Flow of Preformed Particle Gel through Porous Media: A Numerical

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Flow of pre-formed particle gel through porous media: A numerical simulation study based on the size exclusion theory Yongge Liu, Jian Hou, Qingliang Wang, Jingyao Liu, Lan-Lei Guo, Fuqing Yuan, and Kang Zhou Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b03656 • Publication Date (Web): 17 Feb 2017 Downloaded from http://pubs.acs.org on February 18, 2017

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Industrial & Engineering Chemistry Research

Flow of Pre-Formed Particle Gel through Porous Media: A Numerical Simulation Study Based on the Size Exclusion Theory Yongge Liu,,†,‡ Jian Hou,*, †,‡ Qingliang Wang, †,‡ Jingyao Liu, †,‡, Lanlei Guo,§ Fuqing Yuan,§ Kang Zhou, †,‡ †

State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Qingdao, 266580, People’s Republic

of China ‡

School of Petroleum Engineering, China University of Petroleum, Qingdao, 266580, People’s Republic of China

§

Research Institute of Geological Science, Sinopec Shengli Oilfield Company, Dongying, 257015，People’s Republic

of China

Abstract:

After swelling, the pre-formed particle gel (PPG) can be suspended in the aqueous phase, plug small

pores during flowing through the formation or deform under large pressure gradients to move further into the formation, and achieve the in-depth profile control. This paper, based on the size exclusion theory, introduces the concept of the critical pressure gradient to capture the phenomenon that the PPG plugging small pores can deform and pass, and the mean number of bridging particles and mean bridging probability to characterize the bridging of the PPG inside large pores. On this basis combined with the numerical simulation theory, the numerical simulation model of the PPG flowing through porous media has been established and validated with physical simulation experiments. The general regularity of the PPG migrating inside porous media is then studied using the proposed numerical simulation model. Keywords: Pre-formed particle gel; size exclusion theory; numerical simulation; deformation; rigid particle

1. Introduction Migration of suspended particles through porous media is commonly seen in nature, for instance, the migration of rock and soil grains inside unconsolidated formations, filtration of particles of drilling fluids into formations, and the waste water treatment and reinjection.1-8 While the suspended particle is moving, plugging can occur in the small pores (pores with radius smaller than particle radius) due to the incompatibility of sizes between the particle and pore. The permeability of the porous media is hence reduced, which affects the following injection and production process.9 In order to effectively predict the plugging induced by migrating particles, studies on models of grain migration has started since the last century, and correspondingly many scholars have established mathematical models representing the filtration characteristics of particles inside the porous media, among which the most representative is the classical filtration model proposed by Herzig and Payatakes.10-12 This model adopts the filtration coefficient to characterize the retention capability of the porous media to the suspensions and calculate the physical property changes of the porous media and particle retention concentration. Various expressions of the filtration coefficient have been proposed regarding different situations by scholars. However, the large variation in breakthrough times (0.4-100PV) reported in the literature show that the classical model is limited due to its breakthrough concentration always occurs at unity in the classical model. The reason of this paradox lies in two assumptions: one is the particles move forward along with water at the same velocity, and the other is all pores are accessible for the particles.13,14 In order to solve this paradox, 1

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Santos15 and Bedrikovetsky16 proposed an updated model called size exclusion model which presents more detailed physics on the micro scale. This model introduced accessibility and flux reduction factors into the original population balance model so it accounts for the capture of particles when the suspension travels through a porous media. You17 got the analytical solution of the size exclusion model when the retained particle concentration is negligible compared with the vacant-pore concentration. Under this assumption, an experiment was designed and the calculated results showed good agreements with the experimental results. Chalk18 got the steady state solution of size exclusion model and conducted an experiment to measure the outlet concentration of suspended particles. The comparison also shows good agreement between the modeling result and the laboratory measurement. You19 derived the analytical asymptotic model for non-linear deep bed filtration and then the laboratory injection tests were conducted to validate the accuracy of the solution. The pre-formed particle gel (PPG), as a new type of chemical agents, was introduced by Coste in 2000.20 It can swell in water and possesses strong elasticity, so it can deform under a certain pressure gradient, pass the pore and then return to its original shape, when plugging occurs in the porous media. Compared with the rigid particle, PPG possesses the ability to achieve “temporary plugging-deforming and passing-temporary plugging again”. In the meantime, because the crosslinking of the PPG is finished on the ground, its properties remain relatively stable during flowing through the porous media. Consequently, the PPG has been widely used in the petroleum industry. Massive studies on the general regularity of filtration of the PPG inside porous media have been carried out, with its increasingly widespread use. In terms of physical simulation experiments, Bai21-23 measured the property of the PPG and found out that the size and elasticity of PPG particles can both be adjusted artificially. Moreover, the features that the PPG plugs the pore and alters the streamlines of the aqueous phase have been directly observed in his observation on the PPG migration based on the micro model. Cui24 studied the pressure change at different locations during PPG migration using the sandpack model, which indirectly revealed the deep migration mechanism of the PPG. Cao25 combined the PPG with the surfactant to take advantage of the positive effect of the PPG on the swept volume and the improvement of the surfactant to the displacement efficiency. The sandpack model was used in Cao’s research to study the synergistic effect of the two chemical agents and resulted production performance. Yao26-28 observed the whole process of the PPG plugging, deforming to pass, and bridging to plug (the phenomenon that several particles aggregate and plug the pore with radius bigger than particle radius) under the microscope when it was flowing through the porous media, which revealed the migration mechanism of PPG particles inside porous media. In terms of numerical simulation, according to the property of the PPG, Goudarzi29 modified the residual resistance factor and the aqueous solution viscosity to approximately capture the migration characteristics of PPG particles and established the numerical simulation model. Feng30 adopted the classic filtration model to describe the migration characteristics of the PPG migrating inside the porous media, and combined it with the numerical simulation theory to establish the mathematical model of flow of PPG particles, which was used to fit the pressure change recorded in the core flooding experiment. Wang31, combing the classic filtration theory with the probability theory, characterized the probability of plugging induced by PPG particles and established the numerical model of PPG particle migration. It can be concluded from the above that the previous numerical simulation models of the PPG migration are all based on the classic filtration theory. The classic filtration theory uses the filtration coefficient to represent the plugging induced by PPG particles, lacks considerations in the micro scale of compatibility of sizes of the pore and particle, and assumes that all pores are accessible, while the size exclusion theory introduces the concept of the accessible pore of particles and considers the matching of sizes between particles and pores. Hence, the size exclusion theory can better reflect the general regularity of particle migration in the formation. However, so far, in the size exclusion theory the particle is assumed to be rigid with no deformation and the plugging caused by the particle bridging is neglected. Hence, this theory cannot be directly applied to the calculation and simulation of PPG particle migration. Given the reasons mentioned above, this paper aims at quantitatively describing the bridging-induced 2


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plugging and re-flow characteristics of PPG particles on the basis of the size exclusion theory, and establishing the numerical simulation model of PPG migration in the porous media with the numerical simulation theory combined. This paper follows the following structure: Part 2 establishes the kinetic equations of capture, bridging, deposition16 (the phenomenon that one particle deposited on the wall of the pore with radius bigger than particle radius) and deforming to pass of the PPG, the mass balance equation and the conservation equation between the number of plugging particles and the pore number, and meanwhile introduces the numerical simulation model of PPG particle migration through porous media; Part 3 gives the numerical solution to the proposed model and conducts physical simulation experiments, with history matching to validate the model and determine the simulation parameters; at last, Part 4 compares the general migration regularity of PPG and rigid particles inside porous media, based on the history matching.

2. Mathematical model of PPG migration based on the size exclusion theory Figure 1 is the schematic diagram based on the size exclusion theory of suspended particles flowing through pores. As shown in Figure 1a, the porous media is regarded as an interlaced structure of chambers and parallel capillaries. Particles accumulate in the chamber and then flow into each pore. In Figure 1b, where rs represents the radius of the particle and rp stands for that of the pore, shows that if rs < rp, the particle can get through the pore and since the particle has a certain volume, it can only migrate within a specific area in the large pores (pores with radius larger than particle radius), which is shown as the area surrounded by red dashes in Figure 1b. Otherwise it will plug the pore and this kind of plugging caused by particles is called the size exclusion.

(a) Pass and capture (b) Particle in a large pore Figure 1. Schematic plot of suspended particle in porous media In this part, with the introduction of some new concepts, for example, the bridging probability, mean number of bridging particles and deposition probability, the kinetic equations of capture, bridging and deposition of PPG particles are obtained, on the basis of the size exclusion theory. The critical pressure gradient is used to characterize the feature of PPG particles that they can deform to pass pores under high pressure gradients. Based on these equations, the numerical simulation model of PPG particle migration through porous media is established. The basic assumptions of the model are: (1) only water phase conducts isothermal flow in the porous media and PPG is a component which is suspended in the water phase; (2) the sizes of the pores conform to lognormal distribution 14,18,19 (as shown in Figure 2b); (3) PPG will be captured if the size of the particle is larger than that of the pore and one captured particle will block one pore; (4) PPG will enter the pore if the size of the particle is smaller than that of the pore. However, there is a probability that particle deposition or bridging will occur; (5) bridging will totally plug the pore and reduce the effective concentration of the pore. However, particle deposition will not plug the pore and thus it will not affect the effective pore concentration; (6) as the increase of the pressure gradient, the plugged particle will deform gradually and when the pressure gradient exceeds a critical value, the captured or bridging 3



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particles will pass the pore, leading to the recovery of the effective pore concentration;28 (7) due to the large size of PPG particle, diffusion is considered as a very slow process and it is neglected in this paper for simplicity.19,31

(a) Finite element

(b) Concentration of pores with different sizes Figure 2. Element and concentration of pores

2.1 Conservation law We take a finite element which is shown in Figure 2a as the research object and the element is so small that it can be seen as homogeneous. Therefore, the surface concentrations of the left and right cross sections are the same and denoted H 0 . According to size exclusion theory, PPG is only suspended in the accessible area in the pores with large sizes. Therefore, the mass velocity of the element should be

( ρ s vwx fa c ) x , where ρs

is the density of PPG. If we

neglect the density changes of PPG and consider the source/sink terms, the conservation law for particles is obtained according to numerical simulation theory:

∂  vw f a c ( x, t )  ∂ ∂σ −  + qw c0 = ϕ a c ( x, t )  + ∂x ∂t ∂t where, c0 is the injected concentration of PPG;

(1)

σ is the total concentration of captured, deposited and

bridging particles; x is the distance between the survey point and the injection point. According to the classic filtration theory, Feng30 obtained the mass balance equation of PPG in porous media. In comparison with his equation, we can find that the velocity of the aqueous phase vw in Feng’s model is replaced by the velocity of the accessible part vw f a and the porosity ϕ is replaced by the accessible porosity ϕa . In extreme cases, where the radius of the PPG particle equals 0, then f a = 1 ,

ϕa = ϕ , and eq 1 degenerates into the mass

balance equation based on the classic filtration theory. 4


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2.2 Kinetics for particle capture, deposition, bridging and deformation 2.2.1 Particle capture

(a) Capture-plugging

(b) Elastic deformation (c) Recovery of deformation Figure 3. Schematic plot of capture and deformation of PPG

During time interval ∆t , the number of PPG particles with radius rs which enter into the pores with radius rp per unit rock volume is:

1 qw ( rp ) H ( rp , x, t ) C ( rs , x, t ) f a [ H , rs ] ∆t l

(2)

where, l is the average length of the pore. As shown in Figure 3a, PPG capture and plugging occurs when rs is bigger than rp . Therefore, the following equation satisfies:

∂ ∑1' ( rs , x, t ) ∂t

=

C ( rs , x, t ) f a [ H , rs ] l

vw f ns (3)

where, ∑1 is the concentration of captured particles by particle sizes. '

According to the assumption, one captured PPG particle plugs one pore. Therefore, the decreasing rate of the effective pore concentration caused by PPG capture should equal to

∂ ∑1' ( rs , x, t ) . ∂t

2.2.2 Bridging and deposition of particles Figure 4 is the schematic diagram describing the bridging of PPG particles inside pores. It is extremely difficult to precisely elaborate the bridging and deposition of particles from the standpoint of the mechanics. For simplicity, Bedrikovetsky16 adopts the deposition probability to calculate the number of deposited particles. Similarly, this paper uses the deposition probability pa 2 to calculate the potential number of deposited particles, and introduces the bridging probability pa1 to calculate the potential number of bridging particles. Then the following equations satisfies:

∂ ∑'2 ( rs , x, t ) pa1vwC ( rs , x, t ) f a [ H , rs ] ∞ = ∫ k1 ( rp ) H ( rp , x, t ) drp ∂t lk rs

(4)

∂ ∑3' ( rs , x, t ) pa 2 vwC ( rs , x, t ) f a [ H , rs ] ∞ = ∫r k1 ( rp ) H ( rp , x, t ) drp ∂t lk s

(5)

5



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where, ∑2 and ∑3 are the concentrations of bridging and deposited particles by particles sizes respectively; '

'

k1 is the conductivity of a single pore.

(a) PPG aggregation (b) Bridging-plugging-deformation (c) Recovery of deformation Figure 4. Schematic plot of bridging and deformation of PPG As the grain size approaches to the pore size, the probability of contact between the particle surface and the coarse pore wall increases, and hence it is believed that the probabilities of particle deposition and bridging grow. Extremely, if the size of PPG particle is equal to the size of the pore, the particle will be captured and the value of

pa1 or pa 2 should be 1. If the size of pore is much larger than size of particle, the value of pa1 or pa 2 should tend to 0. Therefore, as mentioned by Bedrikovetsky, 16 the deposition probability and bridging probability are the function of rs / rp and they are calculated simply using the following equations in this paper:

r p a1 =  s r  p

n1

  ; 

r pa 2 =  s   rp

  

n2

(6)

where n1 and n2 are the bridging exponent and the deposition exponent. According to assumption (5), deposition causes no plugging while bridging can do so. In the actual process of the flow of PPG particles, the number of bridging particles ranges from several to dozens. Supposing the mean number of particles which can bridge and block a pore is η , the decreasing rate of the effective pore concentration induced by ' 1 ∂ ∑ 2 ( rs , x, t ) bridging can then be described as . η ∂t

2.2.3 Deforming-to-pass of particles The phenomenon that the PPG first plugs the small pore and then deforms to pass is shown in Figure 3b and Figure 3c, and the deforming and then passing behavior of bridging particles in large pores is shown in Fig. 4b and Fig. 4c. Zinchenko31 studied the motion of deformable drops through porous media by microscopic simulation methods and the deforming process of the emulsion was simulated accurately. However, the scale is too small and the result is hardly to be used in the simulation of PPG flow in porous media due to the big difference between the emulsion and PPG. With physical simulation experiments, Wang32 found out that for PPG particles, a re-flow pressure gradient exists, beyond which the PPG can deform to get through the pore and migrate further into the porous media. The critical deformation pressure gradients for captured PPG particles in small pores and bridging particles in large pores can be respectively described as:

∇pg1 = a1 exp(b1

rs ) rp

6


(7)

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∇pg 2 = a2 exp(b2

rs ) rp

(8)

where, ∇pg1 and ∇ p g 2 respectively stand for the critical deformation pressure gradients for captured particles and bridging particles; a1 , a2 , b1 and b2 are constants. Wang31 also found that the rate of deforming-to-pass for the PPG is in direct proportion to the number of plugging particles, fluid velocity and pressure gradient. Therefore, the rate of deforming-to-pass of plugging PPG in small pores can be calculated as below:

ˆ ( r , x, t ) ∂ ∑ 1 s =0  ∂t   ˆ rs ∇p − pg1  ∂ ∑1 ( rs , x, t ) = 1 γ q r H r , x, t , , , r r x t drp ( ) ( ) ( ) ∑ 1 w p p 1 p s ∫0  t l p ∂ g1 

∇p < p g 1 (9)

∇p ≥ pg1

ˆ ( r , x, t ) is the concentration of captured PPG particles in small pores that deform to pass; where, ∑ 1 s

γ 1 is

the deforming-to-pass coefficient. Similarly, the rate of deforming-to-pass of bridging particles inside large pores can be written as:

ˆ ( r , x, t ) ∂∑ 2 s ∂t

=0

∇p < p g 2

ˆ ( r , x, t ) 1 ∞ ∇p − p g 2 ∂∑ s 2 = γ 2 ∫ qw ( rp ) H ( rp , x, t ) ∑ 2 ( rp , rs , x, t ) drp ∂t l rs pg2

(10)

∇p ≥ pg 2

ˆ ( r , x, t ) is the number of bridging PPG particles in large pores that deform to pass; where ∑ 2 s

γ 2 is the

deforming-to-pass coefficient. It is assumed that the plugging occurs in small pores is that one particle plugs one pore, so after the PPG deforms and gets through the pore, the number of PPG particles that have deformed to pass should be equal to the gain of the effective pore number. Therefore, the increasing rate of the effective pore concentration led by the deforming-to-pass of plugging particles can be describes as

particles be written as

ˆ ( r , x, t ) ∂∑ 1 s , while that induced by the deforming-to-pass of bridging ∂t

ˆ ( r , x, t ) 1 ∂∑ 2 s . η ∂t

2.3 Overall conservation law for PPG particles and pores σ 1 ( x, t ) and σ 2 ( x, t ) are the concentrations of net captured particles (concentration of total captured particles minus that of deforming-to-pass captured particles ) and net bridging particles (concentration of total bridging particles minus that of deforming-to-pass bridging particles). Then the following equations are obtained:

7



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∂σ 1 ( x , t )

∞ ˆ ( r , x, t )  ∂ ∑1' ( rs , x, t ) ∂ ∑ 1 s = ∫ −  drs   ∂t ∂t 0 

(11)

ˆ ( r , x, t )   ∂ ∑ '2 ( rs , x, t ) ∂ ∑ 2 s = ∫ −  drs   ∂ t ∂ t 0 

(12)

∂t ∂σ 2 ( x , t ) ∂t

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∞

According to assumptions (3) and (5), one PPG particle plugs one pore with a small size and η particles bridge and plug one pore with a large size. Therefore, for a certain distance x, the decreasing number of the effective pores should equal to the number of plugged pores. Thus the following equation is obtained:

  1 l  σ 1 ( x, t ) + σ 2 ( x, t )  +h ( x, t ) =h0 η  

(13)

where, h0 is the initial surface concentration of pores. The above equation means that the summation of the remaining effective pores and the plugged pores should equal to the initial effective pores and this is the overall conservation law for PPG particles and pores.

2.4 Viscosity model of PPG suspension There are many polymer molecule chains on the particle surface which extend into the aqueous phase leading to a slight increase of the aqueous phase viscosity and this phenomenon has been proven in several literatures. 28, 31 The bigger the PPG concentration is, the higher the PPG suspension’s viscosity is. Similar with polymer solution, the relationship between viscosity and concentration can be described as:

µ PPG = µ w 1 + ( E1c + E2 c 2 + E3c 3 )  where,

(14)

µPPG is the suspension’s viscosity; E1 , E2 and E3 are all coefficients.

2.5 Boundary conditions Due to size exclusion, the injected PPG particles can only enter into the large pores at first. Therefore, the suspension flux entering large pores with the injected concentration is equal to the suspension flux transported by water in accessible pore space and the following equation is obtained:

c ( 0, t ) = c0 ( 0, t )

f a + f nl fa

(15)

From above equation, we can see that the post-inlet concentration is bigger than the injected particle concentration. The producer is assumed at the end of the core (i.e. x=L) and similarly, PPG is only produced from the accessible area due to size exclusion. Therefore, the outlet PPG suspension is diluted in the overall water flux after passing the core outlet. Then the following equation is obtained:

c ( L, t ) f a vw = cou ( L, t ) vw Where, cou ( 0, t ) is the PPG concentration of the produced liquid.

3. Model Solution and validation 3.1 Numerical solution 8


(16)

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Assuming that the parameters at the time step n are known, the parameters at the time step n+1 are to be determined. The pressure p , the aqueous phase velocity vw at the time step n+1 can be obtained by implicitly solving the conventional mass conservation equation of the aqueous phase with the finite difference method. Substitute the calculated p and vw into eq 1, and then the PPG particle concentration c ( x, t ) at the time step n+1 can be calculated explicitly with the four-order Runge-Kutta method. Meanwhile, it can be seen from eq 1 that

f a [ H , rs ] , ϕa [ H , rs ] , H ,

∂σ 1 ( x , t ) ∂t

and

∂σ 2 ( x, t ) ∂t

need to be updated, all of which are the functions of H

or H itself. So it is critical to calculate H and the following is the method. The element shown in Fig. 2a is still used as the research object, which has a length of ∆x and a pore concentration distribution following the logarithmic normal distribution. As is shown in Fig. 2b, the PPG particle radius divides the pore concentration distribution function into two parts: the left part that is corresponding to the small pore is defined as Part I and the right part that is corresponding to the large pore is defined as Part II. For Part I, the change of H is caused by the captured particles, while for Part II, it is caused by bridging particles. Hence the changes in H for the two situations should be calculated respectively. The solution to H of Part I is used as an example and discussed below. As mentioned above, the change of H for Part I is caused by the capture and deforming-to-pass of particles, and hence the equations below can be obtained:

∂H ( rp , x, t ) ∂t

=l

∂H v ( rp , x, t ) ∂t

=−

k1 ( rp ) vw H ( rp , x, t )  ∇p − p g   c ( x, t ) f a [ H , rs ] − γ 1σ 1 ( x, t )  k pg  

r  ∇p − p g  ∂h ( x , t ) vw s = − ∫ k1 ( rp ) H ( rp , x, t )  c ( x, t ) f a [ H , rs ] − γ 1σ 1 ( x, t )  drp ∂t k 0 p g  

(17)

(18)

Eq 17 and 18 can be solved implicitly and the solution of part II also can be obtained in the same way.16 The solution of part I and part II are:  ∇p − pg   − k1 ( rp ) c ( t ) f a [ H , rs ]−γ 1σ 1 ( x ,t )  y1 pg    H ( y , r ) = H e 1 p 0   ∇p − pg   rs − k r  c ( t ) f [ H , r ] −γ σ ( x , t )  y1 pg  h y , r = H e 1 ( p )  a s 1 1 drp ( ) 1 p ∫0 0     ∇p − p g  1 − k1 ( rp )  pa f a [ H , rs ]− γ 2σ 2 ( x ,t )  y2  p g  η   H ( y2 , rp ) = H 0 e   ∇p − p g  1 ∞  − k1 ( rp )  pa f a [ H , rs ]− γ 2σ 2 ( x ,t )  y2 p g  η   h ( y2 , rp ) = ∫ H 0 e drp  rs

(

Combing with eq 13, y1 and y2 can be solved based on the value of h y , rp 19 is obtained.

3.2 Model validation

9


    rs > rp        rs ≤ rp   

)

(19)

and then the new H in eq


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(a) Experimental setup (b) PPG particles Figure 5. Experimental setup and PPG particles In this part, an experiment of PPG migration in porous media was conducted first and then history matching was done using the proposed model. Through history matching, on one hand, the validity of the model was proved. On the other hand, it is nearly impossible to measure all parameters by the experimental method, therefore, some parameters can be determined through history matching. The experimental setup and PPG particle used are shown in Figure 5. The experimental temperature is 22℃. The length and diameter of the core are 17.5cm and 1.95cm respectively. The core is filled by glass beads and the porosity and permeability of the core are 0.31 and 2.9µm2 respectively. The PPG suspension was provided by Shengli oilfield (Sinopec, China) and the mean radius of the PPG particle is 41µm which was measured by HAAKE RS150 Rheometer. The concentration of the PPG suspension is 2000mg/L and the viscosity is 2.1mPa.s in the condition that the shear rate is 7.32s-1. The PPG suspension was first injected into the core with the rate of 0.2mL/min and when the injected slug size of the suspension equals 0.3PV, pure distilled water with no PPG was injected. There are two commonly used measures to simulate the pore shape of glass beads in size exclusion theory. One measure assumes the cross section of the pore is circle shaped and the other assumes the cross section is triangular. 18,19 The former measure is adopted in this paper and the size concentration distribution of pores is assumed to follow the logarithmic normal distribution, namely

f ( rp ) = where, rp is the pore radius;

1 rpτ 2π

( ( ) )

- ln rp -ζ

e

2

2τ 2

(20)

τ is the variance of the pore radius; ζ is the mean value of the pore radius;

f is the density function. Three pressure survey points are set, respectively at the core entrance, one-third-length and two-thirds-length locations. The measured pressures at the three points can well reflect the general migration regularity of the PPG inside porous media and this also is the common widely used experimental method in the research of PPG particles. 28 The experimental and simulation results of the new model (the model based on size exclusion theory) and conventional model (the model based on classic filtration theory) are shown in Figure 6 and the key parameters of the simulation are listed in Table 1. From the experimental result, we can see that the pressure at the inlet increases first and the pressures at other two survey points increases in turn. The pressure does not increase until the injected PPG reaches and plugs or bridging the pores which means the injected PPG particles can move towards the deep portion of the core sample under the differential pressure. In the process of subsequence water injection, the pressure slightly decrease which means the injected PPG particles can not be washed out immediately and the captured or bridging 10


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particles still leads to a high resistance to the injected water. From the comparison of the simulation results, we can see that the pressure increase of the conventional method is later than that of the new method and the larger the distance between the survey point and the inlet is, the later the pressure increase becomes compared with the experimental results. This is because the conventional method assumes that all pores are accessible. While in reality, many small pores are inaccessible to the PPG particles so the migration of the PPG particles simulated by the conventional method is much slower. By comparison of the experimental data and simulation results, we can conclude that the simulation results of the new method can better match the measured data, which proves the validity of the model. Table 1. Key parameters of the simulation

Figure 6. History matching

p1

0.1

p2

0.1

n1

2

n2

2

η

3

a1

0.005

a2

0.003

b1

0.5

b2

0.6

γ1

0.9

γ2

0.8

τ

30

ζ

41

E1

4×10-15

E2

7×10-26

E3

1×10-36

por

0.31

perm

2.9

4. General regularity of PPG migration through porous media Due to the limitation of the physical simulation, the obtained data from the experiment are rare. Therefore, in this part, based on the history matching mentioned above, the change patterns of the pore concentration distribution, concentration of the retained particles and permeability after injecting PPG particles and rigid particles are compared to analyze the migration mechanism of the PPG.

4.1 Change pattern of the pore concentration

(a) PPG (b) Rigid particle Figure 7. Pore concentration distribution at the inlet during the injection process of PPG and rigid particles 11



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(a) PPG (b) Rigid particle Figure 8. Pore concentration distribution at the inlet during the injection of water following PPG and rigid particle suspensions Figure 7 shows the change of the pore concentration distribution at the inlet during the injection process of PPG and rigid particles. Before PPG injection, no plug occurs and therefore the red curve in Figure 7(a) can be seen as the initial pore concentration distribution. For both of the two kinds of particles, it is shown that the effective pore concentration of Part I reduces rapidly, as pores with radius lower than the particle size is quickly plugged. In the meantime, the pore with the radius closest to the particle radius decreases most in concentration, because the probability of particle capture increases in the pore with a large radius, where the flow rate is correspondingly high. As for pores with radius exceeding the particle size (Part II), their concentrations also decrease, due to bridging. The probability of bridging-induced plugging rises, as the pore size reaches the particle size. Therefore, the closer the pore size to the particle size, the larger the reduction of the pore concentration. Compared with the pore concentration reduction due to capture or bridging of rigid particles, that of PPG particles is lower. This is because the injection process of PPG particles is a co-existing process of “plugging, bridging and re-flow”. When massive pores are plugged due to particle capture or bridging, the pressure gradient constantly grows. As the pressure gradient exceeds the critical pressure gradient, the PPG will start to deform and re-flow, and the pores once plugged are open again. However, the rigid particle is incapable of deforming-to-pass, and therefore the permanent plugging of pores occurs continuously during the injection. The concentration of pores smaller than the particle will constantly drop until all the small pores are plugged. Figure 8 shows the change of the pore concentration distribution at the inlet during the injection of water following PPG and rigid particle suspensions. It is shown the PPG particles that have already plugged the pores can constantly get through with deformation under the pressure gradient during the subsequence water injection. Therefore, with no new PPG particles injected, the pore concentration continuously recovers as the water injection proceeds. However, as for rigid particles with no deformation abilities, the plugging and bridging they have caused is permanent and the pore concentration does not change during the subsequence water injection process, which leads to complete overlying of the black, blue and green curves in Figure 8b.

4.2 Change pattern of the retention concentration of particles Figure 9 compares the retention concentrations (the total concentration of captured, deposited and bridging particles) of PPG and rigid particles during the whole process. It shows that after the injection finishes, due to the deformation feature, PPG particles can move further into the porous media than rigid particles, with a lower retention concentration. During the following water injection process, the PPG particles retained at the inlet can still deform to 12


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pass the pore constantly, and hence the retention concentration drops constantly. The position with the maximum retention concentration of PPG particles moves forward along the core with the water injection going, which reflects the effect of “temporary plugging-deforming and passing-temporary plugging again”. In regard to rigid particles, after the particle injection finishes, there are still mobile particles in pores with radius exceeding the particle radius, which can go on to migrate forward during the following water injection and cause permanent capture or bridging afterwards. It should be noticed in Figure 9(b) that the curves of the particle retention concentration of injection of 0.7 PV and 0.9 PV have coincides, so when the subsequence water injection reaches 0.4 PV, approximately all of rigid particles have come to the captured, deposited or bridging status.

(a) PPG

(b) Rigid particle

Figure 9. Retention concentrations of PPG and rigid particles

4.3 Change pattern of the particle concentration

(a) PPG (b) Rigid particle Figure 10. Concentrations of PPG and rigid particles Figure 10 shows the comparison between the concentration of PPG and rigid particles during their migrating through the porous media. It is illustrated that the particle concentration at the inlet is higher than the injection concentration, according to eq 15. During the following water injection, although new capture and bridging of PPG 13



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particles continuously occurs, the particle concentration peak value constantly moves forward, as PPG particles continue to re-flow due to their characteristic of deforming-to-pass and migrate further into the porous media. For rigid particles, with no following particle supply and increasing permanent plugging occurring, the particle concentration decreases rapidly during the water injection. With Figure 9b combined, it can be seen that with injection reaching 0.7 PV, namely the subsequence water injection reaching 0.4 PV, nearly all the particles have been retained due to capture, deposition or bridging, and hence the particle concentrations are almost 0 when the injection volumes have approached 0.7 PV and 0.9 PV.

4.4 Change pattern of the permeability The permeability of the porous media during the injection of PPG and rigid particles is compared in Figure 11. It shows that due to the pore plugging induced by injected particles, the permeability of the porous media decreases after both the injection of PPG and rigid particles. However, due to the deforming-and-passing effect of the PPG, the number of retained particles is lower and so is the resultant permeability reduction. In the meantime, the deforming-and-passing of PPG particles continues in the subsequence water injection period, which leads to continuous recovering of the permeability at the inlet and moving forward of the permeability drawdown cone. Nevertheless, rigid particles cannot deform, and therefore the permeability reduction at the inlet caused by them is permanent. Similarly, because almost all of the rigid particles have been stuck into the permanent blockage when the injection reaches 0.7 PV, the permeability curves with injection of 0.7 PV and 0.9 PV nearly coincide.

(a) PPG

(b) Rigid particle Figure 11. Permeability of the porous media

5. Conclusion This work established the kinetic equations of the plugging, bridging, deposition and deforming to pass of the PPG on the basis of the size exclusion theory. Combining with numerical simulation theory, a numerical simulation model of PPG migration in the porous media was built and validated by the experimental results. The migration regularities of PPG and rigid particles inside porous media were simulated and compared. The major conclusions indicated from this research are as follows: (1) The size exclusion theory assumes that the suspended particle only migrates through the accessible part of the large pores. Therefore, the aqueous phase vw in the conventional model should be replaced by the velocity of the accessible part vw f a and the porosity ϕ should be replaced by the accessible porosity ϕa in the new model. In 14


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extreme cases, where the radius of the PPG particle equals 0, the mass balance equation of the new model degenerates into that of the model based on the classic filtration theory. (2) The suspension flux entering large pores with the injected concentration is equal to the suspension flux transported by water in accessible pore space. Therefore, the post-inlet concentration is bigger than the injected particle concentration. Similarly, PPG is only produced from the accessible area due to size exclusion. Therefore, the outlet PPG suspension is diluted in the overall water flux after passing the core outlet. (3) The conventional method assumes that all pores are accessible. While in reality, many small pores are inaccessible to the PPG particles. Therefore, the migration of the PPG particles simulated by the conventional method is much slower than that by the new method and the simulation results of the new method can better match the experimental results. (4) For rigid particles, the permanent capture and bridging occurs continuously during the injection and therefore the concentration of pores smaller than the particle will constantly drop until it becomes zero. While for PPG particles, the PPG will start to deform and re-flow, and the pores once plugged are open again. Therefore, at the subsequence water flooding period, the permeability and concentration of pores near the inlet will recover and the peaks of the PPG concentration will move towards the deep portion of the core.

ASSOCIATED CONTENT Supporting Information Some basic parameters and relative background knowledge about size exclusion theory.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Phone: +86 139 5467 0741

Notes The authors declare no competing financial interest.

Acknowledgements This work was financially supported by the National Natural Science Foundation of China (Grant no. 51574269), the Indigenous Innovation Program of Qingdao (16-5-1-46-jch), the Foundation for Outstanding Young Scientist in Shandong Province (Grant no. BS2014NJ011), the Important National Science and Technology Specific Projects of China (Grant no. 2016ZX05011-003), the Fundamental Research Funds for the Central Universities (Grant no. 15CX02010A; 15CX08004A) and the National Science Foundation for Distinguished Young Scholars of China (Grant No. 51625403).

Nomenclature a1 = first coefficient of the function of critical deformation pressure gradient for captured particles a2 = first coefficient of the function of critical deformation pressure gradient for bridging particles 15



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b1 = second coefficient of the function of critical deformation pressure gradient for captured particles b2 = second coefficient of the function of critical deformation pressure gradient for bridging particles Bw = volume factor of water C = suspended particle concentration by sizes, m-4

C0 = injected particle concentration by sizes, m-4 c = total suspended particle concentration, m-3 c0 = injected particle concentration, m-3 cin = injected concentration of PPG particles, m-3 cou = PPG concentration of the produced liquid, m-3 D = depth, m E1 = first coefficient of the function of suspension’s viscosity E2 = second coefficient of the function of suspension’s viscosity E3 = third coefficient of the function of suspension’s viscosity f = density function

f a = accessible flow fraction f ns = inaccessible flow fraction in small pores f nl = inaccessible flow fraction in large pores g = gravitational acceleration, m/s2 h = total pore concentration, m-2

h0 = initial total pore concentration, m-2 -3

H = pore concentration distribution by sizes, m H0 = initial pore concentration distribution by sizes, m-3 -4

H v = volumetric pore concentration distribution, m j = jamming ratio

k = permeability, m2

k1 = conductivity of a single pore, m4 l = average length of the capillary, m

n1 = bridging exponent n2 = deposition exponent p = pressure, Pa; pa1 = bridging probability pa 2 = deposition probability por = porosity perm = permeability, µm2 16


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∇pg1 = critical deformation pressure gradients for captured particles, Pa/m ∇p g 2 = critical deformation pressure gradients for bridging particles, Pa/m PV = pore volume, m3

qw = total flow rate, m3/s rs = particle radius, µm rp = pore radius, µm

s1 = cross sectional area of the pore, m2 t = time, s vw = velocity of the accessible flux, m/s vwa = velocity of the total flux, m/s vwns = velocity of the inaccessible flux, m/s x = distance between the survey point and injection point, m Greek Letters

∑ = total concentration of captured, deposited and bridging particles by sizes, m-4 ∑1 = concentration of captured particles by particle and pore sizes, m-5

ˆ = concentration of captured particles in small pores that deform to pass, m-4 ∑ 1 ˆ = concentration of bridging particles in large pores that deform to pass, m-4 ∑ 2

∑1' = concentration of captured particles by particle sizes, m-4 ∑'2 = concentration of bridging particles by particle sizes, m-4 ∑3' = concentration of deposited particles by particle sizes, m-4

σ = total concentration of captured, deposited and bridging particles, m-3 σ ' = total concentration of captured particles, m-3 1

σ 1 = total concentration of net captured particles, m-3 σ 2 = total concentration of net bridging particles, m-3

χ = accessible fraction of a single pore cross-section φa = accessible porosity for particles ω = flux reduction factor γ 1 = deforming-to-pass coefficient for captured particles γ 2 = deforming-to-pass coefficient for bridging particles η = mean number of particles which can bridge and block a capillary

µw = water viscosity, Pa.s µPPG = PPG suspension’s viscosity, Pa.s ρw = water density, kg/m3 17



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δ = impulse function τ = variance of the pore radius ζ = mean value of the pore diameter

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Flow of Preformed Particle Gel through Porous Media: A Numerical

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