Abstract
Background: The subject of well spacing is one of vital
importance in the petroleum industry. The well itself plays
a very significant role in the development of the oil and
gas reservoirs and in the control of the recovery process.
However, determination of appropriate well spacing for
maximum economic oil recovery has been a complicated
and controversial issue in oil field development. Various
studies on the subject have indicated a slim relation
between well spacing and recovery. This study will
however focus on the optimization of field development in
order to maximize returns on investment. A simplified
economic optimization model to optimize well spacing in
oil reservoirs has thus been proposed.
Methodology: The optimization model developed has
been solved to find the optimal well spacing that will
maximize Net Present Value of investment, using NonLinear Programming (NLP) Excel-Based Solver.Data for
case study was from an offshore field. Sensitivity analysis
to determine the extent of changes in optimum Well
spacing with respect to model parameters was carried out.
Results: Cost of field development could be greatly
optimized using NLP. Well spacing has been seen to
decrease with recoverable reserves for maximum Net
Present Value (NPV) until a point where increase in the
number of wells decreases NPV. An optimal solution exits
within this region. Optimal well spacing has been observed
to be far more sensitive to oil price than to other model
parameters such as investment cost per well, production
rate and interest rate.
Conclusion: Intuitive approach to solving well spacing
problem requires current oil industry knowledge and
experience. Economics sets the limit for drilling of
development and infill wells.
Introduction
The ultimate goal underlying the development of the
science of reservoir engineering has been defined by
Muskat(1949) as the attainment of a maximum efficiency
in the exploitation of oil-bearing reservoirs, where the
phrase maximum efficiency is taken to imply the maximum
recovery of oil at a minimum cost. Within these limitations,
the goal in the development of production engineering
could be said to be the attainment of maximum efficiency
in the operation of those producing wells drilled into an oilbearing reservoir; this implies the realization of the
maximum profit from each and every such well.
Maximum utilization of wells to obtain better field recovery
becomes an integral part of sound operating. Efficient
exploitation of oil and gas reserves demands that careful
choice of number of wells, their location and spacing be
made.
Business Objectives
To improve economic
development investment.
returns
on
field
To Increase oil field recovery
Cost effective
optimization
method
for
well
spacing
To protect the right of reserve owners
Technical Objectives
Improve reliability of geological/reservoir data
To advance the frontier of knowledge in well
spacing
¾ Reservoir simulation
¾ CrystalBall
Well spacing optimization software
Well Spacing Optimization Methods
Various methods have been proposed for obtaining Wo.
They include:
•
Graphical solution
•
Analytical solution
•
Global Optimization using Genetic Algorithm
Graphically, the optimum well spacing has been
determined by a plot of economic return versus well
spacing as proposed by Muskat(1949). Garaicochea and
Acuna(1978) approach for determining optimum well
spacing consist of predicting performance of the reservoir
under various spacing schemes graphically. Another
possibility for calculating the optimum well spacing is to
2
prepare a cross plot of Net Present Value versus number
of wells using oil price as a parameter as illustrated by
Martinez (1975). This method assumes a continuous
relationship between variables. However the work of Miller
and Dyes (1959) on reservoirs with solution gas drives
and water drives showed that graphical solutions may
introduce discrete jumps that provide more than one
optimal point. Experience is therefore required to choose
which to use.
A method to determine the optimum well spacing
straightforward without a plot was presented by Tokunaga
and Hise (1966). This method, however, assumes the
production rate of all wells to remain constant over life (no
decline). Corrie (2001) proposed an analytical solution that
the wells initial production rate to decline over the life of
the reservoir (immediately after starting production). He
differentiated the objective function and equated the total
derivative to zero to obtain the optimal solution for well
spacing.
However the value of the optimum number of wells Wo is
selected according to the square root of Q rather than the
first power of Wo. A relationship known as economic order
quantity in inventory control, exits, and decision making
becomes a matter of simple analogies or intuition. The
method also assumes that the function is continuous near
the region of optimality.
Nejad and others (2007) used genetic algorithm to obtain
optimal well spacing for a Middle East onshore oil field.
Although genetic algorithm is holistic, effective in
searching very large space and varied spaces and takes
care of discrete constraints, its model is, however,
complex and application could be limited as highly skilled
manpower is required for both its programming and use.
A much easier, simple and straight forward method is
proposed. This method assumes the problem to be a nonlinear programming problem of constrained optimization
and solution can be obtained using spreadsheet
Optimizers.
Spreadsheet Optimizers
In the 1980s, a major move away from FORTRAN and C
optimization began as optimizers, first linear programming
(LP) solvers, and then NLP solvers were interfaced to
spreadsheet systems for desktop computers. The
spreadsheet has become, de facto, the universal user
interface for entering and manipulating numeric data
(Edgar and others, 2001). Spreadsheet vendors are
increasingly incorporating analytic tools accessible from
the spreadsheet interface and able, through that interface,
to access external databases. Examples include statistical
packages, optimizers, and equation solvers.
The Excel Solver
Microsoft Excel, beginning with version 3.0 in 1991,
incorporates an NLP solver that operates on the values
and formulas of a spreadsheet model. Versions 4.0 and
later include an LP solver and mixed-integer programming
(MIP) capability for both linear and nonlinear problems.
The user specifies a set of cell addresses to be
independently adjusted (the decision variables), a set of
formula cells whose values are to be constrained (the
constraints), and a formula cell designated as the
SPE 140674
optimization objective. The solver uses the spreadsheet
interpreter to evaluate the constraint and objective
functions, and approximates derivatives. The NLP solution
engine for the Excel solver is GRG2.
The generalized reduced gradient (GRG) algorithm was
first developed in the late 1960s by Jean Abadie(Abadie
and Carpentier, 1969) and has since been refined by
several other researchers. For equality constraint, GRG
takes a direct approach to solve the problem. It uses the
equality constraint to solve for one of the variables in
terms of the other.
For inequality constraint, it converts the constraints to
equalities by introducing slack variables.
If s is the slack in a case where x-y ≥ 0, the inequality
becomes x-y-s = 0. We must also add the bound for the
slack, s ≥ 0, giving the new problem.
Economic Model
In this section the economic model which will be used by
the NLP solver in Excel spreadsheet to optimize well
spacing will be derived. This economic model which
relates the Net Present Value of field development
investment with the number of wells to be drilled is called
the objective function.
Objective Function
Assumptions:
In deriving the objective function, which is the applicable
economic model for this problem, the underlying
assumptions are as follows:
The total cumulative oil production(recoverable
reserves) remains constant
The present net value of investment is after tax
All investment are incurred at year zero(although all
the wells will not be drilled simultaneously due to
equipment and personnel constraint and the need to
learn from other wells, the cost will be converted to
present value)
All the wells have the same initial production rate
and decline at the same over life. The decline, D, rate is
a function of the number of wells, initial daily oil
production rate and the total recoverable reserves.
The oil price is netted back to the wellhead.
Rate at economic limit is considered negligible
compared to initial production rate per well.
Assuming oil production decline rate to follow a general
form of exponential equation,
(1)
Where D is the yearly exponential decline factor.
The cumulative oil production, Np which represents a
fraction of the reserves under tested recovery scheme
from time 0 to time “t” is:
365
(2)
SPE 140674
3
Substituting (1) in (2) gives
equation (13) reduces to
365
(3)
Integrating between t = 0 and “t” = t
(14)
N
(4)
The cumulative oil production of W wells is;
(5)
If we neglect the rate at economic limit, Qt, then
For 0≤ i ≤ 0.3 equations (12) and (13) are approximately
equal.
For analytical purposes equation (13) is easily
differentiable.
The Optimization Model
(6)
(7)
The present value of oil production rate PVQt at time, t
is;
(8)
1
i is the interest rate, the opportunity cost of investment.
The present value cumulative oil production PV(Np) from
time, 0 to time, t is;
365
(9)
The problem statement is expressed as:
Maximize
Subject to:
∑
1
0
,…,
,…,
The optimization model is depicted in the cartoon
below(see fig.1)
The constraint imposed on the objective function requires
that the optimal solution, which is the number of wells,
will be an integer of value greater than or equal to zero.
The decision to drill or not to drill is a 1s and 0s decision.
Integrating between time, t = 0 and t = t and neglecting
PVQt at economic limit, the present value of cumulative
oil production of W wells is
(10)
Substituting (7) in (10) gives
(11)
Considering: the present value of all capital investments
per well after income tax’s effect, C, in $; the oil price
netted back to the well after income tax’s effect, V,
$/barrel; the present value of other investments not
dependent on the number of wells after income tax’s
effect, Z, in $.
The Net Present Value (NPV) can therefore be
expressed as:
NPV=(PVNp)V–CW-Z
(12)
Equation (11) is another form of the economic model
proposed by Muskat(1949)
Substituting the appropriate parameters into (11),
(13)
Detailed derivation can be found in Appendix-A
If the limit of equation (12) is taken as
lim
365
365
ln 1
0;
Fig.1: Optimization Model Cartoon
Case Study: Offshore Development
The field I-A is an offshore field composed of two turbidity
currents laying in a NNW direction. The oil trapped is of
good API (>30 API) and reservoir pressures is relatively
high. The lithology is silty sands of medium porosity and
permeability. Studies show that the field can be developed
with associated ultimate recovery of 156MMbbls of oil
recovered in 20 years by means of natural depletion,
water injection and high pressure gas injection. Given the
following information, it is required to estimate the
optimum well spacing required to maximize the field
development Net Present Value.
4
SPE 140674
Expected
production
GOR
initial
oil
$12 million per well
that if the reservoir parameters remain constant, an
increase in oil price after tax netted back to the well, will
favour the drilling of more wells to recover reserves. The
implication on the field life of the project assuming no
government restrictions is a reduction in the field life. For
a heterogeneous reservoir, it could also mean that areas
that were formally not effectively drained due to formation
damage or faults can be produced, as noted by
Bobar(1985), at economic gain and coning can also be
reduced as noted in the work of (Matthews, Carter and
Dake, 1992). Fig.4 shows an increase in the optimum
number of wells and NPV with increase in oil price.
$64 million
(b) Interest Rate
2500 bbls/d per well
75m3/m3
Number of gas injection
Wells
Number of water injection
wells
Oil price at well after
income tax
Present Value of Capital
investment after income tax
Present value of Capital
investment not dependent
on the number of wells after
income tax
Productive Area, A
Interest rate, i
1
3
$40 per bbls
112000 acres
10% per annum
Results and Analysis
From the NLP Excel based solver (See Figures 2a and
2b), the optimum number of wells has been obtained to be
76 for the field under consideration.
The optimum well spacing density in acres/well is obtained
as 1474acre/well
The optimum well spacing assuming a hexagonal well
pattern is obtained in ft as 8600 (see Appendix B).
For comparison, a graphical and analytical solution has
also been obtained to be 76. A plot of NPV vs number of
wells is given in figure (3). The plot shows an increase in
NPV with number of wells until additional investment in
drilling a new well produces a marginal loss. The region of
optimality can be seen to be between 75 and 77 as the
difference in NPV is not much. However, the NLP gives
the exact value of the optimum as 76. The analytical
solution which has been approximated to 76 was originally
75.8. This illustrates the possible error that might arise in
obtaining analytically, the optimal solution especially in
marginal fields where fewer numbers of wells are to be
drilled.
A table of comparison of the exact optimum from the three
methods is given below (Table 1)
Comparison
Method
NLP
Optimum
Number of
Wells
76
of
Different
Graphical
Solution
76
(c) Proration Rate
A very important production and economic parameter
that affects how close or far wells are spaced in a
continuous reservoir is rate of production. Noting the
assumption made earlier in deriving the economic model,
government restrictions on rate(proration), reservoir
properties such as water cut, GOR, sand production,
could affect production rate. The implication here is that a
low production rate will favour more wells but at a much
smaller NPV. If the company is allowed to detect
production rate then a rate that will allow fewer wells to
maximize NPV will be required. Figure 6 shows this
relationship.
Sensitivity Summary
Net Present Value (NPV) = $4,162,355,293
Table 1:
Methods
One other economic parameter that would affect well
spacing in a reservoir is interest rate. As shown in table
3, a higher interest rate will mean that time is very
important as this could force the cost of development and
lifting price of oil per barrel up. If there is a favourable oil
price, companies would want to recover fast their cost of
investment by drilling more wells. However, except
enhanced oil recovery (EOR) methods are employed
alongside infill wells, the NPV from drilling more wells
reduces with increased interest rate making plans on
further development uninteresting. Figure 5 captures this
relationship.
Optimization
Analytical
Solution
75.8 76
Effect of Model Parameters on Optimum Solution
(a) Oil price:
Table 2 which is a reflection of the economic realities of
field development activities in the energy industry shows
Figures 7(a) –(d) shows the probability distributions that
were assumed to hold for each of the parameters in the
economic model that affect well spacing. Figure 8
indicates that well spacing is far more sensitive to oil price
than to other parameters. Increase in oil price favors
drilling of more wells for closer spacing. The same is
applicable for rate. However the inverse is the case for
cost of drilling and lifting of oil per well and interest. The
implication is that before field development decisions are
made it is necessary to know how a change in oil price will
affect proposed field development program
Conclusion
There is an optimum number of wells for each
reservoir
Limit is set by economics
Very efficient, simplified, and cost effective
methods is required
Non-linear programming comes to rescue
SPE 140674
5
With the assumptions and uncertainties that are
inherent in well spacing models, a cost effective
and simplified method has been presented as a
precursor to reservoir simulation
Well spacing is more sensitive to oil price as
against proration rate, interest rate and
recoverable reserves.
Tokunaga, H. and Hise, B. R.(1966): A Method
to Determine the Optimum Well Spacing; SPE
167
Recommendation
Intuitive approach to solving well spacing
problem require
¾ Current oil industry knowledge and
experience
¾ Information on depleted fields as to the
effect of well spacing on recovery
efficiency existence of analog fields
Software for optimum well spacing can be
developed using NLP
Reservoir simulation can be tool used to validate
results on economic model.
References
Abdel Aal, H. K., Baker, B. A., Al Shahlavi, M.
A.(1992):
Petroleum
Economics
and
Engineering, 2nd Edition; Marcel Dekker, New
Delhi.
Bobar, A. R.(1985): Reservoir Engineering
Concepts on Well Spacing; SPE, USMS
015338
COGCC’s Rules and Regulations-500 series.
Corrie, R. D.(2001): An Analytical Solution to
Estimate
the
Optimum
Number
of
Development Wells to Achieve Maximum
Economical Return; SPE 71431.
Edgar T. F., Himmelblau D. M. and Lasdon, L.
S.(2001): Optimization of Chemical Processes,
nd
2 Edition; McGraw-Hill, New York
Fig.2a: Solver Options
Fig.2b: Optimization model definition
Ezeh J. C. and Ezeh G.N.(2000): Fundamentals
of Engineering Economy; M. C. Computers;
pp3.
Matthews, J. D. and Carter, J. N.(1992):
Investigation of Optimum Well Spacing for
North Sea Eocene Reservoirs; SPE 25030.
Mian, M. A.(2002): Project Economics and
Decision Analysis, Vol 1, Deterministic Models;
PennWell Corporation, Oklahoma; pp 95.
Miller, C. C. and Dyes, A. B.(1959): Maximum
Reservoir Worth-Proper Well Spacing; SPE,
Muskat, M.(1949): Physical Principles of Oil
Production; McGraw-Hill Book Company; pp810
Nejad, T. S. A., Aleagaha, A. V. and Salari,
S.(2007): Estimating Optimum Well Spacing in
a Middle East Onshore Oil Field Using a
Genetic-Algorithm-Optimization
Approach;
SPE 105230
Nind, T. E. W.(1964): Principles of Oil Well
Production; McGraw-Hill Book Company, New
York; pp352
Stroud, K. A.(1995): Engineering Mathematics,
4th Edition; Macmillan Press, London; pp183
T.P. 8096
Taha, H. A.(1976): Operations Research; Collier
Macmillan International Editions; pp500
Fig. 3: Graph of NPV vs Number of Wells
6
SPE 140674
Table 3: Effect of Interest rate on Wo
Table 2: Effect of oil price on Well Spacing
V
($/bbl)
Wo
A
10
30
4.27E+16
42,975,000
569,717,277
15
41
8.75E+16
53,012,500
1,095,407,687
20
50
1.42E+17
61,225,000
1,661,030,625
25
57
2.03E+17
67,612,500
2,252,166,389
30
65
2.78E+17
74,912,500
2,861,422,993
35
71
3.54E+17
80,387,500
3,484,432,281
40
77
4.38E+17
85,862,500
4,118,280,390
B
Fig.4: Effect of Oil Price on Wo
NPV($)
I
(per
annum)
Wo
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0
58
76
88
96
103
108
112
115
A
1.40E+01
3.30E+17
4.33E+17
5.01E+17
5.47E+17
5.86E+17
6.15E+17
6.38E+17
6.55E+17
B
NPV($)
0
60,536,266
84,218,388
102,102,863
116,042,163
128,797,894
139,478,825
149,016,316
157,427,169
6,176,000,000
4,695,440,580
4,162,355,293
3,787,521,545
3,494,563,700
3,253,506,131
3,048,927,297
2,871,585,050
2,715,447,219
Table 4: Effect of Proration Restriction on Well
Spacing
Proration
Rate,
Qi(b/d/well)
Wo
A
B
NPV($)
500
124
1.41E+17
37,498,388
2,213,793,874
1000
105
2.39E+17
53,193,388
3,171,821,920
1500
92
3.14E+17
65,238,388
3,649,850,493
2000
83
3.78E+17
75,458,388
3,950,464,837
2500
76
4.33E+17
84,218,388
4,162,355,293
Fig 5: Effect of Interest rate on Wo and NPV
Fig. 6: Effect of Proration rate on Wo and NPV
SPE 140674
7
Fig. 7d: Probability distribution of oil price, V
Fig.7a: Probability Distribution of Np
Fig. 7b: Probability distribution of Q
Fig. 8: Tornado Plot of Vo sensitivity to model
parameters
Fig. 7c: Probability distribution of interest rate, i
Fig. 9: Production Profile of Field
8
SPE 140674
Appendix A
Assuming oil production decline rate to follow a general
form of exponential equation,
(1)
The cumulative oil production, Np which represents a
fraction of the reserves under tested recovery scheme
from time 0 to time “t” is:
365
(10)
Substituting
in (10)
We have;
365
Substituting (1) in (2) gives
365
(3)
365
ln 1
(2)
Rearranging we have that the present value of
cumulative oil produced is;
Integrating between t = 0 and “t” = t
(11)
365
(4)
The cumulative oil production of W wells is;
(5)
Considering: the present value of all capital investments
per well after income tax’s effect, C, in $; the oil price
netted back to the well after income tax’s effect, V,
$/barrel; the present value of other investments not
dependent on the number of wells after income tax’s
effect, Z, in $.
The Net Present Value (NPV) can therefore be
expressed as:
NPV=(PVNp)V–CW-Z
(12)
365
If we neglect the rate at economic limit, Qt, then
365
(6)
ln 1
(13)
(7)
The present value of oil production rate PVQt at time, t is;
(8)
1
as proposed by (Corie, 2001)
It can be shown that;
lim
1
The present value cumulative oil production PV(Np) from
time, 0 to time, t is;
365
365
365
Reduces equation (13) to
(9)
365
1
1
1
Solution to Case Study I-A
Analytical solution of the objective function for optimum
well spacing can be obtained by differentiating equation
(14) and equating it to zero to obtain Wo as:
ln 1
365
(14)
N
Appendix B
Integrating between time, t = 0 and t = t
365
ln 1
ln 1
1
ln 1
1
or
for 0 ≤ i ≤ 0.3
365
ln 1
Neglecting PVQt at economic limit, the present value of
cumulative oil production of W wells is
365
365
75.8
Optimum
1474
well
/
density,
76
112000
76
SPE 140674
9
Well spacing = 224(1474)0.5 = 8600ft
Net Present Value (NPV) = $4,162,355,293
0.01666 per annum
Fig. 10: Well Spacing Illustration from Eclipse