Production systems optimization methods for petroleum fields

Technological Transformation and Competitiveness in the Middle East, XXXX 1

Copyright © XXXX Inderscience Enterprises Ltd.

Production systems optimization methods for petroleum fields

Cheng Seong Khor

Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Chemical Engineering Programme, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, Malaysia E-mail: [email protected] E-mail: [email protected]

Ali Elkamel*

Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 E-mail: [email protected] *Corresponding author

Abstract: In this review, we survey the widespread use of optimisation or mathematical programming approaches in the upstream sector of the petroleum industry, specifically to problems in the area of (1) production systems design and operations, (2) lift gas and production rate allocation and (3) reservoir development, planning, management and optimisation. Early applications have adopted Linear Programming (LP) alongside heuristics-based methods, but the recent ongoing explosion in computing power and advances in optimisation, simulation and computational techniques have enabled the adoption of increasingly complex models. These formulations include non-linear programming and Mixed-Integer Linear (MILP) and Non-Linear (MINLP) programming models. Within these representations, various algorithms and approaches have been employed, for example, metaheuristics such as genetic algorithms to address non-smooth objective functions; techniques for simultaneous decision making in design, planning and scheduling and stochastic programming to handle uncertainty in reservoir information, with the ultimate aim of improving solution quality while reducing computational intensity.

Keywords: optimisation; mathematical programming; oil and gas fields; production optimisation; production systems design and operations; planning and scheduling.

2 C.S. Khor and A. Elkamel

1 Introduction

Oil and gas production from petroleum fields is a widely reported operations problem that is chiefly constrained by the reservoir conditions, flow characteristics of the pipeline network and capacity of the surface facilities. As a consequence, proper determination of the daily optimal operating conditions requires simultaneous consideration of the multitudinous complex interactions among the activities of the reservoir, the wells and the surface network facilities. Various approaches have been proposed in the literature on problems related to this area that broadly encompasses production systems design, operations and control. In general, these methods can be categorised as follows (Kosmidis et al., 2004; Vasantharajan et al., 2006):

• sensitivity analysis by employing simulation tools

• intuition-driven techniques and heuristic algorithms based on rules and learnings derived from past experience and analogies and

• formal optimisation or mathematical programming techniques.

This work emphasises the optimisation approach, which has found applications in virtually all aspects of the hydrocarbon industry. Review papers concerning the application of optimisation tools to problems in the upstream sector of petroleum Exploration and Production (E&P) has been surprisingly scant, considering the enormous importance of the E&P activities to the hydrocarbon enterprise and to the global energy systems and economy on a whole. Moreover, it is an almost well-known fact that the adoption of Linear Programming (LP) (arguably the most established and successful technology of the optimisation approaches) as a prominent operations research tool first found its calling within this industry (although this fact was more evident in the downstream sector of oil refineries) (Ranyard, 1988). To the best of our knowledge, reviews devoted to this scope have been provided previously by Dougherty (1972) and (to a lesser extent) Aronofsky (1983).

In this work, adopting the classification of the doctoral thesis by Wang (2003), whose excellent (although understandably inextensive) literature review greatly influences the overall framework of this paper, we specifically survey problems related to the following areas:

• production systems design and operations

• lift gas and production rate allocation and

• reservoir development, planning, management and optimisation.

The goal of this survey is to provide a unified view of the use of optimisation-based methods to the above problems that are often collectively referred to as ‘production optimisation’ in the petroleum industry (Beggs, 1991). The intention is not so much of providing a comprehensive review with details but rather to shed light on the major directions in which the area has developed and is poised for continued growth.

Production systems optimization methods for petroleum fields 3

2 Tools and techniques for production optimisation problems of petroleum fields

2.1 Intuition-based rules and heuristic algorithms approach

The literature is abound with publications that report the handling of the daily oil production optimisation problems by incorporating intuition-based rules and heuristic algorithms in computer software codes and tools (e.g. see Stackel and Brown, 1979; Wallace and van Spronsen, 1983; Weiss et al., 1990). Heuristic algorithms seek good feasible solutions to optimisation problems under circumstances in which the complexities of the problem and/or the limited time available for the solution generation are considered to be overriding factors than the computation of the exact solution itself (Rardin and Uzsoy, 2001). This is in contrast to employing more formal and systematic approaches such as mathematical optimisation techniques.

The heuristics are typically executed sequentially in well management routines that decompose a pipeline network into several levels (Mattax, 1990). Kosmidis et al. (2004) provide examples of the adoption of these heuristic-based routines; for instance, rules such as ‘close a well if it violates an upper bound of the Gas–Oil Ratio (GOR)’ that are implemented at the well level (with GOR defined as the ratio of gas-to-oil volumetric flow rate).

It is observable then that well management routines, while accounting for pipeline network constraints, are formulated in an ad hoc manner that do not systematically or holistically address the oil production maximisation problem. Major commercial reservoir simulators such as Schlumberger–GeoQuest’s ECLIPSE (Schlumberger, 2000) and Landmark’s (2001) VIP are reported to be based on similar heuristic rules that consider the gas-lift optimisation problem separately from the well rate allocation optimisation.

One of the most widely applied heuristic rules for allocation of gas to gas-lift wells is the ‘incremental GOR’ or IGOR, defined as the amount of gas required by a gas-lift well to produce an additional barrel of oil (Redden et al., 1974; Weiss et al., 1990). The concept states that all wells tied to a common manifold must operate at the same IGOR. This set of heuristic rules has been applied to the operations of Prudhoe Bay and Kuparuk River fields by Barnes et al. (1990) and Stoisits et al. (1992, 1994, 1999), respectively.

However, Kosmidis et al. (2004) stress that a significant drawback to the heuristics approaches is that the necessary condition is derived by analysing a pipeline network under the assumption that all of the wells are tied directly to a fixed-pressure separator. Thus, this practice do not explicitly account for the highly significant and complex non-linear interactions between the wells that share a common flow line, primarily the pressure gradients influence. In fact, this disadvantage underscores the need to adopt a more robust approach, in particular the multivariate optimisation techniques, which is the focus of this paper.

2.2 Optimisation/mathematical programming approach

Multivariable optimisation tools and techniques have been utilised in many aspects of the oil industry to support decisions involved in accomplishing tasks that primarily consist of:


• production systems and facilities design

• resource scheduling

• history matching of reservoir parameters

• well location and trajectory

• production parameter estimation and settings and

• optimisation of the displacement efficiency or recovery factor.

These optimisation initiatives typically involve offline-generated models, either by employing first principles or semi-empirically by using data acquired in the field.

Since the adoption of optimisation methods to develop oil and gas prospects began in the early 1950s, Aronofsky (1988, unpublished) remarks in retrospect that significant progress has been made over the two decades of 1960s–1980s although it is still at the stage in which considerable art and ingenuity is required to set up each modelling situation. The following summarises in general, the progress that has been made as gleaned from both the academic research literature and the industrial practice reported in the trade literature.

• 1950s–1960s: development of LP models in which the reservoir behaviour (i.e. the relationship between flow rates versus pressures) takes on a very simple form; yet, this approach is adequate for many problems and applications dominated by economic-driven optimisation models. Examples include Lee and Aronofsky (1958), Charnes and Cooper (1961), Attra et al. (1961) and Aronofsky and Williams (1962).

• 1970s: formulation of LP models combined with data obtained from numerical reservoir simulation for well development. This approach typically involves several stages as delineated in the following:

− construction of a reservoir simulator based on a preliminary estimate of the reservoir parameters for each block

− employment of an LP model and the preliminary reservoir simulator results to compute improved reservoir parameters that best match production history

− reemployment of the reservoir simulator to generate a set of influence functions describing how the well pressure changes with time when the flow rate abruptly changes from zero to a constant value, based on the improved reservoir parameters and

− construction of a new LP model incorporating the improved influence functions for economic optimisation.

Representative works include Bohannon (1970) and Devine and Lesso (1972).

• 1980s–1990s: use of discrete (integer) variables resulting in Mixed-Integer Programming (MIP) models for improved representation of the non-linear decision variables (DVs) by means of simplified representation of the reservoir parameters. Efforts along this line include Sullivan (1982), Haugland et al. (1988) and Iyer et al. (1998).

• 1990s–present: adoption of the systems approach leading to increasingly simplified yet adequately representative optimisation models, in which the nature of the simplification is to identify only the key DVs to model the


reservoir behaviour that matters, and to include economic optimisation aspects. The models are efficiently structured to enable an analytical solution to be obtained or at least a straightforward computational procedure.

A general conclusion that can be drawn is that the approaches presented have progressed from studies of mere research interest to viable methods for practical field implementations; yet still retain limitations that require informed careful use. In spite of these limitations, there is reason to believe that practical implementations have not been pursued vigorously as these techniques deserve. Fortunately, optimisation is a highly fertile and active area of research, both inside and outside of the petroleum industry, so we can be fairly and assuredly optimistic about the future progress and eventual widespread field applications of these techniques.

Perhaps the most important impact will be on the manner in which numerical reservoir simulation studies will be conducted. If there is a need and justification to develop optimisation models with integrated large-scale reservoir simulations, then consideration should primarily be given to the use of LP for history matching, generation of influence functions and economic optimisation since LP is by far, the most established and successful optimisation technology, with well-understood and mathematically proven properties, capable of handling models of several million constraints and variables (ILOG, accessed on 13 January 2007).

On the other hand, before getting engulfed into unnecessary large-scale simulation studies, it may be worthwhile to consider the balance needed between the complexity of the physical reservoir behaviour and the details for executing economic-based optimisation. It is likely that greater consideration for global production facilities would lead to increased justification in employing an overall systems approach, in which case, detailed reservoir descriptions can be greatly simplified.

3 Application of the optimisation approach to production systems design and operations

The mathematical model for the optimisation of the design (and subsequently, the operations) of an integrated oil production system covers the entire span from the drainage area, the wells, the wellhead assembly, to the surface facilities. The detailed optimisation model is composed of several component models, listed as follows in close correspondence to the graphical description of Figure1:

1 reservoir simulation of the well bore

2 well tubing strings for multiphase fluid flow in the pipelines from the well bores to the well heads, with consideration for artificial lift; two forms of modelling approach are typically adopted here, namely via

3 well choke valve

4 well flow lines from the wellheads to the well pad manifolds

5 well pad manifolds

6 surface flow line for the surface pipeline network systems

7 separator for the separation facilities and

8 business economics goal as the objective function of the optimisation model.


Figure 1 The simulation-based Well Model (WM) for an integrated oil production system


3.1 Nodal analysis/trial-and-error/what-if analyses approach

Traditionally and conventionally, the hydrocarbon industry has employed the simulation tool of the NODAL production systems analysis, originally developed by Schlumberger, to address the optimisation of production systems design and operevolved into such a common adoption over the years, close to a household name within the industry, to the extent that the approach is referred to simply as ‘nodal analysis’ nowadays (Beggs, 1991). Dating back to papers as early as 1954 that propose the use of this technique (Brown and Lea, 1985), nodal analysis generally refers to the systems approach for the optimisation of the production operations of oil and gas wells via a thorough evaluation of the complete well production system. It involves employing correlations to predict multiphase flow behaviour through pipes, well completions, restrictions and the reservoir in order to analyse the flow behaviour in the entire production (Brill, 1987). This is accomplished by repetitively varying the associated optimisation variables to simulate the underlying system.

Ultimately, nodal analysis determines the daily operating policy by forecasting the performance of the various elements that make up a completion and production system. This is executed with the objective of optimising the completion design to suit the reservoir deliverability, to identify restrictions or limitations present in the production system and subsequently, to determine any means of improving the production efficiency (Schlumberger Oilfield Glossary, 2006).

However, Kosmidis et al. (2004) point out that nodal analysis is limited only to oilfields with a small number of wells due to its trial-and-error nature. For instance, a typical execution of nodal analysis requires that by holding all other parameters fixed, a single variable is varied to inspect the value of this variable that produces the optimal value. Quite obviously for multiple DVs, such a procedure is bound to fail due to the large number of function evaluations required to cover the search region. This is precisely one of the principal reasons for the need of applying multivariate (or multivariable) optimisation techniques as advocated in this work.

3.2 Multivariate optimisation techniques

Table 1 provides a summary of the representative works on the application of the optimisation approach for production system design and operations of petroleum fields.

Table 1 Representative works on the application of the optimisation approach to production systems design and operations

Reference Model attribute, DV and constraint Optimisation procedure and solution Carroll and Horne (1992)

• Single-well system • DV: separator pressure,

tubing diameter

• NLP • Modified Newton’s method

(gradient (derivative)-search-based)

• Polytope method (function-value-based)

Fujii and Horne (1995)

• Multiwell system • Two-stage optimisation strategy:

pipeline network simulator as evaluation tool for values of control variables computed by optimisation algorithm

• NLP • Newton-type algorithm • Polytope method • Genetic algorithms (GA)


Table 1 Representative works on the application of the optimisation approach to production systems design and operations (continued)

Reference Model attribute, DV and constraint Optimisation procedure and solution

Palke and Horne (1997)

• Fully compositional model

• Single-well system

• DV: tubing diameter (caused objective function surface to become non-smooth/highly irregular); separator pressure; gas injection volume

• NLP

• Newton-type algorithm (failed for non-smooth objective function)

• Polytope method (depends on good starting point)

• GA (more robust but computationally intensive)

Wang et al. (2002a)

• Based on Fang and Lo (1996)

• Simultaneous optimisation of allocation of well production rates, gas-lift rates and well connections to surface pipeline systems

• Mixed-Integer Non-Linear Programming (MINLP)

• Separable programming

GA

Wang et al. (2002b)

Considers flow interactions among wells in gathering system via pipeline network modelled using tree-like structure

• MINLP

• Sequential/Successive Quadratic Programming (SQP) algorithm

Queipo et al. (2003)

• Integrated oil production systems at design and operational levels comprising reservoir, tubing, choke, separator and economic-based objective

• DV: oil flow rate; diameters of tubing, choke, an pipeline

• NLP

• SQP

• DIRECT global optimisation algorithm

Kosmidis et al. (2004)

• Single reservoir, two-well system

• Constraints: non-linear reservoir behaviour; multiphase flow in pipelines, surface capacity

• Uses binary variables to: impose adjacency condition for a Special Order Set (SOS); linearise non-linear constraints

• MINLP

• Proposes approximate MINLP model based upon: (1) degrees of freedom analysis, (2) well oil rate upper and lower bounds, (3) piecewise linear function approximation of well models

• Solves a sequence of MILP problems

3.3 Commercial simulators with multivariable optimisation capabilities

It has become a standard industrial practice to equip commercial numerical reservoir simulators with multivariate optimisation engines. The following non-exhaustive list serves to highlight some of these simulators:

• ECLIPSE by Schlumberger GeoQuest (2000)

• NETOPT® by Simulation Sciences (SimSci, 1999)

• STARS (Steam, Thermal and Advanced Processes Reservoir Simulator) by Computer Modelling Group (CMG) (accessed on 22 November 2006), popularly referred to as CMG–STARS and

• Codeon Multiflo Simulator (accessed on 17 January 2007), which has been previously used by BP, the British oil multinational.


4 Application of the optimisation approach to rate allocation problems

4.1 Optimisation of the gas-lift allocation problems

A gas-lift optimisation problem is typically posed as the maximisation of daily hydrocarbon production by determining the optimal well production and lift gas rates subject to:

1 pressure and rate constraints in the nodes of the surface pipeline network and

2 the amount of lift gas available.

The mechanism of gas-lift is elaborate, in which an appropriate amount increases the oil production rate, but an excessive injection reduces the oil rate besides being costly due to high gas prices and gas compression costs. Ideally, if there is no restriction in the total amount of injection gas available, sufficient gas could be injected into an individual well until maximum production is attained. However, in most cases, the gas amount is insufficient to achieve the maximum production for each well, especially in depleted fields since the lift gas requirement would have increased compared to the initial facility design specification. Hence, it is necessary to optimise the allocation of a limited amount of injection gas to maximise oil production rate (Buitrago et al., 1996; Wang, 2003; Wang and Litvak, 2004).

The conventional tool for allocating lift gas is based on a gas-lift performance curve, which plots the oil rate versus the lift gas rate for a gas-lift well. When the gas supply is unlimited, the optimal lift gas rate corresponds to the maximum oil production on the performance rate; when limited, the lift gas allocation is typically determined via a formal optimisation routine. The traditional approach applied is the heuristic rule of economic ‘equal slope allocation’ concept, which states that the slope of the gas-lift performance curves should be equal for all wells at the optimal solution. However, in general, this method is not applicable to wells that do not demonstrate an instantaneous response to gas injection (Kanu et al., 1981).

Table 2 summarises the representative papers on the application of the optimisation approach to the gas-lift allocation problem.

Table 2 Representative papers on the application of the optimisation approach to the gas-lift allocation problem

Reference Model attribute and constraint Optimisation procedure and solution Kanu et al. (1981)

Obeys heuristics that slope of gas-lift performance curves are equal for all wells at optimal solution

Geometric/graphical approach (non-optimisation-based)

Nishikiori et al. (1989)

Allocates lift gas to well via gas-lift performance curve

• NLP • Gradient projection with

quasi-Newton method Buitrago et al. (1996)

No restriction in number of wells and well response

• NLP • Global-optimisation-based multistart

algorithm of stochastic domain exploration

• Heuristic to determine descent direction

• Results indicate approach could not properly handle constraints


Table 2 Representative papers on the application of the optimisation approach to the gas-lift allocation problem (continued)

Reference Model attribute and constraint Optimisation procedure and solution

Dutta-Roy and Kattapuram (1997)

Considers:

• Multiphase flow via pressure-balance-based pipeline network simulator

• Flow interactions among wells in gathering network systems sharing a common flow line

• NLP

• Sequential/SQP to handle flow interactions

4.2 Simultaneous optimisation of production rate and gas-lift rate allocation problems

The general rate allocation problem is concerned with the simultaneous allocation of both production rates and lift gas rates of single wells, or the total production rates of multiple reservoirs, with the objective of achieving certain operational goals. Table 3 focuses on the model characteristics of representative works in the literature addressing this problem.

Table 3 Representative works on the application of the optimisation approach to the general rate allocation problem

Reference Model attribute and constraint Optimisation procedure and solution Attra et al. (1961)

• Constraints: well production capacities; reservoir injection requirements; gas compressor capacity limits; gas-lift and pressure maintenance gas requirements; sales contracts

• Omits drilling schedule • Omits reservoir decline

model (because model is time-independent)

LP

Lo and Holden (1992)

• Proposes two well management schemes for rate forecasting of production streams; second scheme reduces computational time, thus more suitable for coupling with reservoir simulator

• Constraints: multiple facility flow rate

• LP • Simplex algorithm

Fang and Lo (1996)

• Constraints: multiple non-linear flow rate

• Assumes: (1) common flow lines, that is, ignores significant flow and pressure interactions among wells possible suboptimal solutions; (2) concave gas-lift performance curve; (3) constant well water-cut and GOR with variable oil rates

• 4-step algorithm: – NLP – reformulation of NLP into

separable program – piecewise linearisation of oil

outflow into MILP – simplex/separable algorithm


Table 3 Representative works on the application of the optimisation approach to the general rate allocation problem (continued)


Fang and Lo (1996)

• Easily modifiable to handle other constraints, for example, capacity of water treatment and separator units

• Disadvantages piecewise linearisation algorithm requires (1) ad hoc treatment of non-concave well profit functions; (2) heuristic decisions of whether to temporarily interrupt certain well operations

• Implemented on a reservoir simulator; simple and efficient in several field case studies

• Demonstrates that dynamic solution of gas-lift optimisation problems could induce substantial profit gains in oilfield exploration

Hepguler et al. (1997)

• Applicable for any general optimisation framework for rate allocation problems

• Coupling of commercial standalone surface pipeline network optimiser (SimSci’s NETOPT 1.0) with commercial reservoir simulator (Schlumberger–GeoQuest’s ECLIPSE 100)

• NLP • SQP (for optimising both general

design and operations)

Davidson and Beckner (2003)

• Integrated facility and reservoir model

• Objective function: maximise oil production rate; minimise water production

• Multiobjective NLP • SQP • Detailed procedure on handling

infeasible conditions

Wang and Litvak (2004)

• Constraints: pressure and rate in surface pipeline network nodes; lift gas amount

• Long-term reservoir development studies

Multiobjective optimisation

4.3 Practical implementations in field development cases and examples

Rate allocation problems are frequently encountered in mature fields for which production facilities are no longer able to meet the conventional field demand. The two Alaskan oilfields at Prudhoe Bay and Kuparuk River are well-documented cases in which the oil production is constrained by the gas processing capacities of the surface facilities comprising the separation units and the central gas plant. Table 4 lists some representative works addressing the rate allocation problem in these actual field development cases.

From Table 3, neural network appears to be the central tool for accelerating the simulation and optimisation procedures in the three studies by Stoisits et al. (1992, 1994, 1999), as advocated in the review by Wang (2003). Nonetheless, it is equally noted that training neural networks is a computationally expensive process; moreover, their reliability primarily depends on the accuracy of the training tools and the behaviour of the simulated system.


Table 4 Representative works on the practical implementations of the optimisation approach to the general rate allocation problems in field development cases


Barnes et al. (1990) (Prudhoe Bay)

Oil rate versus gas rate curve defines optimised oil rate target, which is then broken down to targets for production facilities, flowlines and individual production wells

Applies ‘incremental GOR’ heuristics (introduced in Section 2.1)

Litvak et al. (1997) (Prudhoe Bay)

• Integrated model of reservoir and surface facility gathering systems

• Allocation based on gas-lift tables of GOR, liquid well rate and water-cut

• Applies heuristics to optimise well connections to manifolds, hence possible suboptimal solutions

• Simultaneous solution of reservoir and surface pipeline network flow equations

Stoisits et al. (1992) (Kuparuk River)

• Multiple gas constraints

• Trains neural network using single-well simulation results

• Evaluates effect of additional lift gas compression capacity on production

• NLP adaptive modelling via Neural Network (NN)

• Results show accuracy over non-linear regression of field data


Constraints: gas-lift allocation; selection of non-gas-lifted production well; production allocation to central processing facilities and load balancing between centrifugal and reciprocal compressors

Applies ‘incremental GOR’ and ‘formation GOR’ heuristics


Model comprises well performance, surface line and facility models

• NLP via NN and genetic algorithms

• NN for simulating pressure drop through surface pipeline network to overcome expensive computation via hydraulics simulation

5 Application of the optimisation approach to reservoir development, planning and management

Reservoir management concerns the organisation of relevant technical, operational and business resources from discovery to abandonment to optimally exploit a reservoir in an efficient, safe, responsible and profitable manner by employing data-driven mathematical models coupled with human insight and expertise. The principal goal is to maximise the value of a hydrocarbon asset or a collection of assets by achieving a set of objectives that chiefly consists of:

1 improving the estimates of the hydrocarbons in place

2 increasing the recovery, production and forecasts of the oil and gas

3 minimising the capital investment and operating expenditures and

4 ultimately, maximising the overall profitability.


Accomplishing these objectives requires a holistic consideration of the full spectrum that spans the hydrocarbon assets, ongoing and future development projects and various investments under consideration (Al-Hussainy and Humphreys, 1996; Saputelli et al., 2005, 2006; Vasantharajan et al., 2006).

Development and planning studies in reservoir management has been a perennial area that has benefited enormously from the application of optimisation techniques. Continuously evolving and increasingly sophisticated tools and models have been employed to account for ever more stringent and demanding practical limitations. The representative problems in this field can be broadly categorised as follows (Ierapetritou et al., 1999; Iyer et al., 1998):

• facility location–allocation, which pertains to the decisions on the location (i.e. placement) and the allocation (i.e. number of unit(s)) of production platforms and wells to the well platforms

• production planning and scheduling regarding the following aspects (Lin and Floudas, 2003):

− scheduling of reservoir development

− scheduling of well-drilling operations

− scheduling of production rate

− timing for building a compressor in gas field development and

− annual revenue, capital expenditure and operating expenditure of reservoirs.

5.1 Facility location–allocation problem

Table 5 summarises the major publications that have employed optimisation methods to solve the facility location–allocation problem in reservoir development and planning studies.

Table 5 Representative publications on the application of the optimisation approach to the facility location–allocation problem in reservoir development and planning studies


Devine and Lesso (1972)

• Constraints: platforms capacity, piping cost

• Omits well-drilling scheduling and production planning

• MILP • Two-stage algorithm: fix well

allocation to solve for platform location and its converse

Rosenwald and Green (1974)

• Constraints: production targets from a predetermined subset of possible locations for new wells

• Assumes a specified reservoir production versus time relation

• MILP • Selects a specific number of wells

from candidate solutions and computes the optimal sequence of production rates from the chosen wells

Garcia-Diaz et al. (1996)

Constraints: total number of facilities; various technology constraints

• 0–1 ILP • Branch-and-bound • Lagrangean relaxation

Ierapetritou et al. (1999)

Constraints: vertical well completions

• MILP • Iterative decomposition with model

reformulation at each iteration


Table 5 Representative publications on the application of the optimisation approach to the facility location–allocation problem in reservoir development and planning studies (continued)


Odell and Rosing (1977)

Fixed number and capacity of platforms (valid for this particular model objective but unsuitable for cost minimisation)

Segments oilfield into hexagonal grid; centre of each hexagon represents a potential well target ⇒ reduces modelling complexity and computational time

Grimmett and Startzmann (1988)

Simultaneous selection, sizing and placement of major production facilities (and assignment of wells to these facilities)

ILP

Binary implicit enumeration – guarantees global optimal but excessive computational time and memory even for moderate-size problems

5.2 Production planning and scheduling of petroleum fields

5.2.1 Early applications of LP

Early optimisation studies in reservoir planning and scheduling have featured simple reservoir models with the application of LP techniques as evidenced from Table 6.

Table 6 Representative works on the early applications of LP to production planning and scheduling of petroleum fields


Lee and Aronofsky (1958)

• Single- and two-well system • Time-dependent linear pressure influence

function of reservoir performance surface pressure constraints and well-drilling scheduling

LP

Aronofsky and Williams (1962)

• Maximises non-linear investment rate of return (in objective function) that is approximated by truncated Taylor’s series

• Rounding to approximate non-integer solutions to reasonable integer values

• Parametric LP • Fix well-drilling schedule to

solve for production schedule and its converse

Charnes and Cooper (1961)

• Simplified form of Lee and Aronofsky (1958)

• One-reservoir water injection problem • Assumes fixed production schedule

Specialised semi-manual LP technique

See and Horne (1983)

• Experimental design of model uses multivariable regression analysis for data fitting

• DV: production schedule under fixed producer or injector well locations

• LP • Demonstrates that optimal

control procedure may be applied cheaply and accurately

Lang and Horne (1983)

• DV: injection rates; downhole flowing pressure

• Surrogate model of reservoir performance

• LP • Dynamic programming

(found to be more efficient than LP)


5.2.2 Applications of non-linear programming

LP model formulations, as exemplified in Table 6, require the objective function and constraints to be linear, thus necessitating approximation via linearisation of the non-linear production systems. This restriction greatly limits the application of more representative non-linear reservoir models and simulators for coupling with increasingly efficient and sophisticated optimisation methods for reservoir development. However, the continued explosion in computing power and advances in algorithmic techniques has afforded greater modelling and computational capabilities including various Non-Linear Programming (NLP) approaches. The progress that has been attained is broadly classified in the following tables to document some of the significant works:

• Table 7: early applications of NLP.

• Table 8: applications of NLP to optimal control models for solving dynamic optimisation problems, that is, optimisation models with differential–algebraic equations.

• Table 9: NLP for handling difficulties arising from irregular non-smooth objective functions by adopting artificial-intelligence-based techniques with stochastic search capabilities, in particular, Genetic Algorithms (GA).

Table 7 Representative works on the early applications of NLP to production planning and scheduling of petroleum fields


Huppler (1974)

• Multireservoir gas field

• Constraint: gas sales contract

• Considers risk by stipulating that any investment yield a minimum incremental profit-to-investment ratio

• Iterative (sequential/successive) quadratic programming

• Computes Hessian matrix based on Fletcher–Powell procedure

asdon et al. (1986)

• Constraints: demand schedules and operating constraints

• Employs a single-phase, 2D reservoir simulator

Reduced gradient using interior penalty functions

Table 8 Representative works on the applications of NLP to optimal control model framework for production planning and scheduling of petroleum fields


Rowan and Warren (1967)

Linearises a single differential equation describing rate of average reservoir pressure drop as a function of total water influx and total production rate

• Parameter estimation via least squares fitting to known data

• Fix drilling schedules to obtain production rates and its converse

McFarland et al. (1984)

• 1D tank-type reservoir models to describe reservoir dynamics for both gas reservoir with water drive and three-phase oil reservoir

• Generalised Reduced Gradient (GRG)


Table 8 Representative works on the applications of NLP to optimal control model framework for production planning and scheduling of petroleum fields (continued)


McFarland et al. (1984

• DV: platform size; no. of wells to drill in each time period; production rates; abandonment time

• Model extensions: spatial variation in reservoir; use of grid reservoir models

• DV: platform size; no. of wells to drill in each time period; production rates; abandonment time

• Model extensions: spatial variation in reservoir; use of grid reservoir models

Fathi and Ramirez (1984)

• Optimal injection policy of surfactant slug for Enhanced Oil Recovery (EOR) in 1D chemical flooding

• Simulates multiphase flow in porous medium

• Optimal control of distributed parameter systems

• Steepest descent gradient method

Table 9 Representative works on the applications of NLP to handle difficulties arising from irregular non-smooth objective functions in production planning and scheduling models for petroleum fields

Reference Model attribute and constraint NLP procedure and solution

Bittencourt and Horne (1997)

• Optimal relocation of proposed new wells for oilfield development

• Evaluates objective function from cash flow analysis for production profile obtained from simulation at each iteration executed

• Hybridised Genetic Algorithms (HGA) with polytope method

• Integrates economic analysis, simulation and project design

Pan and Horne (1998)

• Field development scheduling and waterflood project

• Considers sample point distribution design to generate combination of different levels of DVs for simulation runs

• Multivariate interpolation algorithms of least squares and kriging

• Achieves significant reduction in simulation runs

Güyagüler et al. (2002)

• Waterflooding project

• DV: placement of water-injection wells; injection rate

• Adapts HGA of Bittencourt and Horne (1997) with kriging proxy (very efficient) and neural-network proxy (less efficient), resulting in hill climber feature

• Significant reduction in simulations and computational time

Yeten et al. (2003)

• Multilateral well placement problem

• DV: well locations; no. of laterals; trajectory of laterals

• GA with hill climber

• Neural-network


5.2.3 Iterative-based versus simultaneous optimisation of design, planning and scheduling decisions

To ease computational burden, a majority of the earlier works addresses each of the following decisions on a separate basis: the capacities of the PPs and the WPs, the drilling schedules and the production profiles. An approach is to fix a well drilling schedule to determine the corresponding production profile (typically by solving the resulting LP), and to subsequently reverse the procedure by solving the converse problem of fixing the production profile to compute the corresponding drilling schedule, such as to be found in the works of Attra et al. (1961) and Aronofsky and Williams (1962). However, the major drawback of not considering all of the design and planning decisions in a simultaneously solved single model is that important interactions between these variables are not properly accounted for, resulting in the computation and implementation of suboptimal or worse, infeasible solutions. Table 10 surveys some major works that employ the simultaneous optimisation approach.

Table 10 Major works on the simultaneous optimisation of design, planning and scheduling decisions of petroleum fields


Bohannon (1970)

• 15-year development plan

• Multireservoir pipeline system (many reservoirs producing into one or more gathering systems)

• Redetermines linear production rate decline with cumulative oil produced

MILP

Sullivan (1982)

• Four partially depleted North Sea gas fields system

• DV: (1) time-staged platform placement and well drilling; (2) pipeline and compressor capacity installation; (3) production scheduling

• Piecewise linear interpolation via binary variables of non-linear reservoir performance equations

• MILP

• Linearised model ⇒ significant increase in no. of discrete variables ⇒ expensive computational time ⇒ inefficient for real-world problems

Haugland (1988)

• DV: platform capacity; drilling schedule; production planning

• Temporal and spatial discretisation

• MILP

• Applicable to only small model size due to expensive computational time

• Scaling technique to overcome numerical difficulties

Eeg and Herring (1997)

Executes reservoir simulations to obtain production profiles for each well, which are then linearised into discrete differences to be model inputs

• MILP

• Iterative scheme of fixing drilling schedule to solve for production profile and the converse

Currie et al. (1997)

• Redevelopment programme and drilling schedule for Ekofisk field, Norway in North Sea

• MILP

• Considers linear interpolated production profile


Table 10 Major works on the simultaneous optimisation of design, planning and scheduling decisions of petroleum fields (continued)


• Constraints: product-processing requirements; drilling-rig use; well-platform relationships; other physical facilities

Nygreen et al. (1998)

In practical use by Norwegian Petroleum Directorate (NPD) and other major Norwegian oil companies for long-term production planning from new fields and transportation pipelines

• MILP

• Timing for initiation of fields

• Simultaneous design of pipeline systems

Iyer et al. (1998)

Planning and scheduling of investment and operations of offshore oilfield infrastructure

• MILP (multiperiod)

• Bi-level sequential decomposition

• Aggregation–disaggregation of time periods and wells

• No guarantee of optimal solution but generate good feasible solutions and upper bounds

5.2.4 Recent advances in MINLP modelling approach

It is pertinent to deal with non-linearities in reservoir performance in a direct manner so as to avoid the introduction of a large number of integer variables resulting from piecewise linear interpolation, besides the possible loss of accuracy associated with other linearising techniques, in our ultimate aim of improving the solution quality. Hence, this section serves to shed some light in this direction by highlighting the increased modelling power offered by the MINLP approach (Grossmann, 2002). While still largely heuristics based in terms of solution strategy, recent progress in discrete optimisation has spurred the richer although more complex model formulation of mixed-integer non-linear programmes, as reflected in Table 11.

Table 11 Representative publications employing MINLP for production planning and scheduling of petroleum fields

Reference Model attribute, DV and constraints Optimisation procedure and solution

Dawson and Fuller (1999)

• DV (discrete): periodical replacement of pumps at well sites

• Represents non-linear reservoir behaviour as log-linear relationship between oil fraction and cumulative oil produced

• Ignores non-linear surface pressure constraints

• Assumes fixed configuration of PPs, WPs and wells

• MINLP

• Generalised benders decomposition

• Problem size limited to only three wells


Table 11 Representative publications employing MINLP for production planning and scheduling of petroleum fields (continued)

Reference Model attribute, DV and constraints Optimisation procedure and solution

van den Heever and Grossmann (2000)

• Based on Iyer et al. (1998)

• Employs non-linear functions of cumulative oil produced for modelling: (1) reservoir pressures; (2) GOR; (3) cumulative gas produced

• DV: investment of PPs, WPs and wells; drilling schedule; periodic production profiles

• MINLP (multiperiod)

• Iterative aggregation–disaggregation

• Logic-based outer approximation algorithm

• Bi-level decomposition: (1) upper level design; (2) lower level operation and capacity expansion planning

• Dynamic-programming-based aggregation of time periods

van den Heever et al. (2000, 2001)

• Extension of van den Heever and Grossmann (2000) for long-term design and planning of gas field infrastructure

• Objective function considers complex economic metrics of tax, tariff, royalty

• MINLP (multiperiod)

Lin and Floudas (2003)

• Long-term planning of gas field development

• DV: production design; transportation network structure design; field operations over time

• Bi-level formulation and solution framework with complicated economic calculations

• MINLP

• Continuous-time approach with event points concept

• (compared to discrete time model) Substantially reduces computational time and able to solve previously intractable problems

Carvalho and Pinto (2006a,b)

• Applies bilevel decomposition with aggregation–disaggregation techniques of Iyer and Grossmann (1998) and Iyer et al. (1998):

• (1) master problem: platforms assignment to wells; (2) planning subproblem: timing for fixed assignments

• MILP

• Heuristic to reduce solution search space

5.2.5 Planning and scheduling of reservoir operations under uncertainty

Our knowledge of a petroleum reservoir is generally uncertain, thus the practice of reservoir management is inevitably riddled with a multitude of uncertainties involving multiple future outcomes that are difficult to predict in the present (Haugen, 1996). Factors with significant randomness or variability are mainly present in the following parameters: availability and price of resources, size of fields, reservoir properties, development costs, deliverability of production, oil price, price of products and cost of projects. The uncertainty that leads to negative outcomes is associated with the notion of risk. In essence, uncertainty impacts on the economic risk


analysis of the reservoir production forecast and is thus a central concern in any decision-making process that eventually translates to measures of profitability. To mitigate these uncertainties, various approaches have been developed and implemented as reported in Table 12.

Table 12 Representative papers addressing planning and scheduling of reservoir operations under uncertainty

Reference Model attribute, Random Variable (RV), DV and constraint

Optimisation procedure and solution

Jørnsten (1992), Haugen (1996) and Jonsbråten (1998)

RV: demand, production field size and oil price, respectively

Stochastic programming

Güyagüler and Horne (2004)

• Employs utility theory to quantify uncertainty in reservoir simulation forecasts in evaluating well-placement configuration

• Incorporates risk attitude of decision-maker through utility functions

• Based on HGA of Güyagüler et al. (2002)

• Considers random realisations of reservoir properties for each evaluation of well configuration; then performs simulation to evaluate objective function

• Approximate but computationally feasible solution

Cullick et al. (2003)

• Framework to manage uncertainty that comprises three workflows performing flow simulation and economics: (1) global optimiser outer loop; (2) stochastic inner loop made up of scenarios and simulations to describe uncertainty and (3) dispatcher for distributed computing

• Detailed finite difference flow simulator handles non-linear physical model production flow and constraints

• MINLP

• Outer loop: heuristic global-search methods (metaheuristics), for example, Tabu search, scatter search (combined with LP and neural networks)

• Inner loop: Monte Carlo (MC) or quasi-MC sampling that incorporates correlations, production profile reservoir simulator with surface pipeline network and MS Excel economic model

Aseeri et al. (2004)

• Considers financial risk management for optimal operating sequence of platform-building, well-drilling and well-production

• RV: oil price, productivity index

• Stochastic MINLP

• Sample Average Approximation (SAA) (Kleywegt et al., 2001) (to overcome numerical difficulties in evaluating the expected recourse cost function)


Table 12 Representative papers addressing planning and scheduling of reservoir operations under uncertainty (continued)

Reference Model attribute, Random Variable (RV), DV and constraint

Optimisation procedure and solution

Aseeri et al. (2004)

• DV: selection of reservoir and well sites; capacities of well and production platforms; fluid production rates from wells

• Includes budgeting constraints to handle project cash flow, distribution of proceeds and possibility of taking loans against some built equity

• Good solution; proven close to optimality via upper bound risk curve concept

Goel et al. (2006)

• Investment and operational planning of gas fields development

• RV: reserves size and initial deliverability (i.e. maximum production rate) of fields

• DV: production rate • No capacity expansion

• MINLP • Multistage stochastic programming • Lagrangean-duality-based

branch-and-bound algorithm • Improved solutions over

deterministic model • Properties of feasible solutions

significantly reduce model size to aid computation

6 Concluding observations and remarks

In the past, most model-based strategies to the general practice of production optimisation methods of petroleum fields have adopted numerical reservoir simulation models with a case study approach. However, such studies typically focus only on a specific subproblem, evaluating a set of operational plans on the basis of discounted cash flows. Fortunately, as a result of the dramatic increase in computing power that has afforded greater sophistication in modelling and computational techniques, we have witnessed an explosion of activities and efforts within both academic research and industrial practice to adopt formal multivariate optimisation models that are capable of simultaneously handling a multitude of important practical factors, considerations and issues, for greater representation in tackling the reservoir management problem at hand. Furthermore, these approaches involve much simpler reservoir representations than those employed by the simulators. In fact, it is the standard practice today to incorporate simulators for generating reservoir models as a component in the optimisation framework.

Despite widespread use in the refining and petrochemical sectors, Saputelli et al. (2005) argues that multivariable optimisation has not fully penetrated the hydrocarbon industry sector of E&P activities. To the extent that it has been adopted, the approach lacks connection with the real field dynamics. This is evident from its hitherto routine reliance on steady-state conditions for the continuous updating of model parameters, instead of the routine inclusion of the more representative dynamic data, the latter of which is the prevalent practice especially in the refineries.

Additionally, most optimisation models that are coupled with reservoir simulations emphasise the performance characteristics of the reservoirs without due equal attention to the impact of the facility handling and field stimulation injection capacities. In practice,


the latter is an important limiting factor of production rate because of cost considerations in the face of dynamic economic condition and new development opportunities, besides unexpected reservoir performance (Fang and Lo, 1996).

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