Eficient management of wells in oil fields is essential to avoid significant economic losses and maximize oil production. This study presents a stochastic model based on Discrete Time Markov Chains (DTMCs) for the temporal evolution of well states. Specifically, the model estimates the dynamics among well working states and failure states due to diferent operating causes. Moreover, a Monte Carlo method is proposed to simulate future scenarios and forecast oil production losses and potential negative economic performance derived from the unavailability of wells. The approach was applied to an oil development field in Colombia and validated through statistical tests for the DTMCs properties. The proposed model ofers immediate practical benefits for the oil industry in the studied region, as well as the potential for successful application in other fields. This provides a valuable and versatile tool for global oil well availability management.
The oil industry faces major challenges in managing the availability of its wells, as losses associated
with unavailability can be significant. Globally, these losses are estimated to reach millions of barrels
per day, which translates into economic losses in the millions of dollars [
In this sense, the present work seeks to model the transitions between operating states and failure or unavailability states due to diferent operational causes in oil wells. From a reliability engineering perspective, a stochastic model based on Discrete Time Markov Chains (DTMC) is proposed. This approach allows more accurate modeling of the behavior of oil wells, considering the stochastic nature of failure causalities and their dynamics. Additionally, Monte Carlo methods are applied to perform production and economic loss forecasts associated with the unavailability of the wells. This approach allows the simulation of multiple scenarios, taking into account the uncertainty inherent to the failure causes and to the operation and maintenance processes. To illustrate the practical application of the proposed methodology, a case study is carried out in an oil production area located in Colombia.
This article is organized as follows. Section 2 presents work related to failure modeling and availability of infrastructure systems with an emphasis on the petroleum industry. Section 3 addresses business understanding, describing the specific characteristics of the petroleum industry and modeling needs. Section 4 focuses on understanding the data, and explaining the nature and source of the data used in the study. Section 5 describes the preparation of the data, including the cleaning and transformation necessary for analysis. Section 6 presents the modeling approach, detailing the implementation of DTMCs and Monte Carlo methods to enhance production forecasting and economic evaluation. Section 7 presents the results obtained and their discussion. Finally, section 8 exhibits the conclusions of the study, along with recommendations generated from the model and future extensions of the proposed approach.
The analysis of failures in oil and gas stations has employed a variety of methods and models, contingent
upon the availability of data. For instance, Guo-qi Ren et al. [
One of the earliest applications of Markovian models in this context was that of Cochran et al., who
proposed an alternative method for estimating availability with Markov chains for operational failures
in oil refineries. They asserted that the models were more eficient, easier to construct, and provided a
comprehensive range of performance metrics [
Recent research on oil well failures and production losses in Colombia has focused on the development
of predictive models and the improvement of decision-making processes. For instance, Cardona et al.
[
Similarly, the Monte Carlo method has been widely applied in diferent engineering and economic
contexts [
Net oil production from oil fields can be reduced by factors such as evaporation and residues in
production and transportation equipment, among others, which are typical of the extraction process
[
It should be noted that losses refer to unconventional circumstances that negatively afect production.
Usually, these are caused by operational failures, such as electrical failures in generators and components
inside the wells, as well as mechanical and chemical failures in the crude processing plant during the
separation of water and sediments from the oil [
This article focuses on production data from an oil development field in Colombia, where the operator faces the challenge of decreasing daily net production losses due to the aforementioned factors. These
Variables Well, cluster, area, operating day, net oil production, water production, liquid production, lift gas injection, downtime.
Well, cluster, area, operating day, net oil losses, water losses, liquid losses, cause of loss, type of loss, type of uncontrolled loss.
Well, cluster, area, month, expected net production. have contributed to fluctuations in production in recent times, which ultimately have distorted production forecasts and lead to more frequent corrective maintenance activities, resulting in significant economic losses. In total, 61 wells in the field are considered, distributed in 17 geographic clusters, which belong to 3 production areas (x, y, and z). The net daily production of these wells (shown in Figure 1) presents significant fluctuations over the last years, coinciding with the continued development of the field.
The data used in this study comes from a transactional system, which is fed directly by the production
team. This team is responsible for recording, reporting, and categorizing production losses, as well
as recording the daily production of each well. Through this system, detailed control of operations is
performed, allowing precise identification and classification of losses, ensuring that each well has a
reliable record of its performance and events that afect its production. For the application case, three
data sets were used. The first contains information on daily net oil production from each well in the
ifeld over a 2.5-year period, from July 2021 to January 2024. The second one includes the daily net oil
losses per well identifying the cause of each loss during the same time period. The third dataset is the
production forecast, developed by the team of petroleum engineers using the ARPS model [
Deepening into the "Net oil losses" database, the production team has identified 33 diferent causes of loss, taking into account the actors involved and the scope of each cause after an operational on-site inspection. For a preliminary analysis, a classification is proposed by "type of losses", specifically: 1. External causes: These are not due to operational factors, but to external circumstances that afect the operation. They are associated with sporadic events such as the collapse of bridges, landslides that afect the oil output and force the closure of wells, social protests that hinder the operation, and weather factors such as electrical storms, among others. 2. Internal causes: They derive from corporate decisions and are not classified as failures. These decisions are based on economic and financial analysis and corporate standards. Examples include profitability per well, insuficient infrastructure for production, plant shutdowns for preventive maintenance, and corrective maintenance postponed due to strategy or supplier contracting policies. 3. Uncontrolled causes: These refer to failures in internal operating systems that afect production.
The developed model focused on the last cause because it is the most common mechanism of failure. External factors are not included, as they are less frequent in the dataset, making dificult its statistical estimation. However, when they occur they have a significant impact on production, as they usually afect the entire field.
From the "Net oil losses" database the state of each well is identified for each day and labeled as "Working State" (WS) if it is working and "Failure State" (FS) if it is of due to uncontrolled causes. To provide greater specificity the "Failure State" is divided into three states representing the causes of failure: "Mechanical" (Mec), "Electrical" (Elec), and "Chemical" (Chm). The production team validated this categorization, the correct classification of the (uncontrolled) causes of loss, and the reinterpretation of some loss causalities in the original database that presented ambiguities. Tables 2 and 3 show the number of records for the above-described well states.
To prepare the dataset for the simulation of failures over time, it is necessary to transform the production forecast, originally expressed in barrels of average daily production per month. Since the simulation required daily data, values of daily production are replicated for each day in a month. This transformation allows representing the production over the entire month as a daily time series, instead of having the value of the daily average for the month in a single record. This adaptation facilitates overlaying the results with the simulation model in an appropriate manner.
A Discrete Time Markov Chain (DTMC) is a stochastic process {} with a discrete state space Ω and a discrete set of times = 0, 1, . . . , meaning that the temporal measurements progress in individual steps of fixed time intervals. This process satisfies the Markovian property: (+1 = | = , − 1 = − 1, . . . , 0 = 0) = (+1 = | = ), (1) which means that the probability of transition to a future state depends exclusively on the present state and not on previous states.
The complete probabilistic structure of the process can be derived from the 1-step transition probability
matrix P = [ ] with = (+1 = | = ), , ∈ Ω, denoting the probability of transition
from state to state in a single time step [
The approach followed in this paper consists of proposing two DTMCs for each well to model
transitions between 1) States WS, FS, and 2) States WS, Elec, Mec, and Chm. The time is discrete and is
given in days. The methodology applied is detailed below:
1. DTMC fitting: The transition probability matrices for each DTMC model for each well are
estimated from the datasets of well status. This is performed with the library ‘markovchain’ in
the statistical software R and the function ‘markovchainFit’ that uses the maximum likelihood
estimation method [
For this purpose, the ‘steadyStates’ function from the ‘markovchain’ library is used.
3. Verification of the Markovian property: A chi-square test is applied to verify the Markovian
property, equation (1), for each of the fitted models. The test compares the observed frequencies
of transitions with the expected frequencies under the null-hypothesis that the process follows
the Markovian property [
Moreover, stationarity and order properties were also verified using similar test statistics.
Stationarity in this context refers to the property that the transition probabilities remain
constant over time. The stationarity test helps to determine whether the Markov chain process
exhibits this property, which is fundamental for the model to be valid over time. The order of the
process refers to the number of previous states that must be considered to predict the future state
of a well. Determining the proper order is crucial to ensure that the model correctly captures the
dependence relationships between past and future well states. The order test helps identify this
optimal number of past states to consider in the model. These additional tests complement the
verification of the Markovian property and help ensure the validity and accuracy of the DTMC
model used in oil well availability estimation. The statistical tests were performed with the
commands ‘verifyMarkovProperty’, ‘assessStationarity’, and ‘assessOrder’ in R [
Graph representations of the estimated DTMCs and their transition probabilities are shown in Figures 4
and 5 for the model with WS and FS states, and the detailed model with Elec, Mec, Chm, and WS states,
respectively, for a specific well. Note that the sum of the probabilities leaving each node is equal to 1, as
it is required in these models. Figure 4 indicates that when this well is operational, 94% of the time it
remains in this WS state, while 6% of the days results in a transition to an FS. Once in this FS state, in
62% of cases well maintenance lasts one day, transitioning from FS to WS in a single step, but in 38% of
1The natural decline of the oil in the well makes it unprofitable, and it is no longer considered a reserve but a resource [
Transition matrices for all wells are represented as heat maps in Figure 6. Each cell of a heat map indicates the probability of a one-step transition from one state to another, with lighter colors representing higher probabilities. Transitions from any failure state (Mec, Chm, Elec) to the working state (WS) are highly probable, indicating that one-day maintenance is common. Noteworthy, there is variability in transition probabilities between wells, which may reflect diferences in operational or historical conditions specific to each well. In addition, some wells exhibit transitions among failure states, meaning that failure types could be correlated.
Figure 7 shows a scatter plot of the transition probabilities FS→WS and WS→WS for each well. Dot colors represent clusters of wells and sizes indicate the net production of the well. Wells in the upper right of the plot are more reliable as they have high probabilities of remaining in WS being in WS and transitioning to WS from an FS state. Most of the wells are above the probability 0.9 in the WS→WS axis and to the right of 0.7 probability in the FS→WS axis. The variability in net production suggests diferences in operating eficiency between wells. Some clusters, such as x15 and x2, show specific groupings, which may indicate similar failure dynamics of their wells.
Regarding validation of the DTMC’s models, results of the test statistics for the Markovian property, stationarity, and order of the process of the wells are shown in Table 4. These results indicate that all but one of the wells (60) comply with the Markovian and stationarity properties, with an order of 1. The well that does not follow the Markovian property (x-cd) presents a particular failure dynamics. Its higher energy requirement was not conditioned to plant specifications, resulting in a more recurrent WS →Elec transition, as the failure was iterative and went on for a long time until a proper correction was made. On the other hand, the well that does not meet the order property is only three months old since drilling. During this short production period, it has experienced significant operational developments, but not enough to properly evaluate its dependence on past states.
Furthermore, the net production forecast simulation is performed for a well. Unlike the deterministic ARSP forecasting, Monte Carlo simulation considers the stochastic nature of failures and their influence on production. Figure 8 shows a shift of a simulated production curve from the non-failure deterministic curve. Each time a failure occurs, production returns to its pre-failure level and then continues to decrease according to the downward slope given by the ARSP. This process is repeated each time a failure occurs, generating a characteristic pattern in the production curve.
A total of 10,000 MC simulations of the well state give an equal number of production curves for the stochastic scenario. Then, equation (2) is applied to obtain the periodic NPV for each simulation ( ℎ) and with three diferent discount rates to evaluate the sensitivity to this parameter. In addition, a margin loss for each simulated curve is evaluated as: Δ = ℎ − , (3) with corresponding to the NPV of the non-failure deterministic curve. Table 5 reports this value and the mean and standard deviation (SD) of ℎ and Δ from the 10,000 MC simulations and for each discount rate considered. The results show a loss of economic value for the well when considering failures, which increases from 168 to 191 thousand US dollars as the discount rate decreases from 20% to 10%. This can be visualized in the distributions of Δ in Figure 9 that exhibit a moderate variability (see also the SD values) and a significant separation from a null economic loss Δ = 0.
The transition probabilities between well states demonstrated that the most prevalent state was the working state and that maintenance periods are relatively short, indicating a high probability of good availability under optimal conditions. Furthermore, the statistical tests demonstrated that DTMCs are reliable models for the studied situation. The results showed that 98% of the wells complied with the Markovian property and 100% exhibited stationarity. In the event that these properties were not met, it would be necessary to adjust the model accordingly.
In addition, the NPV comparison between the production forecast (with no failures) and the Monte
Carlo simulations that contemplate uncontrolled losses clearly illustrates the diminished profitability
and the prolonged production recovery, which negatively impact the economic performance of the well.
It is important to mention that the delay in production (due to well failures) also afects the profitability
to continue extracting more oil from the reservoir. Longer production periods have greater uncertainty
due to fluctuating Brent oil prices, which is the benchmark for the industry in Colombia. The variability
of these prices, depending on global macroeconomic and political factors, makes an accurate forecast
dificult [
To produce more insightful results in the analysis of operational failures in wells, one option is to enrich the data with external factors. Including climatic variables, such as temperature, rainfall, and thunderstorms, could help identify patterns between weather and production failures. Also, factors such as regional energy demand or external events such as roadblocks or government interventions could provide more context to better understand operational disruptions.
Another key aspect would be to incorporate detailed information on well maintenance. Maintenance history, both preventive and corrective, could provide a clearer picture of the relationship between the frequency of interventions and well availability. In addition, including the costs associated with each intervention would allow the financial impact of these actions to be assessed, comparing maintenance costs versus losses due to operational failures.
At the modeling level, the use of Machine Learning or advanced time series models such as ARIMA could be integrated to complement Monte Carlo simulations, providing more robust production forecasts.
Finally, improving the granularity of the data, moving from daily to hourly data, would allow for ifner analysis, helping to identify patterns in the timing of failures.
This article addresses the challenge of probabilistic estimation of daily net oil production losses in a development field in Colombia, specifically those related to operational uncontrolled loss causes. Based on data of daily net oil production and historical losses, a time series of causes of losses was determined for each well, which allowed modeling its availability using Discrete Time Markov Chains (DTMCs). The ifndings highlight the importance of managing and mitigating uncontrolled causes of losses to maximize value and operational eficiency. The proposed approach could facilitate informed decision-making, enabling the company to improve production management, optimize available resources, and direct further eforts toward preventive maintenance and regular inspections. 10. Future work Further extensions of the proposed approach can be mentioned:
External causes of losses: The data used to build the DTMC model were obtained from a transactional platform, which may limit the availability of certain information or its quality. In this context, the model developed efectively addresses uncontrolled well loss causes, but does not consider external causes, such as infrastructure problems or adverse environmental conditions. These causes can have a significant impact on the total proportion of production losses. Therefore, for a more complete and accurate assessment of well availability and risk management, it would be necessary to integrate these external causes into future analyses and models.
Incorporation of new data and variables: Additional variables can be included to enhance the probabilistic modeling of well failures and production. These variables include, but are not limited to, reservoir type and age, environmental factors, oil price fluctuations, and changes in market conditions. Other variables, such as pressure and electrical voltage in wells infrastructure, may require additional investments for their measurement.
Predictive model development: The availability of more data and variables allows for the development of more sophisticated predictive models that can anticipate operational failures before they occur. These models may include Cox regressions, decision trees, support vector machines, and other supervised and unsupervised machine learning algorithms.
Maintenance optimization: Develop optimized preventive maintenance strategies based on model results, to minimize downtime and production losses.
Long-term economic impact: Perform long-term economic analysis to better understand the impact of diferent loss management strategies on overall project profitability.
Real-time simulation integration: Implement real-time simulation systems that allow companies to quickly adjust their production and maintenance strategies in response to changing well and market conditions.
Expanding the scope of study: Apply the methodology to other oil fields in diferent regions and countries to validate the efectiveness of the model and adapt the approach to diverse geographic and operating conditions.