A Novel Continuous-Time Modeling and Optimization

Optimization and Engineering, 4, 65–95, 2003c© 2003 Kluwer Academic Publishers. Manufactured in The Netherlands

A Novel Continuous-Time Modeling and OptimizationFramework for Well Platform Planning Problems

XIAOXIA LIN, CHRISTODOULOS A. FLOUDAS∗Department of Chemical Engineering, Princeton University, Princeton, NJ 08544-5263, USAemail: [email protected]

Received June 26, 2001; Revised May 13, 2002

Abstract. The long-term planning problem for integrated gas field development is investigated. The key decisionsinvolve both design of the production and transportation network structure and operation of the gas fields over time.A novel continuous-time modeling and optimization approach is proposed, which introduces the concept of eventpoints and allows the well platforms to come online at potentially any time within the continuous horizon underconsideration. A two-level formulation and solution framework is developed to take into account complicatedeconomic calculations and results in mixed-integer nonlinear programming (MINLP) problems. As comparedwith the discrete-time model, the proposed approach leads to more compact mathematical models and significantreduction of the size of the resulting MINLP problems. Even though, the proposed approach in its current formcannot guarantee convergence to the optimal solution, computational results show that this approach can reducethe computational efforts required substantially and solve problems that are intractable for the discrete-time model.

Keywords: long-term planning, gas field development, continuous-time formulation, mixed-integer nonlinearprogramming (MINLP)

1. Introduction

The long-term planning for offshore gas/oil field development is a very challenging problem.It can involve the design and operation of various functional units, such as gas fields, wells,well platforms, producing platforms, pipelines and compressors. Decisions are to be madeconcerning both the network structure and the operation profile, taken into account not onlyphysical restrictions but also economic considerations. The large scale of investment andproduction involved and the long life cycle of such production facilities provide the potentialof great savings a good planning strategy could achieve, which has motivated considerablework to address related problems.

Previous works in the literature vary substantially in the level of detail. Our discussionwill be focused on the following three important aspects: (i) reservoir behavior modeling;(ii) economic considerations; and (iii) discretization of time horizon.

The reservoir behavior of oil/gas fields typically present certain degree of nonlinearity.However, due to the complexity of resulting mathematical problems, most of the previousworks approximate it with linear functions and consequently lead to either linear program-ming (LP) or mixed-integer linear programming (MILP) problems. For instance, Haugland

∗Corresponding author.

66 LIN AND FLOUDAS

et al. (1988) use simple linear constraints to model the oil field reservoir behavior, andIyer et al. (1998) improve the accuracy by applying piecewise linear interpolation withinteger variables. These approximations inevitably reduce the degree of accuracy of thereservoir model and thus may lead to suboptimal solutions. In a more recent work by vanden Heever and Grossmann (2000), the nonlinearities are included directly with the reser-voir pressure and other related quantities expressed as nonlinear functions of the cumulativeoil produced along with an aggregation approach for the time periods.

The construction and operation of offshore oil/gas field production facilities often in-volve complex economic considerations in reality. Most previous works, however, only usesimplified models. van den Heever et al. (2000, 2001) show that including complex rulesfor calculations of tariff, tax and royalty leads to substantial improvement of the solutionto the problem under investigation. However, this also results in significant increase of thecomputational effort required and they address this problem by using a disjunctive program-ming framework in van den Heever et al. (2000) or a heuristic Lagrangean decompositionalgorithm in van den Heever et al. (2001).

All previous publications make use of the discrete-time formulation. The whole horizon isdivided into a number of uniform time intervals and decisions are made for each time interval.Due to the long time horizon that is typical in such problems and the discrete nature of manydecisions concerning network structure and other aspects, the main challenge arises from thesolution of the resulting large-scale mathematical programming problems. Consequently,a lot of research effort has been spent on investigating various specialized approaches toaddress specific problems, which usually explore the special structure of the problemsunder consideration. For example, in van den Heever and Grossmann (2000), an iterativeaggregation/disaggregation approach is proposed to solve the MINLP problems, whichintegrates logic-based methods, decomposition techniques, and dynamic programming. Itshould be pointed out that the discrete-time model has two main limitations. First, it is bydefinition (due to its discrete nature) only an approximation of the actual problem whichdeals with decision making through a continuous time horizon. Second, increasing thenumber of time intervals or reducing the duration of each time interval leads to significantincrease in the size of resulting mathematical problems, which makes it difficult to solvelarge problems and to improve the degree of model accuracy.

In this paper, we address a real long-term planning problem for the development of anintegrated offshore gas fields production network. Both the nonlinear reservoir behaviorand complex economic considerations are incorporated explicitly. We apply the conceptof event points proposed in Ierapetritou and Floudas (1998a, b) and Ierapetritou et al.(1999) to develop a continuous-time formulation which allows the installation of wellplatforms and other events to take place at any time in the time horizon under consideration.A two-level formulation and solution framework is proposed, in which Level 1 featuresapproximate economic calculations and provides an initial sequence of event points forLevel 2, while Level 2 performs economic calculations precisely on a yearly basis anddetermine configurational and operational strategies through an iterative procedure.

The rest of the paper is organized as follows. In Section 2, the problem we investigateis defined and the characteristics of the problem are discussed in details. The difficulty inusing the straightforward discrete-time formulation is briefly analyzed in Section 3. Then,

A NOVEL CONTINUOUS-TIME MODELING AND OPTIMIZATION FRAMEWORK 67

we introduce a novel continuous-time formulation in Section 4, followed by computationalresults presented in Section 5.

2. Problem definition and characteristics

The long-term planning problem we investigate involves the development of a productionand transportation network with up to 15 offshore gas fields. These gas fields are classifiedinto three categories. For all gas fields in Category 1, the platforms were built and wellswere drilled or are to be drilled as specified. Category 2 gas fields are to be developed ina relatively short horizon. Less accurate data on Category 3 gas fields are known and theycan be built within 20 years. Each gas field can have 2 to 8 wells, but exactly one wellplatform is needed for each gas field if it is to be developed. Locations of the well platformsare given as those of the gas fields. There are also other functional units, such as producingplatforms (one to two and one has been in use) and compressors (with up to two stages).Each well platform is connected either to another well platform or directly to a producingplatform before connecting to the gas plant on the shore with or without the compressor.The super-structure of the connectivity is shown in figure 1.

We are given the following information: (i) attributes of gas fields, for example, maximumflow rate, initial pressure, liquid yield, well data, correlation data, tube data, etc; (ii) attributesof well platforms, including location, types, capacities, capital costs and so on; (iii) dataof the producing platform and compressor; (iv) network connectivity restrictions and data;(v) economic data, such as gas price, liquid price, discount factor, tax and royalty data,backstop data, etc; (vi) other specifications, for example, gas plant pressure, productionplateau requirement, etc.

Figure 1. Super-structure of the production and transportation network.

68 LIN AND FLOUDAS

The questions to be answered are:

– Which well platforms should be built? And when?– How should the well platforms and producing platforms be connected?– When, if needed, should the compressor be built?– What is the rate of production of each gas field over time?– What is the revenue, capital expenditure, operating expenditure every year? And what

taxes and royalties to pay?

The objective is to optimize a certain economic performance index, for example, tomaximize the net present value (NPV). The overall model consists of the following threeparts.

2.1. Reservoir model

The reservoir model represents the physical property and behavior of each gas field basedon the bore well model. The variables involved are primarily as follows: gas flow rate, Qf ;field deliverability, Qd f ; cumulative production, Gp; reservoir pressure, Pr ; and surfacepressure, Ps .

The gas flow rate of each gas field is one of the main decision variables. As representedby Eq. (1), it is bounded by the field deliverability, which depends on both the cumulativeproduction and the surface pressure. The cumulative production can be calculated by inte-grating the gas flow rate from when the gas field starts production to the moment of interest,as shown by Eq. (2).

Q f ≤ Qd f (1)

Gp =∫ τ

0Q f dτ (2)

For gas fields in Categories 1 and 2, the reservoir pressure, field deliverability, and surfacepressure are correlated through the following quadratic and cubic equations.

Pr = caG2p + cbGp + Pr,init (3)

(Pr − p)2 − P2s = ra Q3

d f + rb Q2d f + rc Qd f + rd (4)

For gas fields in Category 3, these quantities are correlated indirectly as follows.

Pr = p1G3p1 + p2G2

p1 + p3Gp1 + p4 (5)

Gp1 = Gp

G(1 − r ) + r (6)

Qd f = i1(P3

r − P3f

) + i2(P2

r − P2f

) + i3(Pr − Pf ) (7)

Pb = b1

Q3d f

m3+ b2

Q2d f

m2+ b3

Qd f

m+ b4 (8)

Pf = Pb + d1 P2s + d2 Ps + d3 (9)


Figure 2. Nonlinear reservoir behavior of two example gas fields.

As illustrated with two of the gas fields in figure 2, these equations introduce nonlinearitiesand nonconvexities into the problem.

2.2. Surface model

The surface model involves constraints on the network structure, material balance andpressure balance of the production and transportation network.

The decisions concerning the network structure are: first, whether or not to switch on eachwell platform/producing platform/compressor stage and if selected when to switch them on;second, which of the three types of well platforms to build if selected; third, whether or notto switch on each connection between well platforms/producing platform and if selectedwhen to switch them on.

At the same time, we are to determine the operation across the whole network. Materialbalance and pressure balance equations are formulated around each well platform/producingplatform/compressor stage and the connections between.

For example, concerning a well platform, let’s define the following variables: Qtot(wp),total flow rate through well platform (wp); Qf (wp), flow rate from local gas field to (wp);Q(wp, wp′), flow rate from (wp) to (wp′); Q(wp, pp), flow rate from (wp) to producingplatform (pp); Pin, inlet pressure; Pout, outlet pressure; Ps , surface pressure of local gasfield; Pchk, choke pressure. The following linear relationships need to be met.

Qtot(wp) = Qf (wp) +∑wp′

Q(wp′, wp) (10)

Qtot(wp) =∑wp′

Q(wp, wp′) +∑

pp

Q(wp, pp) (11)

Pin(wp) = Ps(wp) − Pchk(wp) (12)

Pin(wp) = Pout(wp′) − βwp′,wp Q(wp′, wp) − Pchk(wp′, wp) (13)

Pout(wp) = Pin(wp) − γwp Qtot(wp) (14)

70 LIN AND FLOUDAS

Equation (10) states that the total flow rate through well platform (wp) is the sum of itslocal field flow rate and the flow rates from all other well platforms. On the other hand,Eq. (11) expresses that this total flow rate is also the sum of the flow rates to all other wellplatforms and to the producing platforms. In terms of pressure balances, the inlet pressureof well platform (wp) is equal to its local field surface pressure minus the choke pressure,as shown by Eq. (12). If there is a connection from any other well platform (wp′) to (wp),the inlet pressure of (wp) should also be equal to the outlet pressure of (wp′) adjusted bythe pressure drop from (wp′) to (wp), which is proportional to the flow rate, and the chokepressure for flow from (wp′) to (wp), as represented by Eq. (13). The outlet pressure of(wp), as described by Eq. (14), is then equal to its inlet pressure minus the pressure dropacross it which is also proportional to the flow rate.

The other important type of functional unit is the processing platform. The followingvariables are introduced: Qtot(pp), total flow rate into (pp); Qnet(pp), net flow rate out of(pp); C f (pp, cs),fuel consumption in compressor stage (cs) at (pp); Qcmp(pp, cs), flowrate through (cs) at (pp); Qbyp(pp, cs), flow rate bypassing (cs) at (pp); Qshr, total flow rateto the shore; Pin(pp), inlet pressure of (pp); Pout(pp): outlet pressure of (pp); Pbst(pp, cs),pressure boost across (cs) at (pp); Pshr, pressure at shore, specified. They should satisfy theequations as follows.

Qtot(pp) =∑wp

Q(wp, pp) (15)

Qnet(pp) = (1 − s)

[Qtot(pp) −

∑cs

C f (pp, cs)

](16)

Qtot(pp) = Qcmp(pp, cs) + Qbyp(pp, cs) (17)

C f (pp, cs) = αpp,cs Qcmp(pp, cs) (18)

Qshr =∑

pp

Qnet(pp) (19)

Pout(pp) = Pin(pp) +∑

cs

Pbst(pp, cs) (20)

Pbst(pp, cs) ≤ (δpp,cs − 1)Pin(pp, cs) (21)

Pshr = Pout(pp) − [η1 Q2

net(pp) + η2 Qnet(pp)]

(22)

The total flow rate into the producing platform is the sum of the flow rates from all wellplatforms, as shown by Eq. (15). Part of this gas is consumed by the compressor and then theflow rate is further reduced by a certain percentage due to shrinkage, as described by Eq. (16).For each compressor stage, the gas can either undergo the compression or bypass and thefuel consumption is proportional to the flow rate through it, as represented by Eqs. (17) and(18). Finally, the total flow rate to the shore is the sum of the flow rates out of the producingplatforms, as shown by Eq. (19). On the other hand, as far as pressure balance is concerned,Eqs. (20) and (21) state that the outlet pressure of the producing platform is equal to its inletpressure plus the pressure boost across all compressor stages and the pressure boost acrosseach stage is limited by the inlet pressure of this stage times a certain factor. The pressure atshore is then equal to the outlet pressure of the producing platform minus the pressure dropfrom the producing platform to the shore which is a quadratic function of the corresponding


flow rate, as represented by Eq. (22). Note that most of these equations are linear, exceptEq. (22).

2.3. Economic model

The economic model is formulated to perform various economic calculations, includingrevenues, capital expenditures, operating expenditures, taxes and royalties. They are allcalculated on a yearly basis.

2.3.1. Revenues. Revenues come from sales of gas and liquids produced in each year.However, two contract rates are specified and back-stop charges are to be applied if theyare not met, as described by Eqs. (23) and (24).

Rg(t) = φt Qshr(t) + ρt

∑f

l f Qf ( f, t) − ChBS,1(t) − ChBS,2(t) (23)

ChBS,i (t) = υi,t · max{0, qBS,i − Qshr(t)}, i = 1, 2 (24)

where Rg is the gross revenue; Qshr is the flow rate to the gas plant at shore; Qf are gasfield flow rates; ChBS,1 and ChBS,2 are the charges for back stop 1 and 2, respectively; qBS,1

and qBS,2 are the corresponding contractual rates.Note that a set of binary variables is introduced to represent the “max” operator (Floudas,

1995) in Eq. (24). Let’s define binary variable bBS,i (t) to be 1 if Qshr(t) ≤ qBS,i and 0otherwise, and continuous variable QBS,i (t) to represent max{0, qBS,i − Qshr(t)}. Then,Eq. (24) can be formulated as follows:

QBS,i (t) ≥ 0 (25)

QBS,i (t) ≤ qBS,i · bBS,i (t) (26)

QBS,i (t) ≥ qBS,i − Qshr(t) (27)

QBS,i (t) ≤ qBS,i − Qshr(t) + (qmax

shr − qBS,i) · [1 − bBS,i (t)] (28)

ChBS,i (t) = υi,t · QBS,i (t) (29)

where qmaxshr is the maximum possible flow rate to the shore. If bBS,i (t) is equal to one,

Eqs. (25)–(28) result in the relationships of 0 ≤ Qshr(t) ≤ qBS,i and QBS,i (t) = qBS,i −Qshr(t). On the other hand, if bBS,i (t) is equal to zero, the same equations lead to qBS,i ≤Qshr(t) ≤ qmax

shr and QBS,i (t) = 0.Furthermore, due to operation restrictions, the overall production needs to maintain a 10

year plateau once the production level increases, as shown in figure 3. This necessitatesanother set of binary variables. Let’s define b10y(t) to be the binary variable which takes thevalue of one if the overall production level in year (t) is increased from the previous yearand takes the value of zero otherwise. The following constraints need to be introduced toreflect the 10 year plateau requirement.

Qshr(t) ≤ Qshr(t − 1) + qmaxshr · b10y(t) (30)

Qshr(t′) ≥ Qshr(t

′ − 1) − qmaxshr · [1 − b10y(t)], t < t ′ < t + 10 (31)

72 LIN AND FLOUDAS

Figure 3. 10-year production plateau.

2.3.2. Capital expenditures. The capital costs of well platforms, pipelines, compressor,and well development are each distributed uniformly in a corresponding investment horizonwhich is immediately before the unit starts being used. Table 1 shows the horizons ofdifferent investments.

2.3.3. Operating expenditures. The operating costs of production and transportation comefrom well, well platform, pipeline, producing platform, gas plant, etc. They can all becalculated with linear functions similar to Eq. (32).

Xo(t) = c1 · Q(t) + c0 (32)

where Xo is the operating cost and Q is the corresponding flow rate, for example, theoperating expenditure for the gas plant is linearly dependent on the flow rate into it.

2.3.4. Taxes. Taxes are calculated according to a sequence of specific rules which may besophisticated. However, in overall, they are linear functions of the gross revenue, capitalcosts, and operating costs, as represented symbolicly by Eq. (33).

Tax(t) = δr · Rg(t) + δc · Xc(t) + δo · Xo(t) + δ0 (33)

where Tax is the tax to pay for each year; Rg is the gross revenue; Xc are the capital costs;and Xo are the operating costs.

2.3.5. Royalties. For gas fields in Categories 1 and 2, the royalty for each year is calculatedas a whole entity, while it is paid individually for each gas field in Category 3. A multi-phase

Table 1. Capital investment horizons.

Investment horizon (year)

Well platform 3

Pipeline 1

Compressor 3

Well development 1


Table 2. Parameters of the multi-phase royalty model.

Phase rak rgk rnk

1 0% 2% –

2 10.2% 5% –

3 17.7% 5% 30% Categories 1 & 2

25.2% 5% 20% Category 3

4 50.2% 5% 35%

system is applied to determine the royalty, which uses different rates according to differentpay out levels, as described by Eqs. (34)–(37).

Ryl(t) = max{rgk · Rg(t), rnk · Rn(t)} (34)

if POs(t) + Rtcum(t) · rak ≤ 0, k = 1, 2, . . . , K (35)

Simple Payout is:

POs(t) =∑t ′≤t

λr t ′ Rg(t ′) +∑t ′≤t

λct ′ Xc(t ′) +∑t ′≤t

λot ′ Xo(t ′) + λ0t (36)

and Cumulative Return Allowance is:

Rtcum(t) =∑t ′≤t

max{0, POs(t ′)} (37)

where Ryl is the royalty to pay; Rg is the gross revenue; Rn is the net revenue; POs is thesimple payout; Rtcum is the cumulative return allowance; Xc are the capital costs; Xo arethe operating costs.

The parameters for the above multi-phase model are summarized in Table 2. Similarto the “max” operator in Eq. (24), the “max” operator in Eq. (37) also necessitates theintroduction of a set of binary variables. Furthermore, it is required to introduce anotherset of binary variables to determine the rate of which phase should be applied. Let’s definebinary variable bryl(t, k) to be 1 if Eq. (35) is met for phase (k) and 0 otherwise. Then, themulti-phase royalty can be calculated with the following equations.

POs(t) + Rtcum(t) · rak ≥ −M · bryl(t, k) (38)

Rylg(t, k) ≤ (rg,k − rg,k−1) · Rg(t) + Rylmaxg,k · [1 − bryl(t, k)] (39)

Rylg(t, k) ≥ (rg,k − rg,k−1) · Rg(t) − Rylmaxg,k · [1 − bryl(t, k)] (40)

Rylg(t, k) ≤ Rylmaxg,k · bryl(t, k) (41)

Rylg(t, k) ≥ −Rylmaxg,k · bryl(t, k) (42)

Ryln(t, k) ≤ (rn,k − rn,k−1) · Rn(t) + Rylmaxn,k · [1 − bryl(t, k)] (43)

Ryln(t, k) ≥ (rn,k − rn,k−1) · Rn(t) − Rylmaxn,k · [1 − bryl(t, k)] (44)

74 LIN AND FLOUDAS

Ryln(t, k) ≤ Rylmaxn,k · bryl(t, k) (45)

Ryln(t, k) ≥ −Rylmaxn,k · bryl(t, k) (46)

Rylgtot(t) =∑

k

Rylg(t, k) (47)

Rylntot(t) =∑

k

Ryln(t, k) (48)

Ryl(t) ≥ Rylgtot(t) (49)

Ryl(t) ≥ Rylntot(t) (50)

where Rylg(t, k) and Ryln(t, k) represent the royalties to pay in phase (k) in addition to theprevious phase based on the gross revenue and the net revenue, respectively; Rylgtot(t) andRylntot(t) are the overall royalties to pay based on the gross revenue and the net revenue,respectively; M is a positive big number; Rylmax

g,k and Rylmaxn,k are parameters that represent

the maximum possible amount of royalties in phase (k) based on the gross revenue andthe net revenue, respectively. Equation (38) enforces bryl(t, k) to take the value of one ifPOs(t)+Rtcum(t) · rak < 0. If bryl(t, k) is equal to one, Eqs. (39) and (40) lead to Rylg(t, k) =(rg,k − rg,k−1) · Rg(t); otherwise, Eqs. (41) and (42) lead to Rylg(t, k) = 0. Equations (43)–(46) give similar relationships for the net revenue. The overall royalties based on the grossrevenue and the net revenue are then calculated with Eqs. (47) and (48), respectively. Finally,the actual overall royalty to pay is the greater of the two, as shown by Eqs. (49) and (50).

2.3.6. Objective: Net present value. As an example of possible objective function, NetPresent Value (NPV) is the summation of discounted cash flows within the time horizonconsidered and the yearly cash flow is simply calculated as revenues adjusted by capitalexpenditures, operating expenditures, taxes and royalties, as expressed by Eq. (51).

NPV =∑

t

{Rg(t) − [Xc(t) + Xo(t) + Tax(t)] − Ryl(t)} · dt − DCF p (51)

where t are years; dt are discount rates, given as 11.12t ; and DCF p are discounted cash flow

prepaid.

3. Discrete-time formulation

In a discrete-time formulation, the whole time horizon is divided into a number of uniformtime intervals, for example, years. Each physical quantity, for example, the flow rates, isassumed to maintain constant throughout each time interval and changes can only take placeat specific time instances, for example, at the beginning of each time interval, as illustratedby figure 4.

Correspondingly, variables are defined for each time interval, including binary variables,such as whether or not to switch on each well platform in each time interval, and continuousvariables, such as the reservoir pressure of each gas field in each time interval. For example,


Figure 4. Discrete-time formulation.

Eqs. (2)–(4) in the reservoir model are written for each time interval as follows.

Gp( f, t) = Gp( f, t − 1) + Qf ( f, t) · dpy (52)

Pr ( f, t) = ca, f G2p( f, t) + cb, f Gp( f, t) + Prinit, f (53)

[Pr ( f, t) − p f ]2 − P2s ( f, t) = ra, f Q3

d f ( f, t) + rb, f Q2d f ( f, t)

+ rc, f Qd f ( f, t) + rd, f (54)

In the surface model, binary variables need to be introduced for each time interval todetermine the network structure. Logical constraints are written to describe the relationshipsamong these binary variables and to correlate the binary and continuous variables. Forexample, the following variables are defined: bsw(wp, t), binary, switch-on of (wp) in timeinterval (t); bcn(wp, wp′, t) and bcn(wp, pp, t), binary, switch-on of the connection from(wp) to (wp′) and (pp), respectively, in time interval (t); Qtot(wp, t), continuous, total flowrate through (wp) in time interval (t). They are involved in Eqs. (55)–(57), which express therestrictions that each well platform can only be switched on at most once; if it is switched on,one and only one connection from this well platform to the producing platform or anotherwell platform has to be switched on at the same time; and the total flow rate through thewell platform can take positive values only after it is switched on.∑

t

bsw(wp, t) ≤ 1 (55)

bsw(wp, t) =∑wp′

bcn(wp, wp′, t) +∑

pp

bcn(wp, pp, t) (56)

Qtot(wp, t) ≤ Qmaxwp ·

∑t ′≤t

bsw(wp, t ′) (57)

At the same time, material balance and pressure balance across the production and trans-portation network are formulated for each time interval, as illustrated by Eqs. (58) and(59).

Qtot(wp, t) = Qf (wp, t) +∑wp′

Q(wp′, wp, t) (58)

Pin(wp, t) = Ps(wp, t) − Pchk(wp, t) (59)

The economic evaluations are performed for each time interval as described in Section 2.3.It should be pointed out that binary variables are introduced for all gas fields in Categories

76 LIN AND FLOUDAS

1 and 2 as a whole entity and each gas field in Category 3, for each time interval, to modelthe multi-phase royalty calculations, as expressed by Eqs. (38)–(50).

The afore-mentioned discrete-time model inevitably gives rise to large-scale nonconvexMINLP problems. For instance, in a case study with 1 processing platform, 1 compres-sor, 15 well platforms, and a horizon of 15 years, the number of binary variables will beas large as 1,100, and the number of continuous variables and constraints will be around12,000 and 21,000, respectively. MINLP problems of such sizes are intractable with existinggeneral-purpose solvers. There are two ways to address this difficulty. One is to developspecial methods which explore the problem structure, such as the heuristic Lagrangeandecomposition algorithm proposed by van den Heever et al. (2001). This approach isable to reduce the solution time through a heuristic procedure which aims at addressingthe large number of binary variables. This heuristic approach can not guarantee conver-gence or optimality and leads to worse objective function value than that in full space.In this work, we instead focus on the mathematical modeling part and aim at develop-ing a new continuous-time formulation to address the reduction of the number of binaryvariables.

4. Continuous-time formulation

4.1. Basic concepts and assumptions

Ierapetritou and Floudas (1998a, b) and Ierapetritou et al. (1999) proposed a novel andeffective continuous time formulation for the short-term scheduling of chemical processes,which features the concept of event points (EPs). We apply the same concept to addresslong-term planning problems studied in this work. Event points are essentially a sequenceof time instances that can take any values in the continuous domain of time. They canbe classified into two types: fixed event point (FEP), for which the timing is known; andvariable event point (VEP), for which the timing is to be determined. By associating theswitch-on’s of well platforms/compressor stages with VEPs, these key events are allowedto take place at any time in the horizon under consideration, which explains why this iscalled a continuous-time formulation. Economic calculations are associated with the FEPsso that they can be performed for each specified time period, for example, for each year, asrequired. This basic idea is illustrated in figure 5.

Two assumptions are made. First, it is assumed that an initial sequence for well plat-forms to switch on can be determined a priori mainly based on the following information:

Figure 5. Event points in the continuous-time formulation.


capital costs of the well platforms, gas field productivities and restrictions on the networkconnectivity. Second, the objective value of the solution with a given sequence of VEPs andFEPs converges to the optimal value as the sequence gets closer to the optimal sequence.Based on these two assumptions, we have developed a two-level formulation and solutionframework. At Level 1, the formulation includes only VEPs and the goal is to determinetheir approximate timings. The economic calculations are approximated at the VEPs andthus the formulation at Level 1 is regarded as an approximate model. Then at Level 2, theformulation includes both FEPs and VEPs. The economic calculations are performed atthe FEPs accurately on a yearly basis and thus the formulation at this level is consideredas an accurate model in contrast to the approximate model at Level 1. This is the level atwhich the final solution is determined. The timings of the VEPs obtained at Level 1 are usedto form an initial sequence of EPs, including both VEPs and FEPs, and the formulationis solved iteratively to refine the sequence of EPs and the corresponding solution whicheventually gives the optimal network structure design and operation strategy. More detailsof each level will be discussed in the following sections.

4.2. Level 1: Approximate model

Given a sequence of well platforms which are to be switched on in order, each well platformis associated with a VEP and the goal of Level 1 is to determine the timings of the VEPs.Only VEPs are included and changes of physical quantities can take place at the VEPs, asshown in figure 6.

Continuous variables, T (n), are defined to determine the timings of VEPs and theyincrease monotonicly as follows:

T (n0) ≤ T (n1) ≤ · · · ≤ T (nlast) (60)

Variables are defined for each VEP and model equations are formulated for each VEP,including the economic calculations, as will be described in the following section. Becausethese VEPs do not necessarily correspond to the specific times given to perform economicevaluations, the formulation at this level is an approximation of the original problem in termsof the yearly based economic calculations. It is for this reason that we call the formulationat Level 1 an approximate model.

Figure 6. Approximate model at Level 1.

78 LIN AND FLOUDAS

4.2.1. Reservoir model. The gas field property equations are formulated at each VEP, asillustrated by Eqs. (61) and (62) for Categories 1 and 2.

Pr ( f, n) = ca, f G2p( f, n) + cb, f Gp( f, n) + Prinit, f (61)

[Pr ( f, n) − p f ]2 − P2s ( f, n) = ra, f Q3

d f ( f, n) + rb, f Q2d f ( f, n)

+ rc, f Qd f ( f, n) + rd, f (62)

Note that an additional nonlinearity has to be introduced to calculate the cumulativeproduction from the gas flow rate since the duration from one VEP to the next is variable,as shown by Eq. (63).

Gp( f, n) = Gp( f, n − 1) + Qf ( f, n) · [T (n + 1) − T (n)] · dpy (63)

4.2.2. Surface model. The surface model is also formulated for each VEP to determinethe network structure and maintain the material and pressure balances. Compared to thediscrete-time model, there are no such binary variables to determine whether a well platformis to be switched on or not because the switch-on’s of the well platforms are associated withthe VEPs explicitly. The number of binary variables concerning the network structure isthus greatly reduced and a lot of logical constraints are simplified or even eliminated. Thelogical constraint to determine the network structure is now formulated by Eq. (64), whichstates that if a well platform is switched on, one and only one of the connections from thiswell platform to the processing platform or another well platform, which is switched onbefore or at the same time, should be activated.

1 =∑

wp′,nwp′ ≤nwp

bcn(wp, wp′) +∑

pp,n pp≤nwp

bcn(wp, pp) (64)

where nwp and n pp are event points associated with the switch-on of (wp) and (pp),respectively.

The material balance and pressure balance are ensured at each VEP, as illustrated byEqs. (65) and (66).

Qtot(wp, n) = Qf (wp, n) +∑wp′

Q(wp′, wp, n) (65)

Pin(wp, n) = Ps(wp, n) − Pchk(wp, n) (66)

4.2.3. Economic model. Economic quantities are calculated for each VEP to representthe value for the time period from the current VEP to the next one. To facilitate this, somerelevant data need to be represented as explicit continuous functions of time. For example,the gas price is originally given for each year, as shown in figure 7, while we use a piecewiselinear function (Floudas, 1995) to approximate it for any time within the horizon based on


Figure 7. Gas prices.

the original data, as represented by Eq. (67).

pg =

α1τ + α0, if 0 ≤ τ ≤ τ1;

β1τ + β0, if τ1 ≤ τ ≤ τ2;

γ1τ + γ0, if τ2 ≤ τ ≤ τ3.

(67)

4.2.3.1. Revenues. Revenues calculated at each VEP correspond to the time period fromthe current VEP to the next one. The prices of liquid products are given as an explicitfunction of time and thus can be used directly. However, the prices of gas products areapproximated with a piecewise linear function based on linear regression from the datapoints given for successive years, as described previously. Then the revenues are based onthe price at the middle point between two consecutive VEPs. Furthermore, not only theproduction rate but also the duration from the current VEP to the next one are variable,which gives rise to nonlinearities in the calculation of the revenue values, as shown by thefollowing equations.

Rg(n) ={

φn · Qshr(n) + ρn ·∑

f

l f Qf ( f, n) − ChBS,1(n) − ChBS,2(n)

}

· [T (n + 1) − T (n)] (68)

where φn and ρn correspond to parameter values at the middle point between VEPs (n) and(n + 1).

4.2.3.2. Capital expenditures. The capital expenditure calculated at each VEP includesthe costs on installation of those well platforms, compressor and connections that will beswitched on at the next VEP, and on corresponding well development. Each cost item is

80 LIN AND FLOUDAS

discounted with the discount rate(s) at specific moment(s) within the investment horizon toapproximate the original yearly based economic model. Since the discount rate is given asan exponential function of time, it leads to nonlinear terms in these equations. For example,the investment horizon for well platforms is three years and correspondingly the discountedcapital cost can be calculated as follows.

Xcd (wp, nwp − 1) = 1

3Ccwp ·

[1

1.12T (nwp)−2 + 1

1.12T (nwp)−1 + 1

1.12T (nwp)

](69)

4.2.3.3. Operating expenditures. Operating costs are calculated for each VEP to representthose from the current VEP to the next one. They involve only those functional units thathave been switched on before or at the current VEP. Both the production rate and theduration between two consecutive VEPs are relevant to this cost and therefore bilinearterms of continuous variables are resulted in these equations, as summarized by Eq. (70).

Xo(n) = {c1 · Q(n) + c0} · [T (n + 1) − T (n)] (70)

4.2.3.4. Taxes. Taxes calculated for each VEP also correspond to the values from thecurrent VEP to the next one. We apply the same rules as those used in the yearly basedmodel and these equations remain linear, as represented by Eq. (71).

Tax(n) = δr · Rg(n) + δc · Xc(n) + δo · Xo(n) + δ0 (71)

4.2.3.5. Royalties. Royalties are also calculated for each VEP, according to the multi-phase system, as shown by Eqs. (72)–(75). It should be noted that they are formulated ateach VEP only for those gas fields that have been switched on. Therefore, compared tothe discrete model, the number of binary variables introduced to calculate the royalties aresignificantly reduced.

Ryl(n) = max{rgk · Rg(n), rnk · Rn(n)} (72)

if POs(n) + Rtcum(n) · rak ≤ 0, k = 1, 2, . . . , K (73)

where

POs(n) =∑n′≤n

λrn′ Rg(n′) +∑n′≤n

λcn′ Xc(n′) +∑n′≤n

λon′ Xo(n′) + λ0n (74)

and

Rtcum(n) =∑n′≤n

max{0, POs(n′)} (75)

where λrn′ , λcn′ , and λon′ correspond to parameter values at the middle point between VEPs(n′) and (n′ + 1); λ0n corresponds to the parameter value at the middle point between VEPs(n) and (n + 1).


4.2.3.6. Objective: Net present value. After revenues, capital costs, operating costs, taxesand royalties are calculated, the cash flow can be evaluated for each VEP and then the netpresent value is obtained by summing up the cash flows over all VEPs with discounts takeninto account properly, as shown by Eq. (76). Note that the capital costs have been discountedas described previously.

NPV =∑

n

{{Rg(n) − [Xo(n) + Tax(n)] − Ryl(n)} · 1

1.12T (n)+T (n+1)

2

− Xcd (n)

}− DCF p (76)

The equations described above form the approximate model at Level 1. It results in a smallMINLP problem due to the small number of EPs and is solved once to give the approximatetiming of each VEP, which is used to determine an initial sequence of FEPs and VEPs forLevel 2.

4.2.4. Illustrative example. Let’s consider a problem which includes 1 producing platform,PP, 3 well platforms in Category 1, WP1, WP2, and WP3, and 2 well platforms in Category 2,WP4 and WP5. PP, WP1, WP2, and WP3 start production from the very beginning, whileWP4 and WP5 are to be switched on in sequence after three years. The objective is tomaximize the NPV in a time horizon of 7 years. To apply the approximate model at Level 1,a sequence of 4 VEPs is constructed. The existing PP, WP1, WP2, and WP3 are linked tothe first VEP and the timing of this VEP is set to time 0. WP4 and WP5 are associatedwith the second and third VEPs, respectively. The last VEP is set to the end of the wholetime horizon. The reservoir model, surface model, and economic model are formulated asdescribed previously. By solving the resulting MINLP problem, it is found that WP4 is tocome online at time instance 3.0 and WP5 at time instance 5.1, as shown in figure 8.

4.3. Level 2: Accurate model

At this level, we add the FEPs to perform economic calculations accurately on a yearlybasis as required and call the model an accurate one in contrast to the approximate model atLevel 1. The timings of FEPs are set as required for economic calculations, for example, atthe end of each year; while the timings of VEPs, which determine when to switch on the well

Figure 8. Level 1 solution to the illustrative example.

82 LIN AND FLOUDAS

Figure 9. Accurate model at Level 2.

platforms are to be refined. According to the solution from Level 1, which gives the relativepositions of the VEPs among the FEPs, an initial sequence of EPs can be determined. Thegoal of Level 2 is to start with this initial sequence, solve the model iteratively until theoptimal solution is found.

Changes of network operation can be due to either configurational changes, for example,addition of a well platform, or simply time effect. The former is represented at the VEPsand the latter can take place at the FEPs, as illustrated by figure 9.

The relationship between the timings of EPs is as follows:

τn f ≤ T (n) ≤ τn f +1, if n f ≤ n ≤ n f + 1 (77)

4.3.1. Reservoir model. The equations representing reservoir behavior are formulated ateach EP, similar to Eqs. (61)–(63), except that the timings for FEPs are known.

4.3.2. Surface model. The logical constraints concerning the network structure are for-mulated only for VEPs that are associated with the switch-on’s of well platforms and com-pressor, as shown by Eq. (64). While the material balance and pressure balance equationsare written for each FEP and VEP, as illustrated by Eqs. (65) and (66).

4.3.3. Economic model. The economic calculations are performed only at the FEPs sincethey are only required on a yearly basis. However, between two consecutive FEPs, theremay be any number of VEPs, therefore these equations also involve the variables definedfor the VEPs.

4.3.3.1. Revenues. Revenues are evaluated based on the average production levels, whichdepend on the gas flow rates and the timings of VEPs, as illustrated by Eqs. (78)–(80).

Qashr(n f ) = Qshr(n f ) · [T (nv) − τn f

] + Qshr(nv) · [τn f +1 − T (nv)

],

if n f < nv < n f + 1 (78)

Rgas(n f ) = φn f · Qashr(n f ) (79)

ChBS,i (n f ) = υi,n f · max{0, qBS,i − Qashr(n f )}, i = 1, 2. (80)


where Qashr(n f ) is the average flow rate to the shore between fixed event points (n f ) and(n f + 1), and Rgas(n f ) is the revenue from sales of gas between fixed event points (n f ) and(n f + 1).

4.3.3.2. Capital expenditures. The capital cost for each item is distributed uniformlythrough out the corresponding investment horizon. Therefore each of the involved yearsaccount for a fraction of the entire cost proportionally. For example, for a well platform,of which the investment horizon is three years, if the associated VEP (nwp) lies betweenconsecutive FEPs (n f ) and (n f + 1), the capital cost is calculated as follows:

Xc(wp, n f ) = T (nwp) − τn f

3· Ccwp (81)

Xc(wp, n f − 1) = 1

3· Ccwp (82)

Xc(wp, n f − 2) = 1

3· Ccwp (83)

Xc(wp, n f − 3) = τn f +1 − T (nwp)

3· Ccwp (84)

4.3.3.3. Operating expenditures. As represented by Eq. (85), the operating costs calculatedat each FEP are linear functions of the corresponding average processing levels betweenconsecutive FEPs.

Xo(n f ) = c1 · Qa(n f ) + c0 (85)

where Qa(n f ) is the average flow rate between event points (n f ) and (n f + 1).

4.3.3.4. Taxes. As represented by Eq. (86), in overall, tax values linearly depend on thegross revenues, capital costs and operating costs.

Tax(n f ) = δr · Rg(n f ) + δc · Xc(n f ) + δo · Xo(n f ) + δ0 (86)

4.3.3.5. Royalties. Royalties are also calculated at each FEP according to the multi-phasesystem, as shown by Eqs. (87)–(90).

Ryl(n f ) = max{rgk · Rg(n f ), rnk · Rn(n f )} (87)

if POs(n f ) + Rtcum(n f ) · rak ≤ 0, k = 1, 2, . . . , K (88)

where

POs(n f ) =∑

n′f ≤n f

λrn′fRg(n′

f ) +∑

n′f ≤n f

λcn′fXc(n′

f ) +∑

n′f ≤n f

λon′fXo(n′

f ) + λ0n f (89)

84 LIN AND FLOUDAS

and

Rtcum(n f ) =∑

n′f ≤n f

max{0, POs(n′f )} (90)

4.3.3.6. Objective: Net present value. Cash flows are evaluated for all FEPs and then thenet present value is calculated by summing up the discounted cash flows over all FEPs, asshown by Eq. (91).

NPV =∑n f

{Rg(n f ) − [Xc(n f ) + Xo(n f ) + Tax(n f )] − Ryl(n f )} · dn f − DCF p

(91)

The equations described above form the accurate model at Level 2. The resulting MINLPproblem may need to be solved iteratively. We start with the initial sequence of EPs deter-mined by the approximate model at Level 1. If any of the timing variables of VEPs, T (n),hits the upper or lower bounds or if two VEPs coincide, the relative positions of the involvedEPs are exchanged and the model is solved again. This process stops when no bounds arehit or exchange of EPs leads to the same solution. The procedure will be illustrated by anexample in the following section.

It should be noted that by exchanging coinciding VEPs, the sequence for well platformsand other functional units to switch on can be modified through this iterative procedure. Inprinciple, the sequence derived from the results at Level 1 only serves as a starting pointto find the optimal sequence. However, the iterative procedure is based on the assumptionthat the objective value of the solution converges to the optimal value as the sequence ofEPs converges to the optimal sequence. If the model at Level 1 approximates the accuratemodel at Level 2 very well, the initial sequence derived according to the solution at Level1 and used at Level 2 can be very close to the optimal sequence. Then this assumption willhold well and it will take only a small number of iterations to reach the final solution. Inshort, the better the approximation at Level 1 is, the easier it is to find the optimal solutionat Level 2. Therefore, the degree of accuracy of the model at Level 1 is very importantand various aspects of the formulation at Level 1 have been explored to achieve the bestapproximation, as presented in Section 4.2.

4.3.4. Illustrative example. This example follows the one for Level 1. Based on the resultsfrom Level 1, two VEPs are combined with eight FEPs to form an initial sequence of tenEPs to be used in the accurate model. The first VEP, which is associated with WP4, is locatedbetween the FEPS corresponding to year 3 and year 4. Similarly, the second VEP, which isassociated with WP5, is located between the FEPS corresponding to year 5 and year 6. Thereservoir model, surface model, and economic model are then formulated as described inthe previous section. The resulting MINLP problem is solved iteratively as follows.

At the first iteration, the timing variables for the two VEPs both hit their lower bounds,as shown in figure 10, and the resulting NPV is 499.62 units. It is specified that no newwell platform can be switched on before year 3, therefore the first VEP can not be moved;while the second VEP is relocated to between year 4 and year 5. The model is solved with


Figure 10. Iteration 1 for the illustrative example at Level 2.

Figure 11. Iteration 2 for the illustrative example at Level 2.

this new sequence of EPs at the second iteration. Then in the solution, the two VEPs hittheir lower bounds again, as shown in figure 11 and the corresponding NPV increased to518.40 units. This turns out to be the best solution since further moving the second VEP tobetween year 3 and 4 leads to the same solution.

It should be pointed out that due to the significant reduction of the number of binaryvariables, both the approximate model at Level 1 and the accurate model at Level 2 can besolved much more efficiently, as compared to the yearly based discrete-time formulation.

5. Computational results

5.1. Case Studies

The proposed approach is applied to a number of case studies with variations of the timehorizon and the number of well platforms. All the models are implemented with GAMS(Broke et al., 1988) and solved with DICOPT (GAMS Development Corporation, 1999).DICOPT is an MINLP solver based on the extensions of the outer-approximation algorithmwith equality relaxation (Viswanathan and Grossmann, 1990). The NLP and MILP sub-problems are solved with CONOPT2 and CPLEX, respectively.

5.1.1. Case Study 1. This is the illustrative example we presented in the previous section.Within the first 7 years, in addition to three well platforms in Category 1, WP11–WP13,which start production from the very beginning, three in Category 2, WP21–WP23, areconsidered. The network superstructure is as shown in figure 12. PP represents the existingproducing platform. Solid lines are connections already in use and dashed lines are thosethat potentially can be built.

86 LIN AND FLOUDAS

Figure 12. Network superstructure for case studies 1 and 2.

This is a relatively short time horizon. We assume that the capital cost is the dominantfactor in determining which well platforms to switch on and in what order. The productioncapacities of the gas fields are not as important as the capital costs because it is relativelyeasy to sustain the production level in this short time horizon. Table 3 shows the total capitalcost for developing each gas field in Category 2. The order to switch on the well platformsis in the order of ascending costs.

The solution requires introduction of two new well platforms. WP21 is switched on attime instance 3.0 and connected to WP12; WP22 is then switched on at time instance 4.0

Table 3. Capital costs for development of gas fields in Category 2.

Well platform WP21 WP22 WP23

Platform 26.4 90.5 90.5

Pipeline 11.0 74.8 46.9

Well development 71.5 107.25 143

Total cost 108.9 272.55 280.4


Figure 13. Network structure and production profile for Case Study 1.

and connected to PP directly. The objective of NPV is 518.40 units. Figure 13 shows theresulting network structure and the production profile of the flow rates to the shore andthrough the well platforms over time. Note that the overall production rate is at the fullcapacity of the producing platform throughout the whole time horizon of 7 years.

5.1.2. Case Study 2. In this case study, we extend the time horizon to 11 years andintroduce the compressor. The solution requires to introduce all three new well platforms inCategory 2. WP21 is again switched on at time instance 3.0 and connected to WP12; WP22

and WP23 are switched on at time instances 5.3 and 9.2, respectively, and both are connectedto PP directly. The compressor is added at time instance 7.05. This leads to the networkstructure and production profile shown in figure 14 and the resulting NPV is 865.29 units.Note that the overall production rate is lower than that in the solution of Case Study 1 dueto fuel consumption by the compressor and the 10 year plateau requirement, and that theoverall rate drops in the 11th year after the production plateau is maintained for the first10 years.

5.1.3. Case Study 3. Nine more well platforms in Category 3, WP31–WP39, are introducedand the number of all well platforms considered increases to 15 in total. The networksuperstructure is shown in figure 15.

When determining the order to switch on the well platforms, in addition to the relevantcapital costs, we now also take into account other important factors such as production ca-pacity of each gas field and the network connectivity. The capacity of the existing producingplatform is equal to the sum of those of the three existing well platforms in Category 1. Asthe production of the gas fields in Category 1 decreases over time, new well platforms thateither connect to the producing platform directly or connect to the existing well platformsneed to be switched on to sustain the overall production level. As shown in figure 15, onlya small set of connections are allowed, with each new well platform having only one or

88 LIN AND FLOUDAS


Figure 15. Network superstructure for Case Studies 3 and 4.



two candidate connections to other well platforms and/or the producing platform. By analy-zing the network connectivity and then the capital costs as well as the gas field productioncapacities, we are able to propose a set of well platforms which are to be switched on andtheir initial relative order.

Compared to the solution obtained in Case Study 2, a very different design and operationstrategy is generated, as shown in figure 16. Specifically, instead of all three well platformsin Category 2, now one in Category 2 and two in Category 3 are selected, Both WP21

and WP38 are switched on at time instance 3.0 and they are connected to WP12 and WP11,respectively. As the third new well platform, WP36 is then switched on at time instance 6.0and connected to PP directly, with the compressor added at the same time. The NPV isincreased significantly to 908.04 units, mainly because of the lower costs of the selectedwell platforms.

5.1.4. Case Study 4. The time horizon is further extended to 15 years. We assume thatwell platforms in Category 2 are built prior to those in Category 3. The solution obtainedrequires to introduce five new well platforms in total, including three in Category 2 and twoin Category 3. WP21–WP23 are switched on at time instances 4.0, 5.0 and 7.0, respectively;WP21 is connected to WP12 and the other two are both connected to PP. WP38 and WP36

follow at time instances 9.0 and 12.0, respectively, and are connected to WP11 and WP13,respectively. The compressor comes in at time instance 7.0. This leads to the networkstructure and production profile shown in figure 17 and the resulting NPV is 1047.95 units.Note that the overall production rate drops considerably towards the end of the time horizonunder consideration due to border effects.

5.2. Summary and comparisons of results

Table 4 summarizes model properties and computational results for the case studies pre-sented in the previous section. As compared with the discrete-time model, the proposedapproach leads to significant reduction of the size of the resulting MINLP problems.

90 LIN AND FLOUDAS


The numbers of binary variables, continuous variables and constraints are all substan-tially smaller than those of the discrete-time model for all four case studies. This leads tosignificant reduction of the computational efforts required. For Case Studies 1 & 2, the CPUtime required by our approach is less, while for Case Studies 3 & 4, for which even the

Table 4. Summary and comparisons of computational results.

Model NEP NBV NCV NC CPU (sec.) NPV

Case 1 (6 WPs, 7 years)

Discrete-time model 55 1534 2415 59.3 518.40

This approach Level 1 4 18 392 484 9.2 445.5

Level 2 10 27 965 1201 (10–18) × 2 518.40


Discrete-time model 95 1721 2453 596 864.57

This approach Level 1 6 33 664 833 87 739.89

Level 2 16 51 1601 2048 (18–50) × 4 865.29


Discrete-time model 800 8800 15000 ∗ N/A


Level 2 16 117 2182 2927 (320–420) × 2 908.04


Discrete-time model 1100 12000 21000 ∗ N/A


Level 2 22 149 3159 4200 (360–500) × 2 1047.95

NEP: Number of EPs; NBV: Number of binary variables; NCV: Number of continuous variables; NC: Numberof constraints; NPV: Net present value. ∗: First MILP sub-problem can’t be solved after 512 M memory runsout. All models implemented and run with GAMS/DICOPT on an HP J-2240 workstation.


first MILP sub-problem of the discrete-time model can not be solved with 512 M memoryavailable on an HP J-2240 workstation, our formulation can still be solved very efficiently.Furthermore, the objective function value is also improved in Case Study 2.

6. Conclusions

In this work, we study the long-term planning problem for integrated gas field development.Decisions foster around the network structure and the production profile over time. Nonlin-ear reservoir behavior and complex economic calculations are taken into account explicitly.A novel continuous-time modeling and optimization approach is proposed, applying theconcept of event point which allows well platforms to be switched on at any time in thehorizon continuously. A two-level formulation is developed, in which Level 1 approxi-mates the economic calculations and Level 2 formulates them accurately on a yearly basis.Computational results on several case studies demonstrate the effectiveness of the proposedapproach. Smaller size MINLP problems with fewer binary variables can be solved muchmore efficiently and slightly better objective value is achieved for one of the case studies.

The MINLP problems in this work are nonconvex due to the nonconvexities arising fromthe reservoir model and the continuous-time formulation. As such, the proposed approach inits current form cannot guarantee convergence to the global optimal solution. However, themathematical structure of the proposed models can be further investigated so as to developrigorous deterministic global optimization methods through the principles described inFloudas (2000). Our current research includes the development of new approaches forthe global solution of the MINLP problems, as well as mathematical formulation for theswitch-on order of well platforms and the solution of actual problems with larger sizes.

Notation

α1, α0: Parameters, coefficients of linear function correlating gas price with time insegment 1.

αpp,cs : Parameter, fuel consumption per unit of volumetric flowrate through compressorstage (cs) at producing platform (pp).

β1, β0: Parameters, coefficients of linear function correlating gas price with time insegment 2.

βwp′,wp: Parameter, pressure drop from well platform (wp′) to well platform (wp) per unitof flowrate from (wp′) to (wp).

γ1, γ0: Parameters, coefficients of linear function correlating gas price with time in seg-ment 3.

γwp: Parameter, pressure drop across well platform (wp) per unit of flowrate through (wp).δpp,cs : Parameter, compression ratio of compressor stage (cs) at producing platform (pp).δr , δc, δo, δ0: Parameters, coefficients in the calculation of tax from gross revenue, capital

costs, and operating costs.η1, η2: Parameters, coefficients of the quadratic function correlating the pressure drop from

producing platform outlet to the shore and the corresponding flowrate.

92 LIN AND FLOUDAS

λrn′ , λcn′ , λon′ , λ0n: Parameters, coefficients in the calculation of simple payout at eventpoint (n) from gross revenue, capital costs, and operating costs at previous event points.

λrn′f, λcn′

f, λon′

f, λ0n f : Parameters, coefficients in the calculation of simple payout at fixed

event point (n f ) from gross revenue, capital costs, and operating costs at previous fixedevent points.

λr t ′ , λct ′ , λot ′ , λ0t : Parameters, coefficients in the calculation of simple payout in year (t)from gross revenue, capital costs, and operating costs of previous years.

ρn: Parameter, coefficient of total liquid produced for the calculation of gross revenue atevent point (n), related to liquid price at the middle point between event points (n) and(n + 1).

ρt : Parameter, coefficient of total liquid produced for the calculation of gross revenue inyear (t), related to liquid price in year (t).

τ : Continuous variable, time.τ1, τ2, τ3: Parameters, time instances specifying segments of time for piecewise linear func-

tion correlating gas price with time.τn f : Parameter, timing of fixed event point (n f ).υi,t : Parameter, back stop charge per unit of the difference between the flowrate to the shore

and contractual rate (i) in year (t) if the former is below the latter.φn: Parameter, linear coefficient of flowrate to the shore in the calculation of gross revenue

at event point (n), related to gas price at the middle point between event points (n) and(n + 1).

φt : Parameter, linear coefficient of the flowrate to the shore for the calculation of grossrevenue in year (t), related to gas price in year (t).

b1, b2, b3, b4: Parameters, coefficients in the correlation between Pb (intermediate variablefor reservoir property correlations) and field deliverability of a gas field in Category 3.

b10y(t): Binary variable, = 1 if the overall production level in year (t) is increased from theprevious year; = 0 otherwise.

bBS,1(t), bBS,2(t): Binary variables, = 1 if the flowrate to the shore in year (t) is belowcontractual rate 1 and 2, respectively; = 0 otherwise.

bcn(wp, pp): Binary variable, = 1 if the connection from well platform (wp) to producingplatform (pp) is installed; = 0 otherwise.

bcn(wp, pp, t): Binary variable, = 1 if the connection from well platform (wp) to producingplatform (pp) is switched on in year (t); = 0 otherwise.

bcn(wp, wp′): Binary variable, = 1 if the connection from well platform (wp) to wellplatform (wp′) is installed; = 0 otherwise.

bcn(wp, wp′, t): Binary variable, = 1 if the connection from well platform (wp) to wellplatform (wp′) is switched on in year (t); = 0 otherwise.

bryl(t, k): Binary variable, = 1 if royalty is in phase (k) in year (t); = 0 otherwise.bsw(wp, t): Binary variable, = 1 if well platform (wp) is switched on in year (t); = 0

otherwise.c1, c0: Parameters, coefficients of linear function correlating flowrate with operating cost.ca, cb: Parameters, coefficients in the correlation between reservoir pressure and cumulative

production of a gas field in Category 1 or 2.Ccwp: Parameter, total capital cost of well platform (wp).


C f (pp, cs): Continuous variable, fuel consumption in compressor stage (cs) at producingplatform (pp).

ChBS,1, ChBS,2: Continuous variables, charges for back stop 1 and 2, respectively.cs: Index, compressor stage.d1, d2, d3: Parameters, coefficients in the correlation between Pf , Pb (intermediate variables

for reservoir property correlations), and surface pressure of a gas field in Category 3.dn f : Parameter, discount rate at fixed event point (n f ).dt : Parameter, discount rate of year (t).DCF p: Parameter, discounted cash flow prepaid.dpy: Parameter, number of days per year.f : Index, gas field.G: Parameter, used in reservoir property correlations for a gas field in Category 3.Gp: Continuous variable, cumulative production of a gas field.G p1: Continuous variable, intermediate variable for reservoir property correlations for a

gas field in Category 3.i1, i2, i3: Parameters, coefficients in the correlation between field deliverability, reservoir

pressure, and Pf (intermediate variable for reservoir property correlations) of a gas fieldin Category 3.

k: Index, phase of royalty.K : Parameter, number of phases of royalty.l f : Parameter, liquid yield of gas field ( f ).m: Parameter, number of wells in a gas field.M : Parameter, positive big number.n: Index, event point.n f : Index, fixed event point.n pp: Index, event point associated with the switch-on of producing platform (pp).nv: Index, variable event point.nwp: Index, event point associated with the switch-on of well platform (wp).NPV: Variable, net present value.p: Parameter, to adjust reservoir pressure in the correlation between reservoir pressure,

surface pressure, and field deliverability of a gas field in Category 1 or 2.p1, p2, p3, p4: Parameters, coefficients in the correlation between reservoir pressure and

G p1 (intermediate variable for reservoir property correlations) of a gas field in Category 3.Pb: Continuous variable, intermediate variable for reservoir property correlations for a gas

field in Category 3.Pbst(pp, cs): Continuous variable, pressure boost across compressor stage (cs) at producing

platform (pp).Pchk : Continuous variable, choke pressure.Pf : Continuous variable, intermediate variable for reservoir property correlations for a gas

field in Category 3.pg: Continuous variable, gas price as function of time.Pin: Continuous variable, inlet pressure.Pout: Continuous variable, outlet pressure.Pr : Continuous variable, reservoir pressure of a gas field.

94 LIN AND FLOUDAS

Pr,ini t : Parameter, initial reservoir pressure of a gas field.Ps : Continuous variable, surface pressure of a gas field.Pshr: Parameter, pressure at the shore.POs : Continuous variable, simple payout.pp: Index, producing platform.Q: Continuous variable, flow rate.Q(wp, pp): Continuous variable, flow rate from well platform (wp) to producing platform

(pp).Q(wp, wp′): Continuous variable, flow rate from well platform (wp) to well platform

(wp′).qBS,1, qBS,2: Parameters, contractual rates.QBS,1, qBS,2: Continuous variables, the differences between the flowrate to the shore and

contractual rate 1 and 2, respectively, if the former is below the latter.Qbyp(pp, cs): Continuous variable, flow rate bypassing compressor stage (cs) at producing

platform (pp).Qcmp(pp, cs): Continuous variable, flow rate through compressor stage (cs) at producing

platform (pp).Qd f : Continuous variable, field deliverability.Qf : Continuous variable, gas field flow rate.Qnet(pp): Continuous variable, net flow rate out of producing platform (pp).Qshr: Continuous variable, flow rate to the shore.qmax

shr : Parameter, the maximum possible flowrate to the shore.Qtot(pp): Continuous variable, total flow rate into producing platform (pp).Qtot(wp): Continuous variable, total flow rate through well platform (wp).Qmax

wp : Parameter, maximum possible flowrate through well platform (wp).Qa(n f ): Continuous variable, average flow rate between event points (n f ) and (n f + 1).Qashr(n f ): Continuous variable, average flow rate to the shore between event points (n f )

and (n f + 1).r : Parameter, used in reservoir property correlations for a gas field in Category 3.ra, rb, rc, rd : Parameters, coefficients in the correlation between reservoir pressure, surface

pressure, and field deliverability of a gas field in Category 1 or 2.Rg: Continuous variable, gross revenue.Rgas: Continuous variable, revenue from sales of gas.rgk : Parameter, percentage of gross revenue to pay for royalty in phase (k).Rn: Continuous variable, net revenue.rnk : Parameter, percentage of net revenue to pay for royalty in phase (k).rak : Parameter, rate used to determine whether royalty is in phase (k).Rtcum: Continuous variable, cumulative return allowance.Ryl: Continuous variable, royalty to pay.Rylg(t, k): Continuous variable, royalty to pay in phase (k) in addition to the previous phase

based on the gross revenue.Rylmax

g,k : Parameter, maximum possible amount of royalty in phase (k) in addition to theprevious phase based on the gross revenue.

Rylgtot(t): Continuous variable, overall royalty to pay in year (t) based on the gross revenue.


Ryln(t, k): Continuous variable, royalty to pay in phase (k) in addition to the previous phasebased on the net revenue.

Rylmaxn,k : Parameter, maximum possible amount of royalty in phase (k) in addition to the

previous phase based on the net revenue.Rylntot(t): Continuous variable, overall royalty to pay in year (t) based on the net revenue.s: Parameter, shrinkage and fuel fraction across a producing platform.t, t ′: Indices, years.T (n): Continuous variable, timing of event point (n).Tax: Continuous variable, tax to pay in each year.wp, wp′: Indices, well platforms.Xc: Continuous variable, capital cost.Xc(wp, n f ): Continuous variable, capital cost of well platform (wp) invested between fixed

event points (n f ) and (n f + 1).Xcd : Continuous variable, discounted capital cost.Xo: Continuous variable, operating cost.

Acknowledgments

The authors gratefully acknowledge support from the National Science Foundation and theExxon-Mobil Corporation.

References

A. Brooke, D. Kendrick, and A. Meeraus, GAMS: A User’s Guide, South San Francisco, CA, 1988.C. A. Floudas, Nonlinear and Mixed-Integer Optimization, Oxford University Press, 1995.C. A. Floudas, Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic

Publishers, 2000.GAMS Development Corporation, GAMS—The Solver Manuals, 1999.D. Haugland, A. Hallefjord, and H. Asheim, “Models for petroleum field exploitation,” Eur. J. Oper. Res. vol. 37,

pp. 58–72, 1988.M. G. Ierapetritou and C. A. Floudas, “Effective continuous-time formulation for short-term scheduling:

1. multipurpose batch processes,” Ind. Eng. Chem. Res. vol. 37, pp. 4341–4359, 1998a.M. G. Ierapetritou and C. A. Floudas, “Effective continuous-time formulation for short-term scheduling:

2. continuous and semi-continuous processes,” Ind. Eng. Chem. Res. vol. 37, pp. 4360–4374, 1998b.M. G. Ierapetritou, T. S. Hene, and C. A. Floudas, “Effective continuous-time formulation for short-term

scheduling: 3. multiple intermediate due dates,” Ind. Eng. Chem. Res. vol. 38, pp. 3446–3461, 1999.R. R. Iyer, I. E. Grossmann, S. Vasantharajan, and A. S. Cullick, “Optimal planning and scheduling of offshore

oil field infrastructure investment and operations,” Ind. Eng. Chem. Res. vol. 37, pp. 1380–1397, 1998.S. A. van den Heever and I. E. Grossmann, “An iterative aggregation/disaggregation approach for the solution of

a mixed-integer nonlinear oilfield infrastructure planning model,” Ind. Eng. Chem. Res. vol. 39, pp. 1955–1971,2000.

S. A. van den Heever, I. E. Grossmann, S. Vasantharajan, and K. Edwards, “Integrating complex economicobjectives with the design and planning of offshore oilfield infrastructures,” Comp. Chem. Engng. vol. 24,pp. 1049–1055, 2000.

S. A. van den Heever, I. E. Grossmann, S. Vasantharajan, and K. Edwards, “A lagrangean decomposition heuristicfor the design and planning of offshore hydrocarbon field infrastructures with complex economic objectives,”Ind. Eng. Chem. Res. vol. 40, pp. 2857–2875, 2001.

J. Viswanathan and I. E. Grossmann, “A combined penalty function and outer approximation method for minlpoptimization,” Comp. Chem. Engng. vol. 14, pp. 769–782, 1990.

Documents

A Novel Continuous-Time Modeling and Optimization