Copyright 2003, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE International Improved Oil Recovery Conference in Asia Pacific held in Kuala Lumpur, Malaysia, 20–21 October 2003. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A., fax 01-972-952-9435.

Abstract This paper presents a hydraulic fracture treatment design optimization scheme which integrates a hydraulic fracture geometry model, a production model and an economic model. The hydraulic fracture geometry model is used to determine the final fracture geometry and select the treatment parameters for a given stress and reservoir condition. Production from the treated well is estimated for pseudo-steady state condition using a model equivalent to a compositional simulator. A genetic-evolutionary optimization algorithm is used to obtain the optimum treatment parameters for maximum production or NPV. The integrated model has been used to investigate different field scenarios of a multiwell gas-condensate reservoir, including optimization of well locations and hydraulic fracture treatment parameters for any well type, achievement of target production and maximization of NPV with simultaneous minimization of the associated treatment costs. Introduction To increase ultimate recovery with minimum treatment cost, hydraulic fracture treatments are optimized by coupling a hydraulic fracture geometry model, a production model and an economic model. A three-step calculation procedure is then conducted repeatedly to obtain the best combination of hydraulic fracture treatment parameters.

Previous works in hydraulic fracture treatment design optimization were devoted mainly in the development of a search scheme for the optimum design. The main drawbacks of these tools include absence of global optimization procedure and limited number of design variables. In our previous work, we have addressed these shortcomings and we formulated geometric and operational constraints to ensure a reliable optimum treatment design.1 The search scheme used

is a hybrid of genetic algorithm and evolutionary operation. In our current work, we have incorporated an improved objective function into the optimization scheme and applied it to a specific gas-condensate reservoir.

Previous studies simplified the production model by considering the ideal case of a single well, centrally located in a square or circular drainage area. In practice, however, a candidate for a fracture treatment could be part of a multiwell system. To date, several analytical models applicable to multiple wells have been published in literature. Notable is Rodriguez and Cinco-Ley’s2 pioneering work which was further improved by Camacho-V. et al.3 Most recently, Ozkan4 introduced a model that takes into account the transient and pseudo-steady state flow regimes and allows for possibility of any well combination and variable rates. On the other hand, using the matrix approach, Valko et al.5 introduced a simpler model which is applicable for pseudo-steady state flow regime.

In order to obtain the optimum values of the hydraulic fracture treatment parameters, an accurate production estimate for a given reservoir condition is necessary. Some of the published works on the subject coupled a stochastic optimization algorithm with a production model that uses idealized dry gas reservoir or single phase oil reservoir1,6. Others used a reservoir simulator for a more accurate production estimate and employed parametric sensitivity analysis to get the optimum treatment parameters7. The latter gives improved production estimate but does not explore the whole search space for the global optimum design as it is computationally tedious and time consuming whereas the former gives inaccurate production estimate for reservoirs other than the ideal case but efficiently searches for the optimum values of treatment parameters. In order to overcome the foregoing weaknesses, a generalized production model applicable to gas or gas-condensate reservoir is incorporated in the optimization scheme. This guarantees speedier evaluation of the objective function.

Coupling the production model with an efficient optimization algorithm provides solutions to field development problems. Particular attention is given to gas-condensate reservoirs as the need to hydraulically fracture multiple wells in gas-condensate reservoirs arises since exploitation of higher temperature and pressure reservoirs is becoming increasingly important.

SPE 84860

An Integrated Model for the Design and Evaluation of Multiwell Hydraulic Fracture Treatments for Gas-Condensate Reservoirs K.L. Valencia, SPE, Z. Chen, The University of New South Wales, M.K. Rahman, SPE, The University of Western Australia, S.S. Rahman, SPE, The University of New South Wales

2 SPE 84860

Optimization Strategy for Hydraulic Fracturing Design Definition of Objective Function The goal of hydraulic fracture treatment design optimization is to find, for given reservoir conditions, a set of fracture treatment parameters that would maximize the net present value (NPV), maximize cumulative production, or maximize NPV and minimize treatment cost, considering the post-fracture performance of the well. The above three objectives are defined as objective functions.

Formally, the problem is to find the values of hydraulic fracture treatment parameters, such as pumping rate, pumping time, end-of-job proppant concentration and fracturing fluid viscosity which can be represented as x1, x2, x3, … xN, subject to bound constraints (upper and lower limits of each treatment parameter – uN and lN respectively) denoted by

111 uxl ≤≤

M M M , …... ….…… ...……………….……..…(1) NNN uxl ≤≤

and design constraints, such as pressure limitation of surface equipment, burst resistance of tubing and available pump horsepower, expressed as

( ) 111 ul CxCC ≤≤

M M M , .…………….…………….…….(2) ( ) uMMlM CxCC ≤≤

where CM(x) is the design constraint, ClM is the lower bound and CuM is the upper bound. The optimum fracture treatment parameters minimize the objective function (maximum NPV, maximum cumulative production, maximum NPV and minimum treatment cost) represented as f(x1, x2, x3, … , xN). Based on sound engineering practice, the following design variables, with their upper and lower bounds, are used in this study: 0.07Pa-s < viscosity (µ ) < 0.8Pa-s, 0.0264m3/s < pumping rate (qi)< 0.1992m3/s, 59kg/m3 < end-of-job proppant concentration (Pc )< 1797kg/m3 , and 27.432m < fracture half length (xf )< 457.2m.

Optimization Strategy The optimization strategy aims to evaluate the objective function (maximum NPV, maximum cumulative production, maximum NPV and minimum treatment cost) by executing a computational sequence that integrates a hydraulic fracture geometry model, a production model and an economic model. Bound constraints are imposed based on industry practice and design constraints are formulated to cover operational and fracture growth control requirements. This eliminates designs that are unrealistic and difficult to execute in practice due to equipment restrictions. The search space is therefore narrowed down resulting in quicker optimization run.

For a given set of treatment parameters, rock mechanical properties and in-situ reservoir properties, the resulting fracture geometry and conductivity are determined in the hydraulic fracture geometry model. The cumulative

production is then estimated by the generalized production model, which allows handling of a wide array of field development problems. The NPV is then calculated based on production over a period of time and associated well and treatment costs under the economic model. Treatment parameter values are then updated as per the rules of the optimization algorithm.

Optimization Algorithm Optimization starts with the generation of a pre-defined number of random vertices. Each vertex corresponds to a set of treatment parameters (design) satisfying the bound constraints. The coordinates of the random vertex are generated by:

( )iiiii lurlx −+= , i = 1, …, N….........……..………. (3)

which ensures that any randomly generated point complies with the limits enforced by the bound constraints. Here, ri is a pseudo-random deviate rectangularly distributed over the interval (0, 1) which is controlled by a known value, xin, for the i-th coordinate.

To ensure that the random points generated satisfy the imposed design constraints, points violating some of the design constraints are moved towards the centroid, c, successively by a factor of a perturbation, x’:

( )xcx ′+=21 , ..……………...………………..……..…(4)

until the new point, x, satisfies the design constraints. The coordinates of the centroid are calculated using generated points compliant with the design constraints:

∑= ii xn

c 1 , …….………….…..…………...…………(5)

where n is the number of compliant points. The objective function values are calculated from the

points which satisfy the bound and design constraints. The point that corresponds to the maximum objective function is discarded. The rest of the points are preserved and used to define a compound (set of compliant vertices). New random points are generated around the preserved vertices of the compound. Search for the optimum values continues until pre-specified convergence criteria are met.

Figure 1 shows convergence to the optimum design from the generated random vertices of two different objective functions. The optimum design has passed three levels of convergence tests. The first test ensures that a predefined number of consecutive values of the objective function are found identical within the resolution of the convergence parameter. Once the first test has been satisfied, the second test verifies whether the objective function values at all vertices of the current compound are also identical within the resolution of the convergence parameter. Finally, restarts are made using the current optimum point to check if improvement is still possible.

SPE 84860 3

Definition of Models

Hydraulic Fracture Geometry Model From the given formation parameters and treatment design variables, the resulting fracture geometry is calculated. The fracture dimensions define the post-frac productivity index that is used in production estimation. In this work, we have used the Perkins-Kern-Nordgren (PKN) model coupled with the Carter Equation II for leak-off.

The PKN model assumes: (1) vertical plane strain along a fracture with a length to height aspect ratio greater than unity, (2) flow in the lateral direction and (3) an approximate elliptical fracture geometry shape. We use Carter’s formulation of the material balance to account for leak-off 8. At any injection time, t, Carter postulated that the injection rate entering one wing of the fracture is equal to the sum of the different leak-off rates plus the growth rate of the fracture volume:

( )∫ +++

−=

t fpf

Lidt

dwA

dtdASwd

dtdA

tCq

022

2τ

τ.…. (6)

This is further simplified by assuming that width is constant to give:

( )( ) ( ) ( )

−+

+= 12exp

24

2 22 π

βββπ

erfcq

C

SwtA i

L

pf ….(7)

with pf

LSwtC

22

+=

πβ .

In terms of power-law parameters for a non-Newtonian fluid, the maximum width at the wellbore is defined as8:

+−

+

+

+

+

⋅

+

=

221

122

1

2222221

'2

14.2198.315.9

nfn

fn

in

nn

nn

nf

E

xhqK

nnw

............(8)

Coupling with the Carter Equation II, the expression for fracture half length is expressed as8:

( ) ( ) ( )

−+

+= 12exp

24

2 22 π

βββπ

erfcq

hC

Swx i

fL

pff .…(9)

with pf

iLSwtC

22

+=

πβ ,

which can be solved to obtain fracture half length or height. The individual symbols are presented in the nomenclature.

Post-fracturing production model This model calculates the cumulative production of a multiwell system in a gas-condensate reservoir for pseudo-steady state condition.

Gas Condensate Reservoirs. Fevang and Whitson9 developed a simple method to calculate the pseudo-pressure integral in the general volumetric rate equation for a gas-condensate well regardless of geometry. Three flow regions may exist when gas condensate wells are produced and the bottomhole flowing pressure drops below the dewpoint pressure. Region 1 is the inner near-wellbore region where both gas and oil flow simultaneously. This is the main source of deliverability loss in a gas condensate well because the gas relative permeability is reduced due to condensate build up. Net accumulation of condensate occurs in Region 2 where only the gas phase is mobile. Region 3 is the outer most region of single phase gas of constant composition.

The pseudo-pressure integral may be broken down into three parts corresponding to the three possible flow regions:

Region 1: dpRBk

BkP

Pso

oo

ro

gg

rg

wf

∫

+

*

µµ +

Region 2: dpBkdP

P gg

rg∫

* µ + ……………..(10)

Region 3: ( ) dpB

SkR

d

P

P ggwirg ∫

µ1 ,

where P* is the pressure corresponding to po RR 1= , Ro is the solution oil-gas ratio, Rp is the producing gas-oil ratio and Pd is the dewpoint pressure. When P*>PR (reservoir pressure), integration of Region 1 should only be from Pwf to PR and Regions 2 and 3 do not exist. On the other hand, if P*<PR, Region 2 integral should be evaluated from P* to PR and Region 1 should be evaluated from Pwf to P*. If the bottomhole flowing pressure is greater than the in-situ dew point pressure, Region 3 exists. For slightly undersaturated reservoirs, Regions 1, 2 and 3 are evaluated and for rich gas condensate reservoirs, Region 1 dominates.

Multiple wells. In matrix notation, Valko et al.5 showed

that the relationship between production rate and drawdown is as follows:

[ ] [ ]( ) dDAkhq sirr 12 −+= π , .....…………………………...(11)

where dr

is pseudo-pressure integral vector and [Ai] is the influence matrix with elements aij representing the influence of well j on the pressure at the circumference of well i.

[ ]

=

nnnn

n

n

i

aaa

aaaaaa

A

L

MOMM

L

L

21

22221

11211

………...………………….(12)

The influence function is a constant, which depends on the shape of the reservoir and the location of the well. This allows for search of the optimal well locations. Ozkan5 gives the long

4 SPE 84860

time approximations for vertical well, vertically fractured well and horizontal well solutions.

Once a hydraulic fracture is created, most pre-treatment skin effects such as the damage skin, the skin due to partial completion and slant and perforation skin are bypassed and have no impact on post treatment skin performance. Thus, an equivalent skin effect, sf, which is the result of a hydraulic fracture of a certain length and conductivity may be added to the well inflow equation in lieu of the pre-treatment skin effects. Thus, we have for the skin matrix,

[ ]

+

+

+

=

wn

fnfn

w

ff

w

ff

s

rx

s

rx

s

rx

s

D

ln00

000

0ln0

00ln

2

22

1

11

L

O

L

L

….(13)

Production Forecasting. Future gas production forecasting

is done by partitioning the gas reservoir depletion process into successive time steps. In each time interval, the reservoir is in pseudo-steady state flow regime. The wellbore pressures decline by a small value, then the production rates are reduced, and the wellbore pressure jumps by a small value5. Initially, from the given reservoir geometry, well type and configuration, the influence function is calculated for each well and the influence matrix is generated. Using the hydraulic treatment parameter values, the hydraulic fracture geometry created based on the PKN-C model is calculated. The fracture half length is an assumed input and the corresponding fracture height and width are calculated. From the geometry of the created fracture, in-situ reservoir properties and pertinent treatment parameter values, the fracture conductivity is determined and the corresponding diagonal skin matrix is generated.

At the i-th time step, the pseudopressure integral is determined for each well considered in the system. From the definition of the gas-oil ratio, the change in oil saturation is calculated as follows:

( ) ( )

8675

436215

AAAAAAAAAA

dpdSo

−+−+

= …………………...(14)

with

=

dpdB

BSA o

oo 21

1 ,

( )

−−−=

dpdR

BdpdB

BR

SSA o

g

g

g

owco

11 22 ,

+=

dpdB

BR

dpdR

BSA o

o

soso

oo 23

1 ,

( )

dpdB

BSS

A g

g

wco24

1 −−= ,

gg

oo

ro

rgso B

Bkk

RAµµ

+=5 ,

gg

oo

ro

rgo B

Bkk

RAµµ

+= 16 ,

−=

g

o

o BR

BA 1

7 ,

+

−=

o

so

g BR

BA 1

8 .

The flow regions are delineated based on the reservoir pressure, dewpoint pressure, bottomhole flowing pressure and P*, calculated from the gas-oil ratio. Relative permeabilities are determined based on the saturation profile. Production is then calculated from Equation 11 using the pseudo-pressure integral as elements of the pressure vector multiplied by the inverse of the sum of the calculated influence matrix and diagonal skin matrix. The next reservoir pressure is then calculated and refinement of solution is done, by iterating with a new value of oil saturation, 2/ooo SSS ∆−= , until convergence is achieved. The gas rate at the final saturation value is added to the previous production and the time step is incremented.

Care should be taken in the determination of the PVT properties, as accuracy of production estimate is dependent on the integrity of the PVT data. Extended black-oil PVT properties for use in the calculation of the integral are generated using the Whitson and Torp10 or Coats11 method. Economic Model In this model, the NPV is formulated as follows:

( )∑ −

+=

=

NY

ntrn

n Ci

RNPV1 1

, ..……………………...…….(15)

where i is the discount rate. The revenue, Rn, for the year n, is calculated as the product of total production and average gas price. The treatment cost, which is a function of the frac fluid volume, the type of frac fluid, total weight of proppant used and fixed cost to cover equipment hire and other expenses are formulated as follows:

FCHPPWPVPC avpumpprprtflfltr +++= , .………...(16)

where Ctr = treatment cost, $, Pfl = price of frac fluid, $/m3 , Vtfl = volume of frac fluid, m3 , Ppr = proppant cost, $/kg , Wpr = proppant weight, kg , Ppump = pumping cost, $/hp , HPav = hydraulic power of the pump, hp , FC = miscellaneous and fixed cost, $. Model Application and Discussion In this section, different field scenarios are presented to illustrate the application of the optimization scheme. The cases show that we have extended the scope of the original model proposed in our previous studies, to provide solutions to a wide range of reservoir development problems. Different objective functions are executed as deemed appropriate to

SPE 84860 5

provide operators a margin of freedom in making sound decisions.

Hydraulic fracture treatment design is optimized for two vertical wells in a rich gas condensate reservoir. The reservoir has a thickness of 30.48m and a matrix permeability of 5x10-16 m2 (0.5 md). Well 1 coordinates are 488m on the lateral and vertical axes while Well 2 is located at 1673m on the x and y-axes. Reservoir properties, formation properties and well data are presented in Table 1. The wells start production at the same time, t = 0. The extended black-oil PVT properties, generated using the Whitson and Torp procedure, are shown in Figures 2 – 4. Relative permeability curves are given in Figure 5. Proppant and cost data are given in Table 2.

Results show the importance of using an optimization scheme that checks all possible solutions from a wide range of treatment designs. Using parametric sensitivity analysis to obtain the optimum design values of ten treatment parameters is very tedious and time consuming. Furthermore, production engineers may fail to obtain values that may be the best solution from all possible treatment designs. In this example, as shown in Figure 6, the treatment cost is considerably high. This is where the need to consider net present value as an objective function comes in.

Figure 6 compares the treatment cost, production and NPV of the different objective functions during a period of 10 years. A base case is obtained by maximizing cumulative production (Objective Function 1 as shown in Figure 6). A 1% improvement in NPV is achieved when NPV is considered as the objective function (Objective Function 2) to be optimized relative to the base case scenario. This reduced the treatment cost by 28% in comparison with the treatment cost for the base case.

We then tested the merit of further reducing the treatment cost by maximizing NPV and minimizing treatment cost simultaneously (Objective Function 3). This design objective

is formulated as

+

−= 2

21

1min)(min P

DC

PDNPVxZ tr , where Di is

a normalizing factor and Pi is the priority factor. In this example, 47% savings on treatment cost is achieved for an NPV reduction of 5% relative to the base case. The optimum values of treatment parameters for the above case scenarios are presented in Table 3.

Field development planning entails the determination of the optimal placement of wells. This depends on well and surface equipment specifications, reservoir and fluid properties and economic criteria. For placement of several wells, hundreds of combinations must be considered for the achievement of target production or NPV. Various approaches have been discussed in literature such as quasi-Newton algorithm designed for unconstrained minimization4, hybrid optimization technique based on genetic algorithm, polytope algorithm, kriging algorithm and neural networks12 and pure artificial neural networks13. Although these methods adequately provide the optimum well locations, this example illustrates how the well locations are simultaneously optimized with the hydraulic fracture treatment parameters, thus, improving the economics of the project.

Table 4 summarizes the optimum values of the treatment parameters and well locations obtained for Objective

Functions 4 - 6 as defined in Figure 6. The effect of simultaneously optimizing the well locations is clearly seen in Figure 6. When cumulative production is maximized (Objective Function 4), a 55% increase in cumulative production is achieved relative to the base case design. This results in a 63% improvement in NPV and a corresponding 11% reduction in the treatment cost. For maximum NPV as the design objective (Objective Function 5), NPV was improved by 62% and the treatment cost was reduced by 11% relative to the base case. Companies would opt for a compromise between the NPV and the treatment cost. For further improvement, we maximized NPV and minimized treatment cost simultaneously (Objective Function 6). Results show that a 42% treatment cost savings is achieved with a 23% NPV improvement relative to the base case design.

Furthermore, this example illustrates how to achieve target production. This option allows for compliance with the required contracted quantities of gas to be delivered to the sales pipelines. We formulate the objective function as

+

−= 2

21

1

1min)(min P

DC

PD

TGxZ trp , where values of D1 and D2

are adjusted such that the value of both terms at the right hand side of the equation approaches 0.5 and P1 and P2 are set to 1 to assign equal priority to achieving target production and minimizing treatment cost. In this combined objective function, T1 is the target value for the first objective. This design objective is shown in Figure 6 as design objective 7.

For economic reasons, decision to fracture just one well may also be considered. With the methods outlined here, it is possible to handle this scenario by reducing the number of free design variables and setting the influence function of one well as vertical well without fractures. Selection of the well as candidate for a hydraulic treatment job would then be largely dependent on the NPV-treatment cost trade-off.

Results show that if Well 1 is hydraulically fractured and Well 2 left as it is (Objective Function 8), a 54% treatment cost savings is achieved relative to the base case design. This reduces the NPV by 8%. On the other hand, if Well 2 is fractured (Objective Function 9 as shown in Figure 6), a treatment cost savings of 54% results in 20% NPV reduction. The optimum values of the treatment parameters for the above two cases are presented in Table 5. Conclusions On the basis of information presented in this paper, the following conclusions can be reached: 1. Well location and reservoir geometry significantly affect the overall reservoir performance for a multiwell system. Thus, the optimal location of wells and the optimum treatment parameters found by the search scheme greatly improved the economics of the project. 2. The proposed pseudo-steady state model has been efficiently coupled with the optimization algorithm and has given good results for different applications. Extended black-oil PVT data were generated for use in the proposed model to achieve more accurate results. 3. The capability to generate optimum treatment design for any possible scenario, such as different well types (vertical to horizontal wellbores) and any number of wells extends the

6 SPE 84860

application of the model to field development cases which include achievement of production targets and trade-off analysis thus, allowing operators to make robust business decisions.

Nomenclature

A = fracture surface area, m2 Bg = gas formation volume factor, m3/m3 Bo = oil formation volume factor, m3/m3

CL = leak-off coefficient, m.s-1/2

E’ = plane strain modulus, Pa FC = fixed cost h = pay thickness, m hf = fracture height, m k = matrix permeability, m2

krg = gas relative permeability kro = oil relative permeability K = consistency index, Pa.s-n

l = lower bound on design variable n = flow behaviour index, dimensionless NY = number of years q = gas rate, m3/s qi = pumping rate, m3/s ri = pseudo-random deviate rw = wellbore radius, m Rn = revenue for year n, $ Rso = solution gas-oil ratio, m3/m3

sf = skin due to hydraulic fracture, dimensionless Swi = initial water saturation Swc = connate water saturation So = oil saturation Sp = spurt loss coefficient, m ti = pumping time, s v = Poisson’s ratio, dimensionless wf = fracture width, m Wpr = proppant weight, kg xf = fracture half length, m τ = opening time, s µg = gas viscosity, Pa-s µo = oil viscosity, Pa-s

References 1. Rahman, M.M., Rahman, M. K., Rahman, S.S.: “An Integrated

Model for Multi-Objective Design Optimization of Hydraulic Fracturing,” J.of Pet Sci and Eng., (2001), 31, 41.

2. Rodriguez, F. and Cinco-Ley, H.: “A New Model for Production Decline,” paper SPE 25480 presented at the 1993 SPE Productions Operation Symposium, Oklahoma City, Oklahoma, March 21 – 23.

3. Camacho-V., R., Rodriguez, F., Galindo-N., A. and Prats, M.: “Optimum Position for Wells Producing at Constant Wellbore Pressure,” paper SPE 28715 presented at the 1994 SPE International Petroleum Conference and Exhibition, Veracruz, Mexico, Oct. 10 – 13.

4. Umnuayponwiwat, S. and Ozkan, E.: “Evaluation of Inflow Performance of Multiple Horizontal Wells in Closed Systems,” J. of Energy Resources Tech. (March 2000) 122, 8.

5. Valko, P.P., Doublet, L.E. and Blassingame, T.A.: “Development and Application of the Multiwell Productivity Index (MPI),” SPE Journal (March 2000), 5(1), 21.

6. Queipo, N.V., Verde, A., Canelon, J. and Pintos, S.: “ Efficient Global Optimization for Hydraulic Fracturing Treatment Design,” paper SPE 74356 presented at the 2002 SPE

International Petroleum Conference and Exhibition, Villahermosa, Mexico, Feb. 10-12.

7. Aly, A. M., El-Banbi, A.H., Holditch, S.A.; Wahdan, M., Salah, N., Aly, N.M. and Boerrigter, P.: “Optimization of Gas Condensate Reservoir Development by Coupling Reservoir Modeling and Hydraulic Fracturing Design,” paper SPE 68175 presented at the 2001 SPE Middle East Oil Show and Conference, Bahrain, March 17-20.

8. Valko, P. and Economides, M.J.: Hydraulic Fracture Mechanics, John Wiley & Sons Ltd, West Sussex, England (1995).

9. Fevang, O., and Whitson, C. H.: “Modelling Gas Condensate Well Deliverability,” paper SPE 30714 presented at the 1995 SPE Annual Technical Conference and Exhibition, Dallas, U.S.A., Oct. 22 – 25.

10. Whitson, C.H. and Torp, S.B.: “Evaluating Constant Volume Depletion Data,” paper SPE 10067 presented at the 1981 Annual Technical Conference and Exhibition, San Antonio, Texas, Oct. 5 – 7.

11. Coats, K.H.: “Simulation of Gas Condensate Reservoir Performance,” JPT (Oct. 1985) 1870.

12. Centilmen, A., Ertekin, T. and Grader, A.S.: “Applications of Neural Networks in Multiwell Field Development,” paper SPE 56433 presented at the 1999 SPE Annual Technical Conference and Exhibition, Houston, Texas, Oct. 3 – 6.

13. Guyaguler, B., Horne, R.N., Rogers, L. and Rosenzweig, J.J.: “Optimization of Well Placement in a Gulf of Mexico Waterflooding Project,” paper SPE 63221 presented at the 2000 SPE Annual Technical Conference and Exhibition, Dallas, Texas, Oct. 1 – 4.

SPE 84860 7

Table 1 – Well and reservoir properties.

Wellbore radius, m 0.0762

Young’s modulus, Pa 3.50E+10

Poisson’s ratio 0.2

Leak off coefficient, m/s0.5 9.84E-06

Lab-measured fracture conductivity, m2-m 2.04-12

Porosity 0.30 Minimum pressure among the rated pressures of surface equipment, Pa 9.65E+07

Minimum in-situ stress in the pay zone, Pa 4.14E+07

Packed proppant porosity 0.35

Initial reservoir pressure, Pa 2.31E+07

Pwf, well 1,2,Pa 6.89E+06

Depth of the pay zone, m 2286

Temperature, °K 367

Proppant specific gravity 3.1

Connate water saturation 0.16

Burst strength of tubing, Pa 8.96E+07

Conductivity damage factor 0.3

Available horsepower, kW 10444

Pump efficiency factor 0.85

Formation Critical Pressure, Pa 6.72E+07

Proppant Diameter, m 2.07E-03

Specific gravity of proppant, kg/m3 3100 Minimum in-situ stress in the confining zone (shale), Pa 5.31E+07

Maximum in-situ stress in the pay zone, Pa 5.52E+07 Reservoir dimensions 2161mx2161m

Table 2 – Proppant data and material costs

Material Costs

Proppant cost, per lb $1.0 Fracturing fluid cost variable Pumping cost for10444kW, per 0.746kW $20.0 Fixed cost, $ 10,000.0 Gas price, per .0283 m3 $1.0 Discount rate 0.1

Proppant Data Proppant type 20/40 Westprop Specific Gravity 3.1 Diameter, m 6.2992-4 Packed Porosity 0.35 Conductivity @ closure stress (@ 2 lb/ft^2) 2.04-12 m2-m Conductivity Damage Factor 0.6

Table 3 – Results for different objective functions.

Max Production Max NPV Max NPV, Min Cost Variables

Well 1 Well 2 Well 1 Well 2 Well 1 Well 2

Fracture half length, m 444 454 457 233 457 93

Pumping rate, m3/s 0.1079 0.1171 0.1165 0.1100 0.0265 0.0375 EOJ proppant concentration, kg/m3 1489.44 1488.24 1797.40 1425.94 1406.76 599.13

Frac fluid viscosity, Pa-s 0.5507 0.1314 0.7737 0.1361 0.3969 0.1441

Pumping time, s 1127 2221 600 711 600 600

8 SPE 84860

Table 4 – Results for optimized well locations

Max Production Max NPV Max NPV, Min Cost Variables

Well 1 Well 2 Well 1 Well 2 Well 1 Well 2

Fracture half length, m 432 395 425 397.18 106.83 168.36

Pumping rate, m3/s 0.0397 0.048 0.0427 0.0476 0.0928 0.0647 EOJ proppant concentration, kg/m3 1572 1525.4 1646.41 1478.66 830.39 1268.96

Frac fluid viscosity, Pa-s 0.2055 0.1783 0.1673 0.1784 0.1171 0.0921

Pumping time, s 1950 5183 1901 5279 600 621

x-coordinate 1643.44 745.37 1613 728 494.36 1251.94

y-coordinate 1266.24 1266.92 1281 1281 835.5 788.99

Table 5 – Optimum treatment parameters for single well frac jobs

Parameter Well 1 Only Well 2 Only

Fracture half length, m 310 308

Injection rate, m3/s 0.1046 0.1034 EOJ Proppant Concentration, kg/m3 1275 1261

Frac fluid viscosity, Pa-s 0.1914 0.1933

Pumping time, s 1677 1686

Fig. 1 – Convergence to optimum design

Fig. 2 – Solution gas-oil ratio and oil formation volume factor for rich gas.

0

100

200

300

400

500

0 5 10 15 20 25Pressure, MPa

Solu

tion

GO

R, m

3/m

3

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Oil

Form

atio

n Vo

lum

e Fa

ctor

, m3/

m3

Solution GOR

Oil FormationVolume Factor

17.9

18.2

18.5

18.8

19.1

19.4

0 1 2 3 4 5 6 7 8Number of redesign iterations

Net

pre

sent

val

ue, m

$

9.20E+09

9.40E+09

9.60E+09

9.80E+09

1.00E+10

1.02E+10

Cum

ulat

ive

prod

uctio

n, B

scf

Objective Function: NPV

Objective Function: Cumulative Production

SPE 84860 9

Fig. 3 – Gas formation volume factor and condensate gas ratio for

rich gas.

Fig. 4 – Oil and gas phase viscosities.

Fig. 5 – Relative permeability curves.

1 – Max Production (base case) 6 – Optimum Location, 2 – Max NPV Max NPV, Min Cost 3 – Max NPV, Min Cost 7 – Target Production 4 – Optimum Location, Production 8 – Well 1 Only 5 – Optimum Location, NPV 9 – Well 2 Only

Fig. 6 – Results of example application.

0.E+00

1.E+07

2.E+07

3.E+07

4.E+07

5.E+07

6.E+07

1 2 3 4 5 6 7 8 9Different Objective Functions

NPV

, $; T

reat

men

t Cos

t, x1

0$

3700

4400

5100

5800

6500

7200

7900

8600

Cum

ulat

ive

Prod

uctio

n, m

3

NPV

TreatmentCostCumulativeProduction

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Saturation Sw, Sg

Rel

ativ

e pe

rmea

bilit

y

krg krog

krw krow

0.0E+00

1.0E-02

2.0E-02

3.0E-02

4.0E-02

5.0E-02

0 5 10 15 20 25Pressure, MPa

Gas

For

mat

ion

Volu

me

Fact

or, m

3 /m3

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

Con

dens

ate

Gas

Rat

io, m

3 /m3

Gas FormationVolume Factor

Condensate GasRatio

0.0E+00

8.0E-05

1.6E-04

2.4E-04

3.2E-04

4.0E-04

0 5 10 15 20 25Pressure, MPa

Oil

Visc

osity

, Pa-

s

0.0E+00

9.0E-06

1.8E-05

2.7E-05

3.6E-05

4.5E-05

Gas

Vis

cosi

ty, P

a-s

Oil Viscosity

Gas Viscosity