Accepted Manuscript

Pore Size Determination Using Normalized J-function for Different Hydraulic Flow Units

Ali Abedini, Farshid Torabi, Ph.D., P.Eng., Professor, and Program Chair

PII: S2405-6561(15)00034-6

DOI: 10.1016/j.petlm.2015.07.004

Reference: PETLM 25

To appear in: Petroleum

Received Date: 23 April 2015

Revised Date: 6 July 2015

Accepted Date: 8 July 2015

KcAl ISSN: 2405-5816

№-----1 201503

Vol.1, No.l

Petroleum

Please cite this article as: A. Abedini, F. Torabi, Pore Size Determination Using Normalized J-function for Different Hydraulic Flow Units, Petroleum (2015), doi: 10.1016/j.petlm.2015.07.004.

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2 Pore Size Determination Using Normalized J-function for Different Hydraulic

3 Flow Units

4 Ali Abedini and Farshid Torabi*

5 Faculty of Engineering and Applied Science, University of Regina, Regina, SK S4S 0A2, Canada

7 * Corresponding Author:

8 Farshid Torabi, Ph.D., P.Eng.,

9 Professor, and Program Chair,

10 Petroleum Systems Engineering,

11 Faculty of Engineering and Applied Science,

12 University of Regina,

13 Regina, SK, S4S 0A2

14 Canada,

15 Email: farshid.torabi@uregina.ca

16 Phone: +1 306 585 5667

20 Abstract

21 Pore size determination of hydrocarbon reservoirs is one of the main challenging areas in

22 reservoir studies. Precise estimation of this parameter leads to enhance the reservoir simulation,

23 process evaluation, and further forecasting of reservoir behavior. Hence, it is of great importance

24 to estimate the pore size of reservoir rocks with an appropriate accuracy. In present study, a

25 modified J-function was developed and applied to determine the pore size radius in one of the

26 hydrocarbon reservoir rocks located in the Middle East. The capillary pressure data vs. water

27 saturation (Pc-Sw) as well as routine reservoir core analysis include porosity and permeability

28 (k) were used to develop the J-function. First, the normalized porosity (^z), the rock quality index

29 (RQI), and the flow zone indicator (FZI) concepts were used to categorize all data into discrete

30 hydraulic flow units (HFU) containing unique pore geometry and bedding characteristics.

31 Thereafter, the modified J-function was used to normalize all capillary pressure curves

32 corresponding to each of predetermined HFU. The results showed that the reservoir rock was

33 classified into five separate rock types with the definite HFU and reservoir pore geometry.

34 Eventually, the pore size for each of these HFUs was determined using a developed equation

35 obtained by normalized J-function corresponding to each HFU. The proposed equation is a

36 function of reservoir rock characteristics include FZI, lithology index (J*) and pore size

37 distribution index (f). Using this methodology, the reservoir under study was classified into five

38 discrete HFU with unique equations for permeability, normalized J-function and pore size. The

39 proposed technique is able to apply on any reservoir to determine the pore size of the reservoir

40 rock, specially the one with high range of heterogeneity in the reservoir rock properties.

41 Keywords: Pore size, Pore geometry, Hydraulic flow unit, Capillary pressure, J-function

44 1. Introduction

45 Fluid flow through porous media is one of the crucial topics in hydrocarbon reservoir studies.

46 This phenomenon is highly affected by the pore geometry of the reservoir rocks. Reservoir rocks

47 have complex pore geometry which is mainly inspired by and changed through geological and

48 environmental events such as compaction, cementation, fracturing and oxidation [1,2]. To

49 construct a robust model to simulate the fluid flow behavior in the hydrocarbon reservoirs, it is

50 essential to precisely determine the pore size of the reservoir rocks. There are various methods

51 available in the literature to determine pore size and its distribution [3-6]. Among all methods

52 that have been proposed, the capillary pressure curves are more common, accurate and widely

53 used simply; because these curves are directly related to pore throat size and its distribution in

54 reservoir rock [7-12]. The capillary phenomena occur in porous media once two or more

55 immiscible fluids are present in pore spaces and can be quantified by the capillary pressure

56 which is defined as the difference in pressure between the non-wetting and wetting phases:

57 Pc = Pnw - Pw (1)

58 Since oil and water are typically the two main fluids in a water-wet hydrocarbon reservoir, the

59 above equation can be written as:

60 Pcow = Po — Pw (2)

61 Capillary pressure reflects the interaction of rock and fluids and is affected by the pore geometry,

62 pore size distribution, interfacial tension and wettability. As a consequence, the capillary

63 pressure is defined as [13]:

64 p^ = 2gcosq (3)

66 Where y is interfacial tension between oil and water, d is contact angle and r is pore radius.

67 The analysis and interpretation of capillary pressure curves reveals useful information about the

68 reservoir rock properties. However, due to the heterogeneity present in the reservoir rocks, no

69 single capillary pressure curve can be used as a representative of entire reservoir [14]. Therefore,

70 it is desired to normalize capillary pressure curves into a single curve that could be considered

71 for a unique rock types. The capillary pressure curves can be normalized into single curve using

72 a Leverett dimensionless J-function as follows [15]:

w gcos q ]]

74 The square root of k over p is known as rock quality index (RQI), which is used in rock type

75 determination [16]. Therefore, the Eq. (4) can be written as follows:

76 ^ S) = RQI (5)

-cos q

77 As it is shown in Eq. (5), the normalized J-function can be applied for a single rock type which

78 has the uniform rock properties [11]. Due to the dependency of capillary pressure to the pore size

79 radius of reservoir rock, using the J-function would be a reasonable method to determine pore

80 size for a rock with uniform properties.

82 2. Theory and Concepts

83 The mean hydraulic unit radius (rmh) concept is the key to determine the hydraulic units and

84 allows a suitable relationship between porosity, permeability, capillary pressure and geological

85 variation in the reservoir rock [16]. The mean hydraulic unit radius can be defined as the ratio of

86 cross-sectional area to wetted perimeter as follows:

87 rmh = nr2/ 2nr = r / 2 (6)

88 By applying Darcy's and Poiseuille's Laws, a relationship between porosity and permeability

89 can be derived as shown in Eq. (7):

k=t (7)

91 Where (p and t are porosity and tortuosity, respectively.

92 The above simple equation shows that the relationship between porosity and permeability

93 depends on geometrical characteristics of the pore space such as pore size (radius) and pore

94 shape. A tortuosity factor was introduced by Kozeny and Carmen to account for the realistic

95 granular porous medium [17,18]:

k = rj = _j (r)2 =j962L (8)

8t2 2r2 2) 2t 97

98 Expressing the mean hydraulic radius in terms of surface area per unit grain volume (Sgv) and

99 100 101 102

110 111 112

120 121 122 123

porosity results in:

Substituting the definition of rmh into the Kozeny and Carmen relationship results in the following form:

(1 -j) 2

F t2 S 2

Where the term Fst has been referred to as the Kozeny constant.

Dividing both sides of Eq. (10) by porosity and taking the square root of both sides results in:

j (1 -j) _4FsT Sgv _

Finally Eq. (11) can be written as follows [16]:

RQI = jz. FZI (12)

Where ( z and FZI are known as normalized porosity and flow zone indicator respectively and defined as:

v1 -jy

The value of FZI is given at the intercept of a unit-slope line with the coordinate pz = 1, on a loglog plot of RQI versus pz. Each FZI can be attributed to a single hydraulic flow unit (HFU). The hydraulic flow unit is a part of hydrocarbon reservoir that is uniform in horizontal and vertical

124 directions and has similar static and dynamic characteristics. Perhaps each HFU has the

125 consistent pore geometry i.e. pore size and its distribution, along the reservoir rock. As it is

126 mentioned, a unique J-function is able to normalize capillary pressure curves into a single curve

127 for a definite hydraulic flow unit. Hence, substituting Eq. (12) into Eq. (5) results in:

129 J(S. ) = j . FZI (15)

13, —cosq

132 Rearranging Eq. (3) gives:

133 -= - (16)

—cos o r

135 By substituting Eq. (16) into Eq. (15), one can derive:

136 J(Sw)= . FZI (17)

138 For a single hydraulic flow unit with unique FZI value, the J-function can be written as follows

139 [19]:

140 J (Sw ) = J Swn e (18)

141 Where

S — S

142 g = -^ (19)

wn çy v S

143 ^ 1 —

144 J and £ are lithology index and pore size distribution index respectively.

145 Substituting Eq. (18) in Eq. (17) and rearranging gives:

148 r = 2Vz . FZI (20)

149 J Swn e

150 Eq. (18) also can be written in the following form:

151 r = z. Swn e (21)

153 Where Z is pore geometry coefficient which is unique for each hydraulic flow unit. Z is defined

154 as:

- 2 j . FZI

156 Z = -J--(22)

157 Eq. (21) can be applied to each hydraulic flow unit to estimate pore size radius as a function of

158 normalized water saturation.

160 3. Properties of Rock Samples

161 In this study, the rock properties include porosity (p), liquid permeability (k), irreducible water

162 saturation (Swir) and capillary pressure (Pc) data vs. water saturation (Sw) obtained from the

163 routine and special core analyses for one of the Middle East reservoirs were used to apply and

164 examine the proposed technique. The 321 routine core data and 31 complete data sets of

165 capillary pressure vs. water saturation were analyzed. The capillary pressure curves were

166 obtained by mercury injection method. Fig. 1 shows the variation of permeability and porosity of

167 all reservoir core data. As it is shown in this figure, there is a high heterogeneity in the reservoir

168 rock properties. The statistical data of 31 core samples with complete data set is given in Table 1.

169 Fig. 2 depicts the measured capillary pressure curves vs. water saturation. This figure reveals that

170 there is more than one hydraulic flow unit in the reservoir. Therefore, the J-function is unable to

171 normalize all the capillary data into a single curve and it is required to classify the data into

172 separate hydraulic flow units having similar capillary pressure curves.

174 4. Results and Discussion

175 To obtain the optimum number of hydraulic flow units, the RQI vs. pz is plotted. As illustrated in

176 Fig. 3, all data which lie along the straight line correlations with unit slope have the same FZI

177 value. According to this figure, the sample data fall into five discrete rock types confirming that

178 five hydraulic flow units exist. The values of FZI are 276.61, 73.24, 14.93, 5.81 and 1.83 for

179 HFU#1, HFU#2, HFU#3, HFU#4 and HFU#5 respectively. Each of these flow units has the

180 uniform rock characteristics and consistent potential for fluid flow through the porous media. It

181 should be noted that the hydraulic flow unit with higher FZI values will have a better quality to

182 flow the fluids through its pore spaces in the reservoir rock.

184 Fig. 4 depicts the permeability versus porosity as classified by FZI values. This figure confirms

185 that the data are well classified and there is a satisfactory relation between porosity and

186 permeability for each hydraulic flow unit that is obtained through FZI curves.

188 According to the obtained hydraulic flow units, the available capillary pressure curves can be

189 divided into five categories. Fig. 5 shows the five capillary pressure data sets for five hydraulic

190 flow units. Each of these capillary pressure sets can be normalized into a single curve that

191 assigned to a uniform flow unit.

193 The J-function was used to normalize capillary pressure curves for each hydraulic flow unit.

194 Values of J-function vs. normalized water saturation are plotted in Fig. 6. Values of lithology

195 index (J*) and pore size distribution index (f) for each unit are calculated by fitting Eq. (16) on

196 the normalized values of capillary pressure.

198 After determining all parameters associated with Eq. (16), the pore size radius for each hydraulic

199 flow unit was calculated. Fig. 7 depicts the pore size radius vs. normalized water saturation for

200 five flow units of reservoir rock. The fitted relations between the pore radius and normalized

201 water saturation for each unit are also shown in this figure. As it can be observed from this

202 figure, the hydraulic flow unit with higher pore size radius corresponds to the flow unit that has

203 higher values of permeability and FZI. This observation confirms that the hydraulic flow unit

204 with higher values of FZI and pore size radius has the higher rock permeability; which means

205 such a flow unit has a better quality to flow fluids through pore spaces of reservoir rock. Table 2

206 summarized all information include rock characteristics, lithology index, pore size distribution

207 index, pore geometry constant, J-function and pore size radius equations observed for each

208 hydraulic flow unit. The larger value of pore size distribution index shows the narrower

209 distributions. As it is presented in the table, the pore geometry constant of hydraulic flow units

210 having larger FZI is higher than that of those with lower values of FZI. Basically, the pore

211 geometry is the configuration of the pore spaces within a rock or between the grains of a rock.

212 The grain sorting and the pore framework dictate the pore geometry in a rock. If there are finer

213 grains, significant compaction, and differences in grains or framework, it will affect the pore

214 geometry of the rock. Higher pore geometry constant is an indication of the presence of coarse

215 grains, large pore throats, and widespread pore framework in the rock which corresponds to the

216 proper rock quality in term of fluid flow (i.e., higher FZI value).

218 5. Conclusions

219 In this study the pore size of a hydrocarbon reservoir rock was determined by a new proposed

220 technique. The flow zone indicator (FZI) approach was used to classify the reservoir rock into

221 separate zones that have similar rock characteristics; which are known as hydraulic flow units

222 (HFU). The measured capillary pressure curves are sub-grouped into five categories based on the

223 determined hydraulic flow units. Then J-function was employed to normalize all capillary curves

224 that assigned to these flow units. Finally, by performing some mathematics, the new equation

225 was developed to estimate pore size radius for each flow unit. The developed equation is a

226 function of FZI, lithology index (J*), pore size distribution index (f) normalized water saturation

227 (Swn) and normalized porosity (( z). The results of this study indicated that the proposed method

228 to determine the pore size is dependent on the several rock properties and is not restricted to the

229 specific reservoirs; it is able to be applied on any reservoir rocks with diverse range and high

230 heterogeneity in rock properties.

247 Table Captions:

248 Table 1: Rock properties of 31 core samples.

249 Table 2: Rock characteristics and equations obtained for each hydraulic flow unit.

252 Figure Captions:

253 Fig. 1: Permeability vs. porosity for reservoir data.

254 Fig. 2: Capillary pressure curves of 31 reservoir rock samples.

255 Fig. 3: Reservoir quality index vs. normalized porosity.

256 Fig. 4: Permeability versus porosity as classified by FZI.

257 Fig. 5: Five capillary pressure data set for obtained hydraulic flow units.

258 Fig. 6: Values of J-function vs. normalized water saturation for each hydraulic flow unit.

Nomenclatures

Symbols

k J J* Pc

'-'wir

Permeability (md)

J-function

Lithology index

Capillary pressure (psia)

Non-wetting phase pressure (psia)

Wetting phase pressure (psia)

Capillary pressure of oil and water system (psia)

Oil pressure (psia)

Water pressure (psia)

Pore radius (nm)

Mean hydraulic unit radius (nm)

Water saturation

Normalized water saturation

Irreducible water saturation

Greeks

9z 6 £ z

Interfacial tension (dyne/cm) Porosity

Normalized porosity Contact angle

Pore size distribution index Pore geometry coefficient Tortuosity

Abbreviations

Rock Quality Index Flow Zone Indicator Hydraulic Flow Unit

References

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[2] J.M. Kate, C.S. Gokhale, A Simple Method to Estimate Complete Pore Size Distribution of Rocks, Engineering Geology. 84 (2006) 48-69.

[3] C.G. Shull, The Determination of Pore Size Distribution from Gas Adsorption Data, Journal of American Chemistry Society. 70 (1948) 1405-1410.

[4] W.R. Purcell, Capillary Pressure-Their Measurement Using Mercury and the Calculation of Permeability Therefrom, Trans. AIME. 186 (1949) 39-48.

[5] N.C. Wardlaw, Y. Li, D. Forbes, Pore-throat size correlation from capillary pressure curves, Transport in Porous Media. 2 (1987) 597-614.

[6] E.C. Donaldson, N. Ewall, B. Singh, Characteristics of capillary pressure curves, Journal of Petroleum Science and Engineering. 6 (1991) 249-261.

[7] B.K. Mishra, M.M. Sharma, Measurement of Pore Size Distributions from Capillary Pressure Curves, AICHE Journal. 34 (1988) 684-687.

[8] S. Wo, X. Xie, N R. Morrow, A Statistical Model of Apparent Pore Size Distribution and Drainage Capillary Pressure, Colloids and Surfaces. 187 (2001) 449-457.

[9] M.H. Ghazanfari, D. Rashtchian, R. Kharrat, S. Vossoughi, Capillary Pressure Estimation Using Pore Size Distribution Functions, Chemical Engineering Technology. 30 (2007) 862869.

[10] R. Askarinezhad, A new statistical approach to pore/throat size distribution of porous media using capillary pressure distribution concept, Journal of Petroleum Science and Engineering. 75 (2010) 100-104.

[11] A. Abedini, F. Torabi, P. Tontiwachwuthikul, Rock Type Determination of a Carbonate

Reservoir Using Various Approaches: A Case Study, Special Topics & Reviews in Porous Media-An International Journal. 2 (2011) 293-300.

[12] A. Abedini, F. Torabi, P. Tontiwachwuthikul, Reservoir Rock Type Analysis Using Statistical Pore Size Distribution, Special Topics & Reviews in Porous Media-An International Journal. 3 (2012) 97-103.

[13] Amyx, J.W., Bass, D.M. and Whiting, R.L., 1960. Rock Properties: Petroleum Reservoir

Engineering. McGraw-Hill Book Co. Inc., New York.

[14] A. Abedini, Statistical Evaluation of Reservoir Rock Type in a Carbonate Reservoir, Paper No. 152359, presented at the SPE Annual Conference and Exhibition. Denver, Colorado, USA, 30 October-2 November, 2011.

[15] M.C. Leverett, Capillary Behavior in Porous Solids. Trans, AMIE. 142 (1941) 152-169.

[16] F.A. Al-Ajmi, S.A. Holditch, Permeability Estimation Using Hydraulic Flow Units in a Central Arabia Reservoir, Paper No. 63254 presented at SPE Conference. Texas, 1-4 October, 2000.

[17] J. Kozeny, Uber Kapillare Leitung des Wassers im Boden Stizungsberichte, Royal Academy of Science. Vienna, Proc. Class I, 136 (1927) 271-306.

[18] P.C. Carmen, Fluid Flow through Granular Beds, Trans. AIChe. 15 (1937) 150-166.

[19] S.E.D.M. Desouky, A New Method for Normalization of Capillary Pressure Curves, Oil & Gas Science and Technology - Rev. IFP. 58 (2003) 551-556.

Table 1: Rock properties of 31 core samples.

Sample No. V k (md) S . ^wir

1 0.062 0.093 0.221

2 0.083 68.154 0.025

3 0.066 0.161 0.234

4 0.110 230.121 0.026

5 0.037 0.682 0.105

6 0.077 0.160 0.175

7 0.088 0.004 0.370

8 0.141 1.247 0.214

10 0.299 3.212 0.200

11 0.250 1.501 0.313

13 0.042 1.359 0.061

14 0.071 4.212 0.130

15 0.163 0.548 0.334

16 0.142 0.158 0.189

17 0.116 0.504 0.238

18 0.104 0.010 0.394

19 0.093 0.064 0.301

21 0.117 183.061 0.026

22 0.155 797.538 0.015

23 0.246 2419.099 0.035

24 0.195 1180.743 0.028

26 0.103 19.684 0.132

28 0.154 40.599 0.082

29 0.105 0.007 0.424

30 0.133 0.014 0.439

31 0.238 0.119 0.410

Table 2: Rock characteristics and equations obtained for each hydraulic flow unit.

FZI J e Z Permeability Eq. Normalized J-function Eq. Pore size radius Eq.

HFU#1 276.61 0.0333 1.2063 6614.7 k = 12.626e2a253<p J(Sw) = 0.0333Swn -0.829 R = 6614.7Swna829

HFU#2 73.24 0.0411 1.0142 2146.6 k = 1.0271e19'747<f J(SW) = 0.0411Swn -0.986 R = 2146.6Swn°'986

HFU#3 14.93 0.0501 1.0299 404.55 k = 0.0776e16'469<f J(Sw) = 0.0501Swn -0.971 R = 404.55SJ.971

HFU#4 5.81 0.0931 1.0466 68.548 k = 0.0042e2316" J(Sw) = 0.0931Swn -0.956 R = 68.548SJ.956

HFU#5 1.83 0.1861 1.1561 17.345 k = 0.0034e12M3v J(Sw) = 0.1861 Swn -0.865 R = 17.345SWna865

k (md)

° o °

CP o @

<*><?o ° °

O o o o o x> o O eft

o°oS°o

^fTo ou ° O

og^OOQj ° -

k = 0.8681e6-7825? R2 = 0.158

o'frfl »

OO *>of° °

a w « §8 o

• 31 samples with complete

data set 0321 routine core data

Fig. 1 : Permeability vs. porosity for reservoir data.

Pc (Psia) 150

A A A A A

A a A A A a A

A A A % ^ £ AAaA AA/A T A

A A \ Aaa

it A AA A ^ - Aa A/ aA ^ , ¿A A A^ A A^A a A

& A Z vz ^A A A ^ A A AM A L A A & A

Fig. 2: Capillary pressure curves of 31 reservoir rock samples.

♦HFU#1 •HFU#2 HFU#3 AHFU#4 ♦HFU#5

♦ 31 Sample data with data set

Fig. 3: Reservoir quality index vs. normalized porosity.

k (md)

Fig. 4: Permeability versus porosity as classified by FZI.

Pc (Psia) 150

HFU#1: Pc = 1.4277Sw -1.407

R2 = 0.945

HFU#2: Pc = 1.6454Sw -2.148

R2 = 0.919

HFU#3: Pc = 1.9018Sw -3.462

R2 = 0.964

HFU#4: Pc = 2.3639Sw -3.714

R2 = 0.847

HFU#5: Pc = 2.3881Sw -5.239

R2 = 0.901

HFU#1 HFU#2 HFU#3 HFU#4 HFU#5

Fig. 5: Five capillary pressure data set for obtained hydraulic flow units.

5 4.5 4

1 1 1 HFU#1: J(Sw) = 0.0333Swn-0-829 R2 = 0.928 HFU#1 HFU#2

1 HFU#2: J(SJ = 0.0411SWn'a986 R2 = 0.948 HFU#3 HFU#4 HFU#5

1 HFU#3: J(Sw) = 0.0501Swn-a971 R2 = 0.956

s HFU#4: J(Sw) = 0.0931SWn" R2 = 0.963 0.956

HFU#5: J(Sw) = 0.1861Swn~' R2 = 0.915 0.865

№ K*, ,

-I-——r ~ ^ -1

0.4 0.6

"iijii

Fig. 6: Values of J-function vs. normalized water saturation for each hydraulic flow unit.

0 0.2 0.4 0.6 0.8 1

Fig. 7: Pore size radius vs. normalized water saturation for each hydraulic flow unit.