Journal of Applied Mechanics and Technical Physics, Vol. 54, No. 3, pp. 423–432, 2013. c V.V. Kadet, A.M. Galechyan. Original Russian Text 

PERCOLATION MODEL OF RELATIVE PERMEABILITY HYSTERESIS V. V. Kadet and A. M. Galechyan

UDC 532.546

Abstract: A mathematical model of relative permeability hysteresis in drainage and imbibition is constructed on the basis of percolation theory. It is shown that the results are in qualitatively agreement with experimental data. Keywords: relative permeabilities, drainage, treatment, hysteresis, percolation theory. DOI: 10.1134/S0021894413030115

INTRODUCTION In the development of deposits by methods such as cyclic flooding and changing the direction of seepage, displacement of oil by water alternates with displacement of water by oil. This change in the nature of movement affects the dependences of the relative permeabilities on water saturation. This phenomenon is called relative permeability hysteresis for drainage and imbibition. In different sources, by drainage and imbibition are meant different physical phenomena. In this paper, in accordance with the experimental procedure [1], drainage is the displacement of a wetting fluid by a nonwetting fluid under a pressure gradient, and imbibition is the displacement of a nonwetting fluid by a wetting fluid under a pressure gradient. Accordingly, for a hydrophilic sample, drainage is the displacement of water by oil and imbibition is the displacement of oil by water. In the case of hydrophobic rock wetted by oil, the displacement of water by oil is imbibition, and the displacement of oil by water is drainage. In this paper, percolation theory is used to construct a model of relative permeability hysteresis for drainage and imbibition that allows one to describe this phenomenon and explain the mechanism of its occurrence.

1. EXPERIMENTAL STUDY OF RELATIVE PERMEABILITY HYSTERESIS The presence of relative permeability hysteresis was observed in several experimental studies [1–3]. The most convincing results are presented by Wei and Lile [1], whose used the following experimental procedure. Two bulk samples were produced from hydrophilic and hydrophobic sand. The porosity of these samples was 38%. Each sample was fully saturated with a wetting fluid. In the case of the hydrophilic sample initially saturated with water, water was displaced by oil under a pressure gradient until the maximum water saturation (drainage) was achieved. Then, under the action of a gradient pressure, oil was displaced by water until the maximum oil saturation (imbibition) was achieved. The filtration rate was 1.31 · 10−4 m/s. Experimental curves of the relative permeability on water saturation S are presented in Fig. 1.

Gubkin Russian State University of Oil and Gas, Moscow, 119991 Russia; [email protected]; [email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 54, No. 3, pp. 95–105, May–June, 2013. Original article submitted October 29, 2012. c 2013 by Pleiades Publishing, Ltd. 0021-8944/13/5403-0423 

423

koil, kw 1.0 0.8

20

1

0.6 10

2

0.4 0.2

0

0.2

0.4

0.6

0.8

1.0 S

Fig. 1. Experimental curves of relative permeability versus water saturation: drainage (1 and 1  ) and imbibition (2 and 2  ); curves 1 and 2 refer to oil and curves 1  and 2  refer to water.

2. MATHEMATICAL MODEL OF DRAINAGE AND IMBIBITION As the model of the pore space we use a cubic network formed by capillaries [4–6] which are distributed along the radius as a lognormal function qualitatively close to the real capillary radius distribution function:  (ln r − μ)2  1 . exp − f (r) = √ 2σd2 2π σd r Here σd = 0.25 and μ = 2. Considering the granular structure of the medium, in which the pore sizes vary only slightly, the water saturation S can be considered equal to the proportion of capillaries containing water. The occurrence of hysteresis is due to the fact that adsorption of the active oil components on the surface of rock-forming minerals during drainage in hydrophilic rock results in hydrophobization of the surface [7]. We consider the following models of the drainage and imbibition processes. 2.1. Drainage Model In a hydrophilic sample, oil mostly enters large pores since the hydrodynamic drag in them is lower; in addition, in small pores, water is held by capillary forces, which prevents the flow of oil through them. Thus, oil is contained in capillaries whose radius is larger than rk , and water is in capillaries whose radius is smaller than rk . Here rk is the minimum radius of the capillary from which the wetting fluid (water) is displaced under a predetermined difference in pressure between the fluids Δp [8], rk = 2χ cos θ/Δp (χ is the coefficient of interfacial tension and θ is the contact angle). In this case, the radius distribution function for the capillaries filled with oil can be represented as  foil (r) =

424

0,

r < rk ,

f (r),

r  rk .

(1)

Then, the oil relative permeability koil (rk) [4] is described by the analytical expression  rc koil (rk ) =

1 K0

f (r) dr

rc

ν

f (r1 )

r1

8 π

rk

∞ r1

f (r) dr r4

 ∞ f (r) dr

dr1 .

−1

(2)

r1

Here K0 is the absolute permeability of the sample, ν is the index of the correlation radius [4, 5], and the quantity rc is given by the relation ∞

D , z(D − 1)

f (r) dr = Pcb = rc

(3)

Pcb is the percolation threshold through the network of capillaries [4, 5], D is the dimension of the problem, and z is the coordination number of the network. The expression for the absolute permeability K0 is of the form  rc f (r) dr

rc

f (r1 )

r1

K0 = 0

ν

∞

8 π

r1

f (r) dr r4

 ∞ f (r) dr

−1

dr1 .

(4)

r1

The radius distribution function for the capillaries filled with water can be represented as  f (r), r  rk , fw (r) = 0, r > rk .

(5)

Accordingly, the water relative permeability kw (rk ) can be described by the analytical expression [4] r

c 1 kw (rk ) = K0

f (r) dr

rc

 0

ν

f (r1 )

r1

8 π

∞ r1

f (r) dr r4

 ∞ f (r) dr

−1

dr1 .

(6)

r1

Here the quantity rc is given by the relation rk f (r) dr = Pcb .

(7)

rc

In view of the granular structure of the medium, the relationship of dependences (2) and (6) with the water saturation S is determined from the following relation: rk f (r) dr.

S=

(8)

0

Thus, formulas (1)–(8) determine the parametric dependences of the water relative permeability on the saturation of the medium for the above model of the granular structure of the pore space. 425

(a)

(b)

1

I

II

I

II

2

I

II

o2 o2

o1

Fig. 2. Capillary profile in the contact area of oil and water in the cases of drainage (a) and imbibition (b) for 0 < α < 1 (1) and α < 0 (2); I is oil, II is water; arrows show the displacement direction.

f 0.25 0.20 0.15 II 0.10 0.05 I 0

5

I 10

15

r, mm

Fig. 3. Radius distribution for capillaries filled with water (I) and oil (II) in the imbibition mode at a time for α > 0 (arrows show the displacement direction).

2.2. Imbibition Model The passage of oil through the initially hydrophilic porous medium changes the surface properties of some of the capillaries. To describe this phenomenon, we use the percolation model of the medium with microheterogeneous wettability [5, 6] and supplement it by the following parameters: κ is the proportion of capillaries with unchanged surface properties (type 1), 1 − κ is the proportion of capillaries with changes surface properties (type 2), and α = cos θ2 / cos θ1 (θ1 and θ2 are the contact angles in the capillaries of the first and second types, respectively). There are two types of change in the surface properties (Fig. 2): (1) reduction in the degree of hydrophilicity of the surface of capillaries (0 < α < 1); (2) conversion of the surface to the hydrophobic state (α < 0). For 0 < α < 1, water displaces oil primarily in small capillaries. Water is contained in capillaries of the first type, whose radii are smaller than rk1 , and in capillaries of the second type, whose radii are smaller than rk2 (Fig. 3). The radii rk1 and rk2 are defined by the expression [8] rki = 2χi cos θi /pk ,

i = 1, 2,

(9)

where χi is the coefficient of interfacial tension in the capillaries of the ith type and pk is the capillary pressure. In order to reduce the number of parameters, we assume χ1 = χ2 = χ since this parameter is contained only the product χi cos θi . Then, from (9) it follows that rk2 = αrk1 . 426

The radius distribution function for the capillaries filled with oil is of the form ⎧ 0, r < rk2 , ⎪ ⎨ foil (r) = (1 − κ)f (r), rk2  r  rk1 , ⎪ ⎩ f (r), r > rk1 . The oil relative permeability koil (rk1 ) can be described by the analytical expressions koil (rk1 ) = 0, π(1 − κ) koil (rk1 ) = 8K0

2

rc,oil rc,oil  



f (r) dr

ν

r

αrk1

rk1 > rc1 ,

rk1 ∞ rk1 ∞   f (r) f (r) −1  f (r ) (1−κ) f (r) dr+ f (r) dr (1−κ) dr+ dr dr , r4 r4 

r

r

rk1

rk1

rc < rk1  rc1 , koil (rk1 ) =

π

(1 − κ) 8K0

rk1 

(1 − κ)

rk1 rc ν f (r) dr + f (r) dr f (r ) r

αrk1

rk1

rk1 ∞ rk1 ∞   f (r) f (r) −1  × (1 − κ) f (r) dr + f (r) dr (1 − κ) dr + dr dr r4 r4 r

rc  rc +

f (r) dr rk1

r

rk1

ν



∞

f (r )

r

f (r) dr r

rk1

 ∞ f (r) r

r4

dr

−1

 dr ,

0 < rk1  rc , where rc1 is the radius of capillaries of the first type starting from which an infinite cluster (IC) of the two types of capillaries filled with oil is formed; rc,oil is an analog of the radius rc , which is a function of rk1 . The radius rc is calculated by formula (3). The value of rc1 is found from the condition rc1 (1 − κ)

∞ f (r) dr = Pcb ,

f (r) dr +

αrc1

rc1

and rc,oil is given by rk1 (1 − κ)

∞ f (r) dr = Pcb .

f (r) dr +

rc,oil

rk1

In this case, rc,oil varies in the range αrc1  rc,oil  rc , and rc,oil = rc at rk1  rc . The radius distribution function for the capillaries filled with water can take the following values: ⎧ ⎪ ⎨ f (r), r < rk2 , fw (r) = κf (r), rk2  r  rk1 , ⎪ ⎩ 0, r > rk1 . The water relative permeability kw (rk1 ) is described by the analytical expressions π kw (rk1 ) = 8K0

rc,w

  rc,w  k1 rk1  k1 rk1 ν  αr  αr f (r) f (r) −1   f (r) dr f (r ) f (r) dr + κ f (r) dr dr + κ dr dr , r4 r4 0

r

r

rk1 > rc1,w ,

αrk1

r

αrk1

rc,w < αrk1 , 427

f 0.25

f 0.25

(a)

0.20

(b)

0.20

0.15

0.15 II

II

0.10

0.10

0.05

0.05 I

I

I 0

5

10

r, mm

15

0

5

10

15

r, mm

Fig. 4. Radius distribution for capillaries filled with water (I) and oil (II) in the imbibition mode at some time for α < 0: (a) pk < 0; (b) pk > 0; arrows show the displacement direction.

π

kw (rk1 ) = 8K0

αr  k1

0

 k1  k1 rk1  k1 rk1  αr ν  αr  αr f (r) f (r) −1   f (r) dr f (r ) f (r) dr + κ f (r) dr dr + κ dr dr r4 r4 r

r

r

αrk1

αrk1

rc,w

  rc,w rk1 ν  rk1 f (r) −1   +κ f (r) dr f (r ) f (r) dr dr dr , r4 2

αrk1

r

r

r

rc,w  αrk1 .

rk1 > rc1,w ,

Here rc,w is a function of rk1 , which in the analysis of the formation of an infinite cluster of capillaries filled with water—infinite water cluster (IWC) has the same meaning as rc,oil (rk1 ) in the case of an infinite cluster of capillaries filled with oil—infinite oil cluster (IOC). In this case, the function rc,w (rk1 ) varies from zero for some minimum value rk1 = rc1,w to the value of rc for rk1 = ∞. The condition which the function rc,w (rk1 ) is determined is given by αr  k1 rk1 f (r) dr + κ f (r) dr = Pcb , rc,w < αrk1 rc,w

αrk1

or rk1 κ

f (r) dr = Pcb ,

rc,w  αrk1 .

rc,w

The radius rc1,w is determined from the condition αrc1,w

rc1,w





f (r) dr + κ 0

f (r) dr = Pcb .

αrc1,w

We consider the case α < 0. In this case, water first displaces oil from small capillaries of the first type (Fig. 4a). After all the capillaries of the first type are filled with water, it begins to displace oil from large capillaries of the second type (Fig. 4b). Various types of phase filtration are possible, depending on the value of κ. As the capillary pressure pk increases in the range from −∞ to zero, the proportion of the capillaries of the first type in the IOC decreases. For such capillaries, the radius distribution function has the form 428

 foil (r) =

(1 − κ)f (r),

r < rk1 ,

f (r),

r  rk1 .

If rk1 < rc , then rc,oil (rk1 ) = rc , and rk1 rk1 rc rk1 ν  ∞  π   (1 − κ) f (r) dr + f (r) dr f (r ) f (r) dr − κ f (r) dr koil (rk1 ) = 8K0 0

r

rk1



× (1 − κ) r

rc  rc +

f (r) dr rk1

r

rk1

ν

f (r) dr + r4

∞

f (r) −1  dr dr r4

rk1

∞



f (r )

r

r

f (r) dr

 ∞ f (r)

r

r4

r

dr

−1

 dr ,

rk1 < rc . For rk1 > rc , the dependence rc,oil (rk1 ) is of the form rk1 ∞ f (r) dr + f (r) dr = Pcb . (1 − κ) rc,oil

rk1

If 1 − κ < Pcb , then with increasing pk < 0 the IOC should disappear for some pk = pc,oil < 0, since the number of capillaries of the second type is insufficient for the formation of the IOC. If 1 − κ > Pcb , then the IOC does not disappear up to the value pk = 0 (rk1 = ∞) and oil filtration stops only for pk > 0, i.e., for pc,oil = pc2,oil , rc1,oil = ∞. As a result, for koil in the region rc < rk1 < rc1,oil , we have π(1 − κ)2 koil (rk1 ) = 8K0

rc,oil

r

c,oil rk1   ν  ∞   f (r) dr f (r ) f (r) dr − κ f (r) dr

 0

r

rk1



× (1 − κ) r

r

f (r) dr + r4

∞ rk1

r

f (r) −1  dr dr , r4

rc < rk1 < rc1,oil . If 1 − κ > Pcb , then in the case pk > 0, oil will fill only capillaries of the second type with r < rk2 . With increasing pk , the value of koil (rk2 ) will decrease until it becomes equal to zero for rk2 = rc2,oil . The value of rc2,oil is calculated by the formula rc2,oil



(1 − κ)

f (r) dr = Pcb . 0

In the interval 0 < pk < pc2,oil (rk2 > rc2,oil ), the value of koil (rk2 ) is given by the relation π(1 − κ)2 koil (rk2 ) = 8K0

rc,oil

 0

r

c,oil rk2   ν  rk2 f (r) −1  f (r) dr f (r ) f (r) dr dr dr , r4 r

r

r

rk2 > rc2,oil , and the function rc,oil (rk2 ) is found from the condition 429

rk2 (1 − κ)

f (r) dr = Pcb .

rc,oil

The value of kw is calculated similarly. If κ < Pcb , water filtration can begin only after the filling of the capillaries of the first type, which are insufficient for formation of the IWC. With increasing pk , the IWC is formed from the capillaries of the first and second types at some time. In this case, rc2 is calculated by the relation ∞ κ + (1 − κ) f (r) dr = Pcb . rc2

The function rc,w (rk2 ) is determined from the condition rk2 κ

∞ f (r) dr = Pcb

f (r) dr +

rc,w

(10)

rk2

and as rk2 varies from rc2 to zero, it runs from zero to rc . The water relative permeability kw (rk2 ) is calculated by the formulas kw (rk2 ) = 0, πκ 2 kw (rk2 ) = 8K0

rk2 > rc2 ,

rc,w

  rc,w ∞ ν  rk2   f (r) dr f (r ) κ f (r) dr + f (r) dr 0

r

r

rk2

(11)

∞  rk2 f (r) f (r) −1  dr + dr dr , × κ r4 r4 r

rk2

rc < rk2  rc2 ; π

κ kw (rk2 ) = 8K0

rk2 rk2 rc ∞ ν  rk2   κ f (r) dr + f (r) dr f (r ) κ f (r) dr + f (r) dr 0

r

r

rk2

rk2

∞  rk2 f (r) f (r) −1  dr + dr dr × κ r4 r4 r

rc  rc f (r) dr rk2

ν

rk2

f (r )

r

∞ f (r) dr

 ∞ f (r)

r

r4

r

dr

−1

(12)



dr ,

0 < rk2  rc . Let κ > Pcb . Then, the IWC an form even during filling of the capillaries of the first type with water. For −∞ < pk < 0,  κf (r), r < rk1 , fw (r) = 0, r  rk1 . Analytical expressions for kw (rk1 ) are written as kw (rk1 ) = 0, πκ 2 kw (rk1 ) = 8K0 430

rk1 < rc1,w ,

rc,w

  rc,w rk1 ν  rk1 f (r) −1  f (r) dr f (r ) f (r) dr dr dr , r4 0

r

r

r

rk1  rc1,w .

At the time when the capillary pressure will be greater than zero, the IWC begins to include water-filled capillaries of the second type, with all capillaries of the first type being already filled with water. Further calculation of kw (rk2 ) was performed the same as in the case κ < Pcb using formulas (11) and (12), wherein rc2 = ∞ and rc,w is determined from condition (10). The relative permeabilities koil (S) and kw (S) were calculated from the above formulas for different values of the microheterogeneity parameters α, κ. For 0 < α < 1, the water saturation S was defined by the relation αr  k1 rk1 S= f (r) dr + κ f (r) dr, 0

αrk1

and for α < 0, by the relations rk1 f (r) dr, S=κ

S < κ,

0

∞ S = κ + (1 − κ)

f (r) dr,

S > κ.

rk2

Thus, the proposed approach allows us to determine the dependences of the relative permeability on water saturation for the above model of the granular structure of the pore medium with microheterogeneous wettability.

3. RESULTS OF NUMERICAL CALCULATIONS It was shown in the work that for α > 0, the relative permeability curves for the imbibition mode are below the relative permeability curves for the drainage mode for all values of κ. It should be noted that for a constant value of κ with increasing α, the relative permeability curves for the imbibition mode become smoother and approach the relative permeability curves for the drainage mode. This can be explained by the fact that the closer the value of α to one, the smaller the difference between the surface properties of the first and second types and, correspondingly, the smaller the difference between the drainage and imbibition modes. In the limiting case α = 1 (absence of difference in surface properties between the first and second capillary types) the relative permeability curves for the drainage and imbibition modes are the same. If we assume that the value of κ is varying and the value of α is constant, then, with increasing κ (in other words, with decreasing proportion of capillaries with changed surface properties), the maximum deviation of the relative permeability curves for the imbibition mode from the relative permeability curves for the drainage mode is observed for lower relative permeability values. Obviously, for κ = 1 and κ = 0, the relative permeability curves for the drainage and imbibition mode coincide, i.e., if the surface properties of all the capillaries remain unchanged (κ = 1) or if the surface properties of all capillaries change (κ = 0), while remaining hydrophilic (since α > 0), the difference between the drain and imbibition modes disappears. For α < 0 and increasing κ, the relative permeability curves are shifted toward increasing water saturation. For κ ∼ 1 − Pcb , this shift is maximal. With a further increase of κ, the relative permeability curves for the imbibition mode approach the relative permeability curves for the drainage mode, and in the limiting case (κ = 1), they coincide. The conditions of experiment [1] correspond to the case κ = 0.75 and α < 0 (Fig. 5). Consequently, in the drainage mode, the formation of a thin film composed of active oil components leads to hydrophobization of 25% of the pore surface on the walls of the capillaries.

CONCLUSIONS This model is universal and takes into account the different mechanisms of change in the surface properties of the pore space. The best agreement with experimental results is obtained when using the displacement of water 431

koil, kw 1.0 2

0.8

1

20

10

0.6 0.4 0.2

0

0.2

0.4

0.6

0.8

1.0 S

Fig. 5. Relative permeability versus water saturation calculated for the percolation model of relative permeability hysteresis for κ = 0.75 and α < 0: drainage (1 and 1  ) and imbibition (2 and 2  ); curves 1 and 2 refer to oil and curves 1  and 2  to water.

by oil as the main hydrophobic mechanism. The values of the parameters κ and α for a particular process (drainage or imbibition) are determined by comparing the calculated relative permeability dependences on water saturation with experimental data. The proposed method can be used to calculate the relative permeability in any a porous medium based only on data on the mineral composition and the pore radius distribution f (r). Experimental studies with core samples to determine the dependence of the relative permeability on water saturation require considerable time, in contrast to the experimental determination of the dependence f (r). Classifying the types of relative permeability hysteresis for samples with different mineralogical composition and determining the parameters κ and α for these samples, one can experimentally determine the dependence f (r) and calculate the relative permeability hysteresis using the above procedure.

REFERENCES 1. J. Z. Wei and O. B. Lile, Influence of Wettability and Saturation Sequence on Relative Permeability Hysteresis in Unconsolidated Porous Media, http://www.operetro.org/mslib/App/Preview.do?PaperNumber = 00025282& societyCode = SPE. 2. E. M. Braun and R. F. Holland, “Relative Permeability Hysteresis: Laboratory Measurements and a Conceptual Model,” SPE Res. Eng. 10 (3), 222–228 (1995). 3. J. T. Hawkins, A. J. Bouchard, “Reservoir-Engineering Implications of Capillary Pressure and Relative Permeability Hysteresis,” SPWLA. Log Analyst. 33 (4), 415–420 (1992). 4. V. V. Kadet, Methods of Percolation Theory in Underground Fluid Mechanics (TsentrLitNefteGaz, Moscow, 2008) [in Russian]. 5. V. I. Selyakov and V. V. Kadet. Percolation Models of Transport Processes in Microheterogeneous Media (1-yi TORMASH, Moscow, 2006) [in Russian]. 6. S. P. Glushko, V. V. Kadet, and V. I. Selyakov, “Percolation Model of a Two-Phase Equilibrium Filtration in a Medium with Microheterogeneous Wettability,” Izv. Akad Nauk SSSR, Mekh. Zhidk. Gaza, No. 5, 86 (1989). 7. N. S. Gudok, N. N. Bogdanovich, and V. G. Martynov, Determination of the Physical Properties of Oil- and Water-Containing Rocks (Nedra, Moscow, 2007) [in Russian]. 8. L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, 1986; Pergamon Press, Oxford-Elmsford, New York, 1987).

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