flow in porous media

The practical applications of the stochastic processes theory are multiple. This fact is a consequence of the capacity of this theory that can predict the future of a dynamic system by use of its history and its actual state. Between the most famous applications we remember:- the analysis of all movement sorts starting with those from atomic and

molecular level [4.62-4.62] and finishing with those that describe the highsystem evolution such as atmospheric system [4.63-4.64]

- the analysis of dynamic links for networks with stations where the service time is stochastic distributed (computers networks, Internet networks, etc);

- the analysis of virtual experiences given with a stochastic model [4.65]]- the analysis of capitals and finances movement [4.66-4.67];- the analysis and development of all games sorts [4.68- the optimization and the control of all sorts of dynamic systems [4.68]

Curgeri in sisteme multifazice , Lectia 5, 14.11.2011

Flow description by stochastic modeling


Each transformation or phenomena results from one or many

elementary steps or processes. The equilibrium state results from

similar but contrary transport fluxes (Em Bratu, 1974)

The mathematical theory of complete connected chains [4.12-4.16]

could be described with this condensed statement:

The state of a system at time n is a random variable An with values

in a finite space (measurable) (A A ). The state evolution at time n+1

results from the appearance of a Bn+1 result which is also a random

variable with values in a finite space (measurable) (B B). The

appearance of a result signaling the state evolution could be

represented considering a u application of AxB in A and introducing

the following statement: An+1=u(An, Bn+1) for all n>=0 . The Bn+1

probability distribution conditioned by Bn, An, Bn-1, An-1,…….B1, A1,

A0, and symbolized as (P(Bn+1/Bn,An,…)), depends only on the state

An. The group [(A,A),(B,B), u:AxB→A ,P] defines a random system

with complete connections".



Process components and Connection processI) If one elementary particle is participating to a transport phenomena process within a medium with

random characteristics like granular, porous solid etc., the medium will cause random velocity

changes of the particle. In this case the transport process concerning the local velocity are the

named "process components", whereas the transport process changes given by the random

properties of the medium gives the named "connection process

II) The transport phenomena occurs when the displacement of the carriers through different ways

("process components") and the passage from one way to another one is realized by a random

commutation process ("connection process")

III) -A particle (molecule, group of molecules, turbulent group etc) evolves in a medium which

produces its transformation, it means that the process exchange characterizes the particle

evolution. The process occurring before and after the transformation are the "process

components", whereas the transformation itself represents the stochastic evolution "connection

process"

IV) The elementary particles passes randomly from one compartment to another one, the

compartments exchange process form the "connection process" whereas the transformation

realized in each compartment represents the "process components"

V) When a phenomena results on different structures formation, the passage from one structure to

another one is made randomly, in this case the structure formation is the "process component"

and the passing steps are the "connection process"


Upper current

Upper region

Lower region

Pumping region

H

h

E1

E2

S1

S2

S3

Va

Vi

Vm

D

Mechanical mixing of a liquid media

Some process factors

- the turn rate or frequency of the stirrer.

- the configuration and the distribution of the

stirring paddles in the apparatus.

- the physicochemical properties of the medium.

- the position of entry and the exit of the currents in

the apparatus tank.

Basic topology for mixing with mechanical stirrer

2)(

4

2m

aV

hHD

V

224

2m

sum

iV

VVV

hD

V

)5

d

5

DdD(

60

bV

22

m




Upper current

Upper region

Lower region

Pumping region

H

h

E1

E

2

S

1

S2

S3

V

a

Vi

V

m

D


Identification of detailed topology

i) The start conditions are established;

ii) Va, Vm, Vi are calculated with given relations;

iii)If Va>Vi then r=Va/Vi ; for opposite case r=Vi/Va ;

iv) The number of cells is chosen in the smaller region;

ni=nch=1 (frequently but this is a relative consideration)

if the lower region is the smaller one or na=nch=1 when

the upper region is the smaller one;

v)If h/H=0.5 we can consider ni=na=nch ; if h/H>0.5

which is equivalent to Vi>Va we can write na=nch and

ni=r*nch ; however, if h/H<0.5 and Va>Vi we can

consider ni=nch and na=r*nch

),( 5.10 2 fnbdQ

21 QQQ

5.0H/h or 5.0H/h for hH

h

Q

Q

2

1




Upper current

Upper region

Lower region

Pumping region

H

h

E1

E

2

S

1

S2

S3

V

a

Vi

V

m

D


- The system consists on N-1 cells of ideal mixing

each one with its known volume;

- The cells are rejoined by known currents;

- As notations N corresponds to the system exit, Vk

is the volume of k cell and Qkj is the current (flow

rate ) from the k cell to the j cell.

E(n)=[e0 (n), e1 (n);e2 (n)… ek (n)… eN -1 (n),eN (n) ]

N

1kk ,...2,1,0n 1)n(e

Markov conexion: The probability of the particle presence within the k cell after n+1

time (thus τ=n Δτ) is given only by its probability of presence in the j cell after time n

and by the passage probability from the j to the k cell noted like pjk

jk

N

1jjk p)n(e)1n(e




Upper current

Upper region

Lower region

Pumping region

H

h

E1

E

2

S

1

S2

S3

V

a

Vi

V

m

D


For j=1,N and k=1,N the probability pjk is noted as a matrix P

called the stochastic matrix of the process or "the stochastic one“.

NN3N2N1N

N1N12N11N

N2232221

N1131211

p.ppp

p..pp

.....

p.ppp

p.ppp

P

1pN

1j

ij

1NN

p

P*)0(E)1(E

P*)1(E)2(E

P*)n(E)1n(E



Upper current

Upper region

Lower region

Pumping region

H

h

E1

E

2

S

1

S2

S3

V

a

Vi

V

m

D


Which is the information produced with the

assistance of this type stochastic model?

)V

Q

exp(pi

N

ji,1iji

ii

)]V

Q

exp(1[

Q

Qp

i

N

ji,1i

ji

N

ji,1i

ji

ji

ij

1-calculate the system reaction to one disturbance impulse

n ou )n(e)n(e)(F0n

1NN

2- to make a precise estimation for the mean residence time and for

residence time variance around of mean residence time:

))(e1(0n

Nm

2

0nN

0nN

2 ))n(e1(n))(e1(2

3- to appreciate the evolution in time of the function (l) that shows

the intensity of stirring for our topologic cell assembly:

)(e1

)n(e)n(

N

1N



Mechanical mixing of a liquid media: Numeric exemplification

In an elliptic based cylindrical apparatus with D=1 m, H=1 m there is a solution stirred with a 6 paddles stirrer

which has a d=0.4 m and b=0.1 m. The mixer is placed in the tank in a position where the ration h/H=0.2 (see

figure 4.1). It can work at n=0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.5 turns/sec. One reactant with the same physical

properties that the solution is fed close to the liquid surface. The mixture obtained is evacuated by a

connection on the apparatus base. The entry and the exit flows are identical Qex = 0.0048 m3/s. The question

here is to obtain the parameters dependences characterizing this mixing case according to the number of the

stirrer turns.

V1 V2 V3 V4

V5

6

7

Q+Qex

Q2

V1 V2 V3 V4

V5

V6

Qe

x

Q+Qex

Q2+Qex

Q2

Qex

Q1

Q1+Qex

Q+Qex

Topology of the numerical application.

Topology identification

4/18.0/2.0H/h1

H/hr

- Because h/H<0.5 we can affirm that the stirrer is placed

towards the base of the tank and then with nch = 1.

- So we have ni = 1, na = nch / r=4

- The tank contains 6 cells: one in the lower region, one in

the mixture region and four in the higher

)V

QQexp(p

1

ex1

11 )V

QQexp(p

2

ex122

)V

QQexp(p

3

ex133

)V

QQexp(p

4

ex1

44

)V

QQexp(p

5

ex55

)V

QQexp(p

6

ex266

)V

QQexp(1p

1

ex112

)V

QQexp(1p

2

ex123 )

V

QQexp(1p

3

ex134 )

V

QQexp(1p

4

ex145

))V

QQexp(1(

QQ

Qp

5

ex

ex

1

51

))V

QQexp(1(

QQ

QQp

5

ex

ex

ex256

))V

QQexp(1(

QQ

Qp

6

ex2

ex2

265

))V

QQexp(1(

QQ

Qp

6

ex2

ex2

ex

67

,

,

,

,

,

p77=1

V1 V2 V3 V4

V5

6

7

Q+Qex

Q2

V1 V2 V3 V4

V5

V6

Qe

x

Q+Qex

Q2+Qex

Q2

Qex

Q1

Q1+Qex

Q+Qex

Topology of the numerical application.


Mechanical mixing of a liquid media: Numeric exemplification

Non zero transition probalilities


Species displacement and transfer in a porous media- Liquid motion inside the porous media- Deep bed filtration

- for the pores with radius between 10-3-10-5 m it is valid the theory of Poiseuille flow; so the mean force for fluid flow between two planes is expressed by the pressure difference; this can be a consequence of differential actions of external and capillary or/and gravitational forces

-for the pores having mean radius between 10-6-10-

9m we explain the porous body flow by the Knudsen theory; a sort of diffusion coefficient that strongly depends on species dimension shows that this flow has a separation behaviour (two or more species moving inside of porous body present different displacement velocities)

- for the pores or better for the molecular windows, where we have the characteristic dimension smaller than 10-9 m, the movement of one species inside of porous solid occurs due to the molecular interactions between the species and the network of porous body; here for description of species displacement it is used the theory of the molecular dynamics


Species displacement and transfer in a porous media- Liquid motion inside the porous media- Deep bed filtration

An interesting method for the analysis of the species motion inside the porous body can be based on the observation that this motion occurs as a result of two or more elementary evolutions that are randomly connected. It is the stochastic way for analysis of specie motion inside the porous body.

Liquid motion inside the porous media.

Between the classic and stochastic methods that are used for the analysis of liquid flow inside the porous media we have strong liaisons. These liaisons are given by the relations between the parameters of stochastic and classic models of this flow type. We show here that the analysis of this flow type by using of stochastic way explains some parameters of deterministic models such as: parameters contained by the Darcy law ; parameters that appear inside the porous medium flow continuity equation ; parameters used by the models that explain the flow mechanism inside the porous body.

Apparently the parameters of stochastic models are very much different from those of classic (deterministic) models where the permeability, the porosity, the pore radius, the tortuosity coefficient of pore, the specific surface, and the effective species diffusion coefficient represents the most used parameters for porous

media characterisation.


Species displacement and transfer in a porous media- Liquid motion inside the porous media

xx

Figure Fluid motion inside of the uniform porous body

The liquid motion can be described by the motion of liquid element for direction +x that occurs with the probability p respectively by the opposite motion ( -x displacement) where q gives it evolution probability. Here represents the length portion of pore without contact with the nearby pores

=

The probability to

have the fluid

element at time

to x position

The probability that

shows the fluid

element at time

to xx position with

an evolution along of

x for the following

time

+

The probability that

shows the fluid

element at time

to xx position

with an evolution along

of x for the following

time (4.260)

),xx(qP),xx(pP),x(P

A Taylor expansion of ),xx(P and ),xx(P are used in last relation at their right ter

2

22

x

),x(P

2

x

x

),x(Px)qp(

),x(P



x x

The multiplication x

)qp( , that has a velocity dimension

The ratio 2

x 2

has the dimension of one diffusion coefficient (L2T

-1); it is recognized as dispersion

coefficient (D)

It is not difficult to observe that by this simple stochastic model of liquid flow inside the porous body we obtain that the net flow velocity (w) and the dispersion coefficient (D), as model’s parameters, are determined by the porous body structure.

2

2

x

),x(PD

x

),x(Pw

),x(P

More experimental measurements for flow characterization inside the porous body show that the dispersion coefficient has not constant value; for more body structures it frequently presents a time or diffusing species concentration dependency.

More complexes models has been produced for characterization of these situations. One of the first models that intends to produce a response to this problem is recognized as models of motion with states having multiple velocities



x x

With this model the liquid element evolves inside of porous body with random motions having the velocities

m,..1i,vi

.

These random jumps of velocity from a state to other can be explained by random changes of the flow pore

section or by random changes of the flow rate that go in each pore from each pull road coupling of pores from

the porous body

The completion of this description is given by the consideration that accepts a classical states connection. So

here the elementary states connection becomes a Markov type:ij

*

ijijapp .

For this stochastic description the probabilities balance gives the relation:

1,...mi , )vx(Pp),x(Pij

m

1jjii

),x(P),x(Px

),x(Pv

),x(Pj

m

ij,1jjii

m

ij,1jij

i

i

i

Relation (4.265) shows that the time evolution of the fraction of the fluid particles that attain at time the

position x with the velocity i

v is determined by the following particles types :a) the particles having the

specified velocity i

v that leave the position x; b) the particles having the specified velocity i

v that arrive at

position x ; the particles that arrive at position x and change their velocity from j

v to i

v .



x x

For the particular case where we have two evolution states for the fluid velocity ( vv1

, vv2

) the

general model (4.265) comes to the relations set (4.266). Here the consideration 2112

shows that we

consider the case of the isotropic porous body.

),x(P),x(Px

),x(Pv

),x(P21

11

),x(P),x(Px

),x(Pv

),x(P12

22

),x(P),x(P),x(P21

2

22

2

2

x

),x(P

2

v),x(P

2

1),x(PHyperbolic model (new in chemical engineering)

2

22

x

),x(P

2

v),x(PParabolic model (clasic in chemical engineering)

0for x 0

0for x 0)0,x(P

0for x 0

0for x 1)0,x(P

,



x x

The solution for the couple of the model equation and the above specified conditions is obtained by using the Crank observation about the solution existence of an unsteady one dimensional diffusion equation when for a partly finite medium we have a response to a unitary impulse start

0 x

x

impimpimp d),(Pd),(Pd),)x(P),x(P

The particularization of this last expression to the specified problem contains the following observations: a)

,x(P is normalized having values inside of the interval [0, 1]; b) ,x(P is symmetric with respect to the

plane 0x . So we can write:

x

0

imp 0for x d),(P2

1),x(P

0

x

imp 0for x d),(P2

1),x(P

The results for the probabilities ),x(P are given by the following relations:



d)v2

exp(v22

1)P(x, ; 0x

2

2x

02 )

v2

x(erf

2

1

2

1dze

1

2

1

2

z

v2

x

0

2

2

2v2

x(erf

2

1

2

1)P(x, ; 0x Parabolic model

dv

1I

v1

1

v1I

v2

e

2

1)P(x, ; 0x

x

022

2

1

22

222

2

0

vx0for xvv

Ixv

xv2xv

vIe

2

1

1n

222

n

2

n

222

0

vfor x 0),x(P Hyperbolic model


Species displacement and transfer in a porous media- Liquid motion inside the porous media P(x,τ)

x/[v2τ/α]0.

5

-2 -1 0 2 1

0.

5

1

increase

P(x,τ)

x/[v2τ/α]0.5

-2 -1 0 2 1

0.5

1

increase

½ exp(- )

Differences between parabolic (up presented) and

hyperbolic (below presented) ),x(P evolution

This discussion, about the deviation of the stochastic model to the parabolic or to the hyperbolic model for the species transport inside the porous body, shows that this problem presents two solutions; the first solution ( frequently used) to the parabolic model and the second one based on the hyperbolic model. For some practical methods for porous body behaviour characterization such as liquid or gaseous permeation methods, the data interpretation by the hyperbolic model can be of interest

2

v

2

)x(limD

22

0,0x

For the porous body with constitutive elements of height dimension such as the fixed packed bed, where the

characteristic dimension is that of the packed element (diameter d of the packed body), the frequency of the

velocity change is d/v (after each passing over a packed body the local fluid velocity v changes its

direction)

Now if we use this value of in the dispersion coefficient we obtain the famous relation 2D/)vd(Pe .


Species displacement and transfer in a porous media- Deep bed filtration

The deep bed filtration clarifies some suspensions with small content in solid by their flowing into length granular bed formed by fine particles. During the flowing inside the granular bed, between one particle from the suspension and one particle from the granular bed there appear interaction forces that determine the fixing of the particle from the suspension to the particle of the bed. This elementary process occurs in more points placed between suspension input and its bed exit. The quantity of the solid retained at one local section of flow cannot exceed the quantity determined by the granular bed open spaces hold-up.When the retained quantity comes near to the quantity determined by the bed open spaces we affirm that the bed is cloggedAfter clogging we regenerate the granular bed by liquid fluidization when the retained solid goes out from the bed;



i

Micro-particle moving

around of the

deposition bed particle

Micro-

particle

input

Flow direction

Porous bed particle

External liquid

boundary layer

Retained

particle

Deposition

trajectory

Non-

deposition

trajectory

Figure 4.34 Micro-particle retention by one element of the fixed bed structure

- the inertial force, the gravitational force, the diffusion force, the laminar flow force, the electrostatic force, the Van der Waals, the hydrodinamic adhesion force



The filtration coefficient model (Mint model)

This coefficient noted as )c,(ss0

depends on its initial value (0) and on the local concentration of the

retained solid around the bed deposition elements (ss

c ). It is defined as the fraction of the solid retained from

the suspension in an elementary length of the granular bed

dx

1

c

dc)c,(

vs

vs

ss0

ss

f

ss

vs

vsc

w

1c

G

A

dx

dc

0 ; cwc

vs0f

ss

0 ; ccwc

ssvs0f

ss



ss

f

vs0

vs cw

cx

c

00

2

ss

f

vsvs c

w

c

x

c

0x

cc

x

cvsvs

0

vs

2

0c 0x 0vs

0vvscc 0x 0

1nn

1n

0

0

0v

vs )exp(T)!1n(

)x()xexp(

c

c

)!2n(

)(TT

2n

1nn

)exp(T1

1i

2/1

0i

2/i

0

0

0v

vs ])x[(Ix

)x((expc

c


Dintr-un experiment bazat pe teoria Mint s-a stabilit că pentru un strat filtrant valoarea iniţială

a coeficientului de filtrare este 0 = 4 m-1

, iar valoarea coeficientului de detaşare este =

0,057 h-1

. Pentru un filtru lung de 0,1 m realizat în totalitate din acelaşi material, calculaţi

calitatea filtrantului, ca procentaj din valoarea iniţială, la începutul operaţiei şi după 6 ore de

lucru.


Numerical application

At process start: vsvs cx

c0

)exp( 0

0

Lc

c

vs

vs

Cu datele numerice 0 = 4 m-1

, L = 0,1 se obţine %6767,0)1,04exp(0vs

vs

c

c.

n ( 0L)n-1 (n-1)! ( )n-2 (n-2)! Tn exp(- )

1 1 1 1 - - - 1,41 0,715 0,670

2 0,4 1 0,4 1 1 1 0,41 0,715 0,079

3 0,16 2 0,08 0,344 1 0,344 0,066 0,715 0,003

4 0,064 6 0,0107 0,117 2 0,058 0,008 0,715 0,000

)!1(

10

n

Ln

)!2(

)( 2

n

nAfter six hours:

%2,75752,00vs

vs

c

c



Numerical application

In an experimental investigation of a deep bed filtration has been used the retention of Fe(OH)3 gel form water in

fixed bed of sand particles. For a fixed bed with H0 = 20 cm and operating with wf = 2,5 m/h at 30 0 C and c0 = 6.75

mg/l, depending of particle diameter, the following results has obtained: Obtain a dependence of Mint parameters on

sand particle diameter.

D

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.05

0.25

0.65

0.85

0.95

0.99

1.0

1.0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.05

0.25

0.65

0.85

0.95

0.99

1.0

1.0

1.0

1.0

1.0

1.0

0

0

0

0

0

0

0

0

0

0.05

0.25

0.65

0.85

0.95

0.99

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

0

0

0

0

0

0

0.05

0.25

0.65

0.85

0.95

0.99

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

0

0

0

0

0.05

0.25

0.65

0.85

0.95

0.99

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

0

0

0.05

0.25

0.65

0.85

0.95

0.99

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

dp 0.2 0.3 0.4 0.5 0.6 0.7( )

1

1

00

0

)exp()!1(

)()exp(

n

n

n

v

vs Tn

xx

c

c

)!2(

)( 2

1n

TTn

nn)exp(1T

ftd1 D1

D0

x 0.2

f 0 exp 0 x exp( ) 0 x exp( ) 1( )

F 0

1

24

i

f 0 iftd1

i

2

0 50 0.2

R Minimize F 0

R36.049

0.214


Stochastic modeling

A deep bed stochastic model [4.5] identifies for micro-particle evolution in the

filtration bed two elementary processes:

a) a type I process that considers for micro-particles their motion with the

velocity vv1

; this velocity is induced by the surrounding flowing fluid; this

process type physically corresponds with the non deposition of the micro-particle;

b) a type II process that shows the possibility of the micro-particle to take

the deposition way; from the viewpoint of the motion the velocity of this process

is 0v2

.

The stochastic model accepts a Markov type connection between its two elementary states. So with 12

we

define the transition probability from the type I process to type II process. The transition probability from a

type II process to a type I process is21

. By ),x(P1

and ),x(P2

we note the probability to locate the

micro-particle at the position x and time with a type I evolution respectively with a type II evolution

),x(P),x(Px

),x(Pv

),x(P221112

11

),x(P),x(P),x(P

112221

2


0vvsc/),x(c ),x(P),x(P),x(P

21


Stochastic modeling

0),x(P

)(x

),x(Pv

x

),x(Pv

),x(P122121

2

2

2

By considering the combined variable 2/vxz we eliminate the mixed partly differential term from the

equation

z

),2/vz(P)(

2

v

z

),2/vz(P

4

v),2/vz(P12212

22

2

2

0),2/vz(P

)(1221

0)P(z,x z , 0 x, 0

0P)P(z, 0z , 0 x, 0

The important values of the jumping frequencies from one state to other characterize, as specific, the common

deep bed filtration. This observation permits the transformation of the above-presented hyperbolic model into

the parabolic model

0

),2

vz(P

z

),2

vz(P

2

v

z

),2

vz(P

)(4

v

1221

1221

2

2

1221

2

0)P(z,x z , 0 x, 0

0P)P(z, 0z , 0 x, 0



Stochastic modeling

1221

2

1221

12

01221

12

v

vx

erf12

1

P

)P(x, , 0

vx

1221

2

1221

12

01221

12

v

vx

erf12

1

P

)P(x, , 0

vx

Now we go on and complete the validation of discussed model. As it is known only the experimental

investigation can validate or invalidate one process model. For this concrete case we appeal to the

experimental data for the filtration of a dilute Fe(OH)3 suspension (no more than 0.1 g Fe(OH)3 /l ) in a sand

bed with various heights and constitution particle diameters. The considered experiments report the

measurements at constant filtrate flow rate and consist in the time evolution of the Fe(OH)3 concentration at

the bed exit when at their input the solid concentration remains unchanged

Turbidity(%)%)%

2 4 6 8 10

H1

H2

0

100

Time (h)

The response curves at deep bed filtration

),0(P/),H(Pc/),H(cvovs


Deep bed filtration

factorFactor value

Stochastic model parameters

12 21

H [cm]

t=200C

Gv=20cc/min

dg=0.5-0.31mm

C0=6.75mg/lFe(OH)3

2 1.14 326

3 0.592 458.9

5 0.262 420.22

6 17.66 9611.41

t [0C]

H=6cm

Gv=20cc/min

dg=0.5-0.31mm

C0=6.75mg/lFe(OH)3

20 17.95 9805.1

30 2.6 2748.14

35 22.71 1347.5

40 4.337 3028

Gv[cm3/min]

H=6cm

t=300C

dg=0.5-0.31mm

C0=6.75mg/lFe(OH)3

20 3.054 3020.66

30 13.34 6698.96

40 62.08 22605.24

50 118.2 42908.68

C0[mg/l] Fe(OH)3

H=6cm

t=300C

dg=0.5-0.31mm

Gv=50cm3/min

6.75 111.7 40493

13.49 82.7 28999.5

26.98 34.18 10294.41

dg [mm]

H=6cm

C0=6.75mg/lFe(OH)3

Tf=300

Gv=50cm3/min

0.31-0.2 754.98 355456.89

0.5-031 110.65 39773

0.63-0.5 22.409 8795.98

0.85-0.63 23.82 6449.82

The process factors influence on the stochastic model’s parameters


Stochastic modeling

dg28.256C872.2Gv003.375.67012

dg10023.1C1268Gv111110081.3 5

0

4

21


Firstly it is observable that assumption of height values for

2112 and parameters is excellently covered by the experimentally

starting data. Secondly, we find out that all process factors

influence the stochastic model’s parameters values.

flow in porous media

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