Introduction

The need for greater accuracy for the determination of in situ fluid saturations has become more important in recent years because of efforts to develop new technology to enhance oil recovery from old oil fields.

This accurate assessment of the existing reservoir oil saturation is required to project the overall economic feasibility of the new or improved oil-recovery process. This need for greater accuracy and more data from in situ testing has led to the development of new well-logging tools (the repeat formation tester, the 64-electrode microresistivity tool, and the Phasor induction tool).

The formation resistivity factor of a porous medium is a valuable concept in the area of formation evaluation. It has been denned as the ratio of the resistivity of the medium when completely saturated with a conducting fluid to the resistivity of the saturating fluid as follows:

$$ {F}_R={R}_0/{R}_w $$
(1)

Next, Archie cross-plotted porosity versus formation factor on log-log scales, and noted a linear trend that is equivalent to the following:

$$ {F}_R=\frac{1}{\phi^m} $$
(2)

Winsauer et al. 1952 established experimentally a more generalized formation resistivity factor equation, expressed as follows:

$$ {F}_R=a{\phi}^{-m} $$
(3)

Where F is the formation factor (dimensionless), ϕ is the porosity (dimensionless), a is the cementation factor (dimensionless), and m is the cementation exponent (dimensionless).

Several different values, depending mostly on the types of formations selected for the experiments, have been proposed for “a” and “m,” but the curves showing F versus ϕ for these various values of “a” and “m” do not differ much from one another.

Table 1 represents a general equations for computing formation factor as a function of porosity with references to the values of “a” and “m” used in such calculation, and illustrates how lithology affects the formation factor. Also, the complexities of the interplaying factors affecting such parameters (Table 2) have always been preventing a precise estimation of F, especially when indirect methods are used.

Table 1 Coefficients and exponents used to calculate formation factor (F) as a function of porosity
Full size table
Table 2 Factors affecting the formation factor parameters (a and m) (adopted after R. C. Ransom 1974)
Full size table
Table 3 Intervals used for a (0.5 to 1.5) and m (1.5 to 2.5)
Full size table

This paper aims to apply a numerical method to enhance the evaluation of “a” and “m” with the aim of enhancing water and hence hydrocarbon saturations.

Formulation of the problem

It is important to point out which intervals need careful formation evaluation, only intervals with hydrocarbons need wellsite evaluation. With several thousand feet of hole to evaluate quickly, methods were needed to remove the non-hydrocarbon-bearing “wet” intervals and quickly locate possible hydrocarbon-bearing intervals. The five prime techniques used (Best et al. 1978; Bigelow and abt 1972; Fertl 1978) as hydrocarbon indicators are as follows:

  1. 1.

    Apparent water resistivity (R w)

  2. 2.

    F R-R o overlays

  3. 3.

    Movable-oil plot (MOP)

  4. 4.

    The R xo/R t indicator

  5. 5.

    The neutron-density gas overlay

  6. 6.

    BVW method (Mabrouk 2005)

Most of these older techniques are used in current wellsite computer interpretation. However, the present paper depends on F R-R o overlays method where Archie (1942) introduces the following water saturation equation:

$$ {S}_w=\frac{F_R{R}_w}{R_t} $$
(4)

Rewrite equation (4), we get the following:

$$ {R}_t{S}_w={F}_R{R}_w $$
(5)

In water-bearing zones, where Sw = 100 %, Eq. (5) can be represented by Eq. (1) by plotting R o and F R vertically with depth; by knowing R w, we can easily differentiate between water and hydrocarbon bearing zones where:

  1. 1.

    In water-bearing zones, R o is very close to F R.

  2. 2.

    In hydrocarbon-bearing zones, there is a separation between them.

However, in water-bearing zones, R o must be to equal FR and not close to it, but the use of wrong values for a and m in determining FR is the only responsible for that; however, this paper introduce a numerical technique to minimize this error and get more accurate values for a and m based on the following steps:

  1. 1.

    Equation (1) after substituting with Eq. (3) can be written as follows:

    $$ {R}_o=a{\phi}^{-m}{R}_w $$
    (6)
  2. 2.

    Based on Eq. (6), a set of graphs is constructed for R o with knowing ϕ and R w and by taking a = 0.5 and m from 1.5 to 2.5, 0.1 step and changing a by 0.01 till 1.5, as listed in Table 3.

  3. 3.

    From step 2, we get 1,100 graphs of R o and ϕ versus depth.

  4. 4.

    By comparing R o and aϕ−m (or FR), the closest interval depending on standard deviation and root mean square error will be a water-bearing zones with the accurate values of a and m.

  5. 5.

    The selected a and m can be used for the entire well to determine the saturation in the other intervals.

  6. 6.

    Since 1,100 graphs of R o and ϕ and comparing their result to find the standard deviation and root mean square error for all is very hard, so a program is designed to serve the authors’ objective in order to save time and get the required values of a and m easily.

Program description

The program is designed by C++ language in order to read the input data which include the Rw value and the different values of Rt and ϕ for all intervals. There are three steps to run the program.

  1. 1.

    Data gathering: Log data have to be digitized and put into the format shown in Table 4.

    Table 4 Arrangements of the well-logging data
    Full size table
  2. 2.

    Entering Rw value and the range used for a and m from Table 3

  3. 3.

    The output will be in two files, the first file includes the R calculated for different values of a and m with the observed standard deviation and root mean square errors, while the second file contains the final values of a and m, which belong to the closest value of the standard deviation observed between the input and output resistivities and the minimum RMS error between them.

Proposed technique testing and application

The following is mainly concerning with the running of the proposed technique for the following:

  1. 1.

    Synthetic data to explain clearly how to use it

  2. 2.

    Some core samples selected randomly from different wells from Gulf of Suez Basin, Egypt

Test number 1: synthetic data

The authors used synthetic values for Rt to represent water-bearing zones and use it to determine ϕ (Fig. 1) using a = 0.62, m = 2.2, and R w = 0.06 Ω m, through Eq. (6) after rearrangement as follows:

Fig. 1
figure 1

Synthetic values for R t and the corresponding ϕ from Eq. (7)

Full size image
$$ \phi ={\left(\frac{0.62*0.06}{R_t}\right)}^{1/2.2} $$
(7)

Through that, the authors know the actual values for a and m.

The program uses the following Eq. (8) to find the different R calculated at different values of a and m.

$$ {R}_{calculated}=0.06a{\phi}_{Synthetic}^{{}^{-m}} $$
(8)

By using different values of a and m based on Table 3, and by comparing the synthetic values of R t with R calculated from Eq. (8), Fig. 2 represent a set of 10 graphs selected randomly from 1,100 graphs using different values of a and m including the known values which are used in the synthetic data; from Fig. 2, no vision difference can be illustrated, but Fig. 3 can give us the accurate a and m after comparing between all R calculated data using root mean square error and the standard deviation where the lowest RMS and the closest standard deviation to the actual R t belong to a = 0.62 and m = 2.2, which is the used values in the present test.

Fig. 2
figure 2figure 2

A set of 10 graphs selected randomly from 1,100 graphs of R calculated from Eq. (8) using different values of a and m including the known values

Full size image
Fig. 3
figure 3

RMS error between the R t values and R t calculated with the observed standard deviation

Full size image

Test number 2: core samples, Gulf of Suez Basin, Egypt

Ten core samples selected from different wells at different interval from Gulf of Suez basin, Egypt (compiled by Mabrouk et. al. 2013) are used to serve the authors objectives to find a and m with the suggested numerical technique where as follows:

  1. 1.

    R w calculated from chemical composition of uncontaminated connate water produced from these cores.

  2. 2.

    R t calculation is done based on the following general resistance formula:

    $$ R=\left({R}_{\mathrm{t}}\times L\right)/A $$
    (9)

    Where R = resistance (Ω), R t = core resistivity (Ω m), L = length of conducting path (m), and A = cross-sectional area of the path (m).

  3. 3.

    The porosity is calculated from the summation method of the fluid.

  4. 4.

    The resistivity of each water-saturated core is graphed on a log-log graph with a porosity to determine a and m.

    Using the numerical technique to get a and m depending on Table 3, and by using the suggested program, one can easily calculate a and m for each sample. The results of calculated values for each sample get extremely close to the measured one with the neglected amount of error (Table 5).

    Table 5 Measured (Mabrouk 2013) and calculated values for ten core samples, Gulf of Suez Basin, Egypt
    Full size table

Conclusions

The use of wrong values of a and m in Archie’s water saturation equation can yield to overlooking producible zones. This paper introduce a simple numerical technique to calculate a and m accurately which depends on FR-RO overlays method for hydrocarbon indicator. The method is tested using synthetic and real data to represent its ability in determining formation factor parameters a and m with very high degree of accuracy where the amount of error can be neglected.

The paper also introduced a simple program to help apply the technique easily where 1,100 graphs must be held in order to get the final correct values of a and m.