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When allocation “back out” agreements can become “lose out” agreements
Author’s Accepted Manuscript When Allocation “Back Out” Agreements can become “Lose Out” Agreements Phil Stockton
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PII: DOI: Reference:
S0955-5986(17)30333-3 https://doi.org/10.1016/j.flowmeasinst.2018.02.022 JFMI1423
To appear in: Flow Measurement and Instrumentation Received date: 17 August 2017 Revised date: 22 October 2017 Accepted date: 14 February 2018 Cite this article as: Phil Stockton, When Allocation “Back Out” Agreements can become “Lose Out” Agreements, Flow Measurement and Instrumentation, https://doi.org/10.1016/j.flowmeasinst.2018.02.022 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
When Allocation “Back Out” Agreements can become “Lose Out” Agreements Phil Stockton Accord Energy Solutions Limited NOTATION A B BX BY C1 C2 D E F G GDP K k k L M N P P’ ° P P P’ ° P PD PX Q Q’ Q Q” QR QX r R R RP S t V V’ V” VTran Y Z 1 2 0 COP
Constant Back-out balance / constant Back-out balance in payment phase Back-out balance in re-payment phase System parameter defined in (43) System parameter defined in (44) Constant Constant Constant Constant Field Alpha cumulative gas production to date Constant Constant relating Alpha’s reservoir pressure to its gas volume Constant relating Bravo’s reservoir pressure to its gas volume Constant Constant Constant Field Alpha Shared Production reservoir pressure Field Alpha Stand-Alone reservoir pressure Initial Alpha reservoir pressure Field Bravo Shared Production reservoir pressure Field Bravo Stand-Alone reservoir pressure Initial Bravo reservoir pressure Delivery pressure Manifold pressure Field Alpha Shared Production volumetric flow rate Field Alpha Stand-Alone volumetric flow rate Field Bravo Shared Production volumetric flow rate Field Bravo allocated volumetric flow rate Back-out volumetric re-payment rate Back-out volumetric payment rate Discount rate Flow resistance factor Alpha to X (manifold) Flow resistance factor Bravo to X (manifold) Flow resistance factor X (manifold) to delivery point Monetary value of gas per unit volume Time Field Alpha Shared discounted present value Field Alpha Stand-Alone discounted present value Field Bravo allocated allocated discount value Discounted present value of transferred gas Repayment factor Gas compressibility System parameter defined in (41) System parameter defined in (42) Repayment completion time COP time
1
X μ
1 1.1
Cross-over time System parameter defined in (50)
INTRODUCTION Overview
When a newly developed third party field is tied back to existing infrastructure, such as a pipeline system, there is the potential for the new field’s production to cause an increase in back pressure and reduce the flow of the incumbent fields already producing through the system. This is termed “back-out” and in effect defers some of the incumbent fields’ production. Back-out agreements are sometimes put in place, within the allocation system, to compensate the incumbent fields for the reduction in flow they experience. This may take the form of a transfer (or payment) from the third party of hydrocarbons to the incumbent fields so that in effect they do not experience deferment and suffer resultant economic loss. As the production matures, the situation eventually reverses and the transfer (repayment) proceeds in the opposite direction from the incumbents to the third party. The assumption often made is that eventually the transfers will cancel one another and over their lifetime, all fields’ cumulative allocation will equal their cumulative production. There is little published information on back-out agreements as they are generally commercially sensitive and hence confidential in nature. However, the Division of Oil and Gas of the Alaska Department of Natural Resources commissioned and published a report entitled: “North Slope of Alaska, Facility Sharing Study” [3], which presents back-out methodology guidelines. This paper presents the concept of how back-out calculations typically work and presents a simplified mathematical model of a system where an incumbent field is backed out by a new field introduced into a pipeline. The simplified model results in a set of differential equations which can be solved analytically. This simplified model is used to illustrate potential problems with the above concept of back-out, particularly in the repayment period. The simplified model demonstrates that in principle it takes an infinite time for the repayment to be achieved and hence the assumption that all parties are eventually allocated the gas they produce is not valid. In effect there is the very real potential for “backout” to become “lose out” for the new field. The results of the simplified model are compared against the results from more rigorous, complex models (run on commercial software) to confirm that the behaviour of the simplified model adequately reflects the system. The complex models are solved numerically and have to be run in sequential time steps. The advantage of the simplified model is that it is directly solvable for any time and hence the long term behaviour of the system can be analysed directly. Finally the paper discusses methods to overcome the pitfalls exposed by the analysis. In doing so it considers the fundamental question that the back-out payments and transfers are attempting to address; that is: a perceived economic loss due to deferred production is compensated by the transfer of gas quantities without any calculation of the apparent value of the gas. Methods are presented
2
that incorporate the time value of money in order to provide allocation mechanisms that ensure the incumbent field receives the equivalent value of gas as if it was producing alone and still ensures that the third party field is allocated a volume of gas equal to its production over a specified time period. This paper specifically addresses back-out associated with gas reservoirs producing through shared pipeline infra-structure. 1.2
Structure of the Paper
Section 2 describes a typical gas production system in which back-out occurs. This is used as an example to illustrate the issues associated with back-out. Though a fictitious system it is based on typical production rates and parameters from a real system to ensure it is reasonably representative of equivalent real systems in general. The values of the parameters used to generate the flows, pressures, etc. presented in the various charts are given in Section A.10. Section 3 describes a simplified mathematical model that is used to analyse the long term behaviour of the system. It thereby exposes a problem in that it appears it is impossible for the tied back field to be repaid, in any reasonable timeframe, the gas it has transferred to the incumbent field during the early back-out payment phase. Section 4 examines options to address the problem of excessive repayment periods in an effort to ensure all fields receive the gas they have produced over the length of their lives. Section 5 presents conclusions. Appendix A provides a detailed description of the simplified mathematical model employed. 2 2.1
DESCRIPTION OF A SYSTEM WITH BACK-OUT Physical System
Consider the subsea gas pipeline system depicted in Figure 1:
3
Figure 1 – System Configuration The incumbent field has been labelled as Field Alpha (). In order to utilise spare capacity and maximise the use of existing infrastructure it is often mutually beneficial for new developments to tie back into existing transportation and processing facilities. A potential problem for the incumbent field is that the introduction of a new field, denoted Bravo (), increases the total flow down the shared pipeline resulting in a higher back pressure at Field Alpha’s wellhead(s). If Alpha’s wells are off plateau, the increased back pressure will reduce production from the field and this is termed back-out. (Since new field tie-backs into existing infra-structure are usually taking advantage of spare capacity, it is likely that the incumbent fields’ are off-plateau). Unless the increased back pressure has a permanently deleterious impact on the incumbent wells, back-out does not result in lost production, it merely defers it. However, due to the time value of money the incumbent field would normally wish to produce its hydrocarbons sooner rather than later and back-out is considered to have an economic cost to the incumbent. To understand the consequences of back-out, it is instructive first to consider the behaviour of the incumbent field when it is producing alone, i.e. without back-out. 2.2
Stand-Alone Production
When Field Alpha is producing alone and it is off plateau, the pressure drop across the system, from the reservoir to the delivery point at the gas plant, will determine its flow. As the reservoir pressure drops, the driving-force decreases and Alpha’s flow declines. This is illustrated in Figure 2.
4
The dash ’ in Q’ means this is the Stand-Alone volumetric flow for Field Alpha. The horizontal scale ranges from 0 to 4,000 days, almost 11 years of production. The reservoir pressure also declines as shown in Figure 3.
Figure 2 – Alpha Stand-Alone Flow Rate The reservoir acts like a pressurised tank. As gas is produced from the reservoir the pressure in it falls. As the pressure declines the driving force reduces, resulting in a lower flow rate. The rate of pressure decline also falls as can be seen in the figure.
Figure 3 – Alpha Stand-Alone Reservoir Pressure The reservoir pressure tends asymptotically to the delivery pressure at the gas plant, which in this example is set at 6.75 bara. The fall in the reservoir pressure is directly related to the cumulative production which is plotted in Figure 4.
Figure 4 – Alpha Stand-Alone Cumulative Production The above plots represent the evolution of Alpha’s production with time in the absence of a new field being introduced. This is compared in the next section with the scenario of Shared Production with Field Bravo – a new third party tie-back. 2.3
Shared Production and Back-Out
In the Shared Production scenario, Alpha and Bravo are producing together. Alpha experiences a rise in back pressure due to the higher flow and resulting
5
increased frictional pressure drop across the shared section of the subsea flow line. Alpha’s flow rate will therefore be lower when compared with the StandAlone case. The Stand-Alone and Shared Production1 flows are compared in Figure 5:
Back-out Cross-over
Repayment
Figure 5 – Alpha Stand-Alone vs Shared Production Flow Rate The locus of the green dashed line is identical to that in Figure 2. The red line shows the reduced flow experienced by Alpha in the early phase of the profile. The margin narrows until it becomes zero after 1,112 days, when Alpha’s Shared Production exceeds its Stand-Alone level. The reason for this is revealed by comparing the difference in the reservoir pressures for the two scenarios in Figure 6:
1
The Stand-Alone and Shared Production scenarios or cases have been capitalised for convenience of reference throughout the paper.
6
Back-out
Repayment
Cross-over
Figure 6 – Alpha Stand-Alone Minus Shared Production Reservoir Pressure (The difference in pressure is presented as the plots of the two reservoir pressures almost lie over one another). The plot shows that the Stand-Alone reservoir pressure is always lower than in the Shared Production case. This is because more gas has been taken out of the reservoir in the Stand-Alone case. Though the back pressure experienced by Alpha is greater in the Shared case, the driving force at the reservoir is also greater and this eventually results in Alpha producing more in the Shared Production scenario. This reservoir pressure difference between the two cases reaches its maximum at the cross-over point when Alpha’s flow rate is identical in both scenarios. Also in the Shared case the production from Field Bravo is declining with time and therefore further reducing the back pressure. The daily flow for Bravo is presented in Figure 7. In the example, Bravo’s production is less than Alpha’s but its initial reservoir pressure is significantly greater as illustrated Figure 8.
Figure 7 – Bravo Flow Rate
7
Bravo’s initial high reservoir pressure backs out Alpha but it declines rapidly as its reservoir is considerably smaller than Alpha’s.
Figure 8 – Bravo Reservoir Pressure Returning to Alpha, the difference in the Stand-Alone and Shared Production daily flow rates is compared in Figure 9:
Back-out
Cross-over
Repayment
Figure 9 – Alpha Stand-Alone Production Minus Shared Production The difference in the two daily flow rates, which is positive in the first 1,112 days, represents deferred production for Alpha and is termed the back-out. Back-out agreements can be put in place to compensate incumbent fields (such as Alpha) for this deferred production. 2.4
Back-Out Agreement
A typical premise of back-out agreements is to allocate the incumbent field the quantity of gas it would have produced without the new field being present.
8
In the example, this entails calculating Alpha’s deferred production as the difference between Alpha’s measured (or allocated) production when coproducing compared with its estimated Stand-Alone production. Gas is then transferred from Bravo to Alpha to compensate Alpha for the deferment, or back-out, and ensure it does not suffer economic loss. An issue that immediately arises is the Stand-Alone Alpha production. This cannot be directly measured as it did not actually occur and is therefore hypothetical in nature. In order to overcome this, simulation models are used to calculate the hypothetical Stand-Alone production. 2.5
Back-Out Modelling
Back-out agreements usually stipulate the simulation modelling to be under-taken in order to calculate the back-out. These simulations can vary in type and complexity but they usually involve modelling the reservoirs, wellbores and pipeline infrastructure to the point of delivery, e.g. at the inlet to a gas plant where the pressure may be considered fixed. Commercial software programs [1], [2] such as MBAL and Eclipse model reservoirs, PROSPER and OFM model wells and GAP and PipeSim model pipeline networks. These may be integrated under a wider software framework. Such models are relatively sophisticated and require considerable expertise and effort to run. In order to predict the level of back-out normally two models are run:
The Shared Production case with both fields producing, i.e. mimicking reality and; The Stand-Alone hypothetical case with only the incumbent field producing.
The difference between the two then provides the predicted back-out experienced by the incumbent field. At first sight, the model simulating reality appears superfluous. However, it serves two purposes:
It allows the modelling environment to be calibrated against real production, i.e. by history matching the model against real data. The parameters calibrated in the Shared model can then be used in the Stand-Alone model and thereby improve estimates of back-out; It allows calculated daily back-out quantities to be projected into the future, the forecast back-out being used in the daily allocation in the forthcoming period until a new calibration exercise can be undertaken.
There are variations to the above approach but they all tend to include models that simulate both scenarios. 2.6
Deferment Back-Out is different to Hydrodynamic Back-Out
It is worth noting that back-out, in the deferred production sense and addressed in back-out Agreements, cannot be measured directly.
9
At first sight it may appear that the back-out experienced by the incumbent field Alpha could be determined by physically shutting in Bravo and recording the change in Alpha’s production as the back-out. But this is a different kind of backout, termed here as hydrodynamic back-out. Hydro-dynamically, at any instant in time, Bravo is backing out Alpha in the sense that if Bravo is shut in Alpha’s flow will always increase. However, this is not how deferred production back-out is defined in back-out agreements as Alpha’s flow has to be compared with the hypothetical case of Alpha having produced up to this point in time without Bravo producing at all. In such a scenario, Alpha’s reservoir pressure would be lower and it would be producing less on a Stand-Alone basis. When determining back-out in the deferred production sense, the Shared Production case, however it is determined, has to be compared against a parallel, hypothetical scenario in which only Alpha has ever produced. 2.7
Back-Out Balances
There are three phases associated with back-out which are indicated on Figure 5, Figure 6 and Figure 9:
Back-out payment Cross-over Repayment.
During the initial back-out payment phase the transfer of gas from Field Bravo to Alpha is normally recorded and accumulated as a back-out balance. Once the cross-over point is reached the daily back-out payments will fall to zero and the back-out balance reaches its maximum. The repayment phase ensues and Alpha commences transfer of gas back to Bravo. The back-out balance is decremented accordingly. The back-out balance associated with the fictitious example introduced above is presented in Figure 10:
10
Repayment Cross-over
Back-out
Figure 10 – Back-Out Balance The back-out balance reaches its maximum value of 79.1 mcm (million cubic metres) at the cross-over point. This is the cumulative quantity of gas Bravo has transferred to Alpha to compensate for deferred production. After the cross-over point Alpha Field is producing more gas than it would have been if it had produced on its own and the back-out becomes negative and repayment in the opposite direction from Alpha to Bravo takes place. The back-out balance is then decremented by the repayment quantities. When the back-out balance reaches zero the back-out payments will all have been repaid and the back-out calculations cease. However, how long does it take for the balance to return to zero? Examination of Figure 10 shows that it took slightly over 3 years to build up the balance to its maximum value, but after almost a further 8 years, less than half the balance has been repaid. In order to answer the question posed above, a simplified model of the system was developed and this is described, and its behaviour analysed, in the next section. 3 3.1
SIMPLIFIED MODEL Reasons for the Simplified Model
In order to assess the behaviour of the repayment phase and estimate the completion date, a simplified mathematical model of the system and associated back-out was developed. The derivation of the model is presented in Appendix A. This simplified model produces analytical equations that can solve the state of the system at any point in time.
11
Running the rigorous, complex models mentioned in Section 2.5, is considerably more time consuming and being numerical in nature, not so easily extended far into the future in order to examine the long term behaviour of the repayment phase. The analytical solvable equations associated with the simplified model also provide a deeper understanding of the behaviour of the system. 3.2
Model Development
The model is based on the following principles:
the fall in reservoir pressure over a period of time is directly related to the cumulative production over that period the cumulative production is equal to the integrated flow rate over time the flow rate is equal to the rate of change of reservoir pressure.
Two principal sets of equations were developed. The first describes the behaviour of Alpha’s reservoir pressure in the Stand-Alone case:
P' P PD et PD
(1)
Where, P’ P PD t
°
Stand-Alone Alpha reservoir pressure (at time t). (The dash ’ denotes the Stand-Alone case and is not the differential operator) Initial Alpha reservoir pressure Delivery pressure (at gas plant) time
μ is a constant determined by several system parameters; see Appendix A for its definition. The second describes Alpha’s reservoir pressure in the Shared Production case:
P C1e1t C2e2t PD
(2)
Where, P
Shared Production Alpha reservoir pressure (at time t)
C1, C2, 1 and 2 are all constants determined by the system parameters; see Appendix A for their definition. Hence, at any point in time these equations provide the reservoir pressures for the two scenarios. Two analogous equations provide the flow rate at any point in time. Alpha’s flow rate in the Stand-Alone case is given by:
12
Q'
k
P
PD e t
(3)
Where, Q’ k
Stand-Alone Alpha flow rate (at time t) Constant relating Alpha’s reservoir pressure to its gas volume
And for the Shared Production case:
Q
1
C1e1t
k
2 k
C2e2t
(4)
Where, Q 3.3
Shared Production Alpha flow rate (at time t) Cross-Over Point
The cross-over point occurs when Q’ and Q are equal and hence the cross over time (t = X) can be determined by combining equations (3) and (4) to give:
P PD e 1C1e 2C2e X
1 X
2 X
(5)
This equation has to be solved iteratively for X. 3.4 Repayment Completion Point The repayment phase is completed when the back-out balance falls to zero. The back-out balance (B) at any time (t) is calculated as the difference between Alpha’s Stand-Alone and Shared cumulative production:
t
t
0
0
B Q' dt Q dt
(6)
By substituting for the flow rates from equations (3) and (4) and integrating with respect to time:
B
1 k
P P e
D
t
C1e1t C2e2t
(7)
Hence by equating B to zero, the right hand side can be solved for t = 0, the repayment completion time. There are two solutions: 0 = 0 0 = .
13
The back-out starts at zero, indicated by the first solution, but only returns to zero after an infinite time. This explains the shape of the plot of the balance in Figure 10, which starts at zero, goes through a maximum at the cross-over point but then extends out with time. The second solution of the equation completes the picture suggested by the plot in that it shows that the balance returns asymptotically towards zero over an infinite time. To further illustrate the slow rate of decline repayment, in this example after a further 10 years of production, there would still be approximately 20% of the balance unpaid. The analysis illustrates that using this back-out approach, the new third party field (Bravo) is not going to be repaid its gas in any reasonable timeframe. Indeed, the fields will have become uneconomic to produce long before the backout balance is appreciably repaid. A further problem for the new third party field is that the back-out agreement may contain a clause that states that: should the incumbent field’s production cease, or become uneconomic, any liability for back-out repayments ceases. This means that the new field will almost certainly be left with unpaid back-out gas. In effect, the back-out agreement is a “Lose Out” agreement for the new field. 3.5
Comparison with Complex Model
As noted in Appendix A the simplified model does make some simplifying assumptions in order to make the equations solvable analytically and allow the underlying physics of back-out to be analysed, in particular the long term behaviour. However, it is important that the simplified model is reasonably representative of the more complex models (and indeed actual production) in order for the conclusions drawn from it to be credible. Figure 11 is a plot of an incumbent field’s production from a real system though the data is anonymised.
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Shared production starts
Back-out Cross-over
Repayment
Figure 11 – Complex Model Flow Rate Results The real system is different to that in the fictitious example used in the simplified model, which is why the timing of events is not the same when comparing the two. The Shared Production starts after around 500 days and the difference between the forecast Stand-Alone production (red line) and the Shared Production (green line) representing back-out is immediately observed. The data is noisy up to around day 1,300 because the models have been matched to real historical production data up to that point. Beyond that the models are predicting into the future and appear smoother. The cross-over is observed around day 2,100 and the slow repayment behaviour observed after that. The downward spikes to zero represent planned shutdowns which are factored into the more complex models.
15
Figure 12 – Complex Model Back-Out Balance Figure 12 is a plot of the Back-Out Balance for the more complex model. The plot peaks at the cross-over at around day 2,100 before entering the slow payback period thereafter. Though noisier, the more complex model results display all same features as the simpler model. The same slow repayment phase can be observed with the real data but the power of the simplified model is that it shows algebraically that repayment cannot be achieved in practice and certainly not on economic timescales for such gas fields. In the author’s experience this phenomenon does not appear to have been appreciated when these back-out agreements were historically developed. It seems apparent that the focus of the agreements is on the initial back-out experienced by the host fields. The next section discusses options to address the repayment problem. 4 4.1
ALTERNATIVE REPAYMENT OPTIONS Contractual Agreement
Though, once realised, the unreasonably long repayment phase appears correctable in a technical sense. However, if the methodology is written into a contractual agreement it may be extremely difficult to obtain agreement from the incumbent field’s owners to modify the repayment mechanism. The first option presented in Section 4.2 is a relatively straight forward approach. However, it is not necessarily consistent with the premise, frequently encountered in agreements, that: the incumbent field is allocated the quantity of gas it would have produced without the new field being present.
16
The second option in Section 4.3 provides a methodology which is still arguably consistent with the premise. Further options are presented in Section 4.4 that incorporate the discounted value of gas into the back-out payment and repayment mechanisms. 4.2
Fixed Period Repayment
The repayment phase could simply be based on an alternative method. For example running the balance repayments down over the same period they were built up, or some variation of this, and dispensing with the models altogether for the repayment phase. Another option is to retain the model basis but accelerate the repayment so that the back-out gas is repaid to the new field within a specified time. This is accomplished by multiplying the gas transferred from the incumbent field by a factor such that the balance is returned to zero by a specified time (for example COP as discussed in Section 4.3. Prior to cross-over the daily back-out payments are calculated using:
QX Q' Q QX
(8)
Back-out payment rate.
This is the same as the current system and the back-out balance is calculated using Equation (7). At the cross-over time (t = X) the balance reaches its maximum value given by:
BX
1 k
P P e
D
X
C1e1 X C2e2 X
(9)
Where, BX
Back-out balance on cross-over day (at time t = X).
After cross-over, the repayments are accelerated using a modified version of Equation (8) that incorporates a repayment factor Y:
QR Y Q' Q QR Y
(10)
Back-out payment rate in the repayment phase Repayment factor.
The balance equation becomes:
1 BY BX Y k
P P e
D
t
e X C1 e1t e1 X C2 e2t e2 X
And Y is given by:
17
(11)
Y
1 k
P P e
D
BX COP
e X C1 e1 COP e1 X C2 e2 COP e2 X
(12)
This results in the following balance profile:
Figure 13 – Accelerated Back-Out Balance Though pragmatic, the problem with this approach is that the incumbent field’s allocated production is different to that which it would have experienced if it was produced alone as it is paying back the gas faster than the modelling predicts it should be. 4.3
Incorporation of COP into Back-Out
If the incumbent field becomes uneconomic and the repayment liability ceases before the balance is significantly repaid then in such a case the new tie-back field has effectively lost considerable volumes of gas. However, the cessation of production (COP) does offer a route for the new field to receive repayment whilst being consistent with the agreement. The COP represents the point when the incumbent field’s flow is so low that it is uneconomic to produce. So there is a definable COP flowrate. This COP flow rate would have been reached earlier in the Stand-Alone case so all the gas produced by the incumbent field between Stand-Alone COP time and Shared Production COP time should be transferred as repayment. This idea is illustrated in Figure 14:
18
Figure 14 – Alpha Stand-Alone vs Shared Production Flow Rate with COP Nominally, Field Alpha is considered to become uneconomic when its production falls to 1.0 mcm/d (million cubic meters per day) and this occurs on Day 3,562 for the Shared Production case, after which it is shut-in and its flow rate falls to zero. In the hypothetical Stand-Alone scenario this COP flow would have occurred on Day 3,513, 49 days earlier. Hence the entire Alpha production for the last 49 days of its life should be transferred to Bravo as repayment. The difference in the flow rates is plotted in Figure 15.
19
Figure 15 – Alpha Stand-Alone Production Minus Shared Production with COP This plot graphically illustrates the slow repayment after cross over with a relatively short period of high repayment prior to Alpha’s COP. Finally Figure 16 shows the back-out balance rapidly falling to zero just prior to COP:
Figure 16 – Back-Out Balance with COP As can be seen most of the repayment occurs in a single block of production just prior to COP. This could be a significant amount of time after the cross-over date but it is a mechanism by which the new field can obtain repayment, whilst still being consistent with the concept that the incumbent field’s allocation is unaffected by the new field. 4.4
Equivalent Value Method
The justification for the payment of back-out to the incumbent field is that it suffers a deferment of production and an apparent economic loss because of the time value of money. The revenue they would have received for the deferred gas, had they sold it sooner, is greater than that which they would receive at a later date as they could have invested that revenue to accrue interest or in a new project providing additional return on the investment. However, though being cited as the justification for the back-out mechanism, no calculation of this apparent loss of revenue appears in the back-out calculations themselves. ([3] considers back-out to be equivalent to a financial transaction and mentions the time value of money when describing back-out methodology guidelines). In order to incorporate the time value of money into the analysis the revenue is discounted using an interest rate (r); this is discounted cash flow.
20
An alternative is proposed that ensures the incumbent field receives an equivalent discounted cash flow value to that it would have received had it been produced Stand-Alone by a specific time (say COP). The discounted present value of the incumbent field’s (Field Alpha) gas is given by: t
V' S Q' e rt dt
(13)
0
Where, V’ S r
Discounted present value of Field Alpha’s stand-alone cumulative gas production at time t Monetary value of gas volume Discount rate.
This is the continuous form of the discounted cash flow equation. The monetary value of the gas, S, i.e. the gas price, will change over time. However, for simplicity in this analysis it has been treated as a constant. Similarly the value of the gas produced by Field Alpha in the commingled system is given by:
t
V S Q e rt dt
(14)
0
Where, V
Discounted present value of Field Alpha’s cumulative gas production at time t in the Shared system
The difference between V’ and V is the value of gas that Field Bravo should transfer to Alpha to ensure that Alpha receives the same value of gas it would have in the stand alone system by time t. For example if time t is chosen to be when Alpha is estimated to become uneconomic on a Stand-Alone basis, that is Day 3,513 described in Section 4.3, then Bravo needs to transfer gas equivalent in value, when discounted, to this difference:
VTran S VTran
COP
0
Q' e rt dt
COP
0
Q e rt dt
(15)
Discounted present value of Field Bravo’s cumulative gas production transferred to Field Alpha by time t=COP in the Shared system
This is analogous to equation (6) for the back-out balance, but expressed in dollar value rather than standard volume terms. Substituting equations (3) and (4) into (15) and integrating, allows VTran to be determined (see Section A.5). Hence, the value of Bravo’s cumulative gas allocation at time COP is given by:
21
V" S
COP
Q e rt dt VTran
0
(16)
Where, Q V”
Shared Bravo volume flow rate Discounted value of Bravo cumulative allocated gas after back-out payments or repayments.
Q is Bravo’s actual produced flow and is determined from an equation analogous to (4). If Bravo is allocated gas equivalent in value to V”, then Alpha must have been allocated gas equal in value to its Stand-Alone production. Hence Alpha will not have been economically disadvantaged. Further details of the calculation of V” are provided in Section A.6. V” can be expressed in terms of Bravo’s allocated flow:
V" S
COP
0
Q”
Q" e rt dt
(17)
Allocated Bravo volume flow rate after back-out payments or repayments.
Q” is the gas actually allocated to Bravo after payments and repayments of backout gas. In this proposed mechanism, Q” can be any allocated flow profile providing that at time COP, it satisfies equation (17). Hence a flow profile could be chosen such that at time COP, Bravo’s cumulative allocated volume of gas is equal to the cumulative produced gas:
COP
0
Q" dt
COP
0
Q dt
(18)
And hence the volumetric back-out balance returns to zero at COP. Several allocation profiles are proposed below; the simplest is a straight line:
Q" At B
(19)
Where A and B are constants which are determined by solving equations (17) and (18). The details of these calculations are provided in Section A.7. Assuming a values of A system and presented in
discount rate (r) of 10% and that COP occurs on Day 3,513, the and B have been determined. The produced flows in the Shared the allocated flows after back-out transfers for the two fields are Figure 17:
22
Figure 17 – Bravo Straight Line Profile Allocated and Produced Flows Bravo transfers more gas to Alpha (compared with the current back-out mechanism) in the early part of the profile when the discounted value of the gas is higher. The cross-over is reached earlier on Day 866, after which Alpha repays Bravo an equivalent volume of gas, when the gas’s discounted value is lower. The volumetric back-out balance is presented in Figure 18:
Figure 18 – Back-Out Balance Bravo Straight Line Allocation Profile The balance now returns to zero on the specified COP day. Hence, Bravo has been allocated the same cumulative volume of gas that it has produced.
23
Alpha has also been allocated the same volume of gas that it has produced in the Shared system, though this is less than that which it would have produced on a Stand-Alone basis. However, the allocated gas Alpha has received is equal in discounted value to the Stand-Alone gas, meaning it has not been economically disadvantaged as a result of its deferred production. The profile of Alpha’s cumulative gas value in the Stand-Alone and Shared production cases is presented in Figure 19:
Figure 19 – Alpha Allocated Cumulative Value The locus of each line is almost coincide but they converge to the same value on the Stand-Alone COP Day. In addition to the straight line flow profile two further profiles were examined. A one term exponential profile:
Q" De Et
(20)
And two term exponential profile:
Q" Fe1t Ge2t
(21)
D and E, and F and G are constants which are determined by solving equations (16) and (18), with the flow profiles given by equations (20) and (21) respectively. The details of the calculations are provided in Sections A.8 and A.9. The two term exponential profile is included as it is analogous to the actual flow profile given by an equation of the form provided for Alpha in equation (3). The analogous form of the equation for Bravo’s Shared flow profile has the same values of 1 and 2 but different values of the constants C1 and C2. In the
24
proposed two term profile it is these constants that are replaced by F and G, with the terms being retained. The details are provided in Section A.9. The three alternative allocated Bravo flow profiles are plotted in Figure 20:
Figure 20 – Bravo Straight Line Profile Allocated and Produced Flows Also shown for comparison is the Bravo produced flow. The Bravo produced (orange) and Straight Line (purple) are identical to the same coloured lines in Figure 17 labelled: “Shared Bravo” and “Bravo after Transfer”. The above analysis illustrates that it is possible to incorporate the discounted value of the gas into the back-out calculations and simultaneously ensure that fields are allocated the volume of gas they physically produce over a specified time period. The exposure to economic loss is used to justify the very introduction of back-out payments but then does not appear in any quantified manner in the actual system. It is suggested that the alternative approaches presented in this section are more appropriate in a conceptual sense as they incorporate the economic loss directly into the back-out mechanism. 5
CONCLUSIONS
The paper has explored the behaviour of typical currently employed back-out payment and repayment mechanisms. It has attempted to expose potential pitfalls associated with back-agreements and presented potential solutions to address them. It has accomplished this with the use of simplified mathematical models which complement the more complex models but have the advantage that they are more tractable and readily predict the long term behaviour of such back-out systems.
25
The analysis has revealed that the repayment phase is so prolonged that the new third party field may not receive a significant proportion of repayment before the incumbent field’s COP. More profoundly, the simplified model has proved mathematically that the repayment takes an infinite time and hence strongly suggests in principle that the new field cannot be repaid in full. A mechanism has been suggested, that incorporates the COP, and ensures full repayment of gas. It achieves this whilst being consistent with the premise, frequently expressed in back-out agreements, that the incumbent field’s allocated production is no different to that which it would have experienced if it was produced alone. The paper also examines the economic justification for the introduction of the back-out payments. It also identifies that economic valuation calculations do not feature in typical current back-out systems. Further alternative approaches are presented that incorporate the discounted value of the gas into the back-out mechanism. These approaches simultaneously ensure that the incumbent field receives the same value of gas it would have if produced Stand-Alone and that both fields are allocated the same cumulative volume of gas they would have produced by a specified time. When they are written, some back-out agreements appear to focus on the initial back-out experienced by the incumbent field(s). Perhaps this is not surprising as this phase is in the near future. In contrast it appears that the consequences associated with the mechanisms for the repayment phase, are not fully realised. This can result in significant penalty of lost production for the new field.
26
APPENDIX A SIMPLIFIED MATHEMATICAL MODEL A.1
System
Figure 21 depicts the two gas reservoirs ( and ) producing through a shared subsea pipeline to an onshore gas plant.
Figure 21 – System Configuration The pressures and standard volumetric flows at various points in the system are indicated symbolically. A.2
Simplifying Approximations
In order to render the approximations are made.
system
solvable
analytically
some
simplifying
The frictional pressure loss in the pipelines is approximated to be linearly dependent on flow:
Q R P PX
(22)
Q R P PX
(23)
Q Q RP PX PD
(24)
The R terms represent the resistance to flow in each pipe segment and are assumed to be constants.
27
The typical correlation between cumulative gas produced to date (GPD) and reservoir pressure (divided by compressibility) is illustrated in Figure 22 below:
Figure 22 – Reservoir Pressure versus Cumulative Production
Again to simplify the mathematics gas compressibility (Z) is assumed to be constant over the pressure range and reservoir pressure can be expressed as a linear function of cumulative production according to:
P k GPD A.3
(25)
Shared Production Scenario
Cumulative production is given by the integral of the flow rate over time: t
GPD Q dt 0
(26)
Rearranging (25) and substituting for GDP in (26), and rearranging in terms of reservoir pressure gives:
t
P k Q dt 0
Differentiating with respect to time:
28
(27)
dP k Q dt
(28)
A similar relationship can be obtained for Bravo:
dP dt
k Q
(29)
Substituting for flow Q from (22) in (28):
dP k R P PX dt
(30)
k R P PX
(31)
Similarly for Bravo:
dP dt
Rearranging equation (24) in terms of PX and substituting for Q and Q from (22) and (23) results in:
PX
R P R P RP PD R R RP
(32)
This can be substituted in (30) and (31) to obtain:
dP k R P RP P R P R PD RP dt R R RP
(33)
k R P RP P R P R PD RP dt R R RP
(34)
And:
dP
Differentiating (33) with respect to time:
dP dP d 2 P k R R R R P 2 dt dt R R RP dt
(35)
Rearranging (35):
dP dt
1 R
dP d 2 P R R RP RP R 2 dt k R dt
29
(36)
Rearranging (33) in terms of P:
P
1 R
dP R R RP P RP P R PD RP dt k R
(37)
Substituting (36) and (37) in (34) and rearranging produces:
d 2 P k R RP R k R RP R dP k k R R RP dt R R RP dt 2 R R RP k k R R RP PD R R RP
P
(38)
This is a second order linear differential equation in P, which has the solution:
P C1e1t C2e2t PD
(39)
This is an initial value problem, with initial conditions defined when t= 0: At
t 0, P P , P P
(40)
The starting reservoir pressures P° and P° are assumed to be known. The various and C parameters in (39) are:
1
2
k R RP R k R RP R
k R RP R k R RP R
k R R
P
2 2 2 R k R RP R 4k2 R (41 k R
2 R R RP
k R R
P
2
)
R k R RP R 4k2(42 R2 k 2 R2
2 R R RP
2
k R P R P RP R PD RP 1 C1 2 P PD 2 1 R R RP 1 k R P R P RP R PD RP C2 1 P PD 2 1 R R RP
As P is obtained all other variables: P, Q and Q can be determined. A.4
Stand Alone Production Scenario
For the Alpha Stand-Alone case, Q is zero. Equation (24) now becomes:
30
)
(43)
(44)
Q' RP PX PD
(45)
Combining equation (22) and (45) to eliminate PX:
Q '
R P' PD R 1 RP
(46)
This is then substituted in (28) to obtain the first order differential equation in P’: ' dP' k R P PD dt R 1 RP
(47)
With initial conditions defined when t= 0:
At t 0, P P , '
(48)
This has the solution:
P' P PD et PD
(49)
Where,
k R RP RP R
(50)
Now P’ is obtained, Q’ can be determined. A.5
Calculation of Value of Transferred Gas
The discounted value transferred from Bravo to Alpha is given by the cumulative difference in the value of Alpha’s Stand-Alone and Shared produced volumes when t=COP:
VTran S
COP
0
Q' e rt dt S
COP
0
Q e rt dt
(51)
Substituting for Q’ and Q from equations (3) and (4):
VTran S
COP
0
k
P
PD e t e rt dt S
COP
0
1 2 1t 2t rt C1e C2e e dt k k
(52)
Which integrates to:
VTran
S 1C1 S 2C2 S P PD e r t 1 e 1 r t 1 e 2 r t (53 1 ) k r k 1 r k 2 r
31
A.6
Calculation of Required Value of Bravo Allocated Gas
The required value of Bravo’s gas after transfer is:
V" V VTran
(54)
If Bravo is allocated a quantity of gas that is equivalent in discounted value to V ”, then Alpha’s allocated gas in the Shared system must be equal in value to that which it would have received Stand-Alone. The value of Bravo’s produced gas V is given by:
V S
COP
0
1 rt t t C 1e 1 2 C 2e 2 e dt k k
(55)
Which integrates to:
V
S 1C 1 k
e r
1 r t
1
1
S 2C 2 k
e r
2 r t
1
(56)
2
Hence the required discounted value Bravo’s cumulative gas allocation can be obtained by substituting (53) and (56) in (54). A.7
Equivalent Value Back-Out Straight Line Bravo Profile
Assume that Bravo’s allocation profile follows a straight line:
Q" At B
(57)
In order to ensure that Alpha receives the correct discounted value of gas at COP:
S
COP
0
Q" e rt dt V"
(58)
Also, for the back-out balance to return to zero, Bravo has to be allocated an equivalent volume of gas to that which it produced:
COP
0
Q" dt
COP
0
Q dt
COP
0
1 t t C 1e 1 2 C 2e 2 dt k k
(59)
Hence:
COP
0
Q" dt
C 1 k
e
1t
1
C 2 k
e
2t
1 q
(60)
Therefore: COP
At B dt q 0
And,
32
(61)
S
COP
0
At B ert dt V"
(62)
Equation (61) integrates to:
A COP 2 B COP q 2
(63)
And equation (62) integrates by parts to:
1 e r COP A 2 r r
1 e r COP 1 B COP r r r
V S
"
(64)
(63) and (64) are simultaneous equations and can be solved for A and B:
V" q r 2 1 e r COP S COP r A r COP r e r r COP COP COP 1 e 2 2
B
q
COP
A COP 2 2
(65)
(66)
Which are then substituted into (57) to obtain Bravo’s allocated profile. A.8
Equivalent Value Back-Out Single Exponential Term Profile
Assume that Bravo’s allocation profile follows an exponential function of time:
Q" De Et
(67)
Similar to the approach taken in Section A.7, we require the discounted value at COP to be:
S
COP
0
De Et e rt dt V"
(68)
And the cumulative volume to be given by:
COP
0
De Et dt q
(69)
Equation (68) integrates to:
V" D e E r t 1 E r S And equation (69) to:
33
(70)
D Et e 1 q E
(71)
(70) and (71) are simultaneous equations and can be solved for D and E:
E
V" r 1 e E COP
S q e E r COP 1 V"
1 e
E COP
(72)
Which has to be solved iteratively for E and D is then given by:
D
q E
e
Et
(73)
1
Which are then substituted into (67) to obtain Bravo’s allocated profile. A.9
Equivalent Value Back-Out Two Exponential Term Profile
Assume that Bravo’s allocation profile follows a two term exponential function of time:
Q" Fe1t Ge2t
(74)
Similar to the approach taken in Section A.7, we require the discounted value at COP to be:
S
COP
0
Fe
1t rt
e
Ge2t e rt dt V"
(75)
And the cumulative volume to be given by:
COP
0
Fe1t Ge2t dt q
(76)
Equation (75) integrates to:
V" F G e 1 r COP 1 e 2 r COP 1 1 r 2 r S
(77)
And equation (76) to:
e F
1
1 COP
1
G
2
e
2 COP
1 q
(78)
(77) and (78)are simultaneous equations and can be solved for F and G:
V" Mq S K G ML N K
34
(79)
F
q LG
(80)
K
Where,
K
e 1
(81)
(82)
1 COP
1
2 COP
1
1
L
1
2
e
(83)
(84)
M
1 e 1 r COP 1 1 r
N
1 e 2 r COP 1 2 r
And,
Which are then substituted into (74) to obtain Bravo’s allocated profile. A.10 k k ° P ° P PD r R R RP S
Values of Parameters Used in Example System -0.004344 bar/mcm -0.4683577 bar/mcm 48.3 bara 145.3 bara 6.75 bara 10% per annum 0.1047 mcm/bar 0.004911 mcm/bar 0.160105 mcm/bar. 0.197 $/Sm3 (calculated assuming a gas price of £0.4/therm, gas GCV (Gross Calorific Value) of 40 MJ/Sm3 and exchange rate of 1.30 $ per £).
REFERENCES [1] [2] [3]
Petroleum Experts supply MBAL, PROPSER and GAP, http://www.petex.com/ Schlumberger supply Eclipse, OFM and Pipesim, https://www.software.slb.com/products North Slope of Alaska Facility Sharing Study, Prepared for Division of Oil & Gas, Alaska Department of Natural Resources, May 2004, by Bob Kaltenbach, Chantal Walsh, Cathy Foerster, Tom Walsh, Jan MacDonald, Pete Stokes, Chris Livesey and Will Nebesky
Highlights
A simplified model is used to analyse the long term behaviour of back-out systems The model shows that the repayment phase duration is infinite
35
Repayment models are proposed that incorporate the fields’ cessation of production Back-out payment is justified because of the time value of money One alternative incorporates the time value of money into the calculations
36
www.elsevier.com/locate/flowmeasinst
PII: DOI: Reference:
S0955-5986(17)30333-3 https://doi.org/10.1016/j.flowmeasinst.2018.02.022 JFMI1423
To appear in: Flow Measurement and Instrumentation Received date: 17 August 2017 Revised date: 22 October 2017 Accepted date: 14 February 2018 Cite this article as: Phil Stockton, When Allocation “Back Out” Agreements can become “Lose Out” Agreements, Flow Measurement and Instrumentation, https://doi.org/10.1016/j.flowmeasinst.2018.02.022 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
When Allocation “Back Out” Agreements can become “Lose Out” Agreements Phil Stockton Accord Energy Solutions Limited NOTATION A B BX BY C1 C2 D E F G GDP K k k L M N P P’ ° P P P’ ° P PD PX Q Q’ Q Q” QR QX r R R RP S t V V’ V” VTran Y Z 1 2 0 COP
Constant Back-out balance / constant Back-out balance in payment phase Back-out balance in re-payment phase System parameter defined in (43) System parameter defined in (44) Constant Constant Constant Constant Field Alpha cumulative gas production to date Constant Constant relating Alpha’s reservoir pressure to its gas volume Constant relating Bravo’s reservoir pressure to its gas volume Constant Constant Constant Field Alpha Shared Production reservoir pressure Field Alpha Stand-Alone reservoir pressure Initial Alpha reservoir pressure Field Bravo Shared Production reservoir pressure Field Bravo Stand-Alone reservoir pressure Initial Bravo reservoir pressure Delivery pressure Manifold pressure Field Alpha Shared Production volumetric flow rate Field Alpha Stand-Alone volumetric flow rate Field Bravo Shared Production volumetric flow rate Field Bravo allocated volumetric flow rate Back-out volumetric re-payment rate Back-out volumetric payment rate Discount rate Flow resistance factor Alpha to X (manifold) Flow resistance factor Bravo to X (manifold) Flow resistance factor X (manifold) to delivery point Monetary value of gas per unit volume Time Field Alpha Shared discounted present value Field Alpha Stand-Alone discounted present value Field Bravo allocated allocated discount value Discounted present value of transferred gas Repayment factor Gas compressibility System parameter defined in (41) System parameter defined in (42) Repayment completion time COP time
1
X μ
1 1.1
Cross-over time System parameter defined in (50)
INTRODUCTION Overview
When a newly developed third party field is tied back to existing infrastructure, such as a pipeline system, there is the potential for the new field’s production to cause an increase in back pressure and reduce the flow of the incumbent fields already producing through the system. This is termed “back-out” and in effect defers some of the incumbent fields’ production. Back-out agreements are sometimes put in place, within the allocation system, to compensate the incumbent fields for the reduction in flow they experience. This may take the form of a transfer (or payment) from the third party of hydrocarbons to the incumbent fields so that in effect they do not experience deferment and suffer resultant economic loss. As the production matures, the situation eventually reverses and the transfer (repayment) proceeds in the opposite direction from the incumbents to the third party. The assumption often made is that eventually the transfers will cancel one another and over their lifetime, all fields’ cumulative allocation will equal their cumulative production. There is little published information on back-out agreements as they are generally commercially sensitive and hence confidential in nature. However, the Division of Oil and Gas of the Alaska Department of Natural Resources commissioned and published a report entitled: “North Slope of Alaska, Facility Sharing Study” [3], which presents back-out methodology guidelines. This paper presents the concept of how back-out calculations typically work and presents a simplified mathematical model of a system where an incumbent field is backed out by a new field introduced into a pipeline. The simplified model results in a set of differential equations which can be solved analytically. This simplified model is used to illustrate potential problems with the above concept of back-out, particularly in the repayment period. The simplified model demonstrates that in principle it takes an infinite time for the repayment to be achieved and hence the assumption that all parties are eventually allocated the gas they produce is not valid. In effect there is the very real potential for “backout” to become “lose out” for the new field. The results of the simplified model are compared against the results from more rigorous, complex models (run on commercial software) to confirm that the behaviour of the simplified model adequately reflects the system. The complex models are solved numerically and have to be run in sequential time steps. The advantage of the simplified model is that it is directly solvable for any time and hence the long term behaviour of the system can be analysed directly. Finally the paper discusses methods to overcome the pitfalls exposed by the analysis. In doing so it considers the fundamental question that the back-out payments and transfers are attempting to address; that is: a perceived economic loss due to deferred production is compensated by the transfer of gas quantities without any calculation of the apparent value of the gas. Methods are presented
2
that incorporate the time value of money in order to provide allocation mechanisms that ensure the incumbent field receives the equivalent value of gas as if it was producing alone and still ensures that the third party field is allocated a volume of gas equal to its production over a specified time period. This paper specifically addresses back-out associated with gas reservoirs producing through shared pipeline infra-structure. 1.2
Structure of the Paper
Section 2 describes a typical gas production system in which back-out occurs. This is used as an example to illustrate the issues associated with back-out. Though a fictitious system it is based on typical production rates and parameters from a real system to ensure it is reasonably representative of equivalent real systems in general. The values of the parameters used to generate the flows, pressures, etc. presented in the various charts are given in Section A.10. Section 3 describes a simplified mathematical model that is used to analyse the long term behaviour of the system. It thereby exposes a problem in that it appears it is impossible for the tied back field to be repaid, in any reasonable timeframe, the gas it has transferred to the incumbent field during the early back-out payment phase. Section 4 examines options to address the problem of excessive repayment periods in an effort to ensure all fields receive the gas they have produced over the length of their lives. Section 5 presents conclusions. Appendix A provides a detailed description of the simplified mathematical model employed. 2 2.1
DESCRIPTION OF A SYSTEM WITH BACK-OUT Physical System
Consider the subsea gas pipeline system depicted in Figure 1:
3
Figure 1 – System Configuration The incumbent field has been labelled as Field Alpha (). In order to utilise spare capacity and maximise the use of existing infrastructure it is often mutually beneficial for new developments to tie back into existing transportation and processing facilities. A potential problem for the incumbent field is that the introduction of a new field, denoted Bravo (), increases the total flow down the shared pipeline resulting in a higher back pressure at Field Alpha’s wellhead(s). If Alpha’s wells are off plateau, the increased back pressure will reduce production from the field and this is termed back-out. (Since new field tie-backs into existing infra-structure are usually taking advantage of spare capacity, it is likely that the incumbent fields’ are off-plateau). Unless the increased back pressure has a permanently deleterious impact on the incumbent wells, back-out does not result in lost production, it merely defers it. However, due to the time value of money the incumbent field would normally wish to produce its hydrocarbons sooner rather than later and back-out is considered to have an economic cost to the incumbent. To understand the consequences of back-out, it is instructive first to consider the behaviour of the incumbent field when it is producing alone, i.e. without back-out. 2.2
Stand-Alone Production
When Field Alpha is producing alone and it is off plateau, the pressure drop across the system, from the reservoir to the delivery point at the gas plant, will determine its flow. As the reservoir pressure drops, the driving-force decreases and Alpha’s flow declines. This is illustrated in Figure 2.
4
The dash ’ in Q’ means this is the Stand-Alone volumetric flow for Field Alpha. The horizontal scale ranges from 0 to 4,000 days, almost 11 years of production. The reservoir pressure also declines as shown in Figure 3.
Figure 2 – Alpha Stand-Alone Flow Rate The reservoir acts like a pressurised tank. As gas is produced from the reservoir the pressure in it falls. As the pressure declines the driving force reduces, resulting in a lower flow rate. The rate of pressure decline also falls as can be seen in the figure.
Figure 3 – Alpha Stand-Alone Reservoir Pressure The reservoir pressure tends asymptotically to the delivery pressure at the gas plant, which in this example is set at 6.75 bara. The fall in the reservoir pressure is directly related to the cumulative production which is plotted in Figure 4.
Figure 4 – Alpha Stand-Alone Cumulative Production The above plots represent the evolution of Alpha’s production with time in the absence of a new field being introduced. This is compared in the next section with the scenario of Shared Production with Field Bravo – a new third party tie-back. 2.3
Shared Production and Back-Out
In the Shared Production scenario, Alpha and Bravo are producing together. Alpha experiences a rise in back pressure due to the higher flow and resulting
5
increased frictional pressure drop across the shared section of the subsea flow line. Alpha’s flow rate will therefore be lower when compared with the StandAlone case. The Stand-Alone and Shared Production1 flows are compared in Figure 5:
Back-out Cross-over
Repayment
Figure 5 – Alpha Stand-Alone vs Shared Production Flow Rate The locus of the green dashed line is identical to that in Figure 2. The red line shows the reduced flow experienced by Alpha in the early phase of the profile. The margin narrows until it becomes zero after 1,112 days, when Alpha’s Shared Production exceeds its Stand-Alone level. The reason for this is revealed by comparing the difference in the reservoir pressures for the two scenarios in Figure 6:
1
The Stand-Alone and Shared Production scenarios or cases have been capitalised for convenience of reference throughout the paper.
6
Back-out
Repayment
Cross-over
Figure 6 – Alpha Stand-Alone Minus Shared Production Reservoir Pressure (The difference in pressure is presented as the plots of the two reservoir pressures almost lie over one another). The plot shows that the Stand-Alone reservoir pressure is always lower than in the Shared Production case. This is because more gas has been taken out of the reservoir in the Stand-Alone case. Though the back pressure experienced by Alpha is greater in the Shared case, the driving force at the reservoir is also greater and this eventually results in Alpha producing more in the Shared Production scenario. This reservoir pressure difference between the two cases reaches its maximum at the cross-over point when Alpha’s flow rate is identical in both scenarios. Also in the Shared case the production from Field Bravo is declining with time and therefore further reducing the back pressure. The daily flow for Bravo is presented in Figure 7. In the example, Bravo’s production is less than Alpha’s but its initial reservoir pressure is significantly greater as illustrated Figure 8.
Figure 7 – Bravo Flow Rate
7
Bravo’s initial high reservoir pressure backs out Alpha but it declines rapidly as its reservoir is considerably smaller than Alpha’s.
Figure 8 – Bravo Reservoir Pressure Returning to Alpha, the difference in the Stand-Alone and Shared Production daily flow rates is compared in Figure 9:
Back-out
Cross-over
Repayment
Figure 9 – Alpha Stand-Alone Production Minus Shared Production The difference in the two daily flow rates, which is positive in the first 1,112 days, represents deferred production for Alpha and is termed the back-out. Back-out agreements can be put in place to compensate incumbent fields (such as Alpha) for this deferred production. 2.4
Back-Out Agreement
A typical premise of back-out agreements is to allocate the incumbent field the quantity of gas it would have produced without the new field being present.
8
In the example, this entails calculating Alpha’s deferred production as the difference between Alpha’s measured (or allocated) production when coproducing compared with its estimated Stand-Alone production. Gas is then transferred from Bravo to Alpha to compensate Alpha for the deferment, or back-out, and ensure it does not suffer economic loss. An issue that immediately arises is the Stand-Alone Alpha production. This cannot be directly measured as it did not actually occur and is therefore hypothetical in nature. In order to overcome this, simulation models are used to calculate the hypothetical Stand-Alone production. 2.5
Back-Out Modelling
Back-out agreements usually stipulate the simulation modelling to be under-taken in order to calculate the back-out. These simulations can vary in type and complexity but they usually involve modelling the reservoirs, wellbores and pipeline infrastructure to the point of delivery, e.g. at the inlet to a gas plant where the pressure may be considered fixed. Commercial software programs [1], [2] such as MBAL and Eclipse model reservoirs, PROSPER and OFM model wells and GAP and PipeSim model pipeline networks. These may be integrated under a wider software framework. Such models are relatively sophisticated and require considerable expertise and effort to run. In order to predict the level of back-out normally two models are run:
The Shared Production case with both fields producing, i.e. mimicking reality and; The Stand-Alone hypothetical case with only the incumbent field producing.
The difference between the two then provides the predicted back-out experienced by the incumbent field. At first sight, the model simulating reality appears superfluous. However, it serves two purposes:
It allows the modelling environment to be calibrated against real production, i.e. by history matching the model against real data. The parameters calibrated in the Shared model can then be used in the Stand-Alone model and thereby improve estimates of back-out; It allows calculated daily back-out quantities to be projected into the future, the forecast back-out being used in the daily allocation in the forthcoming period until a new calibration exercise can be undertaken.
There are variations to the above approach but they all tend to include models that simulate both scenarios. 2.6
Deferment Back-Out is different to Hydrodynamic Back-Out
It is worth noting that back-out, in the deferred production sense and addressed in back-out Agreements, cannot be measured directly.
9
At first sight it may appear that the back-out experienced by the incumbent field Alpha could be determined by physically shutting in Bravo and recording the change in Alpha’s production as the back-out. But this is a different kind of backout, termed here as hydrodynamic back-out. Hydro-dynamically, at any instant in time, Bravo is backing out Alpha in the sense that if Bravo is shut in Alpha’s flow will always increase. However, this is not how deferred production back-out is defined in back-out agreements as Alpha’s flow has to be compared with the hypothetical case of Alpha having produced up to this point in time without Bravo producing at all. In such a scenario, Alpha’s reservoir pressure would be lower and it would be producing less on a Stand-Alone basis. When determining back-out in the deferred production sense, the Shared Production case, however it is determined, has to be compared against a parallel, hypothetical scenario in which only Alpha has ever produced. 2.7
Back-Out Balances
There are three phases associated with back-out which are indicated on Figure 5, Figure 6 and Figure 9:
Back-out payment Cross-over Repayment.
During the initial back-out payment phase the transfer of gas from Field Bravo to Alpha is normally recorded and accumulated as a back-out balance. Once the cross-over point is reached the daily back-out payments will fall to zero and the back-out balance reaches its maximum. The repayment phase ensues and Alpha commences transfer of gas back to Bravo. The back-out balance is decremented accordingly. The back-out balance associated with the fictitious example introduced above is presented in Figure 10:
10
Repayment Cross-over
Back-out
Figure 10 – Back-Out Balance The back-out balance reaches its maximum value of 79.1 mcm (million cubic metres) at the cross-over point. This is the cumulative quantity of gas Bravo has transferred to Alpha to compensate for deferred production. After the cross-over point Alpha Field is producing more gas than it would have been if it had produced on its own and the back-out becomes negative and repayment in the opposite direction from Alpha to Bravo takes place. The back-out balance is then decremented by the repayment quantities. When the back-out balance reaches zero the back-out payments will all have been repaid and the back-out calculations cease. However, how long does it take for the balance to return to zero? Examination of Figure 10 shows that it took slightly over 3 years to build up the balance to its maximum value, but after almost a further 8 years, less than half the balance has been repaid. In order to answer the question posed above, a simplified model of the system was developed and this is described, and its behaviour analysed, in the next section. 3 3.1
SIMPLIFIED MODEL Reasons for the Simplified Model
In order to assess the behaviour of the repayment phase and estimate the completion date, a simplified mathematical model of the system and associated back-out was developed. The derivation of the model is presented in Appendix A. This simplified model produces analytical equations that can solve the state of the system at any point in time.
11
Running the rigorous, complex models mentioned in Section 2.5, is considerably more time consuming and being numerical in nature, not so easily extended far into the future in order to examine the long term behaviour of the repayment phase. The analytical solvable equations associated with the simplified model also provide a deeper understanding of the behaviour of the system. 3.2
Model Development
The model is based on the following principles:
the fall in reservoir pressure over a period of time is directly related to the cumulative production over that period the cumulative production is equal to the integrated flow rate over time the flow rate is equal to the rate of change of reservoir pressure.
Two principal sets of equations were developed. The first describes the behaviour of Alpha’s reservoir pressure in the Stand-Alone case:
P' P PD et PD
(1)
Where, P’ P PD t
°
Stand-Alone Alpha reservoir pressure (at time t). (The dash ’ denotes the Stand-Alone case and is not the differential operator) Initial Alpha reservoir pressure Delivery pressure (at gas plant) time
μ is a constant determined by several system parameters; see Appendix A for its definition. The second describes Alpha’s reservoir pressure in the Shared Production case:
P C1e1t C2e2t PD
(2)
Where, P
Shared Production Alpha reservoir pressure (at time t)
C1, C2, 1 and 2 are all constants determined by the system parameters; see Appendix A for their definition. Hence, at any point in time these equations provide the reservoir pressures for the two scenarios. Two analogous equations provide the flow rate at any point in time. Alpha’s flow rate in the Stand-Alone case is given by:
12
Q'
k
P
PD e t
(3)
Where, Q’ k
Stand-Alone Alpha flow rate (at time t) Constant relating Alpha’s reservoir pressure to its gas volume
And for the Shared Production case:
Q
1
C1e1t
k
2 k
C2e2t
(4)
Where, Q 3.3
Shared Production Alpha flow rate (at time t) Cross-Over Point
The cross-over point occurs when Q’ and Q are equal and hence the cross over time (t = X) can be determined by combining equations (3) and (4) to give:
P PD e 1C1e 2C2e X
1 X
2 X
(5)
This equation has to be solved iteratively for X. 3.4 Repayment Completion Point The repayment phase is completed when the back-out balance falls to zero. The back-out balance (B) at any time (t) is calculated as the difference between Alpha’s Stand-Alone and Shared cumulative production:
t
t
0
0
B Q' dt Q dt
(6)
By substituting for the flow rates from equations (3) and (4) and integrating with respect to time:
B
1 k
P P e
D
t
C1e1t C2e2t
(7)
Hence by equating B to zero, the right hand side can be solved for t = 0, the repayment completion time. There are two solutions: 0 = 0 0 = .
13
The back-out starts at zero, indicated by the first solution, but only returns to zero after an infinite time. This explains the shape of the plot of the balance in Figure 10, which starts at zero, goes through a maximum at the cross-over point but then extends out with time. The second solution of the equation completes the picture suggested by the plot in that it shows that the balance returns asymptotically towards zero over an infinite time. To further illustrate the slow rate of decline repayment, in this example after a further 10 years of production, there would still be approximately 20% of the balance unpaid. The analysis illustrates that using this back-out approach, the new third party field (Bravo) is not going to be repaid its gas in any reasonable timeframe. Indeed, the fields will have become uneconomic to produce long before the backout balance is appreciably repaid. A further problem for the new third party field is that the back-out agreement may contain a clause that states that: should the incumbent field’s production cease, or become uneconomic, any liability for back-out repayments ceases. This means that the new field will almost certainly be left with unpaid back-out gas. In effect, the back-out agreement is a “Lose Out” agreement for the new field. 3.5
Comparison with Complex Model
As noted in Appendix A the simplified model does make some simplifying assumptions in order to make the equations solvable analytically and allow the underlying physics of back-out to be analysed, in particular the long term behaviour. However, it is important that the simplified model is reasonably representative of the more complex models (and indeed actual production) in order for the conclusions drawn from it to be credible. Figure 11 is a plot of an incumbent field’s production from a real system though the data is anonymised.
14
Shared production starts
Back-out Cross-over
Repayment
Figure 11 – Complex Model Flow Rate Results The real system is different to that in the fictitious example used in the simplified model, which is why the timing of events is not the same when comparing the two. The Shared Production starts after around 500 days and the difference between the forecast Stand-Alone production (red line) and the Shared Production (green line) representing back-out is immediately observed. The data is noisy up to around day 1,300 because the models have been matched to real historical production data up to that point. Beyond that the models are predicting into the future and appear smoother. The cross-over is observed around day 2,100 and the slow repayment behaviour observed after that. The downward spikes to zero represent planned shutdowns which are factored into the more complex models.
15
Figure 12 – Complex Model Back-Out Balance Figure 12 is a plot of the Back-Out Balance for the more complex model. The plot peaks at the cross-over at around day 2,100 before entering the slow payback period thereafter. Though noisier, the more complex model results display all same features as the simpler model. The same slow repayment phase can be observed with the real data but the power of the simplified model is that it shows algebraically that repayment cannot be achieved in practice and certainly not on economic timescales for such gas fields. In the author’s experience this phenomenon does not appear to have been appreciated when these back-out agreements were historically developed. It seems apparent that the focus of the agreements is on the initial back-out experienced by the host fields. The next section discusses options to address the repayment problem. 4 4.1
ALTERNATIVE REPAYMENT OPTIONS Contractual Agreement
Though, once realised, the unreasonably long repayment phase appears correctable in a technical sense. However, if the methodology is written into a contractual agreement it may be extremely difficult to obtain agreement from the incumbent field’s owners to modify the repayment mechanism. The first option presented in Section 4.2 is a relatively straight forward approach. However, it is not necessarily consistent with the premise, frequently encountered in agreements, that: the incumbent field is allocated the quantity of gas it would have produced without the new field being present.
16
The second option in Section 4.3 provides a methodology which is still arguably consistent with the premise. Further options are presented in Section 4.4 that incorporate the discounted value of gas into the back-out payment and repayment mechanisms. 4.2
Fixed Period Repayment
The repayment phase could simply be based on an alternative method. For example running the balance repayments down over the same period they were built up, or some variation of this, and dispensing with the models altogether for the repayment phase. Another option is to retain the model basis but accelerate the repayment so that the back-out gas is repaid to the new field within a specified time. This is accomplished by multiplying the gas transferred from the incumbent field by a factor such that the balance is returned to zero by a specified time (for example COP as discussed in Section 4.3. Prior to cross-over the daily back-out payments are calculated using:
QX Q' Q QX
(8)
Back-out payment rate.
This is the same as the current system and the back-out balance is calculated using Equation (7). At the cross-over time (t = X) the balance reaches its maximum value given by:
BX
1 k
P P e
D
X
C1e1 X C2e2 X
(9)
Where, BX
Back-out balance on cross-over day (at time t = X).
After cross-over, the repayments are accelerated using a modified version of Equation (8) that incorporates a repayment factor Y:
QR Y Q' Q QR Y
(10)
Back-out payment rate in the repayment phase Repayment factor.
The balance equation becomes:
1 BY BX Y k
P P e
D
t
e X C1 e1t e1 X C2 e2t e2 X
And Y is given by:
17
(11)
Y
1 k
P P e
D
BX COP
e X C1 e1 COP e1 X C2 e2 COP e2 X
(12)
This results in the following balance profile:
Figure 13 – Accelerated Back-Out Balance Though pragmatic, the problem with this approach is that the incumbent field’s allocated production is different to that which it would have experienced if it was produced alone as it is paying back the gas faster than the modelling predicts it should be. 4.3
Incorporation of COP into Back-Out
If the incumbent field becomes uneconomic and the repayment liability ceases before the balance is significantly repaid then in such a case the new tie-back field has effectively lost considerable volumes of gas. However, the cessation of production (COP) does offer a route for the new field to receive repayment whilst being consistent with the agreement. The COP represents the point when the incumbent field’s flow is so low that it is uneconomic to produce. So there is a definable COP flowrate. This COP flow rate would have been reached earlier in the Stand-Alone case so all the gas produced by the incumbent field between Stand-Alone COP time and Shared Production COP time should be transferred as repayment. This idea is illustrated in Figure 14:
18
Figure 14 – Alpha Stand-Alone vs Shared Production Flow Rate with COP Nominally, Field Alpha is considered to become uneconomic when its production falls to 1.0 mcm/d (million cubic meters per day) and this occurs on Day 3,562 for the Shared Production case, after which it is shut-in and its flow rate falls to zero. In the hypothetical Stand-Alone scenario this COP flow would have occurred on Day 3,513, 49 days earlier. Hence the entire Alpha production for the last 49 days of its life should be transferred to Bravo as repayment. The difference in the flow rates is plotted in Figure 15.
19
Figure 15 – Alpha Stand-Alone Production Minus Shared Production with COP This plot graphically illustrates the slow repayment after cross over with a relatively short period of high repayment prior to Alpha’s COP. Finally Figure 16 shows the back-out balance rapidly falling to zero just prior to COP:
Figure 16 – Back-Out Balance with COP As can be seen most of the repayment occurs in a single block of production just prior to COP. This could be a significant amount of time after the cross-over date but it is a mechanism by which the new field can obtain repayment, whilst still being consistent with the concept that the incumbent field’s allocation is unaffected by the new field. 4.4
Equivalent Value Method
The justification for the payment of back-out to the incumbent field is that it suffers a deferment of production and an apparent economic loss because of the time value of money. The revenue they would have received for the deferred gas, had they sold it sooner, is greater than that which they would receive at a later date as they could have invested that revenue to accrue interest or in a new project providing additional return on the investment. However, though being cited as the justification for the back-out mechanism, no calculation of this apparent loss of revenue appears in the back-out calculations themselves. ([3] considers back-out to be equivalent to a financial transaction and mentions the time value of money when describing back-out methodology guidelines). In order to incorporate the time value of money into the analysis the revenue is discounted using an interest rate (r); this is discounted cash flow.
20
An alternative is proposed that ensures the incumbent field receives an equivalent discounted cash flow value to that it would have received had it been produced Stand-Alone by a specific time (say COP). The discounted present value of the incumbent field’s (Field Alpha) gas is given by: t
V' S Q' e rt dt
(13)
0
Where, V’ S r
Discounted present value of Field Alpha’s stand-alone cumulative gas production at time t Monetary value of gas volume Discount rate.
This is the continuous form of the discounted cash flow equation. The monetary value of the gas, S, i.e. the gas price, will change over time. However, for simplicity in this analysis it has been treated as a constant. Similarly the value of the gas produced by Field Alpha in the commingled system is given by:
t
V S Q e rt dt
(14)
0
Where, V
Discounted present value of Field Alpha’s cumulative gas production at time t in the Shared system
The difference between V’ and V is the value of gas that Field Bravo should transfer to Alpha to ensure that Alpha receives the same value of gas it would have in the stand alone system by time t. For example if time t is chosen to be when Alpha is estimated to become uneconomic on a Stand-Alone basis, that is Day 3,513 described in Section 4.3, then Bravo needs to transfer gas equivalent in value, when discounted, to this difference:
VTran S VTran
COP
0
Q' e rt dt
COP
0
Q e rt dt
(15)
Discounted present value of Field Bravo’s cumulative gas production transferred to Field Alpha by time t=COP in the Shared system
This is analogous to equation (6) for the back-out balance, but expressed in dollar value rather than standard volume terms. Substituting equations (3) and (4) into (15) and integrating, allows VTran to be determined (see Section A.5). Hence, the value of Bravo’s cumulative gas allocation at time COP is given by:
21
V" S
COP
Q e rt dt VTran
0
(16)
Where, Q V”
Shared Bravo volume flow rate Discounted value of Bravo cumulative allocated gas after back-out payments or repayments.
Q is Bravo’s actual produced flow and is determined from an equation analogous to (4). If Bravo is allocated gas equivalent in value to V”, then Alpha must have been allocated gas equal in value to its Stand-Alone production. Hence Alpha will not have been economically disadvantaged. Further details of the calculation of V” are provided in Section A.6. V” can be expressed in terms of Bravo’s allocated flow:
V" S
COP
0
Q”
Q" e rt dt
(17)
Allocated Bravo volume flow rate after back-out payments or repayments.
Q” is the gas actually allocated to Bravo after payments and repayments of backout gas. In this proposed mechanism, Q” can be any allocated flow profile providing that at time COP, it satisfies equation (17). Hence a flow profile could be chosen such that at time COP, Bravo’s cumulative allocated volume of gas is equal to the cumulative produced gas:
COP
0
Q" dt
COP
0
Q dt
(18)
And hence the volumetric back-out balance returns to zero at COP. Several allocation profiles are proposed below; the simplest is a straight line:
Q" At B
(19)
Where A and B are constants which are determined by solving equations (17) and (18). The details of these calculations are provided in Section A.7. Assuming a values of A system and presented in
discount rate (r) of 10% and that COP occurs on Day 3,513, the and B have been determined. The produced flows in the Shared the allocated flows after back-out transfers for the two fields are Figure 17:
22
Figure 17 – Bravo Straight Line Profile Allocated and Produced Flows Bravo transfers more gas to Alpha (compared with the current back-out mechanism) in the early part of the profile when the discounted value of the gas is higher. The cross-over is reached earlier on Day 866, after which Alpha repays Bravo an equivalent volume of gas, when the gas’s discounted value is lower. The volumetric back-out balance is presented in Figure 18:
Figure 18 – Back-Out Balance Bravo Straight Line Allocation Profile The balance now returns to zero on the specified COP day. Hence, Bravo has been allocated the same cumulative volume of gas that it has produced.
23
Alpha has also been allocated the same volume of gas that it has produced in the Shared system, though this is less than that which it would have produced on a Stand-Alone basis. However, the allocated gas Alpha has received is equal in discounted value to the Stand-Alone gas, meaning it has not been economically disadvantaged as a result of its deferred production. The profile of Alpha’s cumulative gas value in the Stand-Alone and Shared production cases is presented in Figure 19:
Figure 19 – Alpha Allocated Cumulative Value The locus of each line is almost coincide but they converge to the same value on the Stand-Alone COP Day. In addition to the straight line flow profile two further profiles were examined. A one term exponential profile:
Q" De Et
(20)
And two term exponential profile:
Q" Fe1t Ge2t
(21)
D and E, and F and G are constants which are determined by solving equations (16) and (18), with the flow profiles given by equations (20) and (21) respectively. The details of the calculations are provided in Sections A.8 and A.9. The two term exponential profile is included as it is analogous to the actual flow profile given by an equation of the form provided for Alpha in equation (3). The analogous form of the equation for Bravo’s Shared flow profile has the same values of 1 and 2 but different values of the constants C1 and C2. In the
24
proposed two term profile it is these constants that are replaced by F and G, with the terms being retained. The details are provided in Section A.9. The three alternative allocated Bravo flow profiles are plotted in Figure 20:
Figure 20 – Bravo Straight Line Profile Allocated and Produced Flows Also shown for comparison is the Bravo produced flow. The Bravo produced (orange) and Straight Line (purple) are identical to the same coloured lines in Figure 17 labelled: “Shared Bravo” and “Bravo after Transfer”. The above analysis illustrates that it is possible to incorporate the discounted value of the gas into the back-out calculations and simultaneously ensure that fields are allocated the volume of gas they physically produce over a specified time period. The exposure to economic loss is used to justify the very introduction of back-out payments but then does not appear in any quantified manner in the actual system. It is suggested that the alternative approaches presented in this section are more appropriate in a conceptual sense as they incorporate the economic loss directly into the back-out mechanism. 5
CONCLUSIONS
The paper has explored the behaviour of typical currently employed back-out payment and repayment mechanisms. It has attempted to expose potential pitfalls associated with back-agreements and presented potential solutions to address them. It has accomplished this with the use of simplified mathematical models which complement the more complex models but have the advantage that they are more tractable and readily predict the long term behaviour of such back-out systems.
25
The analysis has revealed that the repayment phase is so prolonged that the new third party field may not receive a significant proportion of repayment before the incumbent field’s COP. More profoundly, the simplified model has proved mathematically that the repayment takes an infinite time and hence strongly suggests in principle that the new field cannot be repaid in full. A mechanism has been suggested, that incorporates the COP, and ensures full repayment of gas. It achieves this whilst being consistent with the premise, frequently expressed in back-out agreements, that the incumbent field’s allocated production is no different to that which it would have experienced if it was produced alone. The paper also examines the economic justification for the introduction of the back-out payments. It also identifies that economic valuation calculations do not feature in typical current back-out systems. Further alternative approaches are presented that incorporate the discounted value of the gas into the back-out mechanism. These approaches simultaneously ensure that the incumbent field receives the same value of gas it would have if produced Stand-Alone and that both fields are allocated the same cumulative volume of gas they would have produced by a specified time. When they are written, some back-out agreements appear to focus on the initial back-out experienced by the incumbent field(s). Perhaps this is not surprising as this phase is in the near future. In contrast it appears that the consequences associated with the mechanisms for the repayment phase, are not fully realised. This can result in significant penalty of lost production for the new field.
26
APPENDIX A SIMPLIFIED MATHEMATICAL MODEL A.1
System
Figure 21 depicts the two gas reservoirs ( and ) producing through a shared subsea pipeline to an onshore gas plant.
Figure 21 – System Configuration The pressures and standard volumetric flows at various points in the system are indicated symbolically. A.2
Simplifying Approximations
In order to render the approximations are made.
system
solvable
analytically
some
simplifying
The frictional pressure loss in the pipelines is approximated to be linearly dependent on flow:
Q R P PX
(22)
Q R P PX
(23)
Q Q RP PX PD
(24)
The R terms represent the resistance to flow in each pipe segment and are assumed to be constants.
27
The typical correlation between cumulative gas produced to date (GPD) and reservoir pressure (divided by compressibility) is illustrated in Figure 22 below:
Figure 22 – Reservoir Pressure versus Cumulative Production
Again to simplify the mathematics gas compressibility (Z) is assumed to be constant over the pressure range and reservoir pressure can be expressed as a linear function of cumulative production according to:
P k GPD A.3
(25)
Shared Production Scenario
Cumulative production is given by the integral of the flow rate over time: t
GPD Q dt 0
(26)
Rearranging (25) and substituting for GDP in (26), and rearranging in terms of reservoir pressure gives:
t
P k Q dt 0
Differentiating with respect to time:
28
(27)
dP k Q dt
(28)
A similar relationship can be obtained for Bravo:
dP dt
k Q
(29)
Substituting for flow Q from (22) in (28):
dP k R P PX dt
(30)
k R P PX
(31)
Similarly for Bravo:
dP dt
Rearranging equation (24) in terms of PX and substituting for Q and Q from (22) and (23) results in:
PX
R P R P RP PD R R RP
(32)
This can be substituted in (30) and (31) to obtain:
dP k R P RP P R P R PD RP dt R R RP
(33)
k R P RP P R P R PD RP dt R R RP
(34)
And:
dP
Differentiating (33) with respect to time:
dP dP d 2 P k R R R R P 2 dt dt R R RP dt
(35)
Rearranging (35):
dP dt
1 R
dP d 2 P R R RP RP R 2 dt k R dt
29
(36)
Rearranging (33) in terms of P:
P
1 R
dP R R RP P RP P R PD RP dt k R
(37)
Substituting (36) and (37) in (34) and rearranging produces:
d 2 P k R RP R k R RP R dP k k R R RP dt R R RP dt 2 R R RP k k R R RP PD R R RP
P
(38)
This is a second order linear differential equation in P, which has the solution:
P C1e1t C2e2t PD
(39)
This is an initial value problem, with initial conditions defined when t= 0: At
t 0, P P , P P
(40)
The starting reservoir pressures P° and P° are assumed to be known. The various and C parameters in (39) are:
1
2
k R RP R k R RP R
k R RP R k R RP R
k R R
P
2 2 2 R k R RP R 4k2 R (41 k R
2 R R RP
k R R
P
2
)
R k R RP R 4k2(42 R2 k 2 R2
2 R R RP
2
k R P R P RP R PD RP 1 C1 2 P PD 2 1 R R RP 1 k R P R P RP R PD RP C2 1 P PD 2 1 R R RP
As P is obtained all other variables: P, Q and Q can be determined. A.4
Stand Alone Production Scenario
For the Alpha Stand-Alone case, Q is zero. Equation (24) now becomes:
30
)
(43)
(44)
Q' RP PX PD
(45)
Combining equation (22) and (45) to eliminate PX:
Q '
R P' PD R 1 RP
(46)
This is then substituted in (28) to obtain the first order differential equation in P’: ' dP' k R P PD dt R 1 RP
(47)
With initial conditions defined when t= 0:
At t 0, P P , '
(48)
This has the solution:
P' P PD et PD
(49)
Where,
k R RP RP R
(50)
Now P’ is obtained, Q’ can be determined. A.5
Calculation of Value of Transferred Gas
The discounted value transferred from Bravo to Alpha is given by the cumulative difference in the value of Alpha’s Stand-Alone and Shared produced volumes when t=COP:
VTran S
COP
0
Q' e rt dt S
COP
0
Q e rt dt
(51)
Substituting for Q’ and Q from equations (3) and (4):
VTran S
COP
0
k
P
PD e t e rt dt S
COP
0
1 2 1t 2t rt C1e C2e e dt k k
(52)
Which integrates to:
VTran
S 1C1 S 2C2 S P PD e r t 1 e 1 r t 1 e 2 r t (53 1 ) k r k 1 r k 2 r
31
A.6
Calculation of Required Value of Bravo Allocated Gas
The required value of Bravo’s gas after transfer is:
V" V VTran
(54)
If Bravo is allocated a quantity of gas that is equivalent in discounted value to V ”, then Alpha’s allocated gas in the Shared system must be equal in value to that which it would have received Stand-Alone. The value of Bravo’s produced gas V is given by:
V S
COP
0
1 rt t t C 1e 1 2 C 2e 2 e dt k k
(55)
Which integrates to:
V
S 1C 1 k
e r
1 r t
1
1
S 2C 2 k
e r
2 r t
1
(56)
2
Hence the required discounted value Bravo’s cumulative gas allocation can be obtained by substituting (53) and (56) in (54). A.7
Equivalent Value Back-Out Straight Line Bravo Profile
Assume that Bravo’s allocation profile follows a straight line:
Q" At B
(57)
In order to ensure that Alpha receives the correct discounted value of gas at COP:
S
COP
0
Q" e rt dt V"
(58)
Also, for the back-out balance to return to zero, Bravo has to be allocated an equivalent volume of gas to that which it produced:
COP
0
Q" dt
COP
0
Q dt
COP
0
1 t t C 1e 1 2 C 2e 2 dt k k
(59)
Hence:
COP
0
Q" dt
C 1 k
e
1t
1
C 2 k
e
2t
1 q
(60)
Therefore: COP
At B dt q 0
And,
32
(61)
S
COP
0
At B ert dt V"
(62)
Equation (61) integrates to:
A COP 2 B COP q 2
(63)
And equation (62) integrates by parts to:
1 e r COP A 2 r r
1 e r COP 1 B COP r r r
V S
"
(64)
(63) and (64) are simultaneous equations and can be solved for A and B:
V" q r 2 1 e r COP S COP r A r COP r e r r COP COP COP 1 e 2 2
B
q
COP
A COP 2 2
(65)
(66)
Which are then substituted into (57) to obtain Bravo’s allocated profile. A.8
Equivalent Value Back-Out Single Exponential Term Profile
Assume that Bravo’s allocation profile follows an exponential function of time:
Q" De Et
(67)
Similar to the approach taken in Section A.7, we require the discounted value at COP to be:
S
COP
0
De Et e rt dt V"
(68)
And the cumulative volume to be given by:
COP
0
De Et dt q
(69)
Equation (68) integrates to:
V" D e E r t 1 E r S And equation (69) to:
33
(70)
D Et e 1 q E
(71)
(70) and (71) are simultaneous equations and can be solved for D and E:
E
V" r 1 e E COP
S q e E r COP 1 V"
1 e
E COP
(72)
Which has to be solved iteratively for E and D is then given by:
D
q E
e
Et
(73)
1
Which are then substituted into (67) to obtain Bravo’s allocated profile. A.9
Equivalent Value Back-Out Two Exponential Term Profile
Assume that Bravo’s allocation profile follows a two term exponential function of time:
Q" Fe1t Ge2t
(74)
Similar to the approach taken in Section A.7, we require the discounted value at COP to be:
S
COP
0
Fe
1t rt
e
Ge2t e rt dt V"
(75)
And the cumulative volume to be given by:
COP
0
Fe1t Ge2t dt q
(76)
Equation (75) integrates to:
V" F G e 1 r COP 1 e 2 r COP 1 1 r 2 r S
(77)
And equation (76) to:
e F
1
1 COP
1
G
2
e
2 COP
1 q
(78)
(77) and (78)are simultaneous equations and can be solved for F and G:
V" Mq S K G ML N K
34
(79)
F
q LG
(80)
K
Where,
K
e 1
(81)
(82)
1 COP
1
2 COP
1
1
L
1
2
e
(83)
(84)
M
1 e 1 r COP 1 1 r
N
1 e 2 r COP 1 2 r
And,
Which are then substituted into (74) to obtain Bravo’s allocated profile. A.10 k k ° P ° P PD r R R RP S
Values of Parameters Used in Example System -0.004344 bar/mcm -0.4683577 bar/mcm 48.3 bara 145.3 bara 6.75 bara 10% per annum 0.1047 mcm/bar 0.004911 mcm/bar 0.160105 mcm/bar. 0.197 $/Sm3 (calculated assuming a gas price of £0.4/therm, gas GCV (Gross Calorific Value) of 40 MJ/Sm3 and exchange rate of 1.30 $ per £).
REFERENCES [1] [2] [3]
Petroleum Experts supply MBAL, PROPSER and GAP, http://www.petex.com/ Schlumberger supply Eclipse, OFM and Pipesim, https://www.software.slb.com/products North Slope of Alaska Facility Sharing Study, Prepared for Division of Oil & Gas, Alaska Department of Natural Resources, May 2004, by Bob Kaltenbach, Chantal Walsh, Cathy Foerster, Tom Walsh, Jan MacDonald, Pete Stokes, Chris Livesey and Will Nebesky
Highlights
A simplified model is used to analyse the long term behaviour of back-out systems The model shows that the repayment phase duration is infinite
35
Repayment models are proposed that incorporate the fields’ cessation of production Back-out payment is justified because of the time value of money One alternative incorporates the time value of money into the calculations
36