Technical Notes
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Technical Note # 12.
(or beware of scale problems
when calculating matrix, fracture and vuggy porosity!)
By Maria S. Aguilera, Roberto F. Aguilera and Roberto Aguilera
This is part of articles published by the authors in the JanuaryFebruary
2003 and MarchApril 2004 issues of Petrophysics,
and of a Technical Luncheon Presentation to the Canadian Well Logging Society
(CWLS) in
The analysis of vuggy and fractured reservoirs has been an area of significant interest in the past few years. Several researchers have studied the characterization of these reservoirs using seismic, core, petrophysical and well testing data. The following table shows a brief summary of some of the available formation evaluation methods:

DualPorosity 
TriplePorosity 
Cores 
Borai^{1}
(1985) Lucia^{2}
(1983, 1995) 
Coalson et
al.^{3} (1985) CSPG
Conference on Dolomites^{4} (2004) 
Petrophysics 
Serra^{5}
(1989) Aguilera
and Aguilera^{6} (2003) 
Aguilera
and Aguilera^{7} (2004) 
Well
testing 
Warren and
Root^{8} (1963) Aguilera^{9}
(1987) 
Ershaghi^{10}
(1986) Aguilera^{11}
(1988) 
This note concentrates on the petrophysical aspects of dual and triple porosity systems.
Dual Porosity Reservoirs. This section shows means of estimating values of the dual porosity exponent m for reservoirs made out of (1) matrix and nonconnected vugs, or (2) matrix and fractures. We have found that some of the dual porosity models published in the literature fail particularly for large values of total porosity. The reason for the failures is an improper scaling of the matrix porosity. To avoid this pitfall, it is important to keep in mind the following 2 definitions of matrix porosity: In the first one, matrix porosity (Æ_{b}) is equal to void space in the matrix divided by bulk volume of the “matrix system”. In the second one, matrix porosity (Æ_{m}) is equal to void space in the matrix divided by the bulk volume of the “composite dual porosity system”. Æ_{b} is equivalent to porosity from unfractured core plugs. Total porosity in the chart below is equal to void space in the composite system divided by bulk volume of the composite system. Fracture porosity (Æ_{2}) is equal to volume of the fractures divided by bulk volume of the composite system. Porosity of the nonconnected vugs (Æ_{nc}) is equal to volume of the nonconnected vugs divided by bulk volume of the composite system. Based on these definitions total porosity (Æ) is given by:
for the case of matrix and fractures, and
for the case of matrix and nonconnected vugs.
If for a given interval the cementation exponent (m_{b}) of the matrix is 2.0, total porosity is 10% and fracture porosity is 1%, the dual porosity exponent (m) is read to be approximately 1.73 from the lefthand side of the chart below. If for a given interval the cementation exponent (m_{b}) of the matrix is 2.0, total porosity is 10% and porosity of the nonconnected vugs is 5%, the dual porosity exponent (m) is read to be approximately 2.52 from the righthand side of the chart below. Thus in a matrixfractured reservoir, the dual porosity exponent (m) is smaller than the porosity exponent m_{b} of the matrix. On the other hand, in a matrixvug reservoir, the dual porosity exponent (m) is larger than the m_{b} of the matrix. Similar charts for other cases of practical interest, and the basic equations, have been published by Aguilera and Aguilera.^{6}
Triple Porosity Reservoirs. There are instances where the reservoir is composed mainly of matrix, fractures and nonconnected vugs. In these cases a triple porosity model appears more suitable for petrophysical evaluation of the reservoir. A technique is presented for these types of reservoirs that holds for all combinations of matrix, fracture, and nonconnected vug porosities. At low porosities, the fractures dominate and the m values of the composite system tend to be smaller than the porosity exponent of the matrix (m_{b}). As the total porosity increases, however, the effect of the nonconnected vugs becomes more important and m of the triple porosity system can become larger than m_{b}. This research is inspired by the availability of modern magnetic resonance, microresistivity and sonic image tools that permit reasonable characterization of complex reservoirs.
As in the previous case, the analyst has to be careful with the scaling of porosity. Four different values of porosity are considered in this case. The first is matrix porosity (Æ_{b}), which is equal to void space in the matrix divided by bulk volume of the matrix system. The second is matrix porosity (Æ_{m}) that is equal to void space in the matrix divided by bulk volume of the triple porosity system. The third is fracture porosity (Æ_{2}), which is equal to void space of the fractures divided by bulk volume of the triple porosity system. The fourth is porosity of the nonconnected vugs (Æ_{nc}) that is equal to the void space of the nonconnected vugs divided by bulk volume of the triple porosity system. Total porosity (Æ) of the composite system is given in this case by:
If for a given interval the cementation exponent (m_{b}) of the matrix is 2.0, total porosity is 10%, fracture porosity is 1%, and nonconnected vug porosity is 5%, the triple porosity exponent (m) is read to be approximately 1.9 from the chart below. Similar charts for other cases of practical interest, and the basic equations, have been published by Aguilera and Aguilera.^{7}
References
1. Borai, A. M.: “A New Correlation for Cementation Factor in Low Porosity Carbonates,” SPE Formation Evaluation, 1985, vol. 4, no. 4, p. 495499.
2. Lucia, F. J.: “Petrophysical Parameters Estimated from Visual Descriptions of Carbonate Rocks: A Field Classification of Carbonate Pose Space,” Journal of Petroleum Technology, 1983, vol. 35, no. 3p. 629637.
3. Coalson, E. B. et al.: “Productive Characteristics of Common Reservoir Porosity Types,” Bull. SW Geological Society, 1985, vol. 25, no. 6, p. 3551.
4. CSPG Dolomite Conference, Calgary, Canada (January, 2004).
5. Serra, O.: “Formation Micro Scanner Image Interpretation,” Schlumberger Educational Service, 1989Houston, SMP7028, 117 p.
6. Aguilera, Maria Silvia and Aguilera, Roberto: "Improved Models for Petrophysical Analysis of Dual Porosity Reservoirs", Petrophysics, JanuaryFebruary 2003, volume 44, no. 1, p. 2135.
7. Aguilera, Roberto F. and Aguilera, Roberto: "A Triple Porosity Model for Petrophysical Analysis of Naturally Fractured Reservoirs", Petrophysics, MarchApril 2004, volume 45, no. 2, p. 157166.
8. Warren J. E. and Root, P. J.: “The Behavior of Naturally Fractured Reservoirs,” SPEJ, Sept. 1963, p. 245255.
9. Aguilera, Roberto: “Well Test Analysis of Naturally Fractured Reservoirs,” SPE Formation Evaluation, Sept. 1987, p. 239252.
10. Abdassah, D. and Ershaghi, I.: “Triple Porosity Systems for Representing Naturally Fractured Reservoirs,” SPE Formation Evaluation, April 1986.
11. Aguilera, Roberto and Song, S. J.: “WELLTESTNFR: A Computerized Process for Transient Pressure Analysis of Multiphase Reservoirs with Single, Dual or Triple Porosity Behavior,” Paper 883952 presented at the 39^{th} Annual Technical Meeting of the Petroleum Society of CIM, Calgary, Canada, June 1216, 1988.
© 2004 Copyright by Servipetrol Ltd. All rights reserved.
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