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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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Reservoir Using Various Approaches: A Case Study, Special Topics & Reviews in Porous Media-An International Journal. 2 (2011) 293-300.
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Engineering. McGraw-Hill Book Co. Inc., New York.
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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.
