We consider continuous dynamic model of gas fields with interacting wells. The minimum prime cost of gas extraction is the main criterion for the construction project of the gas field. We analyze dependence of the minimum prime cost on the planning horizon. In our model at some turning point of time one gas field is selected for further development out of two gas fields on the basis of the available projects. The only selection criterion is the minimum prime cost of gas extraction. When the selection is done, any further revision of the choice is impossible. We set two problems about correctness of the made choice under various condition. We completely solve the first problem. We find the conditions when the choice of the gas field remains correct, although the planning horizon changes.
Department of Methods for Designing Developing Systems explores the problem of oil, gas and condensate fields dynamic planning (individually and as a group) for many years. Based on the modern foundations of gas fields exploitation [Vyakhirev, 1998] a number of original mathematical models was developed and a large number of optimal control problems was solved [Margulov, 1992], [Skiba, 2009], [Skiba, 2012], [Khachaturov, 2015]. One of them is a problem of optimal economic growth [Skiba, 1978], whose solution is based on the propositions of K. Arrow [Arrow, 1974].
The Department pays special attention to the computational methods in the sphere of dynamic planning. Thus, a simulation model of the Gas Production Planning System (SPDG) was created and implemented on the computer. This system does dynamic calculation of indicators for any given horizon T > 0.
The economic indicators of gas production are the main subjects to investigation. Profit, income and other indicators related to the market price are usually taken as the main subjects to investigation. However, in the recent years it has become quite clear that the market price is a poorly predictable economic indicator, even given a small time horizon. That is why in this paper we consider prime cost is taken as a target economic indicator.
In recent years, several researches of the prime cost optimization have been made in the adjacent fields of study. For example, shale gas development was investigated in [Li, 2017] and optimal planning and infrastructure development for shale gas production was discussed in [Arredondo-Ramirez, 2016].
In [Elsholkami, 2017] they consider the general optimization model for the energy planning of industries including renewable energy and prime cost optimization of biofuel production was elaborated in [Jong, 2017].
Increased interest in this issue confirms the importance of the topic under the study in this paper. 2
Prime Cost Analysis of the Gas Field In the paper we consider a dynamic model of gas field with interacting wells [Margulov, 1992]. The following notations are used:
q is average flow rate of producing wells;
V˙ = −N q; q˙ = −
N q; = q0 V0 with initial conditions V0 > 0, qo > 0. The following constraint is imposed on the fund of producing wells 0 ≤ N ≤ N¯ :
z + kN¯ Cost = T ∫ qN dt 0
: P rib = c
T ∫ 0
qN dt − z − kN¯ : q˙ = −N q;
z + N¯ Cost = k T ∫ qN dt 0
; P rib = k (c
qN dt − z − N¯ ): T ∫
Capital expenditures are described as a linear function z + kN¯ ; where z and k are constant coefficients. Next, we write out two formulas that determine the prime cost of gas extraction and the profit obtained from its sale. The prime cost is defined as the ratio of capital expenditures during the construction of the field to the extracted volume of natural gas produced for the entire planning period.
N ′ = N; N¯ ′ = N¯ ; V ′ = V = q; z′ = z k ; c′ = c k :
(2) (3) (4) (5) (6) (7) (8) We set two optimal control problems: the first problem is to minimize the prime cost of natural gas extraction, the second problem is to maximize the profit from its sale.
Problem 1. Minimize the functional (7) at differential connection (6), the initial condition q0 > 0 and a constraint on control (2).
Problem 2. Maximize the functional (8) at differential connection (6) the initial condition q0 > 0 and a constraint on control (2).
It is easy to show that the minimum of (7) and the maximum of (8) are achieved under the following control: N (t) = N¯ throughout the whole planning period. Thus, solutions of these problems are to minimize the value of q(T ):
T ∫ N(t)dt q(T ) = q0e− 0
T 0 − ∫ Ndt ≥ q e 0 = q0e−NT :
z + N¯
Cost(N¯ ; T ) = k q0(1 − e−NT ) ;
P rib(N¯ ; T ) = k (cq0(1 − e−NT ) − z − N¯ ):
Next we find the minimum of (9) given a fixed T . So, we take the partial derivative of (9) with respect to N¯ , that should be equaled to zero. After the transformations, we obtain the following equation: eNT − 1 − N¯ T − zT = 0:
We denote the left-hand side of the equation (11) by Φ(T; N¯ ), where T and N¯ are positive variables. In this case, the parameter z is assumed to be positive and fixed. The function Φ(T; N¯ ) is defined and continuous for T > 0 and for N¯ > 0, and this function has continuous partial derivatives @Φ=@T and @Φ=@N¯ for positive values of T and N¯ . Moreover, @Φ=@N¯ = T (eNT − 1) > 0: It is easy to show that for any T0 > 0 there exists a unique N0 > 0 such that Φ(T0; N¯0) = 0. Hence, by the implicit functional theorem for T > 0 there exists a unique ¯ positive-definite function N¯ = N¯ ∗(T ) satisfying the equation (11).
We substitute the function N¯ ∗(T ) into the equation (11). As a result, we arrive at the identity (9) (10) (11) (12) (13) (14) (15)
eN (T )T − 1 − N¯ ∗(T )T − zT = 0: Cost(N¯ ∗(T ); T ) =
eN (T )T : k q0T It is interesting to study the behavior of the functions N¯ ∗(T )T , N¯ ∗(T ) and Cost(N¯ ∗(T ); T ) depending on parameter T > 0. The study of the behavior of these functions will be useful to us in the future. The following three statements are given with proof.
Statement 1. The function N¯ ∗(T )T is defined and continuous for T > 0 and it is strictly increasing from 0 to ∞.
We expand the exponent into the identity (12) in the Maclaurin series. After the transformations, we obtain the following double inequality:
0 < N¯ ∗(T )T < √2zT : In the double inequality (14), we pass to the limit, directing the parameter T go to zero. As a result, we obtain limT →0(N¯ ∗(T )T ) = 0. We differentiate both sides of the identity (12) with respect to T and after the transformations we obtain (N¯ ∗(T )T )′ =
z eN (T )T − 1 > 0: Then the function N¯ ∗(T )T is strictly increasing. The following result follows from the identity (12): limT →∞(N¯ ∗(T ); T ) = ∞. The statement 1 is proved.
Statement 2.The function N¯ ∗(T ) is defined and continuous at T > 0 and is strictly decreasing from ∞ to 0.
Dividing both sides of the identity (12) by T , we differentiate it with respect to the parameter T . After the transformations, we obtain (N¯ ∗(T ))′ = eN (T )T − 1 − N¯ ∗(T )T eN (T )T
T 2(eN (T )T − 1)
−N¯ ∗(T )T eN (T )T (N¯ ∗(T )T )′ < 0: So, the numerator is negative for T > 0. The function N¯ ∗(T ) is strictly decreasing. We divide all the parts of the double inequality (14) by T 0 < N¯ ∗(T ) < √ 2z
T Passing to the limit as T → ∞, we arrive at the following result: limT →∞ N¯ ∗(T ) = 0. We expand the function eN (T )T using the Maclaurin formula of the second order with the remainder term in the Peano form and substitute the resulting expansion into the identity (12). As a result, we obtain (16) (17) (18) (19) (20) We divide both sides of the identity (18) into (N (T )T )2 and taking into account
2 limT →0(N¯ ∗(T )T ) = 0, we pass to the limit as T → 0 (N¯ ∗(T )T )2 2 + o(N¯ ∗(T )T )2 − zT = 0: ( 2z 1 − Tli→m0 (N¯ ∗(T ))2T ) = 0: .
The result is limT →0 N¯ ∗(T ) = ∞. The statement 2 is proved.
Statement 3. The function Cost(N¯ ∗(T ); T ) that describes the prime cost of gas extraction, defined and continuous for T > 0 and it is strictly decreasing from ∞ to kq0z .
We substitute the function N¯ ∗(T ) into the function (9). Next, we differentiate the function Cost(N¯ ∗(T ); T ) with respect to the parameter T and, using the relation (11), we obtain (Cost(N¯ ∗(T ); T ))′ = −kN¯ ∗(T )e−N (T )T
z + N¯ ∗(T ) q0(1 − e−N (T )T )2 < 0:
limits:
z + N¯ ∗(T ) lim Cost(N¯ ∗(T ); T ) = k q0(1 − e−N (T )T ) = ∞; T →0 z + N¯ ∗(T ) z lim Cost(N¯ ∗(T ); T ) = k q0(1 − e−N (T )T ) = k q0 : T →∞
3
Comparative Analysis of Two Gas Fields Prime Costs Suppose one of two gas fields must be selected for further production during the planning period T0. The main criterion for selection is the minimum prime cost of gas extraction. The choice is made on the basis of the submitted projects of field construction. The choice of only one field could be done due to various reasons, for example, the possibility to connect the gas pipeline with the only one out of the two fields.
Obviously, in this situation, every owner of the field wants to win the tender. Therefore, each field is projected taking into account the planning horizon T0 and the minimum prime cost of gas extraction. Thus, the producing wells fund N¯ ∗(T0) and minimum prime cost Cost(N¯ ∗(T0); T0) are determined. The winner of the tender is determined without any possibility of further change.
Let the planning horizon changes from T0 to another value T . There is a natural question of the correctness of the choice made. This section of the paper is devoted to the answer to this question.
In the first case when there is no possibility to reorganize the field construction, the fund of producing wells N¯ ∗(T0) and the capital costs k(z + N¯ ∗(T0)) remain the same. However, the accumulated gas extraction q0(1 − e−N (T0)T ) and the prime cost of gas extraction Cost(N¯ ∗(T0); T ) are changed.
In the second case, the field construction reorganization is possible. Another field construction project is being created with the new planning horizon T . Thus, we get a new value of the fund of producing wells N¯ ∗(T ) and the minimum prime cost of gas extraction Cost(N¯ ∗(T ); T )
Cost(N¯ ∗(T ); T ) ≤ Cost(N¯ ∗(T0); T ); moreover, the equality holds only when T = T0. For any positive quantity N0 there exists a unique value T0 > 0 such that N¯ ∗(T0) = N0. For any positive N ̸= N¯ ∗(T0), a strict inequality holds:
Cost(N¯ ∗(T0); T0) < Cost(N; T0):
Problem 3. We have a set of five positive numbers: N¯1∗(T0), Cost1(N¯1∗(T0); T0), N¯2∗(T0), Cost2(N¯2∗(T0); T0) and T0. The first two numbers of the set refer to the first field. The next two numbers of the set refer to the second field. The last number is the horizon of gas production planning. It is assumed that inequality
Cost1(N¯1∗(T0); T0) < Cost2(N¯2∗(T0); T0)
Cost1(N¯1∗(T0); T ) < Cost2(N¯2∗(T0); T ) Using the new notation, from the inequality ( ref eq21) we obtain a restriction on the value of the function
Cost2(N¯2∗(T0); T0) Cost1(N¯1∗(T0); T0) = a2(1 − e−N1 (T0)T0 ) a1(1 − e−N2 (T0)T0 ) :
Statement 5. The function R(T ) is defined and continuous for T > 0 and it monotonically changes from a2N1 (T0) to aa21 . The function R(T ) is strictly increasing when N¯1∗(T0) < N¯2∗(T0) and is strictly decreasing when a1N2 (T0) N¯1∗(T0) > N¯2∗(T0).
holds and such values of T for which it does not hold.
A set of five numbers is formed from field construction projects. This set is enough to calculate the prime costs of gas extraction for any planning horizon. We show how this is done.
zi + N¯i∗(T0) Costi(N¯i∗(T0); T ) = ki qi0(1 − e−Ni (T0)T ) ; i = 1; 2: ai = Costi(N¯i∗(T0); T0)(1 − e−Ni (T0)T0 ); i = 1; 2: Costi(N¯i∗(T0); T ) = ai 1 − e−Ni (T0)T ; i = 1; 2: R(T ) =
Cost2(N¯2∗(T0); T ) Cost1(N¯1∗(T0); T ) = a2(1 − e−N1 (T0)T ) a1(1 − e−N2 (T0)T ) : (21) (22) (23) (24) (25) (26)
1 − e−N1 (T0)T 1 − e−N2 (T0)T ; (30) f ′(T ) =
N¯1∗(T0)e−N1 (T0)T (1 − e−N2 (T0)T ) − N¯2∗(T0)e−N2 (T0)T (1 − e−N1 (T0)T )
(1 − e−N2 (T0)T )2 (1 − e−N2 (T0)T )2
N¯2∗(T0) =
N¯2∗(T0)e−(N1 (T0)+N2 (T0))T [ N¯1∗(T0) (eN2 (T0)T ] − 1) − (eN1 (T0)T − 1) : At T = 0 the expression in square brackets is zero. Differentiate with respect to T the expression in square brackets in the ratio (29). The result is N¯1∗(T0)(eN2 (T0)T − eN1 (T0)T ). It follows that f ′(T ) > 0 for N¯1∗(T0) < N¯2∗(T0) and f ′(T ) < 0 for N¯1∗(T0) > N¯2∗(T0). In both cases, T > 0. The statement 5 is proved.
Problem 3′. Under the assumption (27) it is necessary to find the values of T when the function R(T ) is greater than 1, and when it is less than 1.
The following theorem (the solution of problem 3) follows from the inequality (27) and the statement 5.
Theorem 1. Case 1. Let N¯1∗(T0) < N¯2∗(T0). In this case, aa12 > 1, then: a) the function R(T ) is strictly increasing on the interval (0; ∞) from aa21NN12 ((TT00)) to aa21 ; b) if aa21NN12 ((TT00)) < 1, then there exists a unique positive number T1∗ less T0 such that R(T1∗) = 1, and when T > T ∗ 1 inequality R(T ) > 1, and when 0 < T ≤ T1∗, it is not executed. c) if aa21NN12 ((TT00)) ≥ 1, then the inequality R(T ) > 1 occurs for all positive values of T .
Case 2. Let N¯1∗(T0) > N¯2∗(T0). In this case, aa21NN21 ((TT00)) > 1, then: a) the function R(T ) is strictly decreasing on the interval (0; ∞) from aa21NN12 ((TT00)) to aa21 ; b) if aa21 < 1, then there exists a unique positive number T1∗ greater T0 such that R(T1∗) = 1, and when 0 < T < T ∗ 1 inequality R(T ) > 1, and if T ≥ T1∗, it is not executed. c) if aa21 ≥ 1, then the inequality R(T ) > 1 occurs for all positive values of T .
Case 3. Let N¯1∗(T0) = N¯2∗(T0). In this case, aa21 > 1, then: a) the function R(T ) on the interval (0; ∞) takes a constant value, equal to aa21 ;
Cost1(N¯1∗(T ); T ) < Cost2(N¯2∗(T ); T ) is performed and, accordingly, in which it fails. 4
In this paper, we considered a continuous dynamic model of a gas field. Within the framework of the model, formulas for calculating profit and prime cost of gas extraction were written out. Profit is considered as the most important economic indicator. Only profit is fully able to reflect the level of efficiency of any enterprise. However, unlike the prime cost of gas extraction , the market price of gas (which is included in the profit calculations) is a poorly predicted dynamic value. This fact greatly complicates the prediction of total profit. The prime cost is defined as the ratio of capital costs during the construction of the field to the volume of natural gas extracted during the planned period.
Capital costs are described as a linear function of one variable with a constant part and a variable part. The variable part depends on the fund of producing wells. For each planning horizon, we find the minimum prime cost of gas extraction. It is proved that the minimum exists. Thus, for each planning horizon, we unambiguously determine the value of the producing wells fund. It is proved that the optimal value of the fund of producing wells is strictly decreasing from infinity to zero with a change in the planning horizon. The minimum value of the prime cost of gas extraction strictly decreases from infinity to a certain positive value. These properties of the fund of producing wells and prime costs of gas extraction do not contradict the basic property of the model.
The following situation is considered. Two fields participate in the tender. One field is selected with the lowest prime cost of gas extraction. Revision of the choice is impossible. The planning horizon is changing. Two cases are considered. In one case, producing wells remain unchanged. In another case, the prime cost of gas extraction for each field is minimized with a new planning horizon. For each case, the problem is posed with the following question. Is the right choice made? Both problems are solved.
However, the paper gives a complete description of the solution of only the first problem. The solution of the second problem is limited only to its formulation. The solutions of the two problems are structurally similar to each other. The existence of no more than two areas is proved. If the area is one, then the choice is made correctly. If there are two areas, then in one area the choice is made correctly, and in another it is wrong.The sequence of areas can be any.
In practice, a conflict situation may arise with the organizers of the tender in the event of the negative answer
to the question posed in the problems. Sometimes a protracted conflict can be resolved only through the courts.
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