Exploring the Impact of Petrophysical Uncertainties on Recoverable Reserves: A Case Study

Abstract


Introduction
Reservoir characterization is a crucial process in the oil industry that involves integrating data with varying quality and quantity to establish a precise geological model that captures the reservoir's inherent characteristics (Mahmood and Al-Fatlawi, 2021;Abdulredah and Al-Jawad, 2022;Singh et al., 2013).However, the scarcity of data and the significant levels of uncertainty associated with it pose substantial obstacles to effective reservoir development (Jassam and Al-Fatlawi, 2023).To overcome these challenges, scientists use various techniques such as seismic and geologic data integration, machine learning, and covisualization of water saturation in the reservoir simulation model with 4D seismic for integrated reservoir characterization (Rosa et al., 2022;Liu et al., 2022).Despite the tradeoffs associated with unconventional oil, it is being pursued at a higher rate due to the scarcity of conventional reservoirs around the world (Cramer, 2008;Dheyauldeen et al., 2022).Factors that affect the quantity of recoverable hydrocarbons in a reservoir include the fluid distribution in the reservoir, initial volumes of fluids in place, reservoir pressure,

Previous Work
Using the Monte Carlo simulation approach, Ericok and Gumrah (2005) assessed uncertainty in predicting the original oil in place (OOIP) of Field-A, a naturally fractured carbonate field.The mean of the reserve distributions nearly matches the deterministic predictions, indicating the approach's resilience.The key elements impacting reserve estimate in various formations are shown by a sensitivity analysis of the inputs.Water saturation and fracture porosity, on the other hand, are critical in assessing reserves.Kok et al. (2006) discussed the uncertainties involved in economic evaluations of oil and gas projects and the need to consider other possible outcomes beyond the "most likely" results.The data used in these evaluations is obtained from various sources, including geophysical surveys, well logs, and core analysis, and the values of these variables are uncertain.The paper highlights the use of Monte Carlo simulation in estimating the volumetric reserves of hydrocarbon reservoirs and discusses how statistical distribution and descriptive statistics of porosity, thickness, area, water saturation, recovery factor, and oil formation volume factor affect the simulated original oil in place values of two different oil fields in Turkey.The paper contributes to the understanding of how Monte Carlo simulation can be applied in assessing uncertainties and generating probability distributions for decision making in the oil and gas industry.Zhao et al. (2014) introduced the Monte Carlo simulation method for calculating reservoir volumes for hydrocarbons in place (STOIIP or GIIP) during the exploration phase.The traditional volume calculation method is not suitable for this phase due to the limited data available.The Monte Carlo simulation method provides several volume results by Monte Carlo sampling, making the resource assessment results a probability distribution rather than a single valuation, which greatly improves the credibility and usefulness of evaluation results.The paper evaluates the S oilfield in Malaysia, and the results show that the P50 STOIIP is 4.82 MMbbl.Vilela et al. (2022) discussed the use of the Monte Carlo method in assessing the impact of uncertainties in input variables on output variables.These uncertainties, such as reservoir permeability and porosity, can affect the future performance of the system and project valuation.Monte Carlo simulation is a procedure that incorporates random uncertainties to generate probability distributions of outcomes.The technique is particularly relevant in problems related to the value of information and value of flexibility.This paper contributes to the understanding of how Monte Carlo simulation can be applied in assessing uncertainties and generating probability distributions for decision making in the oil and gas industry.
As opposed to the conventional sequential approach, Krasova et al. (2013) advocate for a parallel and probabilistic approach to field appraisal and development concept selection.In lieu of waiting for appraisal drilling to corroborate and finalize the reservoir model, the study recommends initiating frontend concept selection work at an earlier stage, despite the model's high level of uncertainty.Using stochastic depth conversion uncertainty analysis to compute structure maps and bulk rock volumes with P10, P50, and P90 probabilities, this method effectively quantifies the uncertainty.Krasova et al. (2013) demonstrate the benefits of employing a probabilistic, parallel approach to field evaluation and concept selection.It is possible to quantify and effectively manage uncertainty if concept selection work is initiated early, even in the presence of uncertainties, and stochastic analysis is utilized (Sen and Limited, 2015;Zou et al., 2021).This strategy enables an optimized appraisal drilling program that concentrates on removing critical uncertainties, resulting in a cost-effective and confident choice of the optimal field development concept.
In order to account for petrophysical uncertainty in field evaluations, Samotorova and Bronnimann (2014) emphasize the importance of incorporating stochastic well log interpretation, specifically the Monte-Carlo simulation method, to improve the precision of hydrocarbon resource estimations.They highlight the primary source of uncertainty in the Termokarstovoye field as saturation.To mitigate the impact of saturation on the cumulative hydrocarbon pore volume (HCPV) distribution, Samotorova and Bronnimann suggest conducting additional formation water salinity analyses and resistivity log inversions, as these variables are believed to have a substantial effect on saturation.
In the field of petroleum engineering, accurately estimating reserves involves considering the inherent risk and uncertainty associated with the estimation process.Traditionally, deterministic methods have been employed to provide a single best estimate of reserves.However, this approach does not account for the variability and uncertainty of parameters involved in the estimation (Alameedy et al., 2022).Both methods can contribute to more precise reserve estimations in oil and gas initiatives (Inuwa Mohammed et al., 2018).

Uncertainty
Uncertainty in well logging includes random uncertainty, systematic uncertainty, and model-based uncertainty.Random uncertainty refers to the variability in well log measurements due to factors such as measurement errors or noise in the data (Feng et al., 2021).Systematic uncertainty arises from biases or inaccuracies in the measurement process, which can lead to consistent errors in the well log data (Bentley and Ringrose, 2021).Model-based uncertainty is associated with the assumptions and limitations of the models used to interpret well log data, which can introduce uncertainties in the predicted properties of subsurface rocks (Katterbauer et al., 2020).These uncertainties can have significant impacts on reservoir modeling and simulation efforts, potentially affecting commercial decisions and the accuracy of well log interpretation (Masoudi et al., 2017).

Random Uncertainty
Random noise in log measurements can contribute to uncertainty in results at a specific depth.This noise is caused by statistical variation in count rates or signal noise (Grana et al., 2012;Scott, 2000).Petrophysical measurements, especially in carbonate formations, involve some degree of uncertainty (Francioli et al., 2019;Salman et al., 2023;Al-Jawad and Kareem, 2016).To account for these uncertainties, Monte Carlo simulation and fuzzy logic can be used to quantify petrophysical parameters (Francioli et al., 2019).Ensemble based well log interpretation and uncertainty quantification can also be used to estimate geomodel properties and their uncertainties (Jahani et al., 2022).Uncertainty analysis plays an important role in petrophysical analysis and provides primary input data for subsurface formation characterization (Moore et al., 2011).To deal with noise and other sources of uncertainty, a full Monte Carlo approach can be introduced to account for uncertainties in quantitative log interpretation and uncertainty propagation of petrophysical properties and facies classification from rock physics modeling and formation evaluation analysis (Grana et al., 2012).Overall, uncertainty analysis and Monte Carlo simulation are important tools for dealing with noise and other sources of uncertainty in petrophysical measurements.
Log measurements can be noisy even when the tool is stationary, which is compounded in heterogeneous formations where the vertical resolution of the tool is greater than the changes in rock type (Marett and Kimminau, 1989;Mondol, 2015).This issue can be addressed through methods such as Monte Carlo simulation, fuzzy logic, and geomechanical approaches, which account for uncertainties and improve the accuracy of petrophysical measurements (Tewari and Dwivedi, 2020;Grana et al., 2012;Francioli et al., 2019).Vertical resolution is a distance that characterizes the ability of a logging tool to resolve changes parallel to the tool axis.
In order to account for random noise in an interpretation, error bars are often applied.An error bar is an amount on either side of a measurement that includes the full range of what that measurement could be.In some applications, error bars are applied at every depth to indicate the possible range (Fig. 1A); however, this can be misleading and often results in a greater error than is realistic.When assessing possible random errors, it is more appropriate to use error bars based on averages over windows of log values (Fig. 1B), as this more clearly indicates the true range.However, in reality, this is difficult to accomplish owing to the complexity and heterogeneity of most formations.When quantifying the uncertainty in the results for a reservoir interval, the issue of random uncertainty is diminished.This is because, by the very nature of random errors, they will be canceled out by the zone averaging process.The main way that random errors can have an effect on interval results is if random noise on a curve pushes it over a cutoff value in the reservoir summary process (Zou et al., 2021).

Systematic Uncertainty
Systematic errors contribute to substantial discrepancies between measurements and their true values.Both the parameters of measurement and interpretation are susceptible to these errors.Calibration errors and environmental effects often introduce discrepancies in well logs, while inadequate understanding of the formation or insufficient data can result in inaccurate parameter estimates.Modeling systematic errors is a common practice when attempting to quantify uncertainty, given their significant impact on the final results (Moore et al., 2011).
Smart grid implementations are rapidly increasing worldwide, with numerous projects underway.Fortum's "intelligent management system of electric consumption" gathers customer consumption data using advanced metering devices and stores and analyzes the information using metering management systems (Camacho et al., 2011).Vattenfall's "automatic household electricity consumption metering system" is another example of a smart grid implementation (Samad and Annaswamy, 2011).These systems improve the accuracy of energy consumption measurements and enable better management of energy resources.Vertical resolution characterizes the ability of a logging tool to resolve changes parallel to the tool axis.Risk assessment methodology can be used to identify and manage risk in oil and gas activities.Monte Carlo simulation is a useful approach for value of information evaluation (Arild et al., 2008).

Model Based Uncertainty
Model based uncertainty refers to situations where the interpretation model used may not be suitable for the specific formation being evaluated.The impact of this type of uncertainty decreases as the interpreter becomes more familiar with the reservoir.However, at the beginning of an evaluation, model-based uncertainty has the greatest impact on the results because it is challenging to quantify and often receives less analysis compared to other types of uncertainty (Uusitalo et al., 2015;Ahmadi et al., 2016).Conventional approaches to uncertainty quantification may overlook large scale uncertainties related to reservoir structure, which are important to consider (Ahmadi et al., 2016).

Potential Causes of Uncertainty
Geological uncertainty arises from hydrocarbons in place, sedimentary depositional configurations, rock types, their heterogeneities, spatial distribution, and particle size.Uncertainties in geophysics exist in migration, picking, fault positioning, and well ties.Petrophysical measurements involve uncertainties, especially in carbonate formations, due to their characteristics.Uncertainties can arise from sparse, noisy, and/or heterogeneous geological observations and rock.Uncertainties in measurements and interpretation also exist.Conventional approaches to uncertainty quantification may overlook large scale uncertainties related to reservoir structure, which are important to consider.Random uncertainties are statistical fluctuations in the measured data (Wellmann and Caumon, 2018;Francioli et al., 2019;Dubrule and Damsleth, 2001;Ahmadi et al., 2016).Additionally, petrophysical parameters such as porosity, water saturation, permeability, fluid contact locations, and mineral or rock volumes are typical petrophysical input data for a reservoir study.However, well logging tools rarely measure these reservoir variables directly; rather, they are derived via a series of processes that may include data acquisition, processing, interpretation, and calibration.The resulting petrophysical data will have some uncertainty and limitations due to the inherent uncertainty in each of these processes (Sen et al., 2015).
For a reservoir study, parameters such as porosity, water saturation, permeability, fluid contact locations, and mineral or solid volumes derived from well logging data are subject to uncertainties and limitations.It is essential to observe, however, that these variables are typically not explicitly measured by well logging equipment.Instead, they are derived via a series of processes, such as data acquisition, processing, interpretation, and calibration.
Various sources of ambiguity are introduced throughout these stages, which can affect the precision and dependability of the derived petrophysical data.Errors in measurements, the limitations of the monitoring instruments, and inherent geological complexities can all contribute to the data's uncertainty.In addition, the interpretation and calibration procedures require the use of assumptions and a variety of models and methodologies, which contribute to the uncertainties.The precision of these processes is contingent on variables such as data quality, the interpreter's skill, and the availability of calibration data.

Modeling Uncertainty with the Monte Carlo Methodology
In petrophysical interpretation, Monte Carlo processing is a widely employed technique to address uncertainty in measurements and parameters.This method involves iteratively executing the interpretation while randomly varying each measurement and parameter based on a predefined statistical distribution.By incorporating this random variation, the Monte Carlo method enables the consideration of the analyst's uncertainty in these variables (Jassam and Al-Fatlawi, 2023).
During each execution of the interpretation, with different randomly generated input values, a result is obtained.By repeating this procedure numerous times, a spectrum of possible answers is generated.This collection of results provides a comprehensive understanding of the range of possible outcomes, taking into account the uncertainties associated with the measurements and parameters involved (Ikotun et al., 2022).
The advantages of using a Monte Carlo based petrophysical uncertainty modeling approach in reserve assessment are that it allows for the quantification of uncertainty and risk associated with reserve estimates, increases investors' confidence, and reduces the risk and uncertainty in reserve estimation (Inuwa Mohammed et al., 2018).This approach takes into account the uncertainties and heterogeneities associated with petrophysical parameters by modeling them using appropriate distributions, such as normal, lognormal, or triangular distributions (Coll, 2019).By using Monte Carlo simulation, the uncertainty in reserve estimates can be quantified, and pessimistic, most likely, and optimistic reserve values can be obtained with their respective certainty (Soni, 2017).This provides a more comprehensive understanding of the range of possible reserve estimates and the associated level of confidence (Nagababu et al., 2019).Overall, the Monte Carlo based approach improves the accuracy and reliability of reserve assessments, making them more economically profitable for oil and gas projects (Lin et al., 2013).
The Monte Carlo approach offers several advantages in petrophysical interpretation.It allows for the exploration of the full range of potential solutions and provides a statistical distribution of the results, including measures such as the mean, standard deviation, and quantiles.This information aids in assessing the level of uncertainty and provides insights into the probability of different outcomes (Amrollahinasab et al., 2023).
Moreover, the Monte Carlo method facilitates sensitivity analysis by examining the impact of individual variables on the interpretation results.By selectively varying specific measurements or parameters, analysts can gain a better understanding of their influence and identify key factors driving the variability in the results (Khatimah et al., 2022).
Overall, Monte Carlo processing in petrophysical interpretation offers a powerful tool to account for uncertainty.It enhances the robustness of the interpretation by incorporating the analyst's uncertainty in measurements and parameters, yielding a spectrum of possible answers and enabling informed decision making in reservoir characterization and development.
The outcome of the Monte Carlo processing provides valuable insights into the distribution of potential results, allowing for a more thorough comprehension of the uncertainty associated with the interpretation.This data enables a more robust analysis of the reservoir or system under consideration by evaluating the potential range of outcomes and their associated probabilities.
The results of the interpretation include zone averages for porosity, water saturations, permeability and bulk volume of fluids, along with counts of net reservoir, net pay, net-gross ratio and equivalent hydrocarbon column (EHC), which is calculated from equation 1 and summed for the interval.

𝐸𝐻𝐶 = 𝜑 × (1 − 𝑆𝑤 ) × 𝑆𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒
(1) The measure of EHC (or equivalent porosity column, if saturations are not computed) provides a means of ranking or ordering the results, as these are direct measurements of the hydrocarbon (or reservoir quality rock) in place in the well.Based on this ordering it is possible to determine the actual Monte Carlo iteration or run which corresponds to the 10 th , 50 th or 90 th percentiles for P90, P50 and P10 results.
An important part of uncertainty modelling is understanding which parameter error bars have the greatest impact on the variability of the results.A sensitivity analysis is performed, where each parameter or log is changed in turn, while all others are kept as the 'base case' value.Monte Carlo processing provides a robust method for quantifying systematic uncertainty and also for capturing the effects of random uncertainty if reservoir summary cutoffs have been used.However, to understand the effects of model-based uncertainty this process needs to be run using different models and the range of results from each model compared.This will be discussed further in a later section.The number of times the interpretation must be run is dependent on the complexity of the model, the number of uncertain parameters and measurements, the amount of potential error assigned to each and the complexity or heterogeneity of the formation.If the number of iterations is high enough, then the distribution of the results will repeat when the complete process is run multiple times.
Depending on the type of variable or measurement, parameter uncertainties are entered either as lowside and high-side errors or as percentages of the measurement value.Three curves, the conventional version plus high and low versions, are used to enter log uncertainties (Fig. 2).Either the user or environmental correction modules can output these curves.The measurement accuracy and uncertainty of environmental parameters can be incorporated into the corrections in newer versions of the environmental correction modules that use Monte Carlo processing.The current set of distribution functions for parameters and logs includes normal/log-normal, asymmetric triangular, and uniform distributions (Fig. 3).During the interpretation process, certain parameters are frequently chosen from log plots or cross plots; these need to be modified to account for the shifting data on the plots in each Monte Carlo iteration.An optional process that automatically modifies the dependent parameters achieves this.Based on the provided cutoffs, a simple reservoir summary is generated.The values of net pay, net reservoir, net gross, and average reservoir properties are provided in addition to conventional results for the mean result, 1P (P90), 2P (P50), and 3P (P10) results.These are based on the equivalent porosity column or equivalent hydrocarbon column, depending on whether Sw has been calculated.

Results and Discussion
In this study, the uncertainty associated with the effective porosity and water saturation of two wells (1A and 2A) in the Oil Field Northern region Iraq was quantified.The interpretation procedures include the logs listed in Table (1).True Matrix Properties by 2 & 4 Mineral Models Lithology For shale volume, we observe that the gamma ray tools provide the lowest and most accurate values compared to the other tools; hence, we favored the gamma ray over the other tools.And for the porosity tools, we utilized all three (sonic, density, and neutron), and after comparing their values to estimate the correct value, we determined that the density / Neutron combination provides the correct value, and then we applied the hydrocarbon correction to the Density-Neutron combination.For lithology, we have used multiple models and optimized them to determine the lithology that best corresponds to the geological description of the drilled cuttings.The well logging processed for wells A1 and A2 is presented in Figs.(4 and 5), respectively.Well A1's computer processed interpretation (CPI) reveals the characteristics of various reservoir zones.The Jeribe zone is approximately 48 meters thick, with a net reservoir thickness of 26.82 meters.The net reservoir has an average shale volume of 33.97 percent.The effective porosity is 11.11 percent, and the total water saturation for the net pay interval is 33.48 percent.These findings suggest that the Jeribe zone has a moderate gas display.The thickness of the Dhiban zone is approximately 41 meters, with a net reservoir thickness of 22.67 meters.The calculated average shale volume is 32.22 percent, and the effective porosity is 12.7 percent.Total water saturation for the net pay interval is 34.37 percent, indicating a moderate gas and oil show in the Dhiban zone.The thickness of the Euphrates zone is approximately 50 meters, with a net reservoir thickness of 40.88 meters.The average volume of shale is determined to be 18.78%, and the effective porosity is 18.72%.The total water saturation for the net pay interval is 32.23 percent, indicating that the Euphrates region is rich in oil.The Euphrates/ Serikgie zone is approximately 19 meters thick, with a net reservoir thickness of 10.9 meters.The average volume of shale is determined to be 26.71%,while the effective porosity is calculated to be 15.23%.In the Euphrates/ Serikgie zone, the total water saturation is high, reaching 85.2%, indicating a weak oil show.
The thickness of the transition beds at well A2 is approximately five meters, with a net reservoir thickness of 3.04 meters.The average volume of shale is estimated to be 24.6 percent, and the effective porosity is 16.45 percent.Total water saturation for the net pay interval is 20.78 percent, indicating a moderate gas occurrence in the transition beds.The thickness of the Jeribe zone of well A2 is approximately 46 meters, but the net reservoir thickness is 33.56 meters, indicating a negative net reservoir thickness.The average volume of shale is calculated to be 20.97%, while the effective porosity is measured to be 16.72%.The total water saturation is 31.27percent, indicating that the Jeribe zone has a moderate gas show.
It is critical to recognize that there are limitations in the interpretation of results, particularly for variables like porosity and saturation.These values are inherently limited to 0 and 1.When computing zone averages, however, unlimited versions of the data are used to avoid biases at the scale's edges.For example, if only limited data were used for averaging, a completely water-bearing zone would produce an average water saturation (Sw) value of less than 100%.This is because random noise above 100% is constrained, whereas random values below 100% are included in the average.As a result, zone average Sw values greater than 1 are possible.
The resulting data is then merged with the previous zone, including pay values and averages.This process entails storing all data for each zone separately and combining only the base case, mean, 1P, 2P, and 3P results in the summary file.
Each iteration for each zone is combined to form the final 1P results when the 1P results are merged.The same principle applies when combining 2P and 3P results.The mean result is the merged data from the particular iteration that is less than or equal to the mean value of the effective hydrocarbon column (EHC) for the respective zone (Fig. 6).
The P90, P50, and P10 values for net pay, porosity, and water saturation were calculated summatively.Table 2 shows the results that there is a wide range of net pay, porosity, and water saturation values.Because of the results' significant variability and potential impact on the analysis as a whole, it is imperative that careful consideration be given to them.The uncertainty in the net pay values is relatively low, with the P10 value being only 17.2% less than the base case value.However, the uncertainty in the porosity and water saturation values is much higher, with the P10 values being 3.1% and 12.4% less than the base case values, respectively.The high uncertainty in the porosity and water saturation values is due to the fact that these values are derived from well logs, A B which are not always accurate.The well logs can be affected by factors such as the drilling mud, the formation fluids, and the heterogeneity of the reservoir .
The high uncertainty in the porosity and water saturation values has a significant impact on the analysis of the two wells.The P90 values for the effective porosity and water saturation are much lower than the base case values.This means that the two wells may not be as productive as originally thought.The high uncertainty in the porosity and water saturation values also makes it difficult to estimate the reserves of the two wells.The P90 values for the reserves are much lower than the base case values.This means that the two wells may not have as much oil as originally thought.
The results of the study show that the two wells have a high level of uncertainty.This uncertainty is due to the fact that the porosity and water saturation values are derived from well logs, which are not always accurate.The high uncertainty in the porosity and water saturation values makes it difficult to estimate the productivity and reserves of the two wells.

Conclusions
There are restrictions on how results from petrophysical analysis can be interpreted, especially when working with elements like porosity and saturation.When computing zone averages, unlimited versions of the data are used to prevent biases at the scale's extremes.This will prevent the constraints from affecting the averaging procedure .
The data, including averages and pay values, are combined with the prior zone.Each iteration for each zone is taken into account when merging the 1P results to create the final 1P results.The 2P and 3P results combined follow the same procedure.The effective hydrocarbon column (EHC) for a given zone's mean value is equal to or less than the mean result, which is the combined data from that iteration.
The interpretation of petrophysical data can provide a more thorough understanding of reservoir properties while accounting for the uncertainties and constraints inherent in the process by taking into account these principles and incorporating suitable merging techniques .

Fig. 1 .
Fig. 1.Error applied at every depth, B: Error averaged over an interval

Fig. 2 .
Fig. 2. The percentage calculated by the process used to calculate log values for iteration n at every depth.

Table 1 .
Interpretation steps and models

Table 4 .
Summary calculations for the P90, P50 and P10 for the net pay, porosity and water saturation.