Resistivity Concepts in Fractured Reservoirs
The following Sections deal with the dual porosity model as defined by Dr Roberto Aquilera in his original paper, “Analysis of naturally fractured reservoirs from conventional well logs”, R. Aguilera, The Journal of Canadian Petroleum Technology, p. 764-772, 1976. Recently Dr Aquilera has published a triple porosity model and Dr Zoltan Barlai has used a “five channel” porosity model with some success, neither of which is discussed here. Dr Aquilera has also published a hard cover textbook on the dual porosity model and fractured reservoirs in general. Students should review his material to augment the material presentrd , which is briefer than the original material.

There Sections contains some amplified or re-defined terminology that contrasts with some of Dr Aquilera’s definitions. Please be aware of these differences when reading both works. Review the definitions provided at the beginning of this Handbook before proceeding.

Quantitative analysis of fractured reservoirs is complicated by the fact that other forms of porosity exist besides that contained by the fractures. Thus a dual porosity model has been proposed to account for both primary and secondary porosity. The theoretical principles behind the dual porosity model have been published previously in the literature by Aguilera and have been used by others with some success in Mexico, Venezuela, the United States, and Canada.

The term "dual porosity" should not be confused with the "dual water" model used for shaly formations. In addition, the fractured reservoir literature uses the phrase "total porosity" to mean the sum of effective matrix porosity plus effective fracture porosity. This is very confusing as the phrase has a different meaning in the shaly sand situation. Since there are fractured shaly reservoirs where the distinction between total and effective porosity is important, we will use the following definitions.


Effective porosity
  PHIe = PHIm + PHIf

Total porosity
  PHIt = PHIe + Vsh * BVWSH

 

WHERE:
  PHIe = effective porosity of dual porosity system (fractional)
  PHIm = effective matrix porosity in dual porosity system (fractional)
  PHIf = effective fracture porosity of dual porosity system (fractional)
  PHIt = total porosity of any rock (fractional)
  Vsh = shale volume (fractional)
  BVWSH = bound water in 100% shale (fractional)

Some fracture literature also uses the term secondary porosity to mean fracture porosity, whereas this term has been used by others to describe the porosity not seen by the sonic log, usually some portion of the vuggy porosity. We prefer to use secondary porosity in its geological sense and use the term fracture porosity in a strictly literal sense.

To develop the dual porosity model, we invoke the basic Archie equations.


Archie’s Laws
  #1:
I
= RESD / (F * RW@FT)
  #2: F = A / (PHIe ^ M)

Rearranged, these become the Pickett plot definition
In Water Zone ONLY
    1: log(RESD) = - M * log(PHIe) + log(A*RW@FT)
    2: M = (log(A*RW@FT) - log(RESD)) / log(PHIe)

In Flushed Zone ONLY
    3: log(RESS) = - M * log(PHIe)+log(A*RMF@FT)
    4: M = (log(A*RMF@FT) - log(RESS)) / log(PHIe)

WHERE:
  A = tortuosity exponent (unitless)
  F = formation factor (unitless)
 
I
= resistivity index (unitless)
  M = cementation exponent (unitless)
  PHIe = effective porosity of dual porosity system (fractional)
  RESD = true )deep) formation resistivity (ohm-m)
  RW@FT = formation water resistivity (ohm-m)

Analysis of equations 2 and 4 indicates that a crossplot of porosity vs resistivity on log-log coordinate paper will result in a straight line with a slope of -M for zones of constant water resistivity (A * RW@FT) and constant resistivity index (I). A constant resistivity index implies that the zone has constant water saturation (Sw), where Sw = (1 / I) ^ (1 / N). This plot has been called the Pickett plot and is widely used to find both A * RW@FT and M for water zones and Sw for hydrocarbon zones in conventional reservoirs.

IMPORTANT: This method is not suitable for shaly reservoirs as no shale term is included in the Pickett plot. Therefore, be sure to exclude shale or shaly zones from a Pickett plot.


Porosity - resistivity crossplot (Pickett plot) identifies M

For reservoirs with fracture porosity, the value of M found from the Pickett plot is smaller than the cementation exponent, M, determined from a primary porosity sample in the laboratory, or estimated from lithological descriptions, or from an un-fractured portion of the reservoir. This is reasonable because fracture porosity results in a reduction in tortuosity and cementation. In addition, fractures can be invaded deeply by drilling fluids, thus reducing RESD and the derived value of M from the crossplot. The lower M may be compensating for invasion as much as for the fractured nature of the rock. In any case, a lower value for M decreases water saturation and this is needed whether the lower resistivity is due to invasion or to lower cementation.

Values of M from the Pickett plot in the range 1.2 to 1.7 can be expected for fractured reservoirs, as opposed to 1.8 to 2.4 for the un-fractured portion of the same rock. The laboratory measurement of M for a well-fractured rock is seldom successful, so there is not much real data to use, except in competent samples with minor micro-fractures.

We can then redefine M to reflect these differences.


Definition of Md and Mb
Md = cementation exponent for dual porosity model, found from a Pickett plot
Mb = cementation exponent for un-fractured matrix rock, found from laboratory measurement, a Pickett plot in an un-fractured zone, or from assumption based on lithology

Choosing Md and Mb

Normally, Md is chosen once for each fractured interval from the Pickett plot, but there is no reason to believe it is a constant because fracture intensity varies dramatically from foot to foot within the reservoir. It is clear that every data point could have a unique value of Md, assuming all are 100% wet.

A method has been proposed by Rasmus whereby Md is calculated and used at each level, thus providing a "variable M" method throughout the zone. It is based on the sonic versus density neutron porosity:

 

Rasmus variable M
    96: Md = log ((1 - (PHIe - PHIsc)) * (PHIsc ^ Mb) + (PHIe - PHIsc)) / log PHIe

Thus for un-fractured rock:
    97: log RESD = - Mb * log (PHIe) + log (A * RW@FT) + log (
I
)

And for fractured rock:
    98: log RESD = - Md * log (PHIe) + log (A * RW@FT) + log (
I
)

The derivation is rather lengthy and not shown here. A fracture tortuosity term has also been omitted because it is often assumed to be 1. This presumes that PHIe >= PHIsc and PHIsc has been adequately corrected for lithology and shale. When there are no fractures, PHIsc = PHIe and Md = Mb.

In many cases, it is possible to carry out the evaluation by crossplotting (DELT - DELTMA) vs RESD, or (DENS - DENSMA) vs RESD on log-log paper instead of PHIe vs RESD. The sonic log method is not recommended if vuggy porosity exists, because the sonic log does not see all this porosity. Thus PHIsc will be too low and Md would be wrong. Likewise, if PHIsc > PHIe, the method should be abandoned.

The value of Md is determined by calculating the slope of the line drawn through the south west points in the Pickett plot, which are assumed to be water bearing levels. If no water bearing points are available because there is no water zone in the fractured interval, it is possible to make the plot by replacing RESD with RESS, the shallow resistivity, based on the assumption that the invaded zone will be more nearly 100% wet than the un-invaded zone. The constant in the equation becomes (A * RMF@FT), but this value is known as well or better than (A * RW@FT).

With the value of Md determined from the crossplot or equation 96 and Mb determined in the laboratory or estimated from lithology, it is possible to complete the evaluation to quantify primary and fracture porosities, as proposed by Aguilera.
 

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