AU2019101643A4 - Analytical Fractal method for Calculating Flowback Capacity of Shale Reservoir - Google Patents

Abstract

The present application discloses an analytical fractal method for calculating flowback capacity of a shale reservoir, which includes the following steps: establishing a flowback model of one single capillary; considering a situation that slip length and viscosity change according to a pore diameter, and correcting the model; considering fractal characteristics of pore diameter distribution, and establishing a flowback model of the porous medium; and collecting relevant parameters and analyzing factors affecting flowback. The present application fully considers the characteristics such as the pore micro-scale effect, the pore channel tortuosity and the diameter distribution satisfying the fractal statistical relationship. Based on the Hagen-Poiseuille equation, considering the material balance principle, the boundary slip effect and the forced external force, the forced flowback model of the single capillary is obtained; on this basis, the effective slip length and effective viscosity are obtained by further correcting the slip length and the viscosity. ........ --- ----- 2 Figure 1 0 L (a) Flowback model without (b) Flowback model slip at boundaries with slip at boundaries Figure 2 - 1/3-

Description

........------- - 2

Figure 1

0 L

(a) Flowback model without (b) Flowback model slip at boundaries with slip at boundaries

Figure 2

- 1/3-

ANALYTICAL FRACTAL METHOD FOR CALCULATING FLOWBACK CAPACITY OF SHALE RESERVIOR CROSS REFERENCE TO RELATED APPLICATIONS

[0001] The present application claims the priority to Chinese Patent Application No. 201910620676.3 titled "CALCULATION METHOD FOR DETERMINING FLOWBACK LENGTH OF SHALE POROUS MEDIUM", filed with the China National Intellectual Property Administration on July 10, 2019, which is incorporated herein by reference in its entirety.

FIELD

[0002] The present application relates to the field of oil-gas field development, and in particular to an analytical fractal method for calculating flowback capacity of a shale reservoir.

BACKGROUND

[0003] There are many nanometer scale pores in shale reservoirs. In order to ensure the recovery efficiency and the economic efficiency of exploitation during mining, horizontal well drilling with multi-stage slick water fracturing is widely used. In a fracturing fluid flowback process, a flowback rate of a shale gas well is mostly between 9% and 50%, and a large amount of fracturing fluid is trapped in reservoir pores. The fracturing fluid spontaneously drawn into the nanometer scale pores is bound thereby and cannot be driven out, which is considered to be a main reason for a low flowback rate of the shale reservoir. At present, there is no research which combines boundary slip, wettability, porosity characteristics and forced external forces in the study of shale reservoir flowback.

[0004] Compared with conventional natural gas reservoirs, the shale reservoirs exhibit characteristics of pore microscale effect, pore tortuosity and diameter distribution satisfying the fractal statistical relationship. 1) Pore Microscale Effect: when a pore is in nanometer scale, the fluid flow in the pore no longer satisfies the macroscopic N-S equation. When a flow characteristic scale of the fluid is approximate to a boundary slip length of the fluid, the N-S equation needs to be corrected by the slip boundary (Zhanhua LI, Xu ZHENG. Problems and progress in micro-nanometer scale flow experimental research [J]. Experimental Fluid Mechanics, 2014, 28(03):1-11). 2) Pore Tortuosity: actual pores in the shale are not linear but irregularly curved, which further improves the complexity of shale fluid flowback research. 3) Diameter Distribution Satisfying the Fractal Statistical Relationship: pore diameters range from 0.7nm to 5[m, which is a very large span. This means that the shale porous medium cannot be simply viewed as a collection of multiple capillaries with same diameters (Mandelbrot B B, Wheeler J A. The fractal geometry of nature [M]. Birkh auser Verlag, 1983). Due to the boundary slip, the wettability and the diameter distribution satisfying the fractal statistical relationship, the flowback length thereby cannot be accurately calculated by the commonly used L-W equation for describing the shale fluid flowback.

[0005] At present, the commonly used method is to introduce the capillary shape factor and the capillary diameter geometric correction factor and consider a situation that the fluid flows vertically. There is no research which combines the boundary slip, the wettability, the porosity characteristics and the forced external forces in the study of shale reservoir flowback (Benavente, D., Lock. Predicting the Capillary Imbibition of Porous Rocks from Microstructure [J]. Transport in Porous Media, 2002, 49, 59-76; Vinogradova, 0.1., Koynov. Direct measurements of hydrophobic slippage using double-focus fluorescence cross-correlation [J]. Physical Review Letters, 2009, 102, 118-302; Wu, K., Chen. Wettability effect on nanoconfined water flow [J]. Proceedings of the National Academy of Sciences of the United States of America, 2017, 114, 33-58.). The microscale effect and tortuosity effect of the shale pores (that is, boundary conditions of the fluid flow differential equations are not corrected, a flowback model of a single capillary is inaccurate, and so is a flowback model of a porous medium) and the diameter distribution satisfying the fractal statistical relationship (that is, the diameter distribution cannot meet an actual situation that the shale pore diameter distribution has a large scale range) are considered by those above methods.

SUMMARY

[0006] In order to solve problems in the conventional technology, the present application provides an analytical fractal method for calculating flowback capacity of a shale reservoir. The technical solution is as follows:

[0007] In one aspect, an analytical fractal method for calculating flowback capacity of a shale reservoir is provided, which includes the following steps: step 1), establishing a flowback model of one single capillary; step 2), considering a situation that slip length and viscosity change according to a pore diameter, and correcting the model; step 3), considering fractal characteristics of pore diameter distribution, and establishing a flowback model of the porous medium; and step 4), collecting relevant parameters and analyzing factors affecting the flowback.

[0008] Further, step 1) includes:

analyzing flow of a fluid in a straight single capillary with circular equal sections.

[0009] When the fluid in the capillary can be deemed as constant, laminar, incompressible and having viscosity, the Navier-Stokes equantion can be simplified as the following equation:

p ddu ]_ dp Ap r dr dr dx L

where p represents viscosity of the fracturing fluid in units of mPa-s; r represents a

distance from any point of a circular capillary to the center thereof in a radial direction in

units m; R represents a radius of the circular capillary in units of m; u represents a flow

velocity of the incompressible fracturing fluid in units of m/s; represents a pressure

gradient when the fluid flows in the capillary in units of MPa/m; Ap represents a flow

pressure difference of the fluid in the capillary in units of MPa; and L represents a capillary

length in units of m.

[0010] The following equation is obtained by developing the equation (1): d 2 u 1 du 1 Ap dr2 + rdr +2 p L -0 (2)

[0011] After considering the boundary slip, the boundary condition becomes as the following equations: du Bd u (3) r =R,u, = LS r=R

[00121 The following equation is obtained by solving the above equations:

u= dp(r2 -R2 -2RL,)- I ( +2RL, r2 (4) 4pdx 4puL

[0013] A flow equation is obtained by a surface integral of the equation (4):

fR 1 Ap 2R4 Ap(4L, q=P(R2 + 2R - r2 )2nrdr -- 1+L )(5 0 4p L 8p L R (5)

[0014] After considering the forced external force, a pressure difference generated during self-priming is equal to a sum of a capillary force Pc and the forced external force PQZ:

40-cos O AP=-Pc APQP +±p = ddos ~ + P, QZ (6) 6

where d represents a diameter of the capillary in units of m.

[00151 Considering the material balance principle, the flow rate must be equal to a volume of the fluid absorbed by the pores per unit time (Benavente, D., Lock, P., Cura, M. A. G. D., and Ord6fiez, S. (2002). Predicting the Capillary Imbibition of Porous Rocks from Microstructure. Transport in Porous Media 49, 59-76.):

ntd 2 dL 4 dt (7)

[00161 The following equation is obtained by a time integral of simultaneous equations (5) to (7) and substituting boundary conditions of t=0, L=0:

d(-4-cosO+dPQz)1+8L,)(8) 16p d

[00171 Further, Step (2) includes:

[0018] (1) Slip length correction:

[0019] Liquid-wall interaction of a closed channel is strongly influenced by surface morphology and physicochemical characteristics of a solid, where the influence of the wettability of the boundary wall at a low shear rate is particularly apparent. The boundary slip occurs at a molecular level and is calculated by a contact angle of a given liquid under a given condition (Wu,K .,Chen .Wettability effect on nanoconfined water flow[J] .Proceedings of the National Academy of Sciences of the United States of America ,2017 ,114 ,33-58.):

L = C/(cosO+1) 2 (9)

where:

L, represents a true slip length of the wall fluid in units of nm;

C represents a constant with a value of 0.41 obtained by MD simulation; and

0 represents a wetting contact angle in units of degree.

[0020] The viscosity of the constrained fluid near the wall is significantly different from the viscosity of the free fluid, resulting in significant slippage at the fluid/fluid interface. In practical applications, a slip length of the constrained fluid which takes into account the true slip and the apparent slip effect should be replaced with effective slip length parameters:

Lle = L,, + Ls = -1 -+ L + Ls (10)

where:

Ls, represents an effective slip length in units of m;

La represents an apparent slip length in units of m;

Ls represents a true slip length of the wall fluid in units of m;

p. represents the viscosity of the free fluid in units of Pa-s; and

Pd represents the effective viscosity of the constrained fluid in units of Pa-s.

[0021] It can be seen from the equation (10), the effective slip length depends not only on wall wettability, but also on the fluid viscosity and the capillary size.

[0022] (2) Viscosity correction:

[0023] When the constrained fluid flows to nanometer scale pores, the viscosity of the fluid near the pore wall will no longer be accurately described by the viscosity of a rock core fluid.

The effective viscosity is highly dependent on the viscosity of the rock core fluid and an interface region. The effective viscosity of the constrained fluid can be represented by a weighted average of a volumetric region and the viscosity of the interface nanometer pores:

Pd iA d + F A Ad Ad]

where:

Pd represents the effective viscosity of the fluid in the volumetric region in units of Pa-s;

pi represents the viscosity of the fluid in the interface region in units of Pa-s;

Ad equals to 2rL d 2- -de , and represents an area of the interface region in

units of m 2;

d. represents a critical thickness of the constrained fluid in units of nm; and

A equals to 2, and represents a total sectional area in units of m 2 .

[0024] The viscosity of the fluid in the interface region is greatly influenced by the interaction of the wall and can be represented by the contact angle obtained by experiments and MD simulation:

P - 0.018 0+3.25 (12)

[0025] It can be seen from equation (12) that, compared with the free fluid, the viscosity of the fluid in the interface region varies greatly with the contact angle.

[0026] For ease of programming, calculating and writing, the expressions for the effective slip length L. and the effective viscosity p, are assigned as following equations:

Define x - - -0.0180 +3.25; y - _d - _ 2dc)2 P. Ad d

Then:

Pd-p [(x-1)Y+] ;

(1-x)y- d+

+ (x-)y+1 8

[0027] After the above correction, the slip length L, and the fluid viscosity p of the capillary force flowback model and the forced flowback model of the single capillary should be replaced with the effective slip length Ls, and the effective viscosity Pd:

d(-4o-cos +8dPz) (13). 16pd d

[0028] Further, step 3) includes:

[0029] (1) Statistical properties of the fractal porous medium

[0030] Many fractal theory studies show that the number of pores and the diameters of pores of the shale satisfy the following fractal power law relationship:

N(d c dDr (14)

where:

d represents a diameter of a unit model in units of m;

N(d) represents the number of unit models contained in the entire model, in units of piece and

Df represents a fractal dimensions of the pore, in a two dimensional plane, which

generally falls in the scale of 1 to 2.

[0031] In the shale pores, the relationship between all of the number of pores with diameter not less than d and the pore diameter satisfies the following relationship:

N(L ;> d) oc (d. / d)D' (15a)

[0032] If the minimum pore diameter of the pore is dmin, then the total number of pores N can be calculated by the following equation:

Nt =(dax/ d )Dr (15b)

[0033] By simultaneously differentiating on both sides of the above equation, the number of pores in the interval d to d+dd can be obtained:

-dN= DfdDd-(D+1)dd (16)

where, -dN>0 represents that the number of pores is inversely proportional to the diameter of the pores.

[0034] By dividing the equation (16) by the equation (15b), a percentage of the number of the pores in the interval from d to d+dd to the total number of the pores can be obtained:

-dN /N, =DfdmDnd-(D+1)dd= f(d)d d (17)

[0035] It can be seen from equations (3) to (7), a probability density function f(d) of the distribution of the number of the pores with respect to sizes of the pores is as follows:

f(d)=DdDd-(Df+) (18)

[0036] The above equation must meet a normalization condition:

d~ f(d)dd=1-(dmidm )Df =1 dmja (19)

[0037] A necessary and sufficient condition for the establishment of equation (19) is as follows:

(d / d.)D0)

[0038] Generally, the shale pores satisfy dmin/dmax<10-2 , at this time, the equation (20) is approximately established, so the fractal theory can be used to describe the shale porous medium.

[0039] An average diameter day of the shale pores is calculated by the following equation:

dav=* df(d)dd= D=d Df -1 (21)

[0040] In the fractal porous medium, a total area Ap of the shale pores is as follows:

2 A =--dm"d dN='ddr D4=d 2- dd D1 (1-#) (22) " d, 4 ( 2 -Df) dmx 4(2- Df)

[0041] (2) Fractal characteristics of tortuous streamlines

[0042] When the fluid flows through a porous medium with a complex spatial structure, an expression for describing a non-uniform pore tortuous streamline is as follows:

Lt( -, LD (23)

where:

c represents a scale of the measurement in units of m;

Lt represents an actual length of the streamline in units of m;

L represents a straight line distance in units of m; and

D, represents a fractal dimension of the tortuosity of the pores, which generally takes 1 to 2 in a two dimensional plane.

[0043] The scale of the measurement in equation (23) is replaced by the capillary diameter:

L,(d)= d'-DLoT (24)

[0044] Two sides of the above equation are differentiated:

vt(d)= Dt(L / d)'-'vo (25)

where, vt=dL/dt represents a velocity of the fluid in a curved channel, vo=dL/dt represents a velocity of the fluid in a straight channel. The equation (25) is integrated with f(d), and an actual average flow velocity, is as follows:

dam (d)f(d)dd= t DTDI d, DTLDO -1 d ( ( DT + DT -1 (26)

[0045] Through a further study about equation (26), DT, r and day satisfy the following expression:

DT =+ In r In(LO / dy) (27)

[0046] L/d is calculated by the following equation:

L /d Ld-- DfI 2p6 O _7 (28) # Df(2 - Df)

[0047] (3) Average tortuosity of pores

[0048] A definition of the tortuosity r is a ratio of a distance Lt by which the fluid flows through a curved flow path to a distance L by which the fluid flows through a straight flow path:

L (29)

[0049] A model of the average tortuosity related to the porosity is as follows:

r-1-1+-R+ (30)

[0050] The model is suitable for describing a case where space of the shale pores are all uniform cubes, not considering an actual situation of the shale pore distribution. Moreover:

K = d2o2 32r (31)

[0051] Through the equation (31), the average tortuosity can be obtained by just obtaining the permeability, porosity and pore radius of the shale. However, the model (31) is derived on the assumption that the shale spatial structure satisfies the uniform capillary bundle model, which cannot be directly applied to the fractal porous medium. Taking into account the limitation of the uniform capillary bundle model represented by the equation (31), the average tortuosity of the porous medium can be obtained by introducing the probability density function of the pore diameter distribution:

r= d"' df (d) d d 3 2K (32)

[0052] By substituting the fractal power law relationship (equation 18) of the pore diameter distribution into the equation (32), the average tortuosity calculation model can be obtained:

,'" df(d) d d= $ Dfd, 32Kd 32K Df -1 (33)

[0053] The average tortuosity calculation model is suitable for the fractal porous medium.

[0054] (4) Flowback model of porous medium

[0055] After considering the correction by the effective viscosity and the effective slip length, the equation (5) is rewritten as follows:

2TR? Ap( 4Ls q=1+± 8 L R ) (34)

[0056] After considering the forced external force, the flowback pressure difference is written as follows:

AP-Pc+ - 2c-cosO + (35)

[0057] Then the following equation can be obtained:

q =8R 14L 2o-cos0 8pdL R R R nd'o- co s0 1+ 2Id~ ~ + 4' u o +8)(6 s O~(36) 32pL d 128 d )

[0058] By integrating the flow rates of all single capillaries with a section area of A, a total flow rate Q can be obtained as follows:

Q=- Jd q(d)dN

[0059] An actual average flow velocity in the capillaries is equal to:

- Q -Av (38)

[0060] The following equation can be derived from the equation (26):

DT +Df -1 - dL DTDf di L?- t dt(39)

[0061] The flowback model of porous medium is obtained by simultaneous equations (36) to (39):

2 o-cosO(DT+Df-1)(2-Df) dF (I p 8Ld,";11((1-p +TD,)ia - _,l-DT-Df, DT 2D

4pd(1-#)DfdImDT 2+DT Df 1+DT-Df L = _ a3i-ItDTD p (DT Df QP ma - 1)(2-Df) +se ax', dina1((1- 3+DT-Dr ) 8L,,dr (1-p2+Dr -D

11 )DfdDT 3+DT -D, 2+D-D

(40).

[0062] Further, Step 4) includes:

[0063] Providing values for the parameters of the equation (40), and analyzing factors affecting the flowback.

[0064] Further, the providing values for the parameters of the equation (40) includes:

[0065] Providing values for the parameters such as wetting contact angle, water phase viscosity, gas-water interfacial tension, average tortuosity, minimum pore diameter, maximum pore diameter, fractal dimension of the pore, fractal dimension of the tortuosity, and porosity.

[0066] The beneficial effects brought by the technical solutions provided by the embodiment of the present application are as follows:

[0067] The present application provides an analytical fractal method for calculating flowback capacity of a shale reservoir based on full consideration of the characteristics such as the pore micro-scale effect, the pore channel tortuosity and the diameter distribution satisfying the fractal statistical relationship. Based on the Hagen-Poiseuille equation, considering the material balance principle, the boundary slip effect and the forced external force, the forced flowback model of the single capillary is obtained; on this basis, the effective slip length and effective viscosity are obtained by further correcting the slip length and the viscosity; and, considering that shale pores also have tortuous features and fractal features, the total flow of the porous medium is obtained by integrating the single capillary flow combined with the probability density function of the pore distribution, and the flowback model of the fractal porous medium is further obtained while considering the forced external force.

BRIEF DESCRIPTION OF THE DRAWINGS

[0068] To illustrate technical solutions in the embodiments of the present application more clearly, the drawings to be used in the description of the embodiments are described briefly hereinafter. Apparently, the drawings described hereinafter are only some embodiments of the present application, and other drawings may be obtained by those skilled in the art based on those drawings without creative effort.

[0069] Figure 1 is a schematic view showing laminar flow in a circular capillary according to embodiments of the present application;

[0070] Figure 2 is a schematic view of correcting boundary conditions by slip length according to the embodiments of the present application;

[0071] Figure 3 is a schematic view of flowback length change according to a wetting angle in embodiments of the present application;

[0072] Figure 4 is a schematic view of the flowback length change according to a forced external force of the embodiments of the present application;

[0073] Figure 5a is a schematic view of a trend of the flowback length change according to a pore maximum diameter of embodiments of the present application; and

[0074] Figure 5b is a schematic view of a trend of the flowback length change according to a pore minimum diameter of embodiments of the present application.

DETAILED DESCRIPTION OF EMBODIMENTS

[0075] In order to make objects, technical solutions and advantages of the present application clearer, the embodiments of the present application are further described below in conjunction with the accompanying drawings.

[0076] The present application provides an analytical fractal method for calculating flowback capacity of a shale reservoir, which is used for determining a unified apparent permeability model of multi-scale different flow states of shale gas. The method mainly includes the following steps:

step 1), establishing a flowback model of one single capillary;

step 2), considering a situation that slip length and viscosity change according to a pore diameter, and correcting the model;

step 3), considering fractal characteristics of pore diameter distribution, and establishing a flowback model of the porous medium;

step 4), collecting relevant parameters such as pore dimensions, fluid viscosity, gas-water interfacial tension, and wetting contact angle, and analyzing factors affecting the flowback.

[0077] In the embodiments, step 1) specifically includes:

[0078] Flow of a fluid in a straight single capillary with circular equal sections is as shown in Figure 1:

[0079] When the fluid in the capillary can be deemed as constant, laminar, incompressible and having viscosity, the Navier-Stokes equantion can be simplified as the following equation:

r dr rdr )]dx L I

where p represents viscosity of the fracturing fluid in units of mPa-s; r represents a

distance from any point of a circular capillary to the center thereof in a radial direction in

units m; R represents a radius of the circular capillary in units of m; u represents a flow

velocity of the incompressible fracturing fluid in units of m/s; represents a pressure

gradient when the fluid flows in the capillary in units of MPa/m; Ap represents a flow

pressure difference of the fluid in the capillary in units of MPa; and L represents a capillary

length in units of m.

[0080] The following equation is obtained by developing the equation (1): d 2u 1 du 1 Ap + -+- -0 (2) 2 dr r dr p L

[0081] After considering boundary slip, the boundary condition becomes as the following equations:

du r =0, = 0 dr du (3) |r =R,u = -L, B~r |r 1=R

[0082] The following equation is obtained by solving the above equations:

u- (r2 -R 2 -2RL)= I-4 (R2+2RL -r2) 4pu dx 4pu L

[0083] A flow equation is obtained by a surface integral of the equation (4):

R 1 Ap 2 q= JlA (R2+2RL -r 2xurdr= R4 Ap 1+ 4L (5 04p L 8p L R (5)

[0084] After considering a forced external force, a pressure difference generated during self-priming is equal to a sum of a capillary force Pc and the forced external force PQZ :

4c-cos O Ap -pc + po d + PQZ (6)

where d represents a diameter of the capillary in units of m;

[0085] Considering the material balance principle, a flow rate must be equal to a volume of the fluid absorbed by the pores per unit time (Benavente, D., Lock, P., Cura, M.A.G.D., and Ord6iez, S. (2002). Predicting the Capillary Imbibition of Porous Rocks from Microstructure. Transport in Porous Media 49, 59-76.):

7rd 2 dL 4 dt(7)

[0086] The following equation is obtained by a time integral of simultaneous equations (5) to (7) and substituting boundary conditions of t=0, L=0:

d(-4-cosO+dpQz)+8L r 16p d

[0087] In the embodiments, step 2) specifically includes:

[0088] (1) Slip length correction:

[0089] Liquid-wall interaction of a closed channel is strongly influenced by surface morphology and physicochemical characteristics of a solid, where the influence of the wettability of the boundary wall at a low shear rate is particularly apparent. The boundary slip occurs at a molecular level and is calculated by a contact angle of a given liquid under a given condition (Wu, K., Chen. Wettability effect on nanoconfined water flow [J]. Proceedings of the National Academy of Sciences of the United States of America, 2017, 114, 33-58.):

L = C/(cos O+1) 2 (9) where:

L, represents a true slip length of the wall fluid in units of nm;

C represents a constant with a value of 0.41 obtained by MD simulation; and

0 represents a wetting contact angle in units of degree.

[0090] The viscosity of the constrained fluid near the wall is significantly different from the viscosity of the free fluid, resulting in significant slippage at the fluid/fluid interface. In practical applications, a slip length of the constrained fluid which takes into account the true slip and the apparent slip effect should be replaced with effective slip length parameters (Wu, K., Chen. Wettability effect on nanoconfined water flow [J]. Proceedings of the National Academy of Sciences of the United States of America, 2017, 114, 33-58.):

L = La + Ls = + L + Ls -p._1|(d (10)

where:

Lse represents an effective slip length in units of m;

Lsa represents an apparent slip length in units of m;

Ls represents a true slip length of the wall fluid in units of m;

p. represents the viscosity of the free fluid in units of Pa-s; and

Pd represents the effective viscosity of the constrained fluid in units of Pa-s.

[0091] It can be seen from the equation (10), the effective slip length depends not only on wall wettability, but also on the fluid viscosity and the capillary size.

[0092] (2) Viscosity correction:

[0093] When the constrained fluid flows to nanometer scale pores, the viscosity of the fluid near the pore wall will no longer be accurately described by the viscosity of a rock core fluid. The effective viscosity is highly dependent on the viscosity of the rock core fluid and an interface region. The effective viscosity of the constrained fluid can be represented by a weighted average of a volumetric region and the viscosity of the interface nanometer pores (Wu, K., Chen. Wettability effect on nanoconfined water flow [J]. Proceedings of the National Academy of Sciences of the United States of America, 2017, 114, 33-58.):

PdiiAAdd ++p F A da ~~L-iAd (11)

where:

Pd represents the effective viscosity of the fluid in the volumetric region in units of Pa-s;

pi represents the viscosity of the fluid in the interface region in units of Pa-s;

Ad equals to 7r d2- -dc , and represents an area of the interface region in

units of m 2;

d. represents a critical thickness of the constrained fluid in units of nm; and

A equals to 2, and represents a total sectional area in units of m 2

.

[0094] The viscosity of the fluid in the interface region is greatly influenced by the interaction of the wall and can be represented by the contact angle obtained by experiments and MD simulation (Wu, K., Chen. Wettability effect on nanoconfined water flow [J]. Proceedings of the National Academy of Sciences of the United States of America, 2017, 114, 33-58.):

P =- 0.0180+3.25 (12)

[0095] It can be seen from equation (12) that, compared with the free fluid, the viscosity of the fluid in the interface region varies greatly with the contact angle.

[0096] For ease of programming, calculating and writing, the expressions for the effective slip length L. and the effective viscosity p, are assigned as following equations:

-- 0.0180+3.25; y_ id _ _ 2dc )2 Define x= - P Ad d

Then:

Pd=P.(X 1)- Y 1]

Lle=(1-x)y (d +L)+L (x-1)y+l 8

[0097] After the above correction, the slip length L, and the fluid viscosity p of the capillary force flowback model and the forced flowback model of the single capillary should be replaced with the effective slip length L,, and the effective viscosity Pd:

L= d(-4-cos+dPz) 1+8L13) 16pd d

)

[0098] In the embodiments, step 3) specifically includes:

[0099] (1) Statistical properties of fractal porous medium

[01001 Many fractal theory studies show that the number of pores and the diameters of pores of the shale satisfy the following fractal power law relationship (Mandelbrot B B, Wheeler J A. The fractal geometry of nature [M]. Birkhlauser Verlag, 1983.; Li K. More general capillary pressure and relative permeability models from fractal geometry [J]. Journal of Contaminant Hydrology, 2010, 111(1): 13-24.):

N(d) oc dDf (14)

where:

d represents a diameter of a unit model in units of m;

N(d) represents the number of unit models contained in the entire model, in units of piece and

Df represents a fractal dimensions of the pore, in a two dimensional plane, which

generally takes 1 to 2.

[0101] In the shale pores, the relationship between all of the number of pores with diameter not less than d and the pore diameter satisfies the following relationship:

N(L > d)oc (d./ d)D (15a)

[0102] If the minimum pore diameter of the pore is dmin, then the total number of pores N can be calculated by the following equation:

N =(d. / d. )D' (15b)

[0103] By simultaneously differentiating on both sides of the above equation, the number of pores in the interval d to d+dd can be obtained:

-dN= DdDId-(D+1)dd (16)

where, -dN>0 represents that the number of pores is inversely proportional to the diameter of the pores.

[0104] By dividing the equation (16) by the equation (15b), a percentage of the number of the pores in the interval from d to d+dd to the total number of the pores can be obtained:

-dN / Nt = Ddmifd-(D+1)dd = f(d)d d (17)

[0105] It can be seen from equations (3) to (7), a probability density function f(d) of the distribution of the number of the pores with respect to sizes of the pores is as follows:

f(d)=DdDnd-(D+1) (18

[0106] The above equation must meet a normalization condition:

d~ f(d)dd=1-(dmidm )Df =1

[0107] A necessary and sufficient condition for the establishment of equation (19) is as follows:

(da / d.)D)

[0108] Generally, the shale pores satisfy dmin/dmax<10-2 , at this time, the equation (20) is approximately established, so the fractal theory can be used to describe the shale porous medium;

[0109] An average diameter day of the shale pores is calculated by the following equation:

dy =jdf(d)dd =Dd Df -1 (21)

[0110] In the fractal porous medium, a total area Ap of the shale pores is as follows (Yu B M, Cai J C, Zou M Q. On the Physical Properties of Apparent Two-Phase Fractal Porous Media

[J]. Vadose Zone Journal, 2009, 8(1): 177-186.):

S=- d2dN= td. D d . 2-D md, (1-#) (22) dm, 4 4(2-D,) d 4(2- Df)

[0111] (2) Fractal characteristics of tortuous streamlines

[0112] When the fluid flows through a porous medium with a complex spatial structure, an expression for describing a non-uniform pore tortuous streamline is as follows (Yu B, Cheng P. A fractal permeability model for bi-dispersed porous media [J]. International Journal of Heat and Mass Transfer, 2002, 45(14): 2983-2993.):

L()=-D LD (23)

where:

c represents a scale of the measurement in units of m;

Lt represents an actual length of the streamline in units of m;

L represents a straight line distance in units of m; and

DT represents a fractal dimension of the tortuosity of the pores, which generally takes 1 to 2 in a two dimensional plane.

[0113] Yu and Cheng replace the scale of the measurement in equation (23) by the capillary diameter (Wheatcraft S W, Tyler S W. An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry [J]. Water Resources Research, 1988, 24(4): 566-578. ):

L,(d)= d-"L'DL (24)

[0114] Two sides of the above equation are differentiated:

vt (d) = Dt (L / d)D, -lvo (25)

where, vt=dL/dt represents a velocity of the fluid in a curved channel, vo=dL/dt represents a velocity of the fluid in a straight channel. The equation (25) is integrated with f(d), and an actual average flow velocity v is as follows (Yu B, Cheng P. A fractal permeability model for bi-dispersed porous media [J]. International Journal of Heat and Mass Transfer, 2002, 45(14): 2983-2993.) : dn = vt (d)f(d)dd= DTDf -T&1V da DD- d(D26LD) d.«» r + , -1(26)

[0115] Through a further study about equation (26), DT, r and day satisfy the following expression (Yu B. Fractal Character for Tortuous Streamtubes in Porous Media [J]. Chinese Physics Letters, 2005, 22(1): 158-160.)

DT =+ In r In(LO / dy) (27)

[0116] L / da, is calculated by the following equation (Xu P, Yu B. Developing a new form of permeability and Kozeny-Carman constant for homogeneous porous media by means of fractal geometry[J]. Advances in Water Resources, 2008, 31(1): 74-81.):

L/d - D 11- t (28) av 2/p # Df(2-Df)

[0117] (3) Average tortuosity of pores

[0118] A definition of the tortuosity r is a ratio of a distance Lt by which the fluid flows through a curved flow path to a distance L by which the fluid flows through a straight flow path:

L (29)

[0119] Boming Yu proposes a model of the average tortuosity related to the porosity as follows (Yu B, Li J. A Geometry Model for Tortuosity of Flow Path in Porous Media [J]. Chinese Physics Letters, 2004, 21(8): 1569-1571)

Y 1- +(1-#)/ 4 1+ +- 1 2 21- 1-R (30)

[0120] The model is suitable for describing a case where space of the shale pores are all uniform cubes, not considering an actual situation of the shale pore distribution. Moreover, Kozeny-Carma obtains (Carman P C. Fluid flow through granular beds[J]. Institution of Chemical Engineers, 1937, 15: 150-166.):

K = d2o2 32r (31)

[0121] Through the equation (31), the average tortuosity can be obtained by just obtaining the permeability, porosity and pore radius of the shale. However, the model (31) is derived on the assumption that the shale spatial structure satisfies the uniform capillary bundle model, which cannot be directly applied to the fractal porous medium. Taking into account the limitation of the uniform capillary bundle model represented by the equation (31), the average tortuosity of the porous medium can be obtained by introducing the probability density function of the pore diameter distribution (Hager J, Hermansson M, Wimmerstedt R. Modelling steam drying of a single porous ceramic sphere: experiments and simulations[J]. Chemical Engineering Science, 1997, 52(8): 1253-1264.): r= A 2K '- df (d) d d 3 (32)

[0122] By substituting the fractal power law relationship (equation 18) of the pore diameter distribution into the equation (32), the average tortuosity calculation model (suitable for the fractal porous medium) can be obtained:

Df dmn F 0- jd~xqd' df (d )d= Dd 32Kd 32K Df -1 (33)

[0123] (4) Flowback model of porous medium

[0124] After considering the correction by the effective viscosity and the effective slip length, the equation (5) is rewritten as follows:

q R4 Ap + 4Lse 8pa L R (34)

[0125] After considering the forced external force, the flowback pressure difference is written as follows:

Ap=-pc P+2o-cosO p R +cros (35)

[0126] Then the following equation can be obtained:

q =8pR 4L 2o-cos0 PQJ 8pdL ( R R

_n d30-cosO 1+8 + nd4 p 41 8L (36) 32pL d 128L

[0127] By integrating the flow rates of all single capillaries with a section area of A, a total flow rate Q can be obtained as follows:

Q - d"_'q(d)dN fdminq~~ (37)

[0128] An actual average flow velocity in the capillaries is equal to:

-Q AP (38)

[0129] The following equation can be derived from the equation (26):

DT +Df -1 - dL O= TLT1v 1 _ (39) DTDfd 1L dt

[0130] The flowback model of porous medium is obtained by simultaneous equations (36) to (39):

2 o-cosO(DT +Df -1)(2-Df) dDDT 1_f2D-f 2 +D-Dr) 8LSedDD-1(1 fl-DT -Df, +- 2 DT

4pd(1 #5)DdiifDT 2+D -Df 1+DT-Df p (DT+Df -1)(2-Df) dr+(1-p3+DT-Dr) 8L,,d, (I,(1-2+Dr -D

16p(1-#)DfdmDT 3+D -Df 2+DT -Df

(40)

[0131] In the embodiments, step 4) specifically includes:

[0132] In order to analyze the factors affecting the flowback, the parameter values of the equation (40) are shown in Table 1:

Table 1 Values of parameters for calculating the flowback model

Wetting contact angle 0 (0) 30 Water phase viscosity p 1 (mPa-s) Gas-water interfacial tension 70 Average tortuosityr 10.73 -(mN/m) Minimum pore diameter dm1 in 1.49 Maximum pore diameter 897 (nm) dmax (nm)

Pore fractal dimension Tortuosity fractal 1.02 dimension (two-dimensional) 1.614 (two-dimensional) Df DT Porosity, % 4.83

[0133] (1) Wetting angle

[0134] Figure 3 shows how the flowback length changes according to the wetting angle. When the forced external force pQz =20MPa, the flowback length increases as the wetting angle increases. When pQz =0, the flowback occurs only when the wetting angle 0 >900, this is because when the pores are only subjected to the capillary force, self-priming occurs in the pores when 0 < 900, (when 0 > 900, wetting reversal occurs, the capillary force reverses and the flowback occurs in the pores).

[0135] (2) Forced external force

[0136] Figure 4 shows how the flowback length changes according to the forced external force. When the forced external force pQz is less than a certain critical value, the pore flowback length is negative (that is, no flowback occurs), this is because when the forced external force pQz is less than the capillary force, self-priming occurs in the pores. Moreover, the larger the pore maximum diameter dmax, the smaller the critical value of the forced external force required (that is, the flowback is easier to occur in a large pore diameter capillary).

[0137] (3) Pore dimensions

[0138] Figures 5a and 5b show trends of the flowback length change according to a pore maximum/minimum diameter of pores. Under a certain forced external force, when the maximum diameter dmax is less than a critical value, no flowback occurs. This is because the capillary force plays a dominant role in self-priming and flowback of capillaries with small pore diameters, while the forced external force plays a dominant role in capillaries with large pore diameters. The larger the forced external force pQz, the smaller the critical value of the required capillary maximum diameter dmax (that is, the larger the forced external force, the easier the flowback of the capillary). The flowback length decreases as the minimum diameter dminincreases, and the larger the forced external force, the more apparent the trend.

[0139] The beneficial effects brought by the technical solutions provided by the embodiment of the present application are as follows:

[0140] The present application provides an analytical fractal method for calculating flowback capacity of a shale reservoir based on full consideration of the characteristics such as the pore micro-scale effect, the pore channel tortuosity and the diameter distribution satisfying the fractal statistical relationship. Based on the Hagen-Poiseuille equation, considering the material balance principle, the boundary slip effect and the forced external force, the forced flowback model of the single capillary is obtained; on this basis, the effective slip length and effective viscosity are obtained by further correcting the slip length and the viscosity; and, considering that shale pores also have tortuous features and fractal features, the total flow of the porous medium is obtained by integrating the single capillary flow combined with the probability density function of the pore distribution, and the flowback model of the fractal porous medium is further obtained while considering the forced external force.

[0141] The above are only preferred embodiments of the present application, and are not intended to limit the present application. Any modifications, equivalents, improvements and the like, which are within the spirit and principle of the present application, should be includes in the scope of the present application.

Claims (6)

1. An analytical fractal method for calculating flowback capacity of a shale reservoir, comprising:

step 1), establishing a flowback model of one single capillary;

step 2), correcting the model in conjunction with a situation that slip length and viscosity change according to a pore diameter;

step 3), establishing a flowback model of the porous medium in conjunction with fractal characteristics of pore diameter distribution; and

step 4), collecting relevant parameters and analyzing factors affecting flowback.

2. The method according to claim 1, wherein, step 1) comprises:

analyzing flow of a fluid in a straight single capillary with circular equal sections;

when the fluid in a capillary satisfies constant, laminar, incompressible and viscosity, a Navier-Stokes equantion is simplified as the following equation:

r dr rdr )]dx L

where p represents viscosity of a fracturing fluid in units of mPa-s; r represents a

distance from any point of a circular capillary to the center thereof in a radial direction in

units m; R represents a radius of the circular capillary in units of m; u represents a flow

velocity of the incompressible fracturing fluid in units of m/s; represents a pressure

gradient when the fluid flows in the capillary in units of MPa/m; Ap represents a flow

pressure difference of the fluid in the capillary in units of MPa; and L represents a capillary

length in units of m;

the equation (1) is expanded to obtain the following equation: d2u 1 du 1 Ap + + -0 (2) dr2 r dr p L after considering a boundary slip, a boundary condition becomes as the following equations: du r =0, =0 dr du (3) r =R,u = -L, l B~r |r 1=R the equations are solved to obtain the following equation:

U=- d r2 -R 2 -2R ) - PR 2 + 2RL -r2) 4 4pu dx 4pu L

a flow equation is obtained by a surface integral of the equation (4):

R 1 Ap 2R 4 Ap q= (R2+2RL -r 2xrdr = 1+ 4L, 04p L 8p L R

after considering the forced external force, a pressure difference generated during self-priming is equal to a sum of a capillary force Pc and the forced external force PQZ :

4c-cos O Ap -pc + po d + PQZ (6)

where d represents a diameter of the capillary in units of m;

considering a material balance principle, a flow rate is equal to a volume of the fluid absorbed by pores per unit time:

2/ddL

4 dt(7) the following equation is obtained by a time integral of simultaneous equations (5) to (7) and substituting boundary conditions of t=0, L=O:

d=- 4 -cos0+dpQz) L= 1+ 8L 0(8). r 16p d

3. The method according to claim 2, wherein step 2) comprises:

(1) slip length correction:

liquid-wall interaction of a closed channel is strongly influenced by surface morphology and physicochemical characteristics of a solid, where influence of the wettability of the boundary wall at a low shear rate is particularly apparent, and the boundary slip occurs at a molecular level and is calculated by a contact angle of a given liquid under a given condition:

L = C/(cosO+1) 2 (9)

where:

L, represents a true slip length of wall fluid in units of nm;

C represents a constant with a value of 0.41 obtained by MD simulation; and

Orepresents a wetting contact angle in units of degree;

viscosity of constrained fluid near the wall is significantly different from the viscosity of the free fluid, resulting in significant slippage at the fluid/fluid interface; in practical applications, a slip length of the constrained fluid which takes into account true slip and apparent slip effect is replaced with effective slip length parameters:

Lse - L,, + L = -1| + L + Ls (10)

where:

Lse represents an effective slip length in units of m;

La represents an apparent slip length in units of m;

Ls represents a true slip length of the wall fluid in units of m;

p. represents the viscosity of the free fluid in units of Pa-s; and

Pd represents the effective viscosity of the constrained fluid in units of Pa-s;

the equation (10) shows that, the effective slip length depends not only on wall wettability, but also on fluid viscosity and capillary size;

(2) viscosity correction:

when the constrained fluid flows to nanometer scale pores, the viscosity of the fluid near the pore wall will no longer be accurately described by the viscosity of a rock core fluid; the effective viscosity is highly dependent on the viscosity of the rock core fluid and an interface region; the effective viscosity of the constrained fluid is represented by a weighted average of a volumetric region and the viscosity of interface nanometer pores:

Pd iAAdd + F A Ad

where:

Pd represents the effective viscosity of the fluid in the volumetric region in units of Pa-s;

pi represents the viscosity of the fluid in the interface region in units of Pa-s;

Ad equals to 21 d 2- -de , and represents an area of the interface region in

units ofin 2;

d. represents a critical thickness of the constrained fluid in units of nm; and

2 Ad equals to 2, and represents a total sectional area in units ofin

the viscosity of the fluid in an interface region is greatly influenced by interaction of the wall and is represented by the contact angle obtained by experiments and MD simulation:

P - 0.018 0+3.25 (12)

the equation (12) shows that, compared with the free fluid, the viscosity of the fluid in the interface region varies greatly with the contact angle;

for ease of programming, calculating and writing, expressions for the effective slip length L, and the effective viscosity p, are assigned as following equations:

defining x - O --0.0180+3.25; y _Aid _ _ _2d)2 P. Ad d

then: pd - x-l)y+] ;

(x-i)y+l 8 after the above correction, the slip length L, and the fluid viscosity p ofthecapillary force flowback model and the forced flowback model of the single capillary are replaced with the effective slip length Ls, and the effective viscosity Pd:

d(-4o-cos +8dPz) (13). 16pd d

4. The method according to claim 3, wherein, step 3) comprises:

(1) statistical properties of fractal porous medium

a plurality of fractal theory studies show that the number of pores and the diameters of pores of the shale satisfy the following fractal power law relationship:

N(d c dDr (14)

where:

d represents a diameter of a unit model in units of m;

N(d) represents the number of unit models contained in the entire model in units of piece and

Df represents a fractal dimensions of the pore, in a two dimensional plane, which

generally takes 1 to 2;

in the shale pores, the relationship between all of the number of pores with diameter not less than d and the pore diameter satisfies the following relationship

N(L ;> d) cc (d. / d)D' (15a)

if a minimum pore diameter of the pore is dmin, then a total number of pores N is calculated by the following equation:

Nt =(dax/ d )Dr (15b) by simultaneously differentiating on both sides of the above equation, the number of pores in an interval d to d+dd is obtained:

-dN=DdDId-(D+1)dd (16)

where, -dN>0 represents that the number of pores is inversely proportional to the diameter of the pores;

by dividing the equation (16) by the equation (15b), a percentage of the number of the pores in the interval from d to d+dd to a total number of the pores is obtained:

-dN /Nt =Ddmifd-(D+1)dd=f(d)dd (17)

equations (3 to 7) show that, a probability density function f(d) of distribution of the number of the pores with respect to sizes of the pores is as the following equation:

f(d)=DdDnd-(D+1) (18

the above equation meets a normalization condition:

d~ f(d)dd=1-(dmidm )Df =1

a necessary and sufficient condition for the establishment of equation (19) is as the following equation:

(da / d.)D)

generally, the shale pores satisfy dmin/dmax<10-2, at this time, the equation (20) is approximately established, and the fractal theory is thereby used to describe the shale porous medium;

an average diameter dayof the shale pores is calculated by the following equation:

dy =jdf(d)dd =Dd Df -1 (21)

in the fractal porous medium, a total area of the shale pores Ap is as the following equation:

L=ld 2dN= td. D dF- 2-D mdD (1-#) (22) dm 4 4(2-D,) d 4(2- Df)

(2) fractal characteristics of tortuous streamlines

when the fluid flows through a porous medium with a complex spatial structure, an expression for describing a non-uniform pore tortuous streamline is as the following equation:

L()=-D LD (23)

where:

c represents a scale of the measurement in units of m;

Lt represents an actual length of the streamline in units of m;

L represents a straight line distance in units of m; and

DT represents a fractal dimension of the tortuosity of the pores, which generally takes 1 to 2 in a two dimensional plane;

scale of the measurement in equation (23) is replaced by the capillary diameter:

L,(d)= d-DL'DL (24)

two sides of the above equation are differentiated:

vj(d)= Dt(L / d)D -lvo (25)

where, vt=dL/dt represents a velocity of the fluid in a curved channel, vo=dL/dt represents a velocity of the fluid in a straight channel; the equation (25) is integrated with f(d), and an actual average flow velocity v is as the following equation:

- dn DTDfI, ID T da t (d)f(d)dd= D D- d7DTLDT-160 d.«» r + , -1(26)

through a further study about equation (26), DT, r and da, satisfy the following expression:

Inr DT =+ In(LO / dy) (27) L / d.a is calculated by the following equation:

L/d -- 1-# (28) 2p 2/d # Df(2-Df)

(3) average tortuosity of pores

a definition of the tortuosity r is a ratio of a distance Lt by which the fluid flows through a curved flow path to a distance L by which the fluid flows through a straight flow path:

L (29) a model of the average tortuosity related to the porosity is as the following equation:

i5 (1- +(1-#)/ 4 1+i+i r-1-1+-R+ (30)

the model is suitable for describing a case where space of the shale pores are all uniform cubes, not considering an actual situation of the shale pore distribution, moreover:

K= d2o 32r 2 (31)

through the equation (31), the average tortuosity is obtained by just obtaining the permeability, porosity and pore radius of the shale; however, the model (31) is derived on the assumption that the shale spatial structure satisfies the uniform capillary bundle model, which cannot be directly applied to the fractal porous medium; taking into account the limitation of the uniform capillary bundle model represented by the equation (31), the average tortuosity of the porous medium is obtained by introducing the probability density function of the pore diameter distribution:

r d'- df(d)dd 32Km (32)

by substituting the fractal power law relationship (equation 18) of the pore diameter distribution into the equation (32), the average tortuosity calculation model is obtained: cd~xq Dfd, d,' df (d )d= #Dd 32Kd 32K Df -1 (33) the average tortuosity calculation model is suitable for the fractal porous medium;

(4) flowback model of porous medium after considering the correction by the effective viscosity and the effective slip length, the equation (5) is rewritten as the following equation:

2TR? Ap( 4Ls q=1+± 8 L R ) (34)

after considering the forced external force, the flowback pressure difference is written as the following equation:

P= Ap=-pc P+2o-cosO -p R °s (35)

then the following equation is obtained:

q =8pR 4L 2o-cos0 8pdL ( R R

__nd'or co s 0 1+ 2Id~ ~ + 4d u o +8)(6 s O~(36) 32pL d 128 d )

by integrating the flow rates of all single capillaries with a section area of A, a total flow rate Q is obtained as the following equation:

Q - d _q(d)dN fdminq~~ (37) an actual average flow velocity in the capillaries is equal to:

-Q (38)

the following equation is derived from the equation (26):

D, + Df -1 - dL DTDfdb 'TL t dt(39)

the flowback model of porous medium is obtained by simultaneous equations (36) to (39): ucosO(D+Df-1)(2-Df) dD 2+De Dr )+8L.dD 1/3 1D-Dr ) 2D 1

4 ODd-T2D-D +T- Df ] tD

~QZ (D D-nFax p (Dr+D,12-D) d,*(1-p3+DT-Dr) , 8L d,"(1p2D) max

( 16p(1-#)Dfd-T L 3+DT -Df 2+DT-Df ] (40).

5. The method according to claim 4, wherein step 4) comprises:

providing values for the parameters of the equation (40), and analyzing factors affecting the flowback.

6. The method according to claim 5, wherein the providing values for the parameters of the equation (40) comprises:

providing values for the parameters such as wetting contact angle, water phase viscosity, gas-water interfacial tension, average tortuosity, minimum pore diameter, maximum pore

diameter, fractal dimension of the pore, fractal dimension of the tortuosity, and porosity.

Figure 1

v=0 Ls

v v

(a) Flowback model without (b) Flowback model slip at boundaries with slip at boundaries

Figure 2

- 1/3 -

Figure 3

Figure 4

- 2/3 -

Figure 5a

Figure 5b

- 3/3 -