Introduction

The determination of water saturation is an essential and continuous process for every well, new or old, throughout the lifetime of a reservoir. Incorrect calculations of this critical parameter result in inaccurate field development decisions that will lead to producing high volume of reservoir water or missing potential hydrocarbon zones.

There are multiple water saturation models developed for field applications. All these models are developed to improve the accuracy of determining this critical parameter, Sw.

Archie (1942) developed the first water saturation model which took the form

$$S_{\text{w}}^{n} = \frac{{aR_{\text{w}} }}{{\emptyset^{m} R_{\text{t}} }}$$
(1)

where (m) and (n) are defined as cementation and saturation exponents, respectively. The Rt and Ф represent the resistivity and the porosity of the reservoir rock at each depth. Archie model does not include corrections for the effect of the clay minerals. Clays are conductive minerals that decease the rock resistivity, Rt. This reduction in rock resistivity, if not corrected for, will falsely show higher water saturation when Archie model is applied. This was later corrected by multiple researchers using different correction approaches.

Simandoux (1982), Waxman and Smits (1968) and Dual Water model (Clavier et al. 1977) are examples of many methods developed to correct for the clay component effects, volume and types. Simandoux used the clay volume Vsh and the clay resistivity Rsh to correct for the clay conductivity effect. Simandoux model takes the form

$$S_{\text{w}} = \frac{{aR_{\text{w}} }}{{2\emptyset^{m} }}\left[ {\sqrt {\left( {\frac{{V_{\text{sh}} }}{{R_{\text{sh}} }}} \right)^{2} + \frac{{4\emptyset^{m} }}{{aR_{\text{w}} R_{\text{t}} }}} - \frac{{V_{\text{sh}} }}{{R_{\text{sh}} }}} \right]$$
(2)

The Waxman and Smits corrected the clay component effect on Archie model using the cation exchange capacity of clays CEC, and Qv. Their model takes the form

$$\frac{1}{{R_{\text{t}} }} = \emptyset_{\text{t}}^{m} S_{\text{wt}}^{n} \left( {\frac{1}{{R_{\text{w}} }} + \frac{{BQ_{\text{v}} }}{{S_{\text{wt}} }}} \right)$$
(3)

The dual water model differentiated between the clay water resistivity Rwb and the formation water resistivity Rwf to perform the correction. The model takes the form

$$\frac{1}{{R_{\text{t}} }} = \emptyset_{\text{t}}^{m} S_{\text{wt}}^{n} \left[ {\frac{1}{{R_{\text{wf}} }} + \frac{{S_{\text{wb}} }}{{S_{\text{wt}} }}\left( {\frac{1}{{R_{\text{wb}} }} - \frac{1}{{R_{\text{wf}} }}} \right)} \right]$$
(4)

Despite that the above models corrected the effect of the clay conductivity on the Archie model, all still require accurate values for the Archie parameters, Rw, m and n. In field applications, when Archie and/or the clay corrected models are used, those parameters are kept constants across the entire reservoir section or change them, by zone, based on the measured laboratory values.

In this paper, a method to determine the water saturation without the need of knowing these parameters is discussed. The scientific basis of the approach is derived using forward modeling of multiple cases including

  1. 1

    Clean formation with variable porosity.

  2. 2

    Formations with variable clay volumes.

  3. 3

    Formations with variable pore structure.

The methodology is also tested on field data for both clastics and carbonates reservoirs and compared to the Archie-based water saturation models.

Basis of the methodology

The methodology is a modification of the Passey (1990, 2010, 2012) Total Organic Carbon model which is widely used to determine the organic richness of the unconventional reservoir.

It is based on characterizing the behavior of two independent measurements that are affected by the existence of hydrocarbon in the rock porosity. These measurements are the rock resistivity and the bulk density or the rock resistivity and the sonic travel time.

The resistivity, bulk density and sonic travel time measurements are affected by the existence of the hydrocarbon in the pore space as follows

  1. 1

    The resistivity of the formation rock increases with the increase in the hydrocarbon volume in response to the high resistivity of the hydrocarbon compared to that of the salty formation water.

  2. 2

    The bulk density of the formation rock decreases with the increase in the hydrocarbon volume since the density of the hydrocarbon is less than that of the formation water.

  3. 3

    The sonic travel time of the formation rock increases with the increase in the hydrocarbon volume since the travel time of hydrocarbon is higher than that of the formation water.

When the resistivity is plotted with either the bulk density or the sonic travel time, in the same plot, a separation is clearly observed and varies with the hydrocarbon saturation. To calibrate this separation, the overlay plot must have a Base Zone with no separation due to the absence of hydrocarbon. This can be achieved using either a clay zone or a water zone where the water saturation in these two zones is a 100%. The overlay technique is investigated using a forward modeling of the resistivity, bulk density and sonic travel time.

The forward modeling of the resistivity, density and sonic travel time for a limestone formation of variable porosity, variable clay volume and variable pore structure are performed. The formation is topped with a Kaolinite clay zone with a resistivity of 3.0 Ω, density of 2.5 g/cc and sonic travel time of 70 us/ft as shown in Fig. 1. The clay zone is the base zone for the overlay where its clay parameters are obtained from service companies tools responses handbooks for the Kaolinite clays.

Fig. 1
figure 1

Illustration model

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The formation resistivity is calculated using the Simandoux equation as

$$\frac{1}{{R_{\text{t}} }} = \frac{{aR_{\text{w}} }}{{4\emptyset^{m} }}\left[ {\left( {\frac{{2\emptyset^{m} S_{\text{w}} }}{{aR_{\text{w}} }} + \frac{{V_{\text{sh}} }}{{R_{\text{sh}} }}} \right)^{2} - \left( {\frac{{V_{\text{sh}} }}{{R_{\text{sh}} }}} \right)^{2} } \right]$$
(5)

while the sonic travel time and the bulk density are calculated using the Wyllie and the bulk density equations, respectively. The water density and resistivity are 1.035 g/cc and 0.131 Ω @75 Fahrenheit for a water salinity of 50 Kppm. The oil density is 0.8 g/cc, and the limestone formation density is 2.71 g/cc. The travel time for water and oil is 189 us/ft and 230 us/ft, while the travel time for limestone is 47 us/ft.

Clean formation case

Table 1 contains the first modeled case, clean formation Vc = 0%, Φ = 30% for variable hydrocarbon saturations from 10 to 90% with a 5% step.

Table 1 Rock responses
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The overlay technique

Figure 2 shows the resistivity–density overlay where the resistivity is plotted on a logarithmic scale, while the density is plotted linearly. The minimum and the maximum values of the density scale are chosen so that the clay points, resistivity and density overlay. The overlay shows clear separation due to the hydrocarbon effect on both measurements. The separation increases with the increase in the hydrocarbon saturation and decreases with the increase in the water saturation.

Fig. 2
figure 2

Density–resistivity overlay, Vc = 0, Φ = 30%

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Similarly, the resistivity and sonic are also overlaid as shown in Fig. 3. The scale of the sonic travel time is reversed since the sonic travel time increases with the increase in the hydrocarbon volume. The scale of the sonic is also chosen to show no separation at the clay zone. The same separation phenomenon is observed where it increases with the increase in the hydrocarbon saturation.

Fig. 3
figure 3

Sonic–resistivity overlay, Vc = 0, Φ = 30%

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Effect of formation porosity

To study the effect of formation porosity on the methodology, the forward modeling for the clean formation is repeated but with low porosity, Φ = 10%.

Table 2 contains the modeled case with variable hydrocarbon saturations from 10 to 90% with a 5% step. Figures 4 and 5 show the separation between the resistivity and the density, and the separation between the resistivity and the sonic. The overlay shows clear separation due to the hydrocarbon effect on both measurements. The separation increases with the increase in the hydrocarbon saturation and decreases with the increase in the water saturation

Table 2 Rock responses
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Fig. 4
figure 4

Density–resistivity overlay, Vc = 0, Φ = 10%

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Fig. 5
figure 5

Sonic–resistivity overlay, Vc = 0, Φ = 10%

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This forward modeled case showed that the values of the resistivity, density and sonic changed due to the change of porosity, but the separation phenomenon is still valid.

Effects of clay minerals

To study the effect of clay minerals on the methodology and its separation phenomenon, the forward modeling for the first clean formation is repeated but with adding variable clay volumes; 25% and 50%, respectively. The clay compaction factor, Cp = Δtsh/100, for the Wyllie equation is used to correct for the effect of clays on the modeled sonic travel time. Table 3 contains the modeled case of a clay volume, Vc = 25%, and variable hydrocarbon saturations from 10 to 90% with a 5% step.

Table 3 Rock responses with, Vc = 25%, Φ = 30%
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Figures 6 and 7 show the separation between the resistivity and the density, and the separation between the resistivity and the sonic respectively for the 25% clay volume case.

Fig. 6
figure 6

Density–resistivity overlay, Vc = 25%, Φ = 30%

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Fig. 7
figure 7

Sonic–resistivity overlay, clay = 25%, Φ = 30%

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Table 4 contains the modeled case of a formation with a clay volume, Vc = 50%, and variable hydrocarbon saturations from 10 to 90% with a 5% step. Figures 8 and 9 show the separation between the resistivity and the density, and the separation between the resistivity and sonic, respectively.

Table 4 Rock responses with, Vc = 50%, Φ = 30%
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Fig. 8
figure 8

Density–resistivity overlay, Vc = 50%, Φ = 30%

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Fig. 9
figure 9

Sonic–resistivity overlay, clay = 50%

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These two modeled cases of variable clay volumes clearly show that the values of the resistivity, density and sonic are changing, but the separation phenomenon still holds.

Effect of pore structure

Focke and Munn (1987) found that the difference between an inter-particle and vuggy pore system for carbonate reservoirs in the Middle East can be seen in the values of the cementation exponent ‘m’. The ‘m’ value can be as large as m = 4 for the vuggy pore structure versus m = 2 for the inter-particle pore structure. This finding is used here to study the effect of the change in the pore structure on the separation phenomenon.

Table 5 contains the modeled case of a carbonate formation with clay volume Vc = 25%, m = 2 representing inter-particle pore structure with variable hydrocarbon saturations from 10 to 90% with a 5% step, while Table 6 contains the modeled case of a formation with clay volume of Vc = 25%, m = 4 representing vuggy pore structure, and variable hydrocarbon saturations from 10 to 90% with a 5% step. Figures 10, 11, 12 and 13 show the separation between the resistivity and the density, and the separation between the resistivity and the sonic respectively for the modeled cases. It is found that the separation phenomenon still holds.

Table 5 Rock responses with clay, Vc = 25% and m = 2
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Table 6 Rock responses with clay effect, Vc = 25%, m = 4
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Fig. 10
figure 10

Density–resistivity overlay, clay = 25%, m = 2

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Fig. 11
figure 11

Sonic–resistivity overlay, clay = 25%, m = 2

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Fig. 12
figure 12

Density–resistivity overlay, clay = 25%, m = 4

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Fig. 13
figure 13

Sonic–resistivity overlay, clay = 25%, m = 4

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Conclusions of the variable modeling cases

All modeled cases using variable porosity values, clay volumes and pore structure showed that the separation phenomenon holds for all cases.

Water saturation determination

The separation between the two measurements, resistivity–density and resistivity–sonic, is quantified to determine the hydrocarbon saturation and accordingly the water saturation. The quantification uses the clay point as the reference point where the separation is zero due to zero hydrocarbon in the clay zones. The base zone has a resistivity Rb, density ρb in case of the bulk density overlay and ΔTb in case of sonic travel time overlay.

The difference between the resistivity at any point and the base resistivity is defined as ΔR and equals

$$\Delta R = \log \left( R \right) - \log \left( {R_{\text{b}} } \right) = \log \left( {\frac{R}{{R_{\text{b}} }}} \right)$$
(6)

Similarly, for the density and the sonic the differences Δρ and Δ(ΔT) equal

$$\Delta \rho = \alpha_{\rho } \left( {\rho - \rho_{\text{b}} } \right)$$
(7)
$$\Delta \left( {\Delta T} \right) = \alpha_{\Delta T} \left( {\Delta T - \Delta T_{\text{b}} } \right)$$
(8)

The factors \(\alpha_{\rho }\) and \(\alpha_{\Delta T}\) are the transformation factors, for the density and the sonic respectively, from linear to logarithmic scale.

They are calculated as

$$\alpha_{\rho } = \frac{n}{{\rho_{\rm max} - \rho_{\rm min } }}$$
(9)
$$\alpha_{\Delta T} = \frac{n}{{\Delta T_{\rm max } - \Delta T_{\rm min } }}$$
(10)

where n is the number of logarithmic decades. For example, n = 3 if the resistivity scale is 0.2–200 and n = 4 if the resistivity scale is from 0.2 to 2000.

The hydrocarbon saturation is proportional to the sum of the separations of the two measurements \(\Delta R\) and \(\Delta \rho\) or \(\Delta R\) and \(\Delta \left( {\Delta T} \right)\) from the base zone as follows

$$S_{\text{hc}} \propto \left( {\Delta R + \Delta \rho } \right)$$
(11)
$$S_{\text{hc}} \propto \left( {\Delta R + \Delta \left( {\Delta T} \right)} \right)$$
(12)

Passey, in his unconventional applications, assumed that the correlation between, Shc, and the separation is linear

$$S_{\text{hc}} \propto \left( {\Delta R + \Delta \rho } \right) \to S_{\text{hc}} = C_{\rho } \left( {\Delta R + \Delta \rho } \right)$$
(13)
$$S_{\text{hc}} \propto \left( {\Delta R + \Delta \left( {\Delta T} \right)} \right) \to S_{\text{hc}} = C_{\Delta T} \left( {\Delta R + \Delta \left( {\Delta T} \right)} \right)$$
(14)

The parameter C in Passey model is determined using the level of maturity, LOM, of the organic material. It takes the form

$$C = e^{2.297 - 0.688*LOM}$$
(15)

The linear assumption of Passey model showed to be applicable in the unconventional reservoirs but not in conventional reservoirs. To prove this conclusion, the separation values and their corresponding water saturation values are plotted using all the forwarded model examples. Figures 14 and 15 show the nonlinear correlations for the case of no clay, m = 2 and Φ = 30% representing clean formation and inter-particle pore structure. The two figures show that the relationship is nonlinear for both cases, resistivity–density and resistivity–sonic for the entire range of Sw or Shc

Fig. 14
figure 14

Shc versus separation for Vc = 0, m = 2, Φ = 0.3

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Fig. 15
figure 15

Shc versus separation for Vc = 0, m = 2, Φ = 0.3

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Figures 16 and 17 show the nonlinear correlations for the case of no clay and m = 2 and low porosity Φ = 10%. representing clean formation and inter-particle pore structure. The two figures show that the relationship is still nonlinear for both cases, resistivity–density and resistivity–sonic.

Fig. 16
figure 16

Shc versus separation for Vc = 0, m = 2, Φ = 0.1

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Fig. 17
figure 17

Shc versus separation for Vc = 0, m= 2, Φ = 0.1

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Figures 18, 19, 20 and 21 show that the nonlinear correlation also holds for the forward modeled cases of Vc = 25%, Vc = 50% and m = 2 representing the inter-particle pore structure case, with variable clay volume.

Fig. 18
figure 18

Shc versus separation for Vc = 025%, m = 2

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Fig. 19
figure 19

Shc versus separation for Vc = 25%, m = 2

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Fig. 20
figure 20

Shc versus separation for Vc = 50%, m = 2

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Fig. 21
figure 21

Shc versus separation for Vc = 0, m = 2

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Similarly, Figs. 22 and 23 show that the nonlinear correlation still holds for the forward modeled cases of Vc = 25% and m = 4 representing the vuggy pore structure.

Fig. 22
figure 22

Shc versus separation for Vc = 25%, m = 4

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Fig. 23
figure 23

Shc versus separation for Vc = 25%, m = 4

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The reason that the linear assumption is acceptable in the unconventional reservoirs is because the range of the TOC is between 0 and 10% at the most. In this range, the relation is linear and far from the nonlinear range.

In conventional reservoirs, the relationship is found to be a nonlinear exponential function. It takes the form

$$S_{\text{hc}} \propto e^{{ - \left( {\Delta R + \Delta \rho } \right)}} \to S_{\text{hc}} = 1 + a e^{{ - \left( {\Delta R + \Delta \rho } \right)}}$$
(16)

The coefficient (a) can be easily determined using a water zone in the reservoir where Shc = 0 and hence

$$a = - \frac{1}{{e^{{ - \left( {\Delta R + \Delta \rho } \right)}} }}$$
(17)

Evaluation of the methodology

The model is evaluated using the forward model example and actual fields data. For the forward modeled example, the calculated hydrocarbon saturation using Eqs. (16) and (17) is compared to the input hydrocarbon saturation.

Figures 24 and 25 show the comparison for the case of Vc = 0, m = 2 and Φ = 30% where excellent agreement is obtained for both cases; resistivity–density and the resistivity–sonic, respectively. The maximum difference observed is 3.0 saturation units. The value of the coefficient (a) using Eq. (17) is determined using the 100% water saturation point, a = − 3.685 and a = − 0.7411 for density and sonic, respectively.

Fig. 24
figure 24

Hydrocarbon saturation from density

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Fig. 25
figure 25

Hydrocarbon saturation from sonic

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Figures 26 and 27 show the comparison for the case of Vc = 0, m = 2 and Φ = 10% where excellent agreement is obtained for both cases; resistivity–density and the resistivity–sonic, respectively. The maximum difference observed is still in the order of 3.0 saturation units. The value of the coefficient (a) using Eq. (17) is determined using the 100% water saturation point, a = − 1.46 and a = − 0.189 for density and sonic, respectively.

Fig. 26
figure 26

Hydrocarbon saturation from density

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Fig. 27
figure 27

Hydrocarbon saturation from sonic

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Figures 28 and 29 show the comparison for the case of Vc = 25%, m = 2 and Φ = 30% where excellent agreement is also obtained for both cases; resistivity–density and the resistivity–sonic, respectively. The maximum difference observed is 4.0 saturation units. Similarly, the value of the coefficient (a) using Eq. (17) is determined using the 100% water saturation point, a = − 4.69 and a = − 0.708 for density and sonic, respectively.

Fig. 28
figure 28

Hydrocarbon saturation from density

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Fig. 29
figure 29

Hydrocarbon saturation from sonic

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Figures 30 and 31 show the comparison for the case of Vc = 50%, m = 2 and Φ = 30% where excellent agreement is also obtained for both cases; resistivity–density and the resistivity–sonic, respectively. The maximum difference observed is 4.5 saturation units. Similarly, the value of the coefficient (a) using Eq. (17) is determined using the 100% water saturation point, a = − 5.99 and a = − 0.67 for density and sonic, respectively.

Fig. 30
figure 30

Hydrocarbon saturation from density

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Fig. 31
figure 31

Hydrocarbon saturation from sonic

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Figures 32 and 33 show the comparison for the case of Vc = 25% and m = 4 for vuggy carbonate pore structure where excellent agreement is obtained for the case of resistivity–density, ± 4 saturation units, while the resistivity–sonic showed higher disagreement, ± 10 saturation units. The value of the coefficient (a) using Eq. (17) is determined using the 100% water saturation point, a = − 1.062 and a = − 0.12 for density and sonic, respectively.

Fig. 32
figure 32

Hydrocarbon saturation from density

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Fig. 33
figure 33

Hydrocarbon saturation from sonic

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This high disagreement in the case of the resistivity–sonic only but not the resistivity–density is attributed to the inapplicability of the forward modeling of the sonic tool response in the case of vuggy formation using the Wyllie equation. This is confirmed by the fining of Baechle et al. (2007).

Field applications

The model is also tested in two actual wells, and comparisons with the Simandoux water saturation model are performed. Figure 34 shows the results of the first well.

Fig. 34
figure 34

Field application—sandstone

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The upper clay zone is used as the base zone where, Rb = 1.66 Ω and ρb = 2.39 g/cc. Track-6 shows the resistivity–density overlay where very clear separation is observed indicating hydrocarbon saturation. The lithology in track-6 and the photo-electric values in track-5 show that the lithology of the formation is sandstone. The fluids saturation and the oil water contact based on Simandoux saturation model are also shown. The Simandoux parameters used in this well are Rw = 0.11 Ω @ 75 °F, m = 2.2 and n = 2.0 as indicated by the core measurements.

The model is applied using Eqs. 16 and 17. There are multiple water zones in this reservoir that are used to calculate the (a) parameter from Eq. (17). The parameter (t) equals to − 1.465 for this reservoir.

Track-7 shows a comparison between the Simadoux hydrocarbon saturation in red dots and the model hydrocarbon saturation in solid line. Excellent agreements are obtained indicating the validity of the method. The calculated Sw using Simadoux model showed an abrupt change in the water saturation at the bottom of the lower zone that cannot be explained by fluids movement. The acceptable explanation is the increase in the clay volume that requires changing the electrical parameters (m, n) to capture this increase in the clay volume effect on these electrical parameters. On the other hand, the resistivity–density separation model did not show that abrupt change. This finding proved that the developed separation model adds values over the Archie-based saturation models

The second application, Fig. 35, is a carbonate reservoir under water injection. Track-4 contains the photo-electric factor, PeF, showing a value of (3) for the top and the bottom zones and a value of (5) for the middle zone. This shows that the top and the bottom zones of the reservoir are dolomite, while the middle zone is limestone.

Fig. 35
figure 35

Field application—carbonate

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Track-5 shows the resistivity–density overlay using the lower clay zone as the base zone where, Rb = 1.1 Ω and ρb = 2.42 g/cc. The overlay shows hydrocarbon separation in the top dolomite zone and in the bottom section of the middle limestone zone, while the upper section of the limestone zone shows no indication of hydrocarbon separation. The reason is the ongoing water injection activities. The upper zone has more sweeping efficiency due to its high intensity of fractures network as indicated by the imaging log in Fig. 36.

Fig. 36
figure 36

Imaging log

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Track-6 shows a comparison between the hydrocarbon saturation using Simandoux saturation model, red dots and the overlay separation model, blue line. The constant (a) is obtained using the water zone, a = − 0.80. The comparison shows some disagreement between the Simadoux and the model in the sweeping zone. This is because of the variation of the formation water salinity Rw in this zone that cannot be correctly handled by the Simadoux model. Also, when the gamma ray increases indicating clay effect, some disagreements between the model and the Simandoux water saturation are observed. This is due to the need for changing the electrical parameters (m, n) as discussed in the first example.

Conclusions

The non-Archie resistivity–density and/or resistivity–sonic overlay method provides an excellent methodology to evaluate the hydrocarbon saturation without the need for the Archie-based models parameters Rw, m and n. The theoretical analysis of the model and the field applications show that the methodology is solid and practical.

Recommendations

In case of highly washed out borehole, it is recommended to use the resistivity–sonic overlay since the sonic travel time tools have deeper depth of investigation compared to the density tools. This will decrease the effect of the washout on the overlaid data and provide more reliable hydrocarbon saturation.