Journal of Technology: Winter 2021, additional articles

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Deep Dielectric-Based Water Saturation in

Freshwater and Mixed Salinity Environments

Dr. Ping Zhang, Dr. Wael Abdallah, Dr. Gong Li Wang and Dr. Shouxiang M. Ma

Abstract /	A low frequency, i.e., KHz, resistivity-based method for water saturation (Sw) evaluation is the desired
	method in the industry due to its deep depth of investigation (DOI) - up to 8 ft. The method becomes
	unreliable if the formation water is fresh or has mixed salinity (SALw). Dielectric permittivity and con-
	ductivity dispersion have been used to estimate the Sw and SALw. The current dielectric dispersion tools,
	however, have a shallow DOI due to their high measurement frequency up to GHz, which most likely
	confines the measurements within the near wellbore mud filtrate invaded zones. It is desirable to evalu-
	ate the possibility of developing a deeper dielectric permittivity-basedSw measurement for various pet-
	rophysical applications.
	In this study, effective medium model simulations were conducted to study different electromagnetic
	(EM) induced polarization effects and their relationships to rock petrophysical properties. Special atten-
	tion is placed on the complex conductivity at 2 MHz due to the availability of current logging tools. It is
	known that the complex dielectric saturation interpretation at the MHz range is quite difficult from
	physics principles, especially when only a single frequency signal is used. Therefore, our study is focused
	on selected key parameters: water filled porosity (ϕ ), SAL		and grain shape, and their effects on the
	modeled formation conductivity and permittivity.	w	w

To simulate field logs, some of the petrophysical parameters previously mentioned are generated randomly within predefined expected ranges. Formation conductivity and permittivity are then calculated using our petrophysical model. The calculated data are mixed with random noises of 10% to make them more realistic - like downhole logs. The synthetic conductivity and permittivity logs are used as inputs in a neural network application to explore possible correlations with ϕw. It was found that while the conductivity and permittivity logs are generated from randomly selected petrophysical parameters, they are highly correlated with ϕw. If new conductivity and permittivity logs are generated with different petro- physical parameters, the correlations defined before can be used to predict ϕw in the new data sets.

We also found that for freshwater environments, the conductivity has much lower correlation with ϕw than the one derived from the permittivity. The correlations are always improved when both conductivity and permittivity were used. This exercise serves as a proof of concept, which opens an opportunity for field data applications.

Field logs confirm the findings in the model simulations. Two propagation resistivity logs measured at

2 MHz are processed to calculate formation conductivity and permittivity. Using independently estimated ϕw, a model was trained using a neural network for one of the logs. Excellent correlation between formation conductivity, permittivity, and ϕw is observed for the trained model. This neural network generated model can be used to predict water content from other logs collected from different wells with a coefficient of correlation (R) up to 96%.

Best practices are provided on the performance of using conductivity and permittivity to predict ϕw. These include how to effectively train the neural network correlation models, and general applications of the trained model for logs from different fields. With the established methodology, deep dielectric-basedSw in freshwater and mixed SALw environments is obtained for enhanced formation evaluation, well placement, and saturation monitoring.

Introduction

A resistivity log was the first downhole log ever run almost 100 years ago - in 1927 - for resources evaluation. This is still the most popular and widely used measurement in formation evaluation, well placement, and reservoir saturation monitoring. To interpret resistivity logs for reservoir saturation requires detailed knowledge of formation water salinity (SALw) as well as rock electric properties. The latter is normally measured from core samples. The former, however, could be hard to know if we have a mixed SALw, a common scenario after water injection in developed reservoirs. In addition, a fundamental assumption for the underling resistivity method is based on

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large resistivity contrast between oil and water. For a freshwater environment, the resistivity difference between oil and water is greatly decreased, leading to an industrywide petrophysical challenge of freshwater environment petrophysics.

Another important rock electric property is permit- tivity, which can be estimated from induction data1, 2. In an effort to incorporate permittivity into petrophys- ical interpretations, an approach that specially targets resistivity and permittivity dispersion properties was proposed in the high frequency range from 10 MHz to GHz3-6. Commercial logging devices based on this approach have been built and successfully used in freshwater environments7. A major limitation for such applications, due to very high frequencies, is their shallow depth of investigation (DOI), only a few inches from the wellbore into the formation.

Current electromagnetic (EM) tools are operating at vastly different frequencies. Induction-type resistivity measurements operate in the KHz range to hundreds of KHz and propagation-type resistivity measurements operate from hundreds of KHz to MHz. Both have much deeper DOI than the GHz dielectric tools. It is therefore desirable to evaluate a possibility of developing a deep dielectric permittivity-based method for petrophysical applications.

Recently, broadband petrophysical models have been developed for clean reservoirs8 as well as shaly sand reservoirs6. These models allow comprehensive studies of relationships between formation permittivity and conductivity with a number of petrophysical parame- ters, such as water-filled porosity (ϕw), SALw, and grain shape. Based on extensive simulation results, substantial knowledge regarding sensitivity and inner dependence of the permittivity on ϕw, SALw, and grain geometry are achieved. Special attention is placed on the model simulations at 2 MHz due to the availability of current logging tools. The main focus of the simulations at 2 MHz is beyond understanding the inner dependence of the permittivity on other parameters, to generate field-like logs, so that a new method can be developed to explore possible solutions of using the permittivity to derive reservoir saturation.

The neural network is selected to explore a possibility of using the permittivity to predict ϕw. The initial results,

after extensive model calculations on different synthetic logs, are very promising. It seems that a strong correlation between permittivity and ϕw makes it possible to estimate water saturation (Sw) using the measured permittivity data. Testing field logs from two different wells further confirm this discovery.

Model Simulations

The broadband EM model that accounts for two key polarization mechanisms present in oil field formations: The polarization on the interfaces between the conductive fluid and nonconductive mineral grains, and the polarization of the electrical double layer present on charged grains. As detailed by Seleznev et al. (2017)8, the model is represented as a collection of spherical inclusions possessing surface charges and spheroidal inclusion without surface charges dispersed in a conductive brine phase, Fig. 1. In addition, the model assumes that the rock is completely water filled, Sw = 1; therefore, ϕw is formation porosity, ϕ.

The model presented in Fig. 1 is most applicable to formations containing grains with a moderate amount of surface charges, e.g., quartz and kaolinite. Quartz grains often have a near-spherical shape and can be reasonably approximated by charged spheres. Variations in the rock tortuosity, or cementation exponent (m), is modeled via the addition of noncharged ellipsoidal inclusions8. The model can be used to calculate rock permittivity and conductivity from pre-defined ϕw, SALw, m, and temperature (T ).

Dispersion Responses

The calculation was first focused on the dispersion effects of permittivity with SALw, ϕw, m, and grain size. Table 1 lists the parameter values used for the calculations. The frequency used to compute dispersion responses is from 102 Hz to 109 Hz. Figure 2 shows the permittivity variations for different SALw levels. Each curve represents one SALw. The values of the remaining parameters are listed on top of the figure, where a is the grain size. Strong dispersions are observed for frequencies below 105 Hz. Lower SALw gives stronger dispersions than the higher SALw. At frequencies above 1 MHz, the dispersions are greatly reduced, but still clearly visible. Based on these results, it is apparent that permittivity has a strong dependence on SALw below 104 Hz, especially as freshwater can substantially impact the dispersion

Fig. 1 Graphical representation of the wideband model.

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Table 1 The parameters used for sensitivity studies of permittivity dispersion.

Salinity (SALw ) (ppk)	Water-Filled Porosity	Cementation Exponent	Radius of Charged
(ϕw )	(m)	Spheres (a) (um)
1, 5, 10, 60, 100	0.1, 0.2, 0.3, 0.4, 0.5	1.5, 1.6, 1.7, 1.8, 2.1	1, 5, 10, 50, 100

characteristic of permittivity.

The dispersion effect due to ϕw is depicted in Fig. 3. Strong dispersions are observed for frequencies less than 104 Hz. In addition, the dispersion curves are clearly separated for each ϕw for the entire frequency range, meaning that permittivity has excellent sensitivity for the ϕw.

Figure 4 shows the dispersions for different rock pore geometries, represented by m. Once again, strong dispersions are observed for frequencies less than 104 Hz. At lower frequencies, larger permittivity values are observed for smaller m. At frequencies above 104 Hz, the dependencies are reversed. It seems that the permittivity is more sensitive to m at high frequencies. The last dispersion plot is related with grain sizes, Fig. 5. Although strong dispersions are shown below 104 Hz,

the permittivity has no sensitivity to the grain size for frequencies above 104 Hz.

Permittivity Responses at 2 MHz

From the dispersion studies (Figs. 2 to 5), it can be observed that at frequencies above the MHz range, the permittivity has greatly reduced dispersion and relatively weak dependence on all modeled parameters except

• w. Consequently, more studies of extracting Sw from permittivity were carried out at the single frequency of 2 MHz, a frequency used in all logging while drilling resistivity tools.

Figure 6 shows permittivity variations as a function of ϕw for five salinities, at two temperatures, 100 °F and 400 °F. Ideally, we would like to see strong correlations between permittivity and ϕw, so that Sw can be derived.

Fig. 2 The effect of differentSALw levels on the permittivity dispersion.

Fig. 4 The effect ofm on the permittivity dispersion.

Fig. 3 The effect ofϕw on the permittivity dispersion.

Fig. 5 The effect of grain size on the permittivity dispersion.

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At the high temperature, 400 °F, a very well confined relationship is observed regardless of the SALw values. At the low temperature, 100 °F, the correlation is not confined. Different SALw value causes noticeable deviations on the curves. In addition, the curves at 100 °F are very different from the ones at 400 °F, indicating that it is not possible to predict ϕw only from measured permittivity without a prior knowledge of SALw and T.

Figure 7 shows similar plots but with different m values. It is even more obvious in this case that the correlations between permittivity and ϕw are more complicated with variable m. Figure 8 shows impacts from grain size. Clearly, the relationship between permittivity and ϕw does not depend on grain size, although it is still affected by temperature.

Predicting Water Filled Porosity

The simulation results of Figs. 6 to 8 clearly show that permittivity at 2 MHz has a strong sensitivity to ϕw. The inter-correlations between them may provide a possibility of estimating ϕw from measured permittivity. SALw, m, and T have huge influences on the correlations, so that a normal regression method cannot be applied to estimate ϕw unless we have prior knowledge of those reservoir parameters. While m can normally be obtained through laboratory measurements on core samples, reservoir temperature can also be measured on-site. It is very difficult to obtain formation SALw9, especially if a reservoir is under waterflooding.

This challenge may open an opportunity of using a machine learning method. In particular, using a neural network to explore the correlations between permittivity and ϕw under mixed SALw conditions, and with different reservoir rock cementation exponents and reservoir temperatures.

Neural Networks

Artificial neural networks, or simply neural networks, are a common technique among the machine learning tools to solve and analyze complex problems - classification and regression. The concept of a neural network, which has found useful applications in function regression, is an adaptation of interconnection of brain neurons to machine, for the nonlinear mapping of input to output10. The neural network architecture, consisting of an input layer, hidden layer, activation function and output layer, controls how the nonlinear mapping of input to output works. The nonlinear mapping of the predictors and target is established by training the neural network. This step is considered an optimization problem with an objective function defined by the standard least-squares method. The optimum neural network parameters to realize the best performance are evaluated with two statistic parameters: the coefficient of correlation (R), and mean square error (MSE):

1

2

The regression values, R, measure the correlation between model outputs, Xp, and targets, Xm. A regression

Fig. 6 The effect ofSALw on the relationships between ϕw and permittivity, where T = 100 °F (left panel) and 400 °F (right panel).

Water Filled Porosity	Water Filled Porosity

Fig. 7 The effect of m on relationships betweenϕw and permittivity, where T = 100 °F (left panel) and 400 °F (right panel).

Water Filled Porosity	Water Filled Porosity

Fig. 8 The effect of grain size on relationships betweenϕw and permittivity, where T = 100 °F (left panel) and 400 °F (right panel).

Water Filled Porosity	Water Filled Porosity

value of 1 means a perfect correlation, and 0 is a random relationship. The MSE is the average squared difference between the outputs and targets. Lower values are better. Zero means perfect predictions, no errors.

Generating Synthetic Logs

The purpose of generating synthetic logs is to explore a possibility of using measured permittivity to estimate

• w. Based on the simulation results, for a clean reservoir, formation conductivity and permittivity depend on ϕw, SALw, m, grain size, and T. At 2 MHz, permittivity is not sensitive to grain size, so for all synthetic logs, the grain size is fixed at 10 micrometers. The synthetic logs are generated within a depth range of 1,000 ft to 2,000

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ft, with a 0.5 ft sampling rate. To closely represent field logs, both ϕw and SALw are generated using a random number generator within predetermined numerical ranges. The m and T, on the other hand, are fixed at a few predefined values.

Table 2 shows the details of the parameters used to calculate the synthetic formation conductivity and permittivity logs.

An example of the synthetic logs is represented in Fig.

1. Track 1 is ϕw, randomly generated within a range of 0.1 to 0.9. Track 2 is SALw, randomly generated within

1. part per thousand (ppk) to 150 ppk. Tracks 3 and 4 are the calculated formation conductivity and permittivity for m = 2 and T = 100 °F, respectively. Then, the cal- culated conductivity and permittivity are added with 10% random noises.

Figure 10 shows the final logs used for the neural network simulations.

Predicting the ϕw for Different m

The purpose of generating synthetic logs is to explore the possibilities of using a neural network to estimate

• w from permittivity. Considering that other parameters (m and T ) are also closely related to permittivity, it is our hope that the neural network, as a general statistic approach, can address this issue.

First, we examine how m affects the neural network predictions with the following steps:

1. The ϕw and SALw in Fig. 9, together with m = 1.5 and T = 100 °F, are used to calculate permittivity and conductivity logs. The calculated logs are used as inputs and ϕw as the target for a three-layer 15 cells' neural network to train a model. The trained model, if successful, can predict the ϕw from the input logs.
2. Calculating new permittivity and conductivity logs with the same parameter setups as step 1, except with m = 1.7 and m = 2. Then the trained model in step 1 is applied to these newly calculated logs to predict ϕw. The predictions are compared with the true ones to assess how well the trained model performs on the new logs.

The regression value and MSE, given in Eqns. 1 and 2, are used to assess the quality of the model predic- tions. Figure 11 shows the results for the trained model and the model predictions. Figure 11a shows the trained model outlined in step 1. In this case, the permittivity and conductivity logs are able to give almost perfect

Fig. 9 Synthetic logs: Track 1 ϕw randomly generated within 0.1 to 0.9; track 2 SALw randomly generated within 1 to 150 ppk; tracks 3 and 4 are the calculated formation conductivity and permittivity for m = 2 and T = 100 °F, respectively.

Water Filled Porosity	Salinity (ppk)	Conductivity (S/m)	Permitivitty

predictions of ϕw (R = 0.99). In other words, if a field condition can be represented by the model parameters, then the measured permittivity and conductivity logs can be used to estimate ϕw, even though we do not have the knowledge of formation SALw (randomly generated within 1 ppk to 150 ppk).

Figures 11b and 11c are the outcomes of step 2, which tries to answer whether the model defined in step 1 is still valid for different m values. If m is increased from

1.5 to 1.7, the estimated ϕw are mostly correct, except the slightly upward bias at the lower end of porosity, Fig. 11b. In a practical sense, the model trained with m = 1.5 can be used to estimate ϕw for m = 1.7. If a further increase of m to 2, the accuracy of the prediction is greatly reduced, Fig. 11c. In this case, more than half of the ϕw are overly estimated.

Predicting the ϕw for Different Temperatures

Reservoir temperature can also impact permittivity and conductivity measurements. Therefore, it is logical to test the influences of the temperatures on the model predictions, using the following steps:

1. The ϕw and SALw in Fig. 9, together with m = 1.5 and T = 100 °F, are used to calculate the permittivity and conductivity logs. The calculated logs are used as inputs and the ϕw as the target for a three-layer

Table 2 The parameters used for calculating the synthetic conductive and permittivity logs.

		Water-Filled	Salinity	Cementation	Radius of	Temperature	Frequency
Name	Depth (ft)	Charged
Porosity (ϕw )	(ppk)	Exponent (m)	(°F)	(MHz)
		Spheres (um)
Range	1,000 to	0.1 to 0.9	1 to 150	1.5, 1.7, 2.0	10	100, 200,	2
2,000	300, 400
Sampling	0.5	Random	Random	Fixed	Fixed	Fixed	Fixed
rate