Optimal production of oil and gas

105

Engineering

Costs and Production Economics, 9 (1985) 105-l 11 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

OPTIMAL PRODUCTION OF OIL AND GAS T.T. Abodunde Organization of the Petroleum

Exporting

Countries

(OPEC),

Vienna

(Austria)

F. Wirl Technical

University

of Vienna,

Vienna

(Austria)

ABSTRACT This paper presents a model of the class of exhaustible resource problems, which focuses on the peculiarities of extracting oil and gas in the form of associated and dry gas. One motivation for such an investigation is the phenomenon that a significant amount of natural gas is flared. For example, about 43% of the gross gas production of the Organization of the Petroleum Exporting Countries (OPEC) was flared in 1981, and more than 62% in 19 72. It is supposed that a non-competitive extractor faces a demand system in the two substitutable commodities which the producer attempts to sell. Such an extractor has two production processes at his disposal - extracting crude oil yielding associated gas as a non-storable

by-product and producing gas from dry basins. All resource basins are known and finite (i.e., phenomena related to “deep gas” are not considered). A major characteristic of the programme is that it might be optimal to flare gas and at the same time to increase the sales of gas (non-trivial control), while the dry basins at these times remain untapped. Hence, the assumption of substitution effects between oil and gas can explain the fact that an exhaustible resource might be wasted. The model solution suggests that during the periods of partial flaring, the availability of natural gas will increase. This implies a compliance with current tendencies in world energy markets.

1. INTRODUCTION

be developed, e.g., Heal [21, Dasgupta [l], Stiglitz and Dasgupta [61. However, the notion of a perfect substitute seems an inadequate description for many problems and competition through an imperfect substitute seems more likely in many markets, e.g., in the energy market where oil, gas, coal and uranium are substitutable primary energy carriers. Most models related to the introduction of a perfect substitute imply a discontinuous transition from the resource use to the backstop technology (for an exception see Hoe1 [ 3]), while it is more likely that a

The literature on exhaustible resources has become increasingly important for economists in recent years, but numerous “real world” phenomena still lack a proper treatment. Since Nordhaus’ introduction of the term “backstop” in 1973, many papers have concentrated on depletion strategies if a perfect substitute (synthetic or non-renewable) can The views expressed in this paper are those of the authors, and not necessarily those of the Institutions with which the authors are associated.

0167-188X/85/$03.30

0 1985 Elsevier Science Publishers B.V.

106 resource (such as oil) will be phased out gradually by an imperfect substitute, as happened in the past to wood and, to a certain extent, to coal, in the development of energy consumption patterns. Since natural gas is currently becoming a resource of increasing interest among energy experts, the marketing problems of substitutable, exhaustible resources will be studied in the following. An application might include a strategy for OPEC of pricing both hydrocarbons (oil and gas) in an interrelated framework. Additionally, the production of oil and gas includes some peculiarities because significant amounts of natural gas are in the form of associated gas, which is a by-product of extracting crude oil. If the associated gas cannot be stored, then the producer could find it rational to flare some amounts to keep the oil price at a desired level. Theoretically he could also flare oil to support his revenues from gas sales. This asymmetry will be assumed in the analysis. Hence, two different aspects are integrated into the following framework: introduction of an imperfect and exhaustible substitute fo a resource, which currently covers the major market share, and an explanation for the phenomenon, that significant amounts of gas are flared, e.g., from OPEC’s gross gas production during 1980 some 43 percent was flared. This was more than 62% in 1972 (OPEC Secretariat [ 51). The model tries to represent the most important aspects of the oil and gas production system, i.e., joint production and separate extractions of imperfectly substitutable and non-renewable resources, see Wirl [7]. Although the subsequent analysis will be purely theoretical, hence general, and also applicable to other problems of related mining industries, the assumptions and the terminology stem from the example indicated in the title. The paper proceeds by introducing the model and the assumptions, investigating the major properties of the optimal strategies and outlines finally the

implications and further research required. Since, during this discussion, in particular in Section 3, the use of mathematics and rigorous proofs are omitted, the derivation of the optimal programme is sketched in the Appendix. 2. THE MODEL Consider a non-competitive producer who is confronted with demand for his two products 1 and 2, which we will assume to be oil and gas, respectively, for the sake of convenience. The producer possesses two production processes: one extracting crude oil, yielding also associated gas; and another, producing gas from dry reservoirs. Demand is supposed to exhibit negative own and positive cross price elasticities, of which the latter capture the substitution effects. Let p’ and I?* denote the resource prices and x1 and xz the corresponding sales and let prices and sales be connected through the inverse demand system: P’

= P’h,,

(1)

x2)

(2)

P z = PZ(X,,X,)

This representation is possible if the own price elasticities dominate the cross price elasticities (Jacobi matrix of the system: demand, as a function of prices, has a positive determinant). The above implies that the first derivatives of relations (1) and (2) are negative and that the determinant of the corresponding Jacobi matrix is positive. For ease of exposition, it is supposed that demand does not shift over time and that extraction costs are negligible. Further requirements, which concern the revenue function implied by the relations (1) and (2) and the already indicated assumptions, are summarized in the following. Assumptions

The inverse

demand

system

(1) and (2) is

107 twice continuously differentiable and has a Jacobi matrix with negative elements and a positive determinant. Furthermore, the implied revenue function n(x1,x2), which is defined as: n(x,,x& = x, P'(X,,X*) +x,P*(x,,x*)

when both resources are flared, but at different points in time, are excluded. If the producer’s aim is to maximize the present value of profits from resources sales then he must solve the following optimal control problem :

(3) max

should be concave and the second derivatives of n should be negative. Production costs at the wellhead are negligible for both activities. The assumptions about concavity for each argument and for both together is standard for optimization studies to ensure “interior” solutions, while the negative cross effect: apz - a2 TT =- ap’ t-+x, ax, ax, ax, ax,

az p’ < 0 (4) + x, $&ax, ax, L 2

needs justification: the inequality (4) states, in economic terms, that the gain from selling an additional unit of oil declines if the sales of gas are simultaneously increased. Aside from the above assumptions, it is also useful to suppose that the backstop prices (p1,p2>, i.e., those prices where consumers are neither willing to buy gas or oil are finite. These choke conditions represent breakeven costs for a competitively produced synthetic, or replacement of oil and gas by another resource not dealt with in this model, e.g., coal or uranium. Arithmetically, they can be computed by inserting zero into the righthand side of (1) and (2). Let xl(t) denote the production and sales of oil yielding ox,(t) amounts of associated gas from which v(t) quantities are sold and the remainder is flared. The gas/oil ratio 0 is supposed to be constant, because it is hard to imagine a relation between 19 and calendar time. However, varying 0 in dependence of cumulative extraction could be a reasonable modification of the model. Let z(t) describe the extraction of gas from dry reservoirs so that x*(t) = v(t) + z(t) yields total gas sales. Hence, oil production equals oil sales at any date: which need not be true for gas. This asymmetry is a loss in generality to that extent only, that policies

T

X,(0 a 0

s 0

e-" n(x,W,x,(t))dt

(5)

0
= x,(r),

X(O)

i(t) = z(t)> Z(O)

= 0, X(T)

= R,

= 0, Z(T) = R,

x,(t) = v(t) +2(r)

(6) (7) (8)

where Y > 0 denotes the discount factor, and X(t) and Z(t) describe the evolution of cumulative productions of crude oil and dry gas, which at the dates of depletion must equal the initial resource stocks R, and R2. T denotes the free date of depleting both reservoirs. The use of the terminal manifolds “depletion” together with the non-negativity constraints for the controls avoids the handling of state inequality constraints, because one easily verifies that leaving resources in the ground is suboptimal. 3. OPTIMAL PRODUCTION OIL AND GAS

AND SALE OF

The optimal policy for exploiting and selling oil and gas consists of four distinct strategies: all 0) oil extraction/flaring associated gas no extraction (ii) oil extraction/partial from dry gas flaring reservoirs (iii) oil extraction/no flaring oil production/no flaring/gas production (iv) from dry reservoirs. The evolution over time proceeds exactly in the above sequence, although the process need not start with the first phase, but could already commence with another strategy.

108

t

(I)

’

(II)

Fig. 1. Production

j

(Ii])

(IV)

and sales of oil and gas.

During the phases (i)-(iii) the reserves of nonassociated gas are not tapped, while at the end of the last phase both production plants are shut down at the same data (if backstop This highlights that a prices are finite). simultaneous demand system provides additional manoeuvreability for the producer, which he intends to keep for the whole planning horizon. Hence, he will be interested to find an integrated framework for pricing both competing fuels. The above four strategies differ also in other aspects. In phases (i) and (iii) oil sales decline and gas sales are constant (zero, (i)) or declining, and hence both resource prices must rise. But in phase (ii) gas sales will increase (hence the share of flaring will be reduced) opposed to declining oil extraction. This increased availability of natural gas in times of partial flaring seems to reflect current tendencies in world energy markets. This suggests that the price of gas relative to oil must decline during this phase, although it need not be the case, and that the gas price must fall in absolute terms (it was not possible to clarify in general the price evolutions in this phase). Hence, current attempts to peg the gas price to oil prices could be a serious obstacle to increasing the market share of natural gas. Similarly, the precise characterization of the strategy during the last phase is ambiguous under the generality of the assumptions, except that sales will ultimately decline to zero at the same date. Despite the differences of the above phases, it is worth

noting that all transitions from one phase to the next are continuous, i.e., no abrupt change takes place. For example, sales of gas start smoothly, production from dry gas reservoirs commences with small quantities and also oil and gas sales terminate continuously. Since ultimate production of dry gas is also zero, a bell shaped production profile for dry gas extraction could emerge. The classification of the ambiguous items and the study of other aspects, like demand shifting over time or concrete policy recommendations, demand more stringent assumptions and/or a parametrical model compatible with emperical evidence. Nevertheless, some critical economic conditions can be derived to qualify the different strategies. The marginal revenues play a crucial role here, similar to related marketing problems. Due to the characteristics of the optimality conditions, which imply a separation of the policy, it is sufficient to discuss marginal revenues from oil and total gas sales (i.e., no separate discussion of the marginal revenue relations for associated and non-associated gas is necessary). In other words, dry gas production can be calculated as the residual: z(t) =

max (0,x,(t)

-0x,(t))

The marginal revenues, which represent the gain from selling an additional (small) quantity, consist of three parts: the price of the commodity minus the loss due to price reduction induced by higher sales and also the effect of price deterioration of the substitutable commodity must be subtracted. In the case of oil production, the implications on gas production must be further added. The optimality condition for oil production is the same in all phases:

namely, that marginal revenues must equal the present value from extracting the last barrel and the corresponding amount of asso-

109 ciated gas. Since both resources are exploited jointly and continuously, the corresponding backstop prices must be charged. In other words, along the optimal policy, the producer must be indifferent between selling a marginal unit today or at any other point in time because, on the contrary, a reallocation could increase the profits and would hence contradict optimality. The marginal revenues from gas sales, however, obey more complex rules over time. During the first phase, where all associated gas is flared, the marginal revenues are negative even for negligible sales:

2


(10)

2

the i.e., the gas price cannot compensate induced decline in oil revenues. In the second phase, where gas is sold and only partially flared, the marginal revenues must equal zero : an -_=o ax,

(lla)

i.e., gas itself is not scarce yet. Equation can be rearranged so that:

(11)

(1 lb)

the additional profit in gas trade for the last sold unit (left hand side in (11 b)) must exactly balance the marginal loss in oil trade. In the third phase, the marginal revenues from selling all associated gas are positive for the first time, but still fall short of the shadow price of dry gas reserves: (12)

It is only in the last phase, that the marginal revenues from total gas sales, i.e., associated plus non-associated gas, equal the present value of selling the last unit of gas at tye date of depletion: an -= ax,

e-r(M$

(13)

Fig. 2. Marginal

revenues

of selling gas vs. oil.

Since in the last phase the Hotelling rule applies, which says that marginal revenues must grow in percentages with the rate of interest, the solution appears as a straight line in the marginal revenues’ plane. The complete solution is sketched in Fig. 2. 4. CONCLUSIONS This study indicates that flaring natural gas could be rational and that, at the same time, the availability of natural gas will increase. The reason for this outcome is that selling additional quantities of gas implies that, due to the substitution effects, less oil can be sold or a lower price must be accepted and hence oil revenues will suffer. If now the revenues from an additional unit of available associated gas cannot offset the loss in oil trade, then the producer will rather flare than sell this unit. However, at some date, flaring will come to a halt and the producer will proceed, with some delay, to extract also from the dry gas reservoirs. But oil production continues as long as gas production and the resources are depleted at the same time. Therefore, the second question addressed at the beginning, whether gas is phasing out oil, must be denied, although temporarily the gas share will increase. However, this conclusion is sensitively related to the supposed market structure and, in case the producer is facing competition from other large gas producers, then a gradual transition from oil to gas use is possible and the market

110 penetration of gas could be speeded up by the availability of associated gas. This remark points at potential extensions of the current model in modifying the theoretical framework. Aside from the above indicated alterations of the assumed market structure (see Wirl [7] ), further interesting issues are related to the question of whether should substitute local oil for gas consumption to increase the availability of the easier transportable oil for exports. A substitution similar to this shift in local consumption patterns can be achieved by utilizing gas for reinjection to increase the amount of recoverable oil. These additional uses of associated g.as seem to be relevant additional strategies for OPEC decision makers. Equally important to such theoretical investigations seems the application of a parametrical emperically estimated and model, where additional detail can be incorporated, e.g., demand shifting over time, different costs, etc. Such research could yield some more concrete policy recommendations for OPEC pricing both of its hydrocarbons in an integrated framework. REFERENCES

Monopoly, Oligopoly & Competition. Discussion papers, Techn. University Vienna, Institute of Energy Economic.

APPENDIX Derivation

The optimal as:

Vienna. Stiglitz, J.E. and Dasgupta, P., 1982. Market structure and resource depletion: a contribution to the theory of intertemporal monopolistic competition. Journal of Economic Theory, 28(l): 128-164. 1983. Joint Production of Substitutable, Wirl, F., Exhaustible Resources or: Is Flaring Gas Optimal? Exhaustible Resources which are Imperfect Substitutes:

of the value Hamiltonian problem (5)-(8) is defined

current control

(1)

H = x,(p’+h’)+(z+y)P2+zh2

if the arguments are omitted. Let, in the following, low case indices denote partial derivatives in respect to sales x1 and x2, e.g.: “,

=

an

-,

ax,

,

P,,

= -

azp'

ax, ax,

1

and so on. Application of Pontryagin’s maxiprinciple provides the following mum conditions for an optimal pronecessary gramme, which are also sufficient because His independent of the state variables: (2)

aH - = 7r2 as aH =

Dasgupta, P., 1981. Resource pricing and technical innovations under oligopoly, a theoretical exploration. The Scandanavian Journal of Economics, 83(2): 289317. Heal, G.M., 1979. Uncertainty and the optimal supply Policy for an exhaustible resource. In: R.S. Pindyck (Ed.), Advances in the Economics of Energy & Resources, Vol. 2. JAI Press, Greenwich, CT, pp. 119-147. Hoel, M., 1983. Monopoly resource extractions under the presence of predetermined substitute production. Journal of Economic Theory, 30(l): 201-212. 1973. The allocation of energy Nordhaus, W.D., resources. Brooking’s Papers on Economic Activity, Vol. 3. OPEC Secretariat, 1982. Annual Statistical Bulletin 1981,

of the optimal programme

az

n2

+

(3)

A2 = 0

(4)

HY < 0 implies Hz < 0, hence at these dates y = z = 0. Similarly, since X2 is negative, for interior and positive y the extraction of dry gas must equal zero: z = 0. Therefore, it is sufficient to study the derivatives rrl and 7~~. The motion over time from phase (i) to phase (iv) is a direct consequence from (2) that marginal revenues of oil, but also gas, must rise, because the costates grow exponentially:

X’

= rh=

The corresponding derived by utilizing

(6)

boundary the condition

values are for optimal

111 stopping time T: H(T) = 0 = x,(p’+epZ+h’)+z(pZ+h2)

(7)

At the terminal date, the choice of oil and gas sales must be interior solutions of (2) and (4), because the contrary, that gas sales continue after the oil exploitation, yields a contradiction, if one analyses optimal stopping time for oil sales conditional on this optimal policy for gas sales. However, the algebra of this argument is omitted here. Since both terms in brackets of the relation (7) are non-negative and vanish only for zero productions, terminal sales must equal zero. Therefore, the backstop prices must be charged and, by the maximum principle, the boundary solution for (5) and (6) are given by: -h’(T) = p’ +ep2

(8)

-Al(T) = p*

(9)

Depletion date T can be calculated as the date where the cumulative sales equal the initial reserves. In order to prove that gas sales increase during the interval of partial flaring, differentiate (2) and (3): (10)

already extracted, oil sales (and hence gas sales) decline. Differentiating eqn. (2) and utilizing dxz/dx, = 0 at these dates gives: n,,x, + nlzxz f %n,,x, + 9rr,,X, =

> 0

(11)

Substituting for Rz = 0 jc, , which must hold in this phase (iii) and collecting properly yields: i, Cn,,+ 2n,,e f n,,e*) > 0

(12)

The term between the brackets represents a quadratic form, which by assumption is negative definite, hence i1 must be negative. The continuous penetration of gas sales and the smooth introduction of dry gas are a consequence of the concavity property of the revenue function. Due to the similarity in the arguments, only the first case will be proven below. Suppose the contrary, that gas sales start with a positive level xl > 0, then by the maximum principle (2), oil production must fall to xl from x;. But, by Weierstrass/ Erdman corner condition, the Hamiltonian must still be continuous: n(x;,

0) + h’

x; = n(x:,x;)

+ A1x,+

On the other hand, concavity

(13)

implies: -n,(x;,O)x;(14)

nOc-,CU-n(x:,x:)

>n,(X;,O) (x:-x;,

and combining

(13) and (14) gives:

-(n,(X;,O)+h’)(X:-X;) and matrix inversion together with the assumptions in Section 2 gives the result. The proposition that oil sales decline in the first three phases is quickly verified because, in the first phase, the rise of marginal revenues demands cut backs in oil production and, during the second phase, the relation (10) implies the proposition. Hence it remains to be shown that at dates, where no associated gas is flared but no dry gas is

-rh’

(15)


But inequality (15) is impossible the maximum principle:

because

of

77,+hl >o

(16)

r* cx;, 0) < 0

(17)

The left hand side in (15) is positive and the right hand side negative, hence there is a and xg = xi = 0, and also contradiction, x: =x1.