Introduction

The thermal neutron porosity tool is a simple tool in design but rather difficult in interpretation (Asquith and Krygowski 2004; Bassiouni 1994; Ellis and Singer 2007; Serra 1988; Wu et. al 2013; Ellis et al 1987). The reason is the complex physics of the neutron transport and interactions with the formation constituents.

The tool design

The thermal neutron tool design consists of three main components, Fig. 1:

  1. 1.

    High energy neutron source (~ 4.5 Mega Electron Volts, MeV)

  2. 2.

    Near thermal neutron detector and

  3. 3.

    Far thermal neutron detector.

Fig. 1
figure 1

Generic thermal porosity tool

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The formation porosity is obtained by correlating the ratio of the near and the far thermal neutron detectors’ count rates with the formation porosity, Eq. 1.

$$\phi = f\left( \frac{N}{F} \right)$$
(1)

All service companies derived the ratio–porosity relationship for clean formations (with no thermal neutron absorbers). If a formation contains elements and/or minerals with high thermal neutron absorption probability, the correlation provides incorrect formation porosity, usually higher than the actual porosity.

The neutron transport parameters

Formation porosity using the thermal neutron tool is controlled by multiple neutron transport parameters. These parameters are the slowing down length, the diffusion length and the migration length.

The slowing down length defines the ability of the formation constituents of reducing the high energy of the neutrons (~ 4.5 MeV) to the thermal energy level (0.025 electron volt, eV). The straight-line connecting the neutron source and the thermalization point represents the slowing down length, Ls, Fig. 2.

Fig. 2
figure 2

Slowing down length

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After the neutron energy reaches its thermal level, the diffusion process becomes dominant and is evaluated by the diffusion length. The thermal neutron diffuses until it is absorbed by the formation elements. The straight-line connecting the start of the thermal energy and the absorption point is the diffusion length, Fig. 3. It is important to mention that the effect of the high thermal neutron absorbers occurs only at this thermal energy level and in turn affects the diffusion length.

Fig. 3
figure 3

Diffusion length

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The combined slowing down length and the diffusion length is the migration length, Lm. The migration length is the straight-line connecting the source and the absorption point, Fig. 4, and is defined by Eq. 2.

$$L_{{\text{m}}} = \sqrt {L_{{\text{s}}}^{2} + L_{{\text{d}}}^{2} } .$$
(2)
Fig. 4
figure 4

Migration length

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Modeling the neutron transport parameters

The neutron transport parameters are modeled by many authors (Ellis et. al 1987) and (Wu et. al 2013). Ellis developed the relationships of the slowing down length, Eq. 3, and the diffusion length, Eq. 4, using the formation porosity and the neutrons absorption probability.

$$L_{{\text{s}}} = c_{1} \left( {\phi + c_{3} } \right)^{{c_{2} }} + c_{4}$$
$$L_{{\text{d}}}^{2} = \frac{{c_{1} \left( {\emptyset + c_{3} } \right)^{{c_{2} }} + c_{4} }}{{\Sigma_{{\text{a}}} }}$$

Tables 1 and 2 contain the coefficients of slowing down length and the diffusion equations for the different formation lithologies. The migration length can then be calculated using Eq. 2.

Table 1 Ls coefficients
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Table 2 Ld coefficients
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Ellis corrected the migration length for the effect of the formation water salinity since it reduces the hydrogen index of the formation water. The correction took the form,

$$L_{{\text{m}}}^{*} = \sqrt {L_{{\text{s}}}^{2} + f\left( {L_{{\text{s}}} } \right) L_{{\text{d}}}^{2} }$$
(5)

where f(Ls) is defined as

$$f\left( {L_{{\text{s}}} } \right) = \sqrt {L_{{\text{s}}} } - 2.5$$
(6)

Finally, the formation porosity is correlated with the migration length Lm. The correlation takes the form,

$$\phi_{0} = C_{1} (\log (L_{{\text{m}}}^{*} ))^{C2} + c_{3}$$
(7)

Table 3 contains the coefficients for the different clean formations (without high neutron absorbers).

Table 3 The Lm − Φ coefficients
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Effects of thermal neutrons absorbers on the transport parameters

To show the effect of the high thermal neutron absorbers on the measured neutron porosity, the slowing down length, the diffusion length and the migration length are calculated for the three clean lithologies: limestone, sandstone and dolomite using Eqs. 2–6 as shown in Table 4 for porosity ranging between 15 and 25 PU. The absorption cross section, Ʃa, in Eq. (4) is calculated as,

$$\Sigma_{{\text{a}}} = \phi \Sigma_{{\text{w}}} + \left( {1 - \phi } \right)\Sigma_{{\text{m}}}$$
(8)

where Ʃw is 22.2 capture units for fresh water, and Ʃm is 7.03, 4.55 and 4.3 capture units for clean limestone, sandstone and dolomite, respectively.

Table 4 The neutron parameters-clean formation
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The same calculations are performed for a formation having additional thermal neutron absorbers to show their effect. Equation 8 is modified to account for such effect.

$$\Sigma_{{\text{a}}} = \phi \Sigma_{{\text{w}}} + \left( {1 - \phi } \right)\Sigma_{{\text{m}}} + \Sigma_{x}$$
(9)

where Ʃx is the additional absorption cross section caused by the neutron absorbers. Table 5 shows the neutron parameters for a formation having two additional capture units (Ʃx) caused by the neutron absorbers. Comparison between the migration lengths of the clean formation and the formation with the additional neutron absorbers clearly shows the absorption effects.

Table 5 The neutron parameters- for formation with two additional capture units
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For example, the 15 PU limestone in Table 5 is showing a migration length of 15.34 cm. This value shows 16 PU using the clean limestone in Table 4. This comparison illustrates that only 2 additional capture units in a 15 PU limestone formation increase the neutron porosity by one porosity unit causing a 6% error in the neutron porosity.

The correction methodology

A methodology to correct for the effect of the neutron absorbers on the thermal neutron porosity is developed. The methodology requires measuring the formation absorption cross section and the neutron porosity tool. This can be obtained either by running the geochemical logging tool (Schweitzer et. al 1987) or by the sigma tool (Allan et al 1986).

The correction steps are:

  1. 1.

    Calculate the Apparent migration length, Lm1 using the thermal neutron porosity, Φtool, in Eq. 10. The calculated Lm1 using this equation assumes that the formation is clean (has no additional neutron absorbers)

    $$L_{m1} = 10^{{\left( {\frac{{\phi_{{{\text{tool}} - C_{3} }} }}{{C_{1} }}} \right)^{{\frac{1}{{C_{2} }}}} }}$$
    (10)
  2. 2.

    Use Eqs. 3–6 to calculate the migration length, Lm2, by iterating over porosity, using the measured formation sigma absorption, Ʃa. The porosity is increased by ΔΦ until the migration length Lm2 equals to the calculated apparent migration length Lm1. Figure 5 is the flow chart of the algorithm.

Fig. 5
figure 5

The correction algorithm

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Fields applications

The correction methodology is applied in two fields; the first is the Unayzah field in Saudi Arabia using the published mineralogy and elemental core data (Khalifa et al. 2018), while the second is a well in North Africa where the sigma absorption log is acquired.

Application I: Unayzah field

Khalifa et al. (2018) reported the existence of gadolinium and other rare elements in the Unayzah sandstone formation in Saudi Arabia. The mineralogy and elemental core analysis for eight wells showed variable concentrations of gadolinium, Table 6. The average concentration varied between a minimum of 1.29 ppm and a maximum of 3.03 ppm. The absorption cross section, Ʃa, is calculated as,

$$\Sigma = W_{i} \frac{{\rho_{i} A_{v} }}{{A_{i} }}\sigma_{i}$$
(11)

where Wi = weight of the element; ρi = density of the element; σi = microscopic absorption cross section; Ai = atomic mass number of the element; Av = Avogadro’s number.

Table 6 Gadolinium concentration in Unayzah field
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The absorption cross sections for all elements are included shown “Appendix.” The sigma absorption of gadolinium for variable concentrations in the Unayzah field is calculated and shown, Table 7.

Table 7 Gadolinium Ʃa for different concentrations
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The corrected porosities versus the measured porosities for different concentrations of gadolinium: 1, 2 and 3 ppm are shown in Table 8. The results clearly show that very low concentration of this rare element, 3 ppm, for example, will increase the thermal neutron porosity by 8%.

Table 8 Thermal neutron porosity correction for Unayzah sandstone
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Application II: North Africa well

The second application is in North Africa, Fig. 6.

Fig. 6
figure 6

North Africa well

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The reservoir is clean sandstone as indicated by the GR and the photoelectric value of 2, in Track-1. The sigma tool was run, Track-2, while Track-3 shows the overlay of the neutron and the density where no sand separation is observed due to the existence of the neutron absorbers. The measured sigma absorption in this reservoir varies between 16 and 20 capture units. Based on the core analysis, the formation contains the siderite mineral which has an absorption cross section of 52 capture units. The volume of siderite in this formation ranges between 10 and 15% by volume. The siderite resulted in an increase in the absorption cross section and a false increase in the thermal neutron porosity which masked the sand separation in the neutron—density overlay. The correction algorithm is used to correct the thermal neutron porosity. The corrected porosity is then overlayed with the density tool, track-4. The corrected porosity and the density overlay restored the sandstone separation.

Conclusions

The thermal neutron porosity tool is an old technology that is routinely acquired and used in formation evaluation. The tool is simple in design but rather difficult in interpretation due to its complex physics. The neutron transport in formation rocks is controlled by many factors including the probability that the formation constituents absorb the thermal neutrons. Some rare elements and formation minerals have very high probability of absorbing thermal neutrons resulting in a false increase in the thermal neutron porosity measurements. Example of these rare elements is the gadolinium. The effect of these rare elements is recently reported in the Middle East. Also, some minerals, especially the iron-rich ones, such as siderite, glauconite and chlorite have the same effect. It is essential to correct the thermal neutron porosity before using in the evaluation of the reservoir parameters. None of the existing commercial software(s) have modules to correct the neutron tools for such effects.

In this paper, a correction methodology and an algorithm are developed based on combining the neutron transport physics and the thermal neutron tool design. The model is applied in two fields for two cases: rare elements and iron-rich minerals. The first application quantified the effect of few parts per million on the thermal porosity, while the second restored the sand separation between the neutron and the density logs which disappeared because of the increase in the thermal neutron porosity due to the existence of the siderite mineral.