An+overview+on+nonlinear+porous+flow+in+low+permeability+porous+media.pdf

7/28/2019 An+overview+on+nonlinear+porous+flow+in+low+permeability+porous+media.pdf

1/8

THEORETICAL & APPLIED MECHANICS LETTERS 3, 022001 (2013)

An overview on nonlinear porous flow in low permeability porous media

Yanzhang Huang, Zhengming Yang,a) Ying He, and Xuewu WangThe Institute of Porous Flow and Fluid Mechanics, Chinese Academy of Sciences, Hebei Langfang 065007, China

(Received 10 January 2013; accepted 21 January 2013; published online 10 March 2013)

Abstract This paper gives an overview on nonlinear porous flow in low permeability porous media,reveals the microscopic mechanisms of flows, and clarifies properties of porous flow fluids. It shows

that, deviating from Darcys linear law, the porous flow characteristics obey a nonlinear law in alow-permeability porous medium, and the viscosity of the porous flow fluid and the permeabilityvalues of water and oil are not constants. Based on these characters, a new porous flow model,which can better describe low permeability reservoir, is established. This model can describe variouspatterns of porous flow, as Darcys linear law does. All the parameters involved in the model, havingdefinite physical meanings, can be obtained directly from the experiments. c 2013 The ChineseSociety of Theoretical and Applied Mechanics. [doi:10.1063/2.1302201]

Keywords low permeability porous media, nonlinear porous flow, porous flow equation, porous flowfluid

I. INTRODUCTION

Low permeability reserves cover a very large pro-portion of the proven reserves worldwide. And Chinasdeveloped low permeability oilfields are also tremen-dously increasing in the recent ten years.1 For provid-ing guidance to optimal exploitation of low permeabil-ity oilfields, it is an urgent demand to have an in-depthundersanding of the porous flow law of fluids in low per-meability porous media. The porous flow theory basedon Darcys linear law has a history of over 100 years.With pioneering contributions of great many outstand-ing scientists, the porous flow theory has been system-atically developed and applied in various engineeringproblems. However, engineers have faced challenges

with the progress of production, such as low perme-ability oilfield development. In low permeability porousmedia, people found that the flow characteristics re-markably deviate from Darcys linear law and show anonlinear flow pattern. Applications of porous flow the-ory based on Darcys law had become infeasible for low-permeability reservoirs. It was timely to advance thenew theories and new methodologies for more effectivelyexploit the low-permeability reservoirs.

The early investigations on fluid flow through low-permeability porous media were connected mainly withwater conservancy projects,26 showing that there wasa threshold pressure gradient and non-Newtonian be-havior of fluid in small channel. Marhasin paid great

attention to examing the variation of physi-chemicalcharacter in oil boundary layer.7 In China, professorYan Qinglai et al. set up the experiment of porous flowin low-permeability reservoirs,8 and identified how theflow characterization drifted off the Darcys linear law.Huang Yanzhang proposed a new concept of porous flowfluid along with a physical occurrence model in porousmedia, explained the basic principles of porous flow fluidand gave equations of viscosity and porous flow law.9

a)Corresponding author. Email: [email protected].

In a porous medium, the fluid is a porous flow fluidrather than a bulk fluid. It contains both bulk fluid and

boundary fluid. Boundary fluid refers to that whoseproperties are affected by the liquidsolid interfacialforce. It abuts against the pore wall to form a bound-ary layer. It is non-flowable or can only be derivenunder a considerable pressure gradient. The propertiesof the porous flow fluid are based on the properties ofthe bulk fluid, boundary fluid, porous medium and thepressure gradient. In a porous medium, one part of theporous flow fluid is flowable and the other part is non-flowable. The same fluid presents different viscosity val-ues in pores of different sizes. In a large pore channel, itsviscosity is lower, while in a small pore channel, its vis-cosity become higher. In addition, the boundary fluidshows non-Newtonian flow characteristics. The viscos-ity of the porous flow fluid is not a constant, and it isan important factor affecting the nonlinear flow in lowpermeability porous media.

It is also necessary to note a special feature of theporous media, that is, its selectivity for fluid tryingto pass through them. In a low permeability porousmedium, its selectivity for fluid passing through it isincreasing if the throat size is reduced to a certain de-gree. It allows gas to pass but prevents water and oilfrom passing or can only allow water and oil to passunder a higher pressure gradient. The lower the perme-ability is, the more obvious the characteristic is. Thisprompts people to pay more attention on throat dis-

tribution in porous media. This physical mechanismaffects numerous flow parameters.10 It is an importantfactor causing the fluid to follow the nonlinear flow pat-tern in low permeability porous media. This shows thepermeability values of water and oil are not constantsin a low permeability porous medium.

A lot of experiments show that, due to the influ-ence of the above physical mechanisms, the porous flowin a low permeability porous medium has the follow-ing features:1115 nonlinear porous flow pattern withthreshold pressure gradient, basic flow curve including anonlinear segment and a quasi-linear segment, and with
http://dx.doi.org/10.1063/2.1302201http://dx.doi.org/10.1063/2.1302201


2/8

022001-2 Y. Z. Huang, Z. M. Yang, and Y. He, et al. Theor. Appl. Mech. Lett. 3, 022001 (2013)

0 2 4 6

0 2 4 6

6.0

5.5

5.0

4.5

4.0

3.5

3.0

H/mm

C

24

20

16

12

8

H/mm

C

Experimental pointsFitting curve


(a) Asphaltene

(b) Colloid

Fig. 1. Asphaltene and Colloid content distributions. H isthe distance from pore wall.

a minimum threshold pressure gradient (true thresholdpressure gradient) and average threshold pressure gra-dient (quasi-threshold pressure gradient). Many Chi-nese scientists in this field have proposed their porousflow equations.1619 At present, non-linear porous flowtheories and mathematical simulation method based onnon-linear porous flow law have found wide applicationsin the exploitation of low-permeability oil fields.20

II. MICROSCOPIC MECHANISM OF FLOWS INLOW-PERMEABILITY POROUS MEDIA

Key factors affecting the vary of porous flow laware properties of the porous flow fluid, properties of theporous media and the flow conditions. Change of onlyone of them may affect the vary of flow law. Lets knowwhat is changed in key factors? What are microscopicmechanisms of nonlinear porous flow?

A. Concept of porous flow fluid

In a porous medium, there is a considerable spe-cific surface, and therefore, the intermolecular force atthe liquidsolid interface is very obvious. Under the in-termolecular force, the liquid molecules near the liquidsolid interface are orderly distributed. A boundary layeris developed on the liquidsolid interface. The proper-ties of the fluid within the boundary layer are differentfrom those of the bulk fluid. Its viscosity is higher than

0 20 40

0 20 40

0.12

0.10

0.08

0.06

0.17

0.15

0.13

0.11

0.09

0.07

th

th

Viscosity/(Pa.s)

Viscosity/(Pa.s)

(b)

(a)

2.1 mm gap

20.3 mm gap

2.1 mm gap

20.3 mm gap

Fig. 2. Crude oil viscosity vs. time.

that of the bulk fluid. Markhasin7 carried out studyin this aspect. He pointed out that colloid and asphal-tene of crude oil are orderly distributed in pores andchannels (Fig. 1).

He also introduced how to use the energy transfer

approach to measure the viscosity values of crude oilin gap of different sizes. He stated that, the same oilsample has a lower viscosity in a large gap and a higherviscosity in a small gap (Fig. 2). This shows that, theproportion of boundary-layer crude oil in smaller gap ishigher than that in larger gap.

Thus, we propose a new concept of porous flow fluidalong with a physical occurrence model in porous media(Fig. 3).9

The porous flow fluid contains both bulk fluid andboundary fluid. The boundary fluid abuts against thepore wall to form a boundary layer. Bulk fluid appearsin the central section of the large pore.

Basic concepts of porous flow fluid are as follows areas follows:(1) The porous flow fluid is a fluid in the porous

flow environment, and it is different from a general bulkfluid.

(2) The porous flow fluid is composed of bulk fluid(free fluid) and boundary fluid (bound fluid).

(3) Bulk fluid refers to a fluid whose properties arenot affected by the interface phenomenon, while bound-ary fluid refers to a fluid whose properties are affectedby the interface phenomenon. The boundary fluid isevolved due to the interaction between molecules at the


3/8

022001-3 An overview on nonlinear porous flow Theor. Appl. Mech. Lett. 3, 022001 (2013)

Pore channel wall

Boundary fluid

Bulk fluid

Fig. 3. Occurrence of porous flow fluid in a pore channel.

liquidsolid interface.(4) The bulk fluid is at the central section of the

large pore and the boundary fluid is on the pore wall toform a boundary layer, as shown in Fig. 3. One part ofthe porous flow fluid is flowable and the other part isnon-flowable.

(5) The properties of the boundary fluid have itsspecial patterns. The pore system of a porous mediumis filled with fluid, some molecules of which interactwith the molecules on the surfaces of the pores. Thus,the concentration of molecules on the pore surfaces ismuch higher than that of molecules away from the poresurfaces. The molecular concentration varies with thedistance of molecules away from the pore surface. Thisresults in variations of other physical and chemical prop-erties. So, in a porous flow environment, the propertiesof a porous flow fluid have its special patterns.

(6) The properties of the porous flow fluid are based

on the properties of the bulk fluid, boundary fluid,porous medium and the pressure gradient. So, the vis-cosity of the porous flow fluid is not a constant.

B. Viscosity of porous flow fluid9

The viscosity of oil in pore channel can be ex-pressed by the following equation

= A1 + (1 A) 2, (1)where 1 is the viscosity of boundary fluid, 2 is the

viscosity of bulk fluid, and A is the ratio of volumeof boundary oil to total volume of oil and it can beexpressed as

A = 2h

r0(

h

r0

)2, (2)

where h is the boundary layer thickness, and r0 is thepore radius. The pore radius has the following relationwith the permeability K

r0 = 0.35

K, (3)

0 10 20 30 40 50 60

1.0

0.8

0.6

0.4

0.2

0

Gas permeability/mD

Water/gas

permeabilityratio


Fig. 4. Water/gas permeability ratio vs. gas permeability.(1 mD = 103 m2)

thus

A =2h

0.35

K(

h

0.35

K

)2. (4)

To sum up, the viscosity of crude oil can be expressedas follows

= 2{ 2h

0.35K ( h

0.35K)2 (

12

1)

+ 1

}. (5)

C. Selectivity of porous medium for fluid trying to passthrough it

Porous media have a special feature, that is, its se-lectivity for fluid trying to pass through them. Whenthe throat radius reduces to a certain degree, its selec-

tivity for fluid passing through it increases, allowing gasrather than water and oil to pass through it or allowingwater and oil to pass under a higher pressure gradient.The lower its permeability is, the more obvious the fea-ture is, as shown in Fig. 4. This physical mechanismshows that, in a low permeability porous medium, thepermeability values of water and oil are not constants.

Figure 4 has at least two pieces of information toremind us: (1) When the gas permeability decreases,the water permeability rapidly decreases, indicatingthat the porous mediums selectivity for fluid passingthrough it increases obviously. In this case, if we usegas permeability to evaluate the reservoir, it will bring

about considerable deviation; (2) In the low permeabil-ity part, the discrete degree of experiment points in-creases, indicating that the effect of the pore structureand pore throat distribution on the flow process be-comes greater.

The properties of porous media affect the displace-ment process and efficiency (Fig. 5).

In Fig. 5(b), s/p are degrees of recovery degreevalues at unit pressure drop, cores with different perme-ability values have their respectively peak values, andlower permeability means higher displacement pressure(higher pressure gradient) required to achieve the peak


4/8


0 1 2 3 4

70

60

50

40

30

20

10

0

908070605040302010

0

Displacement pressure/MPa

0 1 2 3 4

Displacement pressure/MPa

0.028md

0.031md

0.120md

0.162md

0.374mdRecovery/%

s/p

(a) (b)

Fig. 5. Impact of properties of porous media on displacement process.

100 90 80 70 60 50 40 30 20 10 0

102

101

100

10-1

10-2

SHg

Pc

MPa

Fig. 6. Constant pressure mercury injection curve.

value. This physical mechanism affects numerous flowparameters, prompting us to carry out more in-depthinvestigation into properties of porous media. In addi-tion to porosity and permeability characteristics, pore

structure and pore throat distribution along with theirimpacts on the porous flow process also need to be fur-ther studied.

D. Pore structure characteristics of low permeabilityporous media

Low or ultra-low permeability reservoirs mainlyhave small pores and thin pore throats. In such porousmedia, pores are very small, throats are very thin, andthe pore/throat ratio is very high. Such reservoirs gen-erally have high initial water saturation and low initial

oil saturation. The mercury injection test shows a lowmercury withdrawal efficiency, indicating that the hard-to-recover reservoirs account for a considerable propor-tion of original oil in place (OOIP). In case of very lowpressure, i.e., when the pressure gradient increases ob-veously, mercury withdrawal begins as shown in Fig. 6.It also shows that the flow resistance is substantial. Inaddition, there is a threshold pressure gradient.

The above mercury injection data can be used tomake a table and a graph showing the volumetric pro-portions of pores of different sizes (Fig. 7). Figure 7shows that, lower permeability corresponds to higher

Table 1. Volumetric proportions of minute pores in reser-voir.

Permeability/ Volumetric proportions of various pores

mD < 1 m < 0.75 m < 0.5 m

> 1000 < 18 < 16 < 13

1 000 18 16 13

500 21 18 14

200 25 21 17

100 30 25 20

50 35 29 24

20 43 35 30

10 50 40 33

5 60 47 37

3 70 53 43

1 88 70 57

0 50 100 150 200

100

80

60

40

20

0

Permeability/mD

Volumetricproportions

ofminutespores/%


5/8


0 2 4 6 8 10 12

Throat radius/mm

14

12

10

8

6

4

2

0

Frequency/%

k=0.13k=1.56k=2.17k=2.81k=3.37k=18.7k=47.08k=73.67k=102.35k=128.6

Fig. 8. Throat distribution curves of porous media of dif-ferent permeability values.

to water/oil permeability.This technology can also be used to identify the

throat distribution differences between rock cores of thesame permeability and evaluate their flow capabilities(Fig. 9).

These lab findings show that, even with regard toreservoirs with similar permeability values, the differ-ence in throat distribution leads to the difference inoil/water permeability and thus affects development ef-fectiveness of development.

E. Relations between porous media properties andflow parameters

Properties of porous media affect numerous flowparameters,10 as shown in Fig. 10.

Figure 10 shows that, as the permeability decreases,the parameters adverse to the flow process increase,which will affect the change in porous flow pattern.

III. THE POROUS FLOW LAW AND MATHEMATICALEQUATIONS

A. Description of porous flow characteristics andporous flow law

A lot of experiments show that, the porous flowcharacteristics and porous flow law of low permeabilityreservoirs can be generally described as follows: suchreservoirs have a nonlinear porous flow law with thresh-old pressure gradient, as shown in Fig. 11.

Their basic porous flow characteristics are:(1) When the pressure gradient is lower than a cer-tain value, the fluid does not move. The certain pressuregradient is called minimum threshold pressure gradient(true threshold pressure gradient), expressed as a.

(2) Within the range of low pressure gradients, theincrease in flow velocity follows the downcurved seg-ment.

(3) When the pressure gradient is high, the increasein flow velocity is linear.

(4) The straight line segment extends to cross thepressure gradient axis at some point, but does not go

through the origin of coordinates. The point is referredto as average threshold pressure gradient (quasi- thresh-old pressure gradient), expressed as c.

(5) The pressure gradient corresponding to thepoint D is maximum threshold pressure gradient, ex-pressed as b.

B. Flow pattern of low-permeability nonlinear flow

The Reynolds number Re can be expressed by usingthe following equation

Re =4

2K

3/22f, (6)

where is the porosity, is the porous flow velocity, isthe density, and f is the viscosity factor. The resistancecoefficient in case of the porous flow in low-permeabilityporous media can be written as

=23/2K

2

(p

L i

), (7)

where p is flow pressure difference, L is the length,and i is threshold pressure gradient, i = a, b, c. Theabove equations can be converted into

Re =8

2K

2f

(p

L i

), (8)

where

f =[5.714hK

8.613h2

K](1

2 1) + 1, (9)Substitute the porous flow equation

=K

2f

(p

L i

)(10)

into Eq. (8), and one gets

Re = 11.2, (11)

which indicates that in case of low-permeability andlow-velocity, the porous flow follows the laminar flow

pattern. It is generally considered that when the prod-uct of the Reynolds number and resistance coefficientis a constant, the laminar flow pattern exists and theporous flow follows the linear law. Nowadays, researchresults have showed that, in a low permeability porousmedium, porous flow follows a nonlinear law, but theproduct of the Reynolds number and resistance coef-ficient is still a constant and the flow pattern is still alaminar flow pattern. This tells us that, in a low perme-ability porous medium, the viscosity-related equationsare still applicable even though the porous flow followsa nonlinear law.


6/8


0 1 2 3Throat radius/mm

0 1 2 3Throat radius/mm

0 1 2 3 4 5Throat radius/mm Throat radius/mm

0 2 4 6 8 10 12

50

40

30

20

10

0

50

40

30

20

10

0

20

10

0

10

5

0

Di

stributionfrequency/%

Distributionfrequency/%



Daqing cores (0.22 mD)

Changqing cores (0.18 mD)







(a) (b)

(c) (d)

Fig. 9. Throat distribution differences between rock cores from different oil regions.

Capillary pressureThreshold pressure gradient

Non-Darcy's flow area Darcy's flow area

Average throat radius

Mobility

Oil saturation

Movable fluid percentage

Boundary layer thickness

Reservoir

Cap rock Ultra-low K Special-low Low K Medium K High K

10-3

10-2

10-1

100

101

102

103

K/mD

Fig. 10. Relations between porous media properties and flow parameters.

C. Nonlinear porous flow equations

1. Preexisting nonlinear porous flow equations

Huang9 put forward Eq. (8) and recommended asegmented process

v = 0,p

L a, (12)

v =K

f

(p

L a

), a

p

L b, (13)

v =K

(p

L c

),

p

L b, (14)

where v is flow velocity, a is the minimum thresholdpressure gradient, b is the maximum threshold pres-sure gradient, and c is the average threshold pressuregradient.


7/8


a c b

Pressure gradient/(MPa. m-1)

Flow

velocity/(m.

s-1)

A

E

tan =( )ko

Fig. 11. Pressure gradient vs. flow velocity.

Yao et al.16 put forward equations as followsIn ultra-low velocity area

p

L= 0. (15)

In low velocity area

v = c

(p

L

)1/(2n), (16)

where c is a parameter determined by experiment.In Darcy flow area

v =k

p

L. (17)

The above equations are based on a segmented function,which is difficult to apply although it is accurate. So,nonlinear porous flow equations based on a continuousfunction were developed.

Deng et al.17 put forward

v

[a1 +

a21 + b1v

]= p, (18)

where a1, a2, and b1 are experimental parameters.Yang et al.18 put forward

v =k

p

(1 1

a + b|p|)

, (19)

where a and b are experimental parameters.

Jiang et al.19

put forward

=K

1 1dp

d r

12dp

d r

(dp

d r 2

) dpd r , (20)

where 1 and 2 are experimental parameters. Theabove equations are based on a continuous function, andare applicable within a certain scope. However, it is re-quired to carry out matching of parameters therein, andsome parameters have indefinite physical meanings.

10-1

100

101

20

16

12

8

4

0

Permeability/mD

Permeability/mD

0 1 2 3 4

a b c

a b c

a

b

c

a

b

c

102

101

100

10-1

10-2

Fig. 12. a, b, and c values of rock cores of differentpermeability values.

2. New nonlinear porous flow equation

We put forward a new nonlinear porous flow equa-

tion

v =

(k

)o

dp

d x

(1 c

dp/ d x + c a

)(21)

where (k/)o is the quasi-linear slope. It is a nonlin-ear porous flow equation based on a three-parametercontinuous function, all of those parameters are fromexperiments and have definite physical meanings. Val-ues in Fig. 11 are parameters of the equation. They canbe used to make parameter charts for direct reference,as shown in Fig. 12.

The nonlinear porous flow equation has been ver-ified (Fig. 13). Figure 13 gives several discrete points

based on experimental data. According to the distribu-tion of the discrete points, we can conclude: minimumthreshold pressure gradient, average threshold pressuregradient, and quasi-linear slope are calculation param-eters of the nonlinear porous flow equation. It is nec-essary to substitute them into the equation and makecalculation to check if the calculated curve is in compli-ance with the experimental data. It can be seen thatthe curve is in perfect compliance with the experimentalpoints.

Equation (21) is a nonlinear porous flow equationbased on a three-parameter function. It can describe


8/8


0 0.5 1.0 1.5 2.0 2.5

11.6

10.2

8.5

6.8

5.1

3.4

1.70

Pressure gradient/(MPa. m-2)

Fl

ow

velocity/(m.

s-1)

ExperimentCalculation

10-5

Fig. 13. Comparison of calculated curve and experimentaldata.

the nonlinear porous flow pattern with threshold pres-sure gradients. It is a general equation. In case ofa = c = 0, the equation turns into a porous flowequation describing Bingham fluid. In case of a = 0,

the equation turns into a nonlinear porous flow equa-tion with average threshold pressure gradients; in caseof a = c = 0, it turns into Darcy linear flow.

IV. CONCLUSIONS

(1) Production developments such as water conser-vancy projects and exploitation of low-permeability oil-field promote the development of porous flow theories.In a low-permeability porous medium, the porous flowcharacteristics evidently deviate from Darcys linear law

and exhibit nonlinear features.(2) Fluid property study shows that the fluid in

a porous media belongs to a porous flow fluid ratherthan a bulk fluid. It contains both free bulk fluid andbound boundary fluid. We proposed a physical occur-rence model of porous flow fluid in porous media andmathematical equation of viscosity. The viscosity of theporous flow fluid is no longer a constant.

(3) Each porous medium shows the selectivity forfluid to pass through it. In a low permeability porousmedium with the throat radius reduces to a certain de-gree, its selectivity becomes notable, allowing gas ratherthan water and oil to pass through it or allowing thewater and oil to pass under a higher pressure gradient.

The lower its permeability is, the more obvious the fea-ture is. This physical mechanism shows that in a lowpermeability porous medium, the permeability values ofwater and oil are variable.

(4) The new nonlinear porous flow equation we pro-posed is very general. Through parameter conversion, itcan describe various porous flow laws: such as nonlinearporous flow law with minimum and average threshold

pressure gradient, nonlinear porous flow law with an av-erage threshold pressure gradient, a quasi-linear porousflow law with an average threshold pressure gradient,and also Darcys linear law.

1. W. R. Hu, China Petroleum Enterprise 6, 54 (2009).

2. G. I. Barenblatt, Journal of Applied Mathematical Mechanics24, 1286 (1960).

3. W. von Engelhardt, and W. L. M. Tunn, HeidelbergerBeitrage zur Mineralogie und Petrographie 2, 12 (1954).

4. D. Swartzendruber, Geophys. Res. 13, 5205 (1962).

5. R. J. Miller, Proc. Soil. Sci. Soc. Am. 27, 606 (1963).

6. G. H. Bolt, and P. H. Groenevelt, L. A. S. H. 2, 14 (1969).

7. E. L. Marhasin, The Physical Chemistry Mechanism of OilReservoir(Petroleum Industry Press, Beijing, 1987).

8. Q. L. Yang, Q. X. He, and X. J. Ren, et al., Technology ofExploration and Development of Low Permeability Oil/gasReservoir(Petroleum Industry Press, Beijing, 1993).

9. Y. Z. Huang, Porous Flow Mechanisms in Low-permeabilityReservoir(Petroleum Industry Press, Beijing, 1998).

10. Z. M. Yang, Study on Porous Flow Mechanisms in Low Per-meability Reservoir and Its Application [Ph. D. Thesis]. Bei-

jing: University of Chinese Academy of Sciences, 2004.

11. Y. Z. Huang, Special Oil & Gas Reservoirs 4, 9 (1997).

12. F. Hao, L. S. Cheng, and O. Hassan, et al., Petroleum Scienceand Technology 26, 1204 (2008).

13. H. Q. Song, W. Y. Zhu, and M. Wang, et al., PetroleumScience and Technology 28, 1700 (2010).

14. Q. Lei, W. Xiong, J. R. Yuan, et al., in: Behavior of Flowthrough Low Permeability Reservoirs, Sixth SPE/DOE IORSymposium, Tulsa, Oklahoma, USA, April 19-23, 2008.

15. H. Liu, H. Zhang, and Z. Wang, Petroleum Science and Tech-nology 29, 898 (2011).

16. Y. D. Yao, and J. L. Ge, Xinjiang Petroleum Geology 21, 213(2000).

17. Y. E. Deng, and C. Q. Liu, Petroleum Journal 22, 72 (2001).

18. Q. L. Yang, Z. M. Yang, and Y. F. Wang, et al., Drilling &Production Technology 30, 52 (2007).

19. R. Z. Jiang, and R. F. Yang, Nonlinear Percolation Theoryand Numerical Simulation Techniques in Low PermeabilityReservoirs (Petroleum Industry Press, Beijing, 2010).

20. Z. M. Yang, X. G. Liu, and Z. H. Zhang, et al. The Grad-ing Evaluation for the Ultra-Low Permeability Reservoir andthe Technique of Numerical Simulation in the Well PatternOptimization (Petroleum Industry Press, Beijing, 2012).

Documents

An+overview+on+nonlinear+porous+flow+in+low+permeability+porous+media.pdf