Optimum exploration and extraction in a petroleum basin

Resources and Energy 8

OPTIhIUM

( 1986)

2 19-230.

North-Holland

EXPLORATION AND EXTRACTION PETROLEUikI BASIN

The Case of Simultaneous Tore NILSSEN

Field Developmen&*

and Arild N. NYSTAD

Chr. Michelsen Inmtutr,

Received November

IN A

N-5036

FantojI.

Nomu_~

1985. linal version received March

1986

In this paper we consider a petroleum basin characterized by a need for simultaneous lield developments. With this assumption we are able to develop a model for calculating the optimum exploratory elthrt. the optimum exploration time. and the optimum plateau production rate. Results from a numerical example are reported. showing how the optimum is alTected by changing condltlons and assumptions. In particular. we analyze the effect of introducmg a dlxtortive tax system into the model.

I. Introduction The purpose of this paper is to invcstigatc a sequential decision problem which arises in connection with the exploration for and extraction of pctrolcum resources. We analyze the three-fold problem of how much one should explore in a given basin, for how long one should cxplorc bcforc dovclopmcnt of the discovcrcd resources is started, and, finally, at what rate the available rcscrvcs should bc extracted. We consider a petroleum basin (i.e., a collection of adjacent petroleum fields having common geological propcrtics) which is characterized by a certain functional relationship between accumulated discovered reserves and accumulated exploratory effort. The basin is also characterized by a need for simultaneous field developments to make it economically viable.’ The literature on production models for the petroleum industry has grown vast over the years, as has the literature on oil and gas exploration. Attempts in the literature to handle both exploration and production in the same model are fewer. Liu and Situnen (1982), Pakravan (1977), Peterson (1978), *Partial Cnanclal support was provided by StatoIl. Thanks to Rolf Wideroe for computer assistance. ‘This assumption is realistic for basins such as Tromsollaket on the continental shelf outside Northern Norway. where mostly gas has been discovered and where therefore economies of scale in design of the transportation system may force the operators to coopcra~c in a set of simultaneous developmcnls.

0165-0572iX6.!S3.50

cj 19X6. Elsevier Science Publishers

B.V. (North-Holland)

“0

T Ntlssen

c.-

wui AN.

Nysrud.

Opttmum

exploration

and extrucrion

and Pindyck (1978) develop models in which the producers must simultaneously determine their (worldwide) exploration activity rates and their production rates, Thus, the main differences between these models and the one presented below are, first, that we here consider one petroleum basin only, and second that we assume that exploration and production are performed in sequence. Thirdly, we consider the case of one price-taking company only and leave out any discussion of the implications of different market conditions. We proceed in section 2 with a detailed description of our model. In section 3 we present a numerical example, including a sensitivity analysis. Here we also analyze the effects of taxation. Some tentative conclusions from this section are summed up in the concluding section 4. 2. Model description We consider a profit-maximizing price-taking oil company (or group of companies)’ that is contemplating an exploration venture in an as yet unexplored petroleum basin. It is assumed that any development and subsequent production of the reserves will take place only after exploration is tcrminatcd. It is further assumed that the amount of discovered rcscrves is an increasing function of the exploratory effort. and that the exploration exhibits diminishing returns. It is also assumed that the unit cost of exploration, i.e., the cost of drilling a well, is an increasing function of the intensity of exploration. Prices arc assumed to rise cxponcntially but only up to some limit imposed by the so-called ‘backstop’ technology. When exploration is finished, the ticld dcvclopments, if any, will start simult~~ncot~siy. Dcvelopment is done instantaneously, an assumption that is madr in order to simplify without distorting the results. Wc assume economics of scale in the tlcvclopmcnt technology. The production is assumed to bc constant for some time. followed by a phase of exponential decline. Production stops when operating costs exccwd gross rcvcnucs. To bc more specific, we assume the comp~~ny to confront the following maximization problem: IL

F,;:H=

-i

C,e-“df+

~(p(r+r,)y(r)-C,)e-“dr-C,]e-rfL,

(I)

where x is the net present value of the venture: fE is the exploration time; CE is the rate of exploration costs, or the exploratory effort rate measured in money terms; 7” is total duration of production; p(t) is the petroleum price at time r from start of exploration; q(t) is the production at time t from production start; Co and Cn are operating and investment costs, respectively; ~Thr~~~h~ut the p;tpcr. ‘company’ n-scans ‘company or group ol compltnk’, cooperation among comp;:nics if mote than one are operttttng in the basin.

thus altowing for

T Nilsen

and AN. Nystud.

Opclmum

cxplurumn

wd extrucrion

221

B is the accumulated exploratory effort, i.e., the rate of exploratory effort times exploration time;3 k is the rate-reserves ratio during plateau production; and r is the discount rate. The relationships between the variables in the objective function (I) can be formulated through a number of equations. We assume a relation to be between accumulated discovered reserves and accumulated exploratory effort, the so-called ‘finding function’, which is often described by some S-shaped curve. The necessary realism is, however, obtained by defining the discovered reserves Q as a monotonically increasing, concave function of exploratory effort B. Let Qmax be the amount of discoverable reserves in the ground. A simple relationship that fultils the requirements Q’(B) > 0, Q”(B) ~0. Q(0) = 0, and lim B-. ~QQ(B)=Qmaw is the following, also used by Arps et al. ( 1971):

where (zo is a measure of the exploration productivity. The higher ~zo is. the more petroleum is discovered with the same exploratory effort. We assume an increase in costs per exploration well as the rate of exploratory effort incrcnscs and, further. that this relationship is linear. i.e., that the supply price of a well is an increasing linear function of the rate of effort: C,/A = (zt:+ h,,cl

for

n > 0.

(3)

whcrc A= B/t,. is the rate of exploratory effort mcasurcd in real terms. (1,: and h,: i\K exploration cost par:lmctc~S; h,:, tcrmcd ‘activity sensitivity coctlicicnt’. mcasurcs how scnsitivc the unit cost is to the lcvcl of exploration activity, while (I~ mcasurcs the level of the unit cost. A similar relationship is proposed by Uhler (1979). The price is assumed to grow exponentially until it rcachcs a predcfincd ‘ceiling’, and from then to be constant:

p(r) = min (pueu”‘,p,,,). whcrc

(4)

p. is the price at start of exploration, 11,. the price growth rate. and the price ceiling. The argument for introducing such a ceiling is that the P mrlx price for petroleum will not exceed the cost of producing alternative ener$y sources (the so-called ‘backstop technology’). The ceiling is set at what this altcrnativc cost is expected to be. Note that p(t +lE) is the price at time t from start of production.

31, __

?: Nilsse~ and R.N. Nystad, Optimm

~.~plor~~on luuf rxtracdon

Like most process industries [see e.g.. Manne (1967)J. petroleum ptoduction exhibits economies of scale. This is reflected in our development cost function:

co=u~(~Q)bD,

(3

where liQ is the plateau production rate (see below), and an and b,, together describe the production technology. For a given plateau production rate, a high value for a, implies a high-cost technology, and a low value a tow-cost technology. When bn is in the interval (0, i), the technology exhibits economies of scale. When b n= 1, there are no economies of scale. The smaller 6, is, the greater are the economies of scale. The production profile is modelled with a plateau phase, i.e., the aggregate production from all the fields is held constant until well productivity can no longer uphold the necessary pressure and exponentia1 decline starts through~ out the basin. This is in line with Warren (1978) and Nystad (1985a), although we disregard here the initial production build-up phase, Let I, be the length of the plateau production period and thus the time when dcciinc starts. Up to this point, the peod~lction rate is: f{(f)= kQ.

O
(6)

Thus, k is the rate-rcscrvcs ratio during the plateau production phase. Let G hc total accumulated production at I, as a fraction of total discovcrcd rcscrvcs. WC can now find a simple rcl~ltionship bctwcon t,, G, and k:

G is it useful paramctcr which can bc considered to bc exogcnously given by the type of reservoirs being dcplctod. A collection of gas reservoirs will gcncratly have a higher G than a collection of oil reservoirs. The higher G is, the longer will the plateau last, and the more rapidly will production fall during the decline phase. The Iattcr is seen by rccogni~ing that we have two expressions for the cumulative (and, in theory, infinite) production in the decline phase These must be equal, thus: [Qc-“dt=(L-G)Q. Solving for S, we obtain $ = k/( I - G). We scc that i$S/i’G> 0 and A?/% >O.

(8)

T: Nilsen

ad

A.N. Nyrfud. Optrmum expioration and exrraction

223

The production behavior is now fully described with the introduction of a production cut-off rule. The rule we apply says that production is stopped when operating costs equal gross revenues. i.e., the cut-off time T, as measured from start of production, must satisfy the following equality:

PtT+ ad 7-J= co.

(10)

The operating costs Co are simply proportional to development costs:

regarded

as constant

over

co = Cl&,,

time

and

(11)

where u. is the operating cost coeffkient. Operating costs will not be reduced when production enters the decline phase. i.e., operating costs per unit of production increase through the decline phase. Hence, the cut-off rule is well applicable and the cut-off time uniquely defined. The production profile can now be fully described:

q(r)= kQ

if

Ocfsf.

if

r,
(13 =kQe In the next section

-ItA-(i)lWt,)

this model is illustrated

by a numerical

cxamplc.

3. A numcricul cxamplc The model described above can be solved by implementing it on a computer system. However, it is impossible to solve it continuously. Instead, WC must dcterminc in advance the possible values for each of the decision variables B. t,, and k, and thus solve it discretely. In our computer programmc WC have chosen first to fix B and t, and find the optimum k and corresponding value of n by going through all possible values for k. Thereafter, a new value of rE is fixed and the process is reiterated. When the set of possible fa values is exhausted, the combination of I, and k yielding the highest n is stored, and the programme moves to a new value of B, and so on until also all possible values of B have been explored, and the optimum {LI,t,, k) combination is reported. Below we present the results from our numerical analysis. This presentation is divided into three parts. First, we present the base case. Next, we report on the sensitivity analysis that was carried out to see which parameters have the largest influence on the final result. Finally, we study the implications of introducing taxes in this model.

224

T: Nilssen and AN.

Nysrad. Optmnam esplorution

un.d earaction

3.1. The base case In table I we present the parameter values used in the base case run of our model, and the optimum values of the decision variables. In addition, the vatues of some other key variables are induded. Care should be taken in the interpretation of this table. A base case run has little value without the extra information that a sensitivity analysis provides. Rather than dwell upon table 1, we will defer comments until subsequent sections, where the topics are sensitivity analysis and the introduction of taxes respectively.

3.3. Srnsiticity analysis The sensitivity analysis is summarized in table 2. The table reports the qualitative effects of changes in one parameter while aft other parameters retain their base case values. The model has been tested for changes in most

Table I The base ease: Parameter

Notation

vrtluca. oplimum WIUCS nf decision other key variables.

reserves pr~~uelivlIy

Expl0rd.m

unit cost parameter

Discounl

measure

coeflicient

per well

I O.WO 0.05

mill. tJSS,‘wcll

15.0

mtll. USS/well”

3.0

“;,/year

rate

7.0

Imt~al price

US/W.

R&e of price change

“,/year

Price ceiling

USS;bbl.

60

US

30,000

Development Economics

cost regime parameter

of scale paramcler

Accumul;ltcd

PQ

mdl. bhl.

Exploration

Activity scn\ltlvlty

is

values of

UIIII of me;~rcmen1

N;lItlC Dr5covemhle

A

vnr~uhles, and implied

production

Operating

cost coeflicient

25 2.0

0.7 60

at start of decline

IO

Optimum

exploralory

eiTort

WCllS

70

Optimum

exploration

time

Optimum

pllttcrtu mte-reserves

years “, .u

IO

Rate of exploratory

3.0

wells/year

23.3

Net present value

mdl. USS

58.093

Dlsecovercd reserves

mdl. bbl.

Plstrau

production

effort

ratio

fute

mzll. bbl./year

9,698 970

1: Nhsen

and A.N. Nystacl. Opti-

c.xplorutton und r.xtraction

Table Summary

Q

ml..

225

Z

of sensttlvity analysts.’

+

+

0

0

+

-

0

-

+b + -

+ + 0 + -

+

%

b, r

+ + + +

PO UP *D b, G 00

+

‘Explanation of parameter notation 1s found in table 1. The entries refer to the effect of an increase in the parameter to the left on the value of the variable on the top. + = increase; - = decrease; 0 = practtcally no change. k. -’ for larpc values of r.

of the pnramctcrs listed in the first part elaborate on the results of this analysis.’

of table

I. In the following

we

Di.wmrd.de

ravws. Higher amounts of discoverable reserves imply that the exploration should be intensified (increasing A). There ilre illso clear incentives to shorten the exploration time and to increase the plateau ratcreserves ratio. The former of these may be due to ;L decreasing willingness to wait for an ever increasing rate of production. productiuiry. Higher exploration productivity implies a fall in the accumulated exploratory effort. But due to ;L corresponding fall in the optimum exploration time, the rate of effort is practically unaffected by changes in uo. Explorufion

Acliairy sensitivity. A higher activity sensitivity coeflicicnt implies a decrease in A, which is achieved partly by increasing rE and partly by decreasing !I. Discount

rate. A higher discount rate means that the company puts a higher value on time and therefore tends to shorten the exploration time and increase the production rate. But this increased haste will also affect the total ‘A XI of tables prwnting obtained from the authors.

the results from

thr

wznsitlvity analysis

m more

dctad

can be

226

‘I: Nilsen

and A.N. Nysrd.

Opmum

exploration

und u.waction

exploratory effort, such that more of the discoverable undiscovered.

reserves will remain

lniriul price. For a given up. a change in pO can be seen as a vertical, though not parallel, shift in the time development of the price. An increase in p. creates in effect much the same haste as will be observed if the discount rate increases. Exploration time is shortened and the production rate is increased. But the total exploratory effort is practically unaffected. since the increase in exploration time is offset by a corresponding increase in the rate of exploratory effort. Price growrh rate. Exploration time and exploratory effort rate are increased and decreased. respectively, when ~1~increases. A price increase serves to counteract the effect of the discount rate and thus to reduce the perceived time preference. In addition, the total number of wells drilled is positively affected by a higher price increase. Production technolopp. An increase in the cost regime parameter u,, can be seen as equivalent to moving from shallow to deep waters, for example. Variations in ho signify different economies of scale. Together, these two paramctcrs describe the production technology. Higher costs and smaller economics of scale both imply a longer exploration time, a reduction in the rate of exploration activity, and a lower production rate. Gtrs-oil ratio.

A high G, i.e., the accumulated production at decline start, as proportion of discovered reserves, is equivalent to a long plateau production period and a steep production decline, and vice versa. The higher G is, the shorter will be the exploration time, the more exploration wells will be drilled each year, and the lower will be the plateau production rate. while the total exploratory effort stays practically unchanged. The decrease in optima1 exploration time may be explained as follows. The increased G implies an increase in the decline rate S, so that the production ‘tail’ is shortened. This has two implications: first. less of the technically possible production falls beyond the economically optimum cut-off time I: and second, the total length of the production profile will decrease for a given k. In other words, more will be produced in a shorter time, and this creates an increased incentive to hurry up in the exploration phase. Recall that a high G often occurs when the gas content of the basin is relatively high. From our analysis WCthus see that expectations of a high gas content will speed up exploration and lower development expenditures.

Op~ru~i~~ COSCS.A change in the operating cost coefficient ~lo will affect the cut-off time T and thereby the total accumulated production. in the case of

7: Nilssen and A.N. Nystad, Optimum explorution

ao=O.

T

does

exploration

not

time

even

and

exist.

the

An

increase

in

and extractron

a,

implies

127

that

plateau

production

period

we have deliberately

overlooked

so far is that

both

are stretched

the

out

(tz

increases and li* decreases).

3.3. Tuxes A fact that profits

are subject

to taxation

selects the exploration present

value

divergences

c+

and development tax.

from

by the government

The

the pre-tax

strategy

introduction

the company’s

and that

the oil company

that

maximize

of taxes

will

will

in general

its net result

in

optimum.’

We present here a tool for analyzing the effects of taxation on both exploration and development decisions. based on the model presented in section

2. It is not particularly complicated to introduce a simplified into this model. Adapting from Nystad (1985b). we define

regime following

tax

parameters:

z=the

fraction

of the exploration

costs that

company covers (0 cz < I): /j= the fraction of the revenues the company (O
covers (O
the objective

function

(I) so as to maximize

(13)

taxes is to scale down

the net

cm”b.

0

r =/I=;1, value

the only

cffcct of introducing

n. The tax system

is nrulrc~l in the scnsc that

the company’s decisions. Actually, this is just a tax on the net present is equivalent

to the cash

Smith (1963). Nystad petroleum production base case of table

flow

the after-

1

j /~(p(t+r,,,)q(r)-C,,)c~~“tlf-yC,,

When

the

that accrues to costs that the

I).

WC can now rewrite tax net prcscnt value:

prcsont

tax the

it does not affect

with equal values on the tax parameters, value of the company’s cash flow. Thus, it regime

tax

advocated

by Brown

(1948)

and

(198Sb) dicusses the cash flow tax in the context of and also proposes a way to implement it. Using the

1, an example

is provided

in table

3 (row

I) for %=[I=

‘/ = 157;. As an illustration consider /I=:l.

of a tax system

the Norwegian

The royalties

petroleum

in the Norwegian

rules affect the tax parameters so-called

‘uplift’

allowance

‘SIX the discussion on taxlltion (1979. ch. 12).

that

is not

tax system.”

neutral

tax system along

in net present

on the Special of non-rrnzwabls

in the above

In Norway,

value Tax

rrwurcrs

with

sense,

it is unlikely

terms such that /I
tends

that

the depreciation

to make

The

fl>;,

at low

in general in Daqup~a

and Hcd

T: Ndsrcn and AN. Nystad, Oprrmum rxplorntton and extraction

7: Nrfsscn and A.N. Nystud. Optrmum exploratron

and extraction

229

discount rates. An increase in a company’s debt-equity ratio will also lower 7 relative to B_ The composite effect at an appropriate discount rate is. however, that p 7. Hence, we can describe the Norwegian tax system in terms of the above defined tax parameters as one where *>r>fl. The effects of having such a system are discussed below. Examples are given in table 3. Consider first the case of a>y = j (rows 2 and 3 in table 3). tg increases as LXincreases. but B* and k* remain unchanged. The exploration time is stretched to reduce the effect of higher after-tax exploration costs. Next, consider the case of z=y>fi (rows 4 and 5). A decrease in /? affects all three decision variables. The exploration time is increased, while the exploratory effort and the plateau production rate are decreased. This reflects the gcncral worsening of the economics of undertaking any business in the basin as the after-tax share of revenues is reduced. The picture guts even clcarcr when WC, finally, combine the above two casts and consider the cast of r>y >/j (rows 6 and 7). What we see now only cmphasizcs the tcndcncy: under this kind of tax rcgimc the companics will drill fcwcr exploration wells, and postpone production start relative to what in is optimum from society’s point of view. The tax analysis is summarized table 4. Table 4 Summary

z=p=y z>y>p ‘See

of tax analysis.’

B

tt.

A

k

0

0

0

0

-

+

-

-

footnote a. tublc

2.

4. Conclusions On the basis of our results to make trrttcttiue conclusions,

as they are presented some of which are:

in section

3, it is possible

(I) The effects of having more petroleum in the ground are increases in both the rate of and accumulated exploratory effort. In addition, exploration time is shortcncd and plateau production rate is increased.

230

T. Nd.ssrn und A.N. Nystad.

Oprrmum

exploration

and extwctton

(2) The rate of exploratory effort and the plateau production rate are not affected by exploration productivity to a great extent. (3) If the company and the state make use of different discount rates in their calculations, the one with the highest rate will want to explore less and more intensely and produce more rapidly. (4) The gas-oil ratio in the basin will affect the decisions which are made. The more gas, the higher will G be, resulting in more intense exploration. shorter exploration time and a lower production rate. (5) Under a tax regime where z >y > j?, the companies will decrease their exploratory effort, increase exploration time. and decrease production rate, compared to the pre-tax levels. Under a neutral tax regime, i.e.. where a= /I=;‘, there are no changes from the pre-tax to the post-tax analysis.

References Arps. J.J.. M. Mortada and A.E. Smith, 1971, Relationship between proved reserves and exploratory effort. Journal of Petroleum Technology. June. 671-675. Brown. E. Cnry, 1948. Business-income taxation and investment Incentives. in: Income. employment and puhhc pohey: Essays in honor of Alrm Cl. Hunxen (Norton. New York) 30&316. Daspupta. Partha S. and GeolTrey M. Heal. 1979. Economic theory and exhauattble resources (Cambridge University Press. London). Fl3m. Spur D. and Gunnur Stcnsl;md, 19X5, Exploratum and tax.ltlon: Some normative issues. Energy Economics, Oct., 237-240. Kemp. Alex.mdcr G. and David Rose, 19X3. Petroleum tar andy\cb. Sorth SL’J (Financial Times Buclnc\\ Infi~rm;dion. London). Liu. P.T. and J.G. Situncn. 19X2. On the hchavtor of optlmd exploration and extraction rates for non-rcncwahlc reb<>uree stocks. Resources and Energy 4. 145 .I62 Mantle. Alan S.. 1967. CalculJtlons for a single producing arca. m: Al:m S. Marine, ed.. Invcdmcnts for capacity exp.insion: Silt. locatIon. and time-phasing (Allen & Unwin. London) 28 4X. Nys~ad. Add N.. 19X5:1. Rcservt)ir-economic optimiratum. Prc\cntcd at the Society of l’drolcuni I:ngtncsrs I9XS tlydrocarhon Economics dnd l\valu.~t~on Symposium. Dallas, fX, IJ I5 Ml.~rch. Paper Sl’t: 13775. Ny\~.d, Add N.. I9XSh. Petroleum IIXCY and optunal rc>ource recovery, Energy Policy, Aug.. 3XI~4OI. Pdravdi. Karim. 1977, A model of oit production, dcvrlopmcnt, and exploration. Journal of Energy and Development 3. no. I. 143-152. I’ctcr\on. Frcdcrick M.. 197X. A model of mining and explormg for exhaust&de resources. Journd of F.nvironmental Economics and Management 5. 236-251. Pdyck, Roberl S., 197X. The optImaI explorauon nnd production of nonrenewable resources, Journal of Poht~cal Economy X6. no. 5. X41-X61. Smith. Vernon L.. 1963. Tax depreciation pohcy and mvatmrnt theory. Intrrnattonul Economic Rcv~ew 4, no. I. X&91. Uhlrr. Russell S.. 1979. The rate of petroleum exploration und extractlun. Advances in the Economics ol Energy and Resources 2, 93-l IX. Warren. Joseph E.. 197X. The development drclsion for frontier areas: Thr North Sea. Presented at the European Offshore Petrolrum Conference and ExhIbItion. London, 25-27 Oct., Paper EUR 11-l.