface, then from h = p + z, and the h values

given, we have p = 0 at both boundaries. The soil in the flow field has an anisotropic

conductivity.

The influence of anisotropy on the nature of groundwater flow nets is illustrated

in Figure 7.6 for the previous problem. The streamlines are shown as the solid lines, the

lines of equipotential are dashed. Note, the most important feature of the anisotropic

flow nets is the lack of orthogonality.


7.6

Figure 7.6 Flow distributions for various permeability anisotropies

(Freeze and Cherry, 1979)

7.3 2D flow, superposition

Consider horizontal (xy) plane flow in a homogeneous media, with upper and

lower impermeable boundaries as illustrated in Figure 7.7. Permeability is anisotropic

and aligned with the principal directions. In this example, no vertical flow exists; i.e., vz

= 0.

impermeable

impermeable

y

x

z

vz = 0

Figure 7.7 Schematic of horizontal flow in a homogeneous media

4

1

yk

xk

1

yk

xk

4

yk

xk


7.7

From the definition of potential,

p

op

gzdp

it is evident that pressure and density are dependent on the vertical direction, z, therefore

gdz

dp (7.14)

The continuity equation for this problem becomes,

0

y

pyk

yx

pxk

x

(7.15)

where the horizontal components of velocity are given by Darcys Law.

For the ideal liquid case the pressure becomes, gz)0,y,x(

p)z,y,x(

p , where z=0

occurs at the bottom of the flowing layer.

To simplify the problem, apply the following coordinate transformations,

y

yk

xk

y

xx

(7.16)

The result is a Laplace-type equation,

02

y

p2

2x

p2

(7.17)

This solution is valid on the xy plane at a given depth, z. However, pressure is a function

of z as well, therefore to account for this dependency, introduce a new variable U

(Collins, 1961).

gzpy

kx

k

U (7.18)

The solution is:

02

y

U2

2x

U2

(7.19)


7.8

Both Eqs. (7.17) and (7.19) are Laplace-type equations. Numerous solution techniques

have been presented in the literature for these equations and the associated boundary

conditions.

Multiwell systems

The Superposition principle defines the net potential at any point in the multiwell

system as the sum of the potential contributions from each well computed as if it were

alone in the system. The linear combination of n source/sink terms in an isotropic and

homogeneous media can be expressed as;

Ci

yyi

xxn

ii

qkh

yx

2)(2ln

14),(

(7.20)

For example, Figure 7.8 illustrates a two well system, a source and a sink. The desired

location for potential is at (x,y).

+

Source

Sink

(x,y) (x1,y1)

(x2,y2)

Figure 7.8 Schematic of a two well system for the principle of superposition

The flux in the x and y directions can be determined by using Darcys Law,

?

x

kxv

(7.21)

Flow potential near a discontinuity

As a special 2D problem consider a horizontal layer of uniform thickness and

infinite in areal extent. This layer is separated by a vertical plane; with differing

permeability on opposite sides. Figure 7.9 illustrates the problem and the differential

equations on the surface of the discontinuity and in the two regions. The fluid is

incompressiblethus density and viscosity are invariant.


7.9

q

(-d,0) (d,0)

cq

kb ka

d

At discontinuity:

x

bb

kx

aa

k

ba

)0,0()0,0(

Figure 7.9 Schematic of a 2D problem with a linear discontinuity

A well of fixed point source strength q is located at (d,0). To satisfy the boundary

conditions at the discontinuity, another point source of strength cq is located at (-d,0) in

medium (b). This procedure is known as the method of images. From the principle of

superposition, the system of equations becomes,

22ln22ln

4),( ydxcydx

hak

qyxa

(7.22)

Dydxh

bk

qyx

b

22ln4

),(

(7.23)

where c and D are constants to be determined.

Applying the boundary conditions at the discontinuity (y=0) we have

DC

and

bk

ak

DC

1

1

combine and substitute in Equations (7.22) and (7.23), results in

02

2

2

2

y

a

x

a

02

2

2

2

y

b

x

b


7.10

22ln

1

122ln4

),( ydxakbk

akbkydxhak

qyxa

(7.24)

ak

bk

ak

bk

ydxh

bk

qyx

b 1

222ln4

),(

(7.25)

Two limiting examples illustrate the utility of the above equations. If kb = 0, there exists

no potential in region (b) and therefore the discontinuity reduces to an impermeable

boundary. If kb , then the discontinuity becomes an equipotential boundary.

PET524-6-steadystate

Documents

Transcript of PET524-6-steadystate