Central Limit Theorem-based Stochastic Economic Evaluation (CLT-SEE) Model for Evaluating Oil Wells: Case Study from Niger Delta, Nigeria

— Oil and natural gas production has been highly contributing to world economy and is some countries’ economy root. After the discovery of a new oil/gas field, the operator has to decide whether or not to develop that field. Such decisions rely on the economic evaluation of potential oil/gas fields development when they will be discovered and of the proven oil/gas reserves. The economic indicators used for that purpose are actually computed with deterministic and/or stochastic methods. Deterministic models show limitations while stochastic ones reduce the risks and doubts in the decision making. Stochastic models require the knowledge of the probability distribution of the model inputs, what is costeous in terms of software, data and conditions to be satisfied. Our study proposes a technique, called “Central Limit Theorem-based Stochastic Economic Evaluation (CLT-SEE) Model’’ that eases projects NPV probability distribution determination and the computation of P10, P50 and P90 of projects NPV, IRR and PI. A case study is carried out on a Nigerian’s Niger Delta onshore oil well. The results show the well NPV, IRR and PI are respectively MM$ 84.112, 24.5%, 1.169. The well project P10(NPV), P50(NPV) and P90(NPV) are respectively MM$ 96.4, MM$ 84.16 and MM$ 71.89; P10(IRR), P50(IRR) and P90(IRR) are respectively 27%, 24.75% and 22%; P10(PI), P50(PI) and P90(PI) are respectively 1.34, 1.17 and 1. These stochastic outputs show that the company has 90% of chance to earn at least MM$ 71.89 which is its investment and the likelihood that the project IRR be more than 22% is 0.9. As a result, the use of CLT-SEE model for oil wells economic evaluation offers much more chance and confidence to oil companies to decide righteously in field and well development projects.


I. INTRODUCTION
Oil and natural gas production has been highly contributing to world economy and is some countries' economy root.For example, from 2000 to date Nigeria's economy depends strictly on oil and gas exportation.[7] have shown that in 2013 about 70% of government revenue is derived from oil and over 90% of new investments are associated with oil and allied products.According to [19], Russia's oil and natural gas production (including coal) provides 27% of the land's 2015 GDP and 43% of oil and gas revenues to the budget account for the fuel and energy complex.
Due to the contribution of oil and gas production activities to these countries' GDP coupled with the fact that oil and gas remain the main world energy source, the discovery and development of new oil/gas fields become a crucial aim for both international oil and gas companies and host countries although the need of energy transition.
At the exploration stage the oil and gas international companies need to decide whether to go into an exploration contract on a block.Also, after the discovery of a new oil/gas field, the operator has to decide whether or not to develop that field, that is, going ahead in the expenses (well drilling and completion, production facilities, flow system, etc.).The decision-making tools on investors' hands must prove then that the income received by the operator must be sufficient to repay all investment expenditures, cover production costs including taxes and still yield an acceptable return to investment; otherwise, the investment will not be undertaken.Making such decisions is always a very complicated endeavor [8].In fact, these projects are impacted by many high-risk factors associated with the petroleum industry, such as relatively high initial investment requirements, long term investment horizons, volatile oil/gas prices, costs, production rates and reserves [10].
Such decisions rely on the economic evaluation of potential oil/gas fields development when they will be discovered and of the proven oil/gas reserves.The economic evaluation models of an oil and gas field development project are decision-making tools which help oil companies in deciding whether to invest in a given project or not [12].Economic models are abstractions that characterize real economic systems and are typically just detailed enough to roughly approximate the outcomes of interactions between economic agents.The process for economic evaluation is the same for a whole field development project and a single well.For [11] some of the questions the company wants to answer with the economic evaluation of a well, are how much does the well require?Will this well be viable to develop?Will the rate of return on the investment on this well higher than other investment alternatives?Another key question that can also be asked is how long will the company get back into its investment?
The economic evaluation of oil and gas projects passes through the assessment of some economic indicators (also called economic decision-making tools) thereof.In the literature the common economic decision-making tools used for that purpose are the cumulative cash flow (undiscounted), the cumulative Discounted Cash Flow (DCF) or Net Present Value (NPV), the payback or payout (POT) period, the Internal Rate of Return (IRR) or Rate Of Return (ROR), the Investment Return Index (IRI) or Profitability Index (PI) or Cost-Benefit Ratio or Net Present Value Index (NPVI) or Return On Investment (ROI) [1], [6], [11]- [13], [19]- [22].
The economic indicators are actually computed with deterministic and/or stochastic methods [6].The deterministic techniques are used by [1], [6], [11]- [12], [20] in their studies.They return single values and do not say about the random aspect of these indicators, that is, the probability of getting other values than these single ones.Due to this limitation of deterministic economic evaluation models, these last decades investors rely heavily on probabilistic models through statistical distributions and simulations.With the use of statistical distributions and simulations, the economic evaluations will say more about the risks and uncertainties of the project [14].For instance, for a given oil and gas field/well development project, the oil company gets fund from fund providers/banks with a preset interest/discount rate.As rule of thumb if the project Internal Rate of Return (IRR) is less than the discount rate proposed by fund providers (or the inflation rate), the project will be abandoned.With a deterministic economic model, the project may be unrighteously abandoned whereas a stochastic model would show the likelihood that the IRR be greater than the single value got from the deterministic model and therefore gives the confidence and chance to company to invest in that project.[13], [17], [19], [21] have relied on stochastic approaches (Monte Carlo Simulation) for the economic evaluation made in their studies with the commercial software such as Cristall Ball Package (Oracle crystal ball) and Risk Excel add-in.
The application of stochastic model for oil and gas filed/well economic evaluation requires the knowledge of the probability distribution of some of all the model inputs.The search for model inputs probability distributions in [14] research work shows how courteous and hard this task is.Our study aims to propose a technique, called "Central Limit Theorem-based Stochastic Economic Evaluation (CLT-SEE) Model'' for oil field or well development.This model is a detailed stochastic technique and not designed for only oil and gas upstream projects but for the economic evaluation of any project with a ''sufficient'' lifespan.It eases projects NPV, IRR and PI probability distributions determination and provides single values, P10, P50 and P90 for these three project economic indicators.A case study will be performed for a Nigeria's Niger Delta onshore development oil well.

II. CLT-SEE MODEL ECONOMIC INDICATORS AND NPV COMPUTATION STRUCTURE FOR OIL AND GAS PROJECTS
Based on the requirements of CLT-SEE Model root which is the Central Limit Theorem (CLT), only three project economic indicators can be gotten through that model: • The project Net Present Value (NPV); • The project Internal Rate of Return (IRR); • The project Profitability Index (PI).

A. Net Present Value
The understanding of some key economic indexes will help in presenting the net present value.These indexes are net cash flow (NCF) and the discounted cash flow (DCF).
The Net Cash Flow (NCF) of a project at the end of the year n (n ≥0) is the cash outflow (or costs) of the project withdrawn from it cash inflow (or revenue) [2] and [8].The formula of net cash flow calculation is the one of (1).NCF of an oil and gas project is a function of oil and gas prices, total oil and gas production, development expenditure, operating expenditure, abandonment expenditure, real discount rate and government taxes among other factors [21].
The discounted cash flow is a technique which translates the time value of money by discounting the future cash flow to a present value reference [8].The discounted cash flow at the end of the year n is given with (2).
where n is the number of the year, D the discount rate and  ! the net cash flow at the year n.
The choice of a discount rate to be used to calculate the net present value is a matter of corporate policy.Some companies prefer a fixed discount rate over the project lifetime while other companies use a declining discount rate.However, there are some methods which can be used to determine the appropriate discount rate for a project, such as the weighted average cost of capital (WACC) method, which is one of the most commonly used methods [8].
According to [8], the net present value is the most significant economic indicator of petroleum projects evaluation that says about the profitability of projects and other indicators relative to projects viability is linked to it.It makes a comparison between the value of a dollar today and its value in the future, putting into consideration inflation and returns [16].A project net present value (NPV) is the algebraic or cumulative sum of discounted annual net cash flows associated with the project over its lifespan [8], [12], [14], [21].It is computed from (3).
where N is the project lifespan, n is the numbers of the years and  !discounted cash flow at the year n.The use of NPV indicator is based on decision rule that a project with positive NPV should be undertaken while the one with negative NPV should be rejected because it is not profitable [21].The higher the net present value of all cash flows during the life cycle of a well, the more profitable the well is [12].

B. Internal Rate of Return
The internal rate of return (IRR) of a project is the value of the discount rate which equates its NPV to zero [8], [21].In other word, the internal rate of return is the interest rate at which the net present value of all the cash flows (both positive and negative) from a project or investment equals zero [9].It is given by (4).
Fig. 1 shows the graphical way of determining the internal rate of return.It represents the discount rate for which the net present value curve of the project crosses the x-axis.If the IRR is greater than the weighted average cost of capital then the NPV is positive, and if the IRR is less than the weighted average cost of capital then the NPV is negative [8].
The higher the PI of the project the greater the profit from the implementation of that project.

D. NPV Computation Structure for Oil and Gas Projects
As stated above the NPV computation relies on the net cash flows of the project over its exploitation years.The structure of cash flow depends upon the type of the contract [18].Our study will focus on the structure of a Production Sharing Contract (PSC) since the case study is carried out for a well under a Nigerian PSC.The features of a PSC involved in the net cash flows calculation are shown by Fig. 2.
The resources and required information to build cash inflow and cash outflow and therefore the net cash flow of petroleum projects is summarized in Fig. 3.
The cash inflow(revenue) arises from petroleum production sales (oil, gas or condensate) in addition to other activities such as money received from asset sales or interest on the provisions (abandonment, capex recovery and others).
The cash yearly outflows of a petroleum project are the algebraic sum of different costs.The structure of field costs is shown in Fig. 4.This figure highlights the information related to Algerian petroleum operations cost and the average contribution of these costs to the overall costs.For the Nigerian fiscal regime, the abandonment cost is not part of the CAPEX.
Based on the production sharing contract structure we propose the flowchart of Fig. 5 for the contractor and the IOC net cash flows computation.Table I and Table II summarize the input parameters and the formulae to be used for that purpose.Apart from the oil production and cost, the other features of Table I belong for the fiscal regime of the petroleum activities host country.

A. Materials
The application of the approach proposed in this study, CLT-SEE model, to oil and gas filed or well projects, requires economic data (capex, opex, abandonment provision, inflation rate for prices and costs, project discount rate, oil and gas prices), technical data (production profile) and fiscal regime data (royalty rate, profit split rates, taxes structure).

B. Methods
The essence of our approach, called "Central Limit Theorem-based Stochastic Economic Evaluation (CLT-SEE)'' model is an application of Monte Carlo Simulation to the Discounted Cash Flow (DCF) method for projects economic evaluation.DCF method is widely used in the literature for oil and gas upstream projects economic evaluation as noticed in the studies carried out by [1], [6], [11]- [13], [19]- [22].CLT-SEE model sets a step-by-step procedure for determining three projects economic indicators as well as P10, P50 and P90 thereof.These economic indicators are NPV, IRR and PI.
The model is a two-step procedure: (1) computation of projects deterministic NPV, IRR and PI; (2) computation of P10, P50 and P90 of projects NPV, IRR and PI.

1) Computation of Projects Deterministic NPV, IRR and PI
At this stage, the project NPV and PI will be calculated using the formulae of (3) and (5) as well as those of Table I.The IRR must also be determined using the graphical technique explained in section II.It consists of plotting the NPV as function of discount rate and identifying the abscise value of the cross point of that curve with the x-axis.
Through this process single values of NPV, IRR and PI are got and will be used at the next stage.
2) Determination of P10, P50 and P90 of Projects NPV, IRR and PI Since single values of the concerned economic indicators are computed, it remains the determination of their P10, P50 and P90.They will be gotten with Monte Carlo Simulations.Since Monte Carlo Simulations is performed on well-defined random variables, we must first determine the probability distributions of concerned random variables.Frequency and cumulative density function (CDF) analyses coupled with a statistical test of distribution (Chi-square test for instance) are used for probability distribution determination [5].Frequency and CDF analyses consist respectively of plotting the empirical frequency and CDF and checking which theoretical probability density function (PDF) and CDF comes closest the empirical one.
In case of CLT-SEE model, the P10, P50 and P90 computation of the three economic indicators requires the knowledge of the project NPV probability distributions.The Central Limit Theorem (CLT) comes and eases the determination of NPV probability distribution function.The reasons and the conditions of that significant fact are developed as follows: i

. Central Limit Theorem and Conditions of Application
The Central Limit Theorem (CLT) states that for a huge number of independent random variables  & ,  9 , … ,  !identically distributed under a probability law of average μ and standard deviation , the random variable  = = : " ': # '⋯': !!follows approximatively the normal (Gauss) law of mean μ and variance < # !, even though the variable X does not follow the normal law [4].That is, A random variable  is said to follow a normal (Gauss) law of mean µ and variance  9 and is noted ~ (µ ,  9 ), when its probability distribution function (PDF) is defined by ( 6); its cumulative distribution function (CDF) is given by ( 7) [4].
CLT is widely used in mathematics and statistics for statistical variables probability distributions determination when the variable of interest is the average of another given variable.Its application requires huge-size samples, but mathematicians have proved that CLT can be used for samples of size of 30 or more or close to 30. ii

. Project NPV Probability Distribution
As shown by (3), a project NPV is given by with  !=  *  ! .As a result, CLT can be applied for NPV probability distribution determination since NPV is the average of the statistical variable  =  * , where DCF is the discounted cash flow.Due to the random aspect of some oil and gas projects DCF inputs such as production rate, oil price, inflation rate, etc.,  =  *  can be considered as a random variable.Therefore, According to CLT-SEE model, when an oil and gas field project life span N is 20 years or more, the NPV follows the normal (Gauss) law of mean  & and variance " " !# .That is, ,, with  ( ' the empirical variance of the sample ( & ,  ' , … ,  ! ).A simple mathematical transformation leads to (8).
The empirical variance of a random variable X sample  & ,  ' , … ,  ! is defined by (9).It is an unbiased estimator of the population variance.
where  & , the sample average.

iii. P10, P50 and P90 Computation
As highlighted above, the three economic indicators P10, P50 and P50 will derive from the one of NPV.Then, NPV P10, P50 and P90 must be calculated first before the others.The procedure of computing them is as follows.

iv. P10(NPV), P50(PNV) and P90(PNV) Computation
For a random variable X, P10, P50 and P90 are respectively defined as the value for which any realization of X has 10 percent, 50 percent and 90 percent of chance to be greater than.They are got by solving ( 10)- (12).
( ≥ 10) = 0.5 (11) ( ≥ 10) = 0.9 [5] have pointed out that the advantage of computing P10, P50 and P90 is that they give the expected minimum values of the parameter of interest with respectively low, moderate and high chances.
The following is the algorithm for computing NPV P10, P50 and P50 [5].Step 2: Determine 90th, 50th and 10th percentile of the generated G.
Step 2: Plot P10(NPV) curve as function of discount rate using S and SP10 items.Let's note this curve CP10(NPV).
Step 3: Plot P50(NPV) curve as function of discount rate using S and SP50 items.Let's note this curve CP50(NPV).
Step 4: Plot P90(NPV) curve as function of discount rate using S and SP90 items.Let's note this curve CP90(NPV).

IV. CASE STUDY ON A NIGERIA'S NIGER DELTA ONSHORE OIL WELL
The case study is performed on a Nigerian onshore oil filed well.The NPV computation has been carried out with Microsoft Excel and the stochastic indicators computation with Jupyther Notebook using Python programming language.
The field was discovered in 2015 and is under the Nigerian Petroleum Mining License (PML).The well is drilled in 2022 and is planned to start producing in 2023.The well performance analysis has proved that the well will produce over 25 years, from 2023 to 2047.
The well economic information provided are as follows.
• The well capital expenditure (Capex) breakdown is MM$ 28.6 for the exploration expenditure (Expex) and MM$ 43.38 for the development expenditure (Devex).The overall Capex is MM$ 71.98.• The operating cost will be calculated with an Opex per barrel model set at US$ 3/STBO the first production year.• The cost inflation rate is set at 2%.• The abandonment cost to development Capex ratio is set at 10% which is discounted at 12% discount rate.• The costs will be recovered with a depreciation of five years and an uplift of 15%.• The oil price used is US$ 70 per barrel.
• The operation cost, the abandonment provision as well as the well production profile are those of table III.• The technical information required for the identification of the fiscal regime items are the following.• The filed will produce an overall amount of crude oil less than 500 MMSTBO (table III).• The filed production range will be 5,765 to 38,609 bopd over its life span.on the above and according to [15] and its summary made by PwC in August 2021, the fiscal regime items used for the case study are as follows.
• The royalty rate is 5% for the first 5,000 bopd, 7.5% for the next 5,000 bopd and 15% for the remaining of the production over 10,000 bopd.• The additional royalty rate got by linear interpolation is 7% since the oil price used is US$ 70.• The National Hydrocarbon Tax (NHT) rate is 15 %.• The Investment Tax Credit (ITC) is US$ 7 per barrel.
• The Company Income Tax (CIT) rate is 12%.• The Cost Price Ratio limit (CPR) is 65%.
• The contractor take is 95% up to and including 50MMSTO cumulative production, 90% for over 50MMSTO and up to including 100MMSTO cumulative production and 85% for over 100MMSTO and up to including 350MMSTO cumulative production.

V. RESULTS AND DISCUSSION
A. Well Project Deterministic NPV, IRR and PI On the basis of the approach defined by CLT-SEE model for deterministic petroleum project NPV calculation and the economic data, technical data as well as fiscal regime data, the computations made for the well NPV determination are summarized in Table IV and Table V.The IRR is determined through graphical method.The PI is computed in accordance with (5).Fig. 6 shows the NPV trend and the identified project IRR.
As a result, we noticed that: • the project NPV is MM$ 84.112; • the project IRR is 24.5%; and • the project PI is 1.169.
These results show that the well development project is viable since the NPV is positive.The IRR of 24.5% highlights the fact that a discount rate greater than 24.5% leads to a negative NPV and therefore the project abandonment.Since the actual discount rate in the petroleum upstream is around 12%, the project has the chance to be implemented by the company.The PI of 1.169 shows that at the discount rate of 12% the company will make a profit of the order of 1.619 times its investment.Therefore, the NPV has 10% of chance to be higher than MM$ 96.4,50% of chance to be higher than MM$ 84.16 and 90% of chance to be more than MM$ 71.89.On the basis of that the company has much more chance to earn MM$ 71.89 implementing the project.P10, P50 and P90 of the project IRR are determined on the basis of the graphical method set by CLT-SEE model in the methodology section.The project P10(IRR), P50(IRR) and P90(IRR) identified are shown in Fig. 7.They are respectively 27%, 24.75% and 22%.We can notice that the project is economically flexible since the likelihood that its IRR be higher than 27%, 24.75% and 22% are respectively 0.1, 0.5 and 0.9.This offer much more chance and confidence to the company to get fund with discount and interest rates more than 12% if there is any opportunity.
The project P10(PI), P50(PI) and P90(PI) are respectively 1.34, 1.17 and 1. Though, the company has 50% of change of earning 1.17 times its investment and 90% of chance of earning exactly what it will invest in the project.

VI. CONCLUSION AND RECOMMENDATION
Oil and natural gas production has been highly contributing to world economy and is some countries' economy root.After the discovery of a new oil/gas field, the operator has to decide whether or not to develop that field.Making such decisions is always a very complicated endeavor which relies upon the economic evaluation of potential oil/gas fields development when they will be discovered and of the proven oil/gas reserves.The economic indicators are actually computed with deterministic and/or stochastic methods.Due to this limitation of deterministic economic evaluation models, investors rely heavily on probabilistic models through statistical distributions and simulations.The application of stochastic model for oil and gas filed/well economic evaluation requires the knowledge of the probability distribution of the model inputs.

Fig. 3 .
Fig. 3. Resources of necessary information to build cash flow of petroleum projects [8].

Fig. 6 .
Fig. 6.Well project IRR determination.B. P10, P50 and P90 of Well project NPV, IRR and PI According to CLT-SEE model, the well NPV follows a normal distribution of average MM$ 84.112 and standard deviation MM$ 9.537.100,000 random numbers of that distribution have been generated and P10, P50 and P90 are determined.The results show that the well project P10(NPV), P50(NPV) and P90(NPV) are respectively MM$ 96.4,MM$ 84.16 and MM$ 71.89.

Fig. 7 .
Fig. 7. Well project P10(IRR), P50(IRR) and P90(IRR) determination.The search for model inputs probability distributions is costeous in terms of software, data and conditions to be satisfied.The technique proposes by the current study, called "Central Limit Theorem-Based Stochastic Economic Evaluation (CLT-SEE) Model'' for oil field or well development economic evaluation eases projects NPV, IRR and PI probability distributions determination and provides single values, P10, P50 and P90 thereof.The results of the case study carried out on a Nigeria's Niger Delta onshore oil well show that the use of CLT-SEE model for oil fields or wells projects economic evaluation offers much more chance and confidence to oil companies to decide righteously in fields and wells development projects.Therefore, we recommend the use of CLT-SEE model for economic evaluation of petroleum upstream projects with longer life span when a large amount of DCF inputs is not available.

TABLE I :
INPUTS OF NET CASH FLOW COMPUTATION MODEL

TABLE II :
INPUTS OF NET CASH FLOW COMPUTATION MODEL

TABLE III :
PRODUCTION PROFILE, FILED CUMULATIVE PRODUCTION, OPEX AND ABANDONMENT PROVISION OF THE PROJECT