Richard D. Corrie, SPE — Inepetrol, S.A.

Abstract

The preliminary evaluation of development well counts is a crucial element in the exploitation of oil fields, aiming to find a balance between maximizing recovery and adhering to economic efficiency constraints. In 2001, SPE 71431 presented an analytical solution to determine the Optimum Number of Wells (Wo) based on firsthand data available at the time, i.e. an estimate of the field's ultimate recovery, the initial production rate per well, the development cost per well and the price of crude oil. Now, twenty-five years later, this paper reviews the original analytical model to assess its relevance in the context of modern industry challenges, including volatile energy markets and the complexities of infill drilling in mature assets.

While the original equation provides a robust mathematical peak for Net Present Value (NPV), this review demonstrates that the sensitivity of this peak to economic variables necessitates a more strategic approach to field development. It introduces the concept of the Near-Optimum Number of Wells (Wno), a sound development strategy that operates within a financial safety zone by applying marginal economics.

Our findings suggest that by selecting a slightly more conservative number of wells (Wno) than the optimum number of wells (Wo), operators can significantly preserve capital and reduce exposure to geological and oil-price risks without jeopardizing the long-term value of the project. Supported by contemporary literature on optimization under uncertainty and case studies in infill drilling, this updated methodology offers decision-makers a framework that prioritizes capital efficiency and operational resilience in an increasingly unpredictable energy scene. An illustrative example is provided to show that the Wno criterion results in a significantly reduced well count, leading to wider well spacing and a more economically justifiable investment.

This review includes three searches assisted by artificial intelligence as appendices, which present a list of 25 papers that have referenced SPE 71431, detailing the reasons and context of their citations, as well as the current state of the art regarding optimal well spacing worldwide.

Keywords: Well Spacing Optimization, Net Present Value (NPV), Infill Drilling, Capital Risk Management, Economic Resilience, Field Development Strategy

Introduction

Following the discovery and appraisal of an oilfield, the critical next step involves an economic viability analysis for the field's development. A key factor required at this stage is an estimation of the Optimum Number of Wells (Wo) necessary to maximize the Net Present Value (NPV) of the project.

SPE 71431 presented a simplified analytical economic solution to address this issue, providing a quick-screening tool that yields Wo and, in turn, the optimal well spacing (OWS). The model assumes exponential production decline and, fundamentally, that cumulative oil production (Np), i.e. primary EUR, is independent of well spacing.

This paper reviews SPE 71431 by proposing a change in focus — from targeting the optimum number of wells (Wo) to estimating, by numerical solution, the near-optimum number of wells (Wno) under optimal conditions by applying the principle of marginal economics. This approach provides a financially more robust development strategy than seeking the maximum NPV for the project.

1. SPE 71431 Equations

SPE 71431 presented an equation to estimate the Net Present Value of the project as a function of the number of wells — Corrie (2001).

The optimum number of wells (Wo) was determined by setting the first derivative of with respect to equal to zero, i.e. :

Where:

• — cost per well after income-tax effect, $/well
• — discount rate, fraction, 1/year
• — cumulative oil production during project life (primary Estimated Ultimate Recovery, EUR), barrels
• — NPV as a function of the number of wells , after income-tax effect, $
• — initial oil production rate per well, barrels/day per well
• — crude-oil price netted back to wellhead, $/barrel
• — number of wells, dimensionless
• — optimum number of wells, wells
• — present-value cost of other investments independent of , after income-tax effect, $

The graph below shows the optimum number of wells Wo at the peak of the curve, using the example provided in SPE 71431.

From this equation, the optimal well spacing (OWS) and the maximum economic return were derived:

Where:

• — productive area, acres
• — maximum economic return, $
• — optimal well spacing, acres/well

Although the equation for Wo is independent of , the economic feasibility of the oilfield development expressed by depends on the value of .

2. Key Assumptions

The key assumptions in SPE 71431 were:

• The reservoir is homogeneous and behaves as a tank model — zero-dimensional.
• Np is fixed, equal to EUR, independent of the number of wells and limited to primary recovery.
• Uniform performance — all wells have the same initial oil production rate and decline exponentially at the same rate throughout project life. (The exponential decline rate is the most conservative and simplest equation of the decline-curve family — Arps, 1956.)
• Production decline rate D is a function of , and :

• Investments C and Z are incurred at year zero.
• Oil price V is netted back to the wellhead, free of operating costs.
• Taxation — , and are expressed after the income-tax effect.

3. Shifting Focus

This review proposes to shift the focus from the maximum economic return derived from the Optimum Number of Wells (Wo) to the more practically robust concept of the Near-Optimum Number of Wells (Wno). This approach is a clear application of marginal economics — the law of diminishing returns.

Wno is defined as the largest integer number of wells such that the incremental NPV for the last well drilled is equal to or greater than the discounted capital cost of that well () — this being the point of maximum capital efficiency. Wno is derived by numerical methods using any value of Wo as a seed.

The near-optimum NPV is then calculated:

This principle is more conservative and economically defensible, providing an economic threshold compared to Wo.

4. Illustrative Example

Using the same input parameters as presented in SPE 71431, the difference between Wo and Wno can be demonstrated. In this example, Wno was derived by the numerical method provided by What-If Analysis in MS Excel, using Wo as a seed.

Given:

• bbl
• bbl/d
• $/bbl
• fraction, 1/year (10% p.a.)
• $
• $
• acres

Calculation results:

Optimal well criteria	Wo	Wno
Number of wells	18	11
Well spacing, acres/well	111	182
Production decline rate, %	25	16
Capex, M$	73	56
NPV(10%), M$	94	88
Investment efficiency	1.29	1.56
Internal Rate of Return, %	79	67

The Wno criterion suggests a significantly lower number of wells, leading to much larger well spacing and higher capital efficiency — i.e. a slightly lower NPV with significantly lower capital exposure.

The graph below shows NPV(W) versus the number of wells , marking Wo = 18 wells and Wno = 11 wells.

The following graph — Diminishing Returns: incremental NPV versus well number — shows the incremental NPV of each additional well. It reaches the cost of a well ( M$) at Wno = 11, and falls to zero at Wo = 18.

5. Sensitivity Analysis

The Wo equation — and by extension Wno — is sensitive to the economic and reservoir parameters, as summarized below:

• Oil price — higher oil prices justify drilling more wells.
• Well cost — higher well costs mean fewer wells are needed to drain the field profitably.
• EUR — a larger justifies drilling more wells.
• Initial well production rate — a higher means fewer wells are needed to drain the field.
• Production decline rate D — higher and induce a higher decline rate, while a higher induces a lower decline rate.
• Discount rate i — affects NPV non-linearly, taking Wo in the example from 18 wells at 10% p.a. to a maximum of 23 wells at approximately 50% p.a.

6. Critiques and Limitations

The most significant critiques from the SPE reservoir-engineering community highlight that a simple equation like Wo should not be used to solve such complex problems, because it lacks geology, physics and reservoir simulation.

It is important to reiterate that the Wo equation embodies an economic model that provides speed and economic clarity that is essential during the early stage of oilfield-development decision-making. It should not be used in tight or unconventional reservoirs, which are governed by transient flow and where a hyperbolic production-decline model — instead of an exponential one — may be more appropriate.

7. Concluding Remarks

The equation to estimate the optimum number of wells (Wo), along with its revised focus on estimating the maximum capital-efficiency near-optimum number of wells (Wno) by numerical method using Wo as a seed, offers a swift economic insight that is necessary during the initial decision-making phase of an oilfield-development project developed by vertical or horizontal wells — and should not be used in tight or unconventional reservoirs. It acts as the analytical foundation for studies that seek to optimize profitability in the initial planning phase (greenfields) or in infill-drilling projects.

Wo treats the NPV curve as a target, whereas Wno treats it as a limit. The equation keeps a volumetric balance between the rate of reservoir production (), the reservoir production decline rate () and a fixed EUR equal to .

It is worth noting that not every economic decision necessitates a complex analysis. No matter how complex the model, it is still a simplification of reality. Often, a simple model — or a single equation — can adequately address the issue at hand. The equation presented in SPE 71431 is grounded in Muskat's concept of economic ultimate recovery, which acknowledges that there exists an optimum number of wells between two extremes: too few wells, which results in lower costs but delayed revenue; and too many wells, which generates quicker revenue but incurs excessive costs — Muskat (1949).

The appendices that follow were assisted by artificial intelligence. Appendix A lists 25 papers (via Google Scholar) that have cited SPE 71431. Appendix B summarizes a Google Gemini search on why and in what context SPE 71431 was cited by those 25 papers. Appendix C summarizes a Gemini search on the state of the art of optimum well spacing.

8. References

• Arps, J.J.: "Estimation of Primary Oil Reserves," Petroleum Conference — Economics and Valuation, Dallas, Tex., Mar. 1956.
• Corrie, R.D.: SPE 71431, "An Analytical Solution to Estimate the Optimum Number of Development Wells to Achieve Maximum Economic Return," SPE ATCE, Sep. 2001.
• Muskat, M.: "Physical Principles of Oil Production," McGraw-Hill Book Company, Inc. (1949) 810–904.

9. SI Metric Conversion Factors

• acre × 4.046 873 × 10³ = m²
• barrel × 1.589 873 × 10⁻¹ = m³
• ft × 3.048 × 10⁻¹ = m

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Appendix A — 25 Papers That Cited SPE 71431

(Search provided by Google Scholar.)

1. Assisted process for design optimization of oil exploitation strategy — A.T.F.S. Gaspar, C.E.A.G. Barreto, D.J. Schiozer. Journal of Petroleum Science & Engineering, 2016 (Elsevier).
2. Number of development wells: a decision under uncertainty — M.H. Al-Harthy. The Engineering Economist, 2010 (Taylor & Francis).
3. Application of assisted optimization to aid oil exploitation strategy selection for offshore fields — A.T. Gaspar, C.E. Barreto, E.O. Muñoz Mazo et al. SPE Latin America & Caribbean, 2014 (onepetro.org).
4. The optimal parameters for oil field development — M. Khasanov, O. Ushmaev, S. Nekhaev et al. SPE Russian Oil & Gas, 2012 (onepetro.org).
5. Selection of optimal parameters for an oil field development system — М.М. Хасанов, О.С. Ушмаев, С.А. Нехаев et al. Нефтяное хозяйство, 2012 (elibrary.ru).
6. Methods to determine drainage area in shale formations produced by stimulated horizontal wells using reservoir simulation modelling — W. Szott, K. Miłek. Nafta-Gaz, 2015 (academia.edu).
7. Optimal parameters for oil field development — М.М. Хасанов, О.С. Ушмаев, С.А. Нехаев et al., 2012 (researchgate.net).
8. Estimating optimum well spacing in a Middle East onshore oil field using a genetic-algorithm-optimization approach — S.A. Tabatabaei Nejad, A.V. Aleagha et al. SPE Middle East Oil & Gas, 2007 (onepetro.org).
9. A stochastic approach to well spacing optimization of oil reservoirs — U.M.P. John, S. Ibukun, J. Pius, O. Kayode. SPE Nigeria Annual, 2011 (onepetro.org).
10. Non-linear programming for well spacing optimization of oil reservoirs — U.M. John, M.O. Onyekonwu. SPE Nigeria Annual International, 2010 (onepetro.org).
11. A model to predict the performance of drilling operation using SVM approach in one of the Iranian oilfields — M. Kakoli, A. Ebrahimabadi, B. Mirshekari et al. Int. J. Pet. & Petrochemical Eng., 2018 (academia.edu).
12. Case study of the structure of the process for production strategy selection — A. Gaspar, E.O.M. Mazo, D.J. Schiozer. International Journal of …, 2011 (researchgate.net).
13. Uso de metamodelos na seleção de estratégias de produção e avaliação econômica de campos de petróleo — G.D. Avansi. Dissertation (Master's degree), 2008 (repositorio.unicamp.br).
14. A methodology for reservoir management via production strategy optimization — L. Nakajima, R.F. Martini, D.J. Schiozer, 2004 (osti.gov).
15. Increased oil production by unconventional wells — short-radius, horizontal, and multilateral in the Hassi Messaoud: thick multilayer Cambrian formation, Algeria — B. Youcef, D. Tiab. SPE Unconventional Resources / Gas, 2008 (onepetro.org).
16. Speeding up oil recovery by infill drilling prior to considering implementation of a full-field WAG process — El Furrial field, Venezuela — L.G. Leonardo, A. Rivas, J. Bello et al. SPE Latin America & Caribbean, 2007 (onepetro.org).
17. Análise estratégia de locação de poços de petróleo utilizando conceitos de mapa de qualidade — N. Battisti (researchgate.net).
18. Mathematical modeling of economic indices for oil field development — Y.K. Mizyakin, V.A. Mizyakina, N.A. Petrov. Journal of Applied and Industrial Mathematics, 2016 (Springer).
19. Well spacing index, a new approach for planning the oilfield development — L. González. SPE Latin America & Caribbean, 2015 (onepetro.org).
20. Performance evaluation of non-conventional wells in Hassi-Messaoud field, Algeria — B. Youcef, B. Kamel. Abu Dhabi International Petroleum Exhibition, 2008 (onepetro.org).
21. Mathematical modeling of economic indicators for oil field development — Ю.К. Мизякин, В.А. Мизякина, Н.А. Петров. Сибирский журнал …, 2016 (mathnet.ru).
22. تعيين تعداد بهينه چاه‌ها در يکي از ميادين نفتي ايران — Determination of the optimal number of wells in one of Iran's oil fields using Monte Carlo models and genetic algorithms — Keramati Moezabad Mohammad, Ghasem Mohammad (sid.ir).
23. Determination of optimum well number in one of the Iranian oil reservoirs using Monte Carlo and genetic algorithm models — M. Keramati Moezabad et al. Journal of …, 2015 (Research Institute of Petroleum Industry).
24. Distancia óptima para la perforación de pozos en campo Boscán — R.P. González, 2015 (Universidad del Zulia).
25. Seleção da estratégia de explotação para um campo petrolífero sob restrições operacionais e incertezas geológicas e econômicas — A. Ravagnani, E.O.M. Mazo, D.J. Schiozer. Rio Oil & Gas Expo and Conference, Rio de Janeiro, 2010.

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Appendix B — Why and in What Context SPE 71431 Was Cited

(Search provided by Google Gemini.)

It is genuinely interesting to analyze the impact of the original 2001 work (SPE 71431) on current technical literature. After reviewing the report and the list of the 25 citing papers, a clear set of patterns emerges for why and in what contexts the methodology remains relevant to the industry today. The work serves as the analytical foundation for studies seeking to optimize profitability in the initial planning phase (greenfield) or in infill-drilling projects.

1. Citation contexts and applications

The 25 reports fall into three major categories based on how they use the research (referencing the numbering in Appendix A):

A. Exploitation strategy optimization (Cases 1, 3, 12, 13, 14, 25). Works led by researchers such as D.J. Schiozer and A. Gaspar use the analytical solution as the starting point or base model. They develop complex systems (such as ATFS or metamodels) for offshore fields with high uncertainty, using the equation to quickly establish the optimum number of wells before moving on to expensive numerical simulations.

B. Application of advanced and stochastic algorithms (Cases 8, 9, 10, 22, 23). Researchers in Iran and Nigeria (such as Tabatabaei Nejad or M. Keramati) adopted the model to integrate it with AI and statistical tools, using the formula as the objective function within genetic algorithms or Monte Carlo simulations. The analytical solution provides the mathematical rigor needed to feed these algorithms and search for the maximum NPV under risk scenarios.

C. Pure economic and mathematical modeling (Cases 4, 5, 7, 18, 21). Works by Russian authors such as M. Khasanov and Y.K. Mizyakin focus on the mathematical formality of petroleum economics. They validate the derivation of the differential equation to demonstrate how economic variables (oil price and interest rate) directly affect the geometry of field development. It serves as an authoritative reference on the Economic Ultimate Recovery concept proposed by Muskat.

2. Technical relevance matrix

Context of use	Key references (from the list)	Value provided by SPE 71431
Numerical simulation	6, 17, 25	Provides the base case to compare and validate simulator results.
Unconventional wells	15, 20	Cited to adapt spacing logic to horizontal and multilateral wells.
Local case studies	11, 16, 19, 24	Direct application of the formula in specific fields (Boscán, El Furrial, Hassi-Messaoud).

3. Why is the report still cited after 25 years?

Unlike models that depend exclusively on simulation software ("black boxes"), the report stands out for three reasons that authors value:

• Independence from graphics. The analytical solution can be implemented directly in spreadsheets or programming code without the need for visual interpretation — a direct solution compared to Muskat's traditional graphical method.
• Inclusion of exponential decline. Many earlier models (such as Tokunaga and Hise) assumed constant rates. The integration of actual decline is what allows the method to be used today in mature fields (infill drilling).
• Transparency of variables. By using clear variables (Q, V, C, i), it is easy for other researchers to perform sensitivity analyses — something clearly observed in computer-aided optimization works.

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Appendix C — State of the Art of Optimum Well Spacing

(Search provided by Google Gemini.)

Determining optimum well spacing is one of the most critical aspects of oil-field development planning. The historical evolution of this discipline can be broadly categorized into three distinct phases.

1. The classical physical and economic foundation

Early research — most notably by Craze and Buckley (1945) and Muskat (1949) — focused on the distinction between physical ultimate recovery and economic ultimate recovery.

• Physical standpoint. The primary focus was defining a minimum number of wells () required for pressure maintenance and maximum reservoir drainage.
• Economic standpoint. Muskat introduced the concept of the Economic Optimum, acknowledging that beyond a certain density the cost of additional wells outweighs the incremental recovery, thereby negatively impacting the project's NPV.

2. The analytical shift (1960s – 2000s)

The industry recognized that graphical methods for determining optimum well spacing (plotting NPV vs. well spacing) were inefficient for rapid sensitivity analysis. Tokunaga and Hise (1966) pioneered the move toward direct mathematical solutions. However, these initial analytical models were constrained by the assumption of constant production rates (no decline).

SPE 71431 (Corrie, 2001) addressed this gap by integrating the exponential production-decline model directly into the economic NPV function. This allowed for a more realistic estimation of reserves () and capital return, providing an explicit equation for Wo that accounted for initial production rates (), time-dependent decline rates (), and discount factors related to project life ().

3. Modern optimization and risk management (2010s – present)

Contemporary research has shifted from finding a single "point optimum" to managing optimization under uncertainty. As reflected in the literature (e.g. Al-Harthy, 2010; Schiozer, 2016), the industry now uses:

• Stochastic modeling — Monte Carlo simulations and genetic algorithms to account for geological and economic volatility.
• Response surfaces — mapping the NPV function to demonstrate that the peak is often flat. This has led to the emergence of near-optimum well spacing, a strategic choice that prioritizes capital resilience and risk mitigation over the pursuit of a theoretical maximum.
• Infill-drilling contexts — modern approaches now account for well interference and the complexities of mature fields, where the optimum is frequently constrained by existing infrastructure and operational limits rather than pure reservoir theory.

Summary of the current state

Today, the state of the art is no longer restricted to calculating a single optimum; it is defined by resilient optimization. Modern decision-making leverages analytical foundations (like those established in SPE 71431) as the baseline, which are then refined through stochastic sensitivity analysis to identify a development plan that remains profitable across a wide range of future economic scenarios.