spe-152837 new methods to predict inflow performance of multiply fractured horizontal wells under...

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SPE 152837

New Methods to Predict Inflow Performance of Multiply FracturedHorizontal Wells under Two-Phase Condition and Optimize Number ofFracture StagesHe Zhang, Guoqing Han*, Fabien Houeto, Rodney Lessard, Wenhao Wang, and Jun Li, Schlumberger

Copyright 2012, Society of Petroleum Engineers

This paper was prepared for presentation at the North Africa Technical Conference and Exhibition held in Cairo, Egypt, 2022 February 2012.

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 have not beenreviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, itsofficers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission toreproduce 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 SPE copyright.

AbstractA new method is presented to calculate the total inflow and associated productivity index (PI) under two-phase conditions by

a sum of contributions from matrix andN multiple fractures using semi-analytical methods. Maximizing the net Present

present value (NPV), the new inflow performance relationship (IPR) models can determine the optimal fracture stage

number. Furthermore, the deliverable PI is used as an input to a network model to calculate the operating rate, taking intoaccount the network constraints such as the node pressure, allowed flowrate, or choke in place. Through this analysis, one

may determine that the horizontal well of interest may not flow at the maximum PIit suggests that the network constraints

dissipate the contribution from mstage(s) fractures. The number of fracture stages is then reset to (Nm), the IPR model isrerun, and an updated PI is imported into the network model. This process is iterated until the final optimization is achieved.

This method can quickly optimize the number of fracture stages for horizontal wells under two-phase solution-gas conditions,

assisting operating companies in planning field development.

IntroductionA very high percentage of wells drilled today are horizontal and most of those require stimulation. Because of the high cost

of fracturing, the optimal number of fracture stages needs to be determined as soon as possible when drilling is being

completed. Ideally, a numerical reservoir and fracturing simulation would give the most accurate results, but such simulationis time consuming and sometimes impractical as it requires logging data, fluid sampling, and pressure transient analysis

(Taylor et al.2011).

Inflow performance relationship (IPR) models can be used to quickly estimate the productivity index (PI) and inflow rate.

However, none of the IPRs in literature is satisfactory for a horizontal fractured well under two-phase conditions. Thomaset

al.(1998) presented a method to calculate the IPR for a horizontal well by considering non-Darcy flow effects in terms of an

additional skin. Their model integrates a production forecasting dynamic reservoir model, which enables the calculation of

the PI over time. However, the innovative skin calculation is for the existing single-phase steady/pseudosteady state models

(Joshi 1991; Babu and Odeh 1989; Economides et al. 1996). This indeed suggests that such approach is restricted to single-phase flow only. Retnanto and Economides (1998) presented a generalized dimensionless IPR curve for a horizontal and

multilateral well by a nonlinear regression in a similar form to Vogels model (1968). It is specifically useful for a solution-gas drive reservoir. This empirical equation employs a pseudosteady PI proposed by Economides et al.(1996) earlier. Yildiz

(2001) presented an analytical IPR curve for a perforated multisegment horizontal well in a 3D anisotropic reservoir. This

work focused on the completion damage by calculating the perforation total pseudoskin across different segments of thehorizontal well, but again this analytical equation is for single phase flow. Billiter et al. (2001) developed the fundamental

IPR for an unfractured, horizontal gas well including non-Darcy effects from the Babu and Odeh (1989) model. Furthermore,

with a nonlinear regression, they developed an empirical multivariable-dependent dimensionless IPR curve based upon 384

cases generated by the fundamental IPR. However, this work focused on the gas reservoir only. Wiggins and Wang (2005)

presented new forms of generalized IPR for a horizontal well in a solution-gas drive reservoir based on numerical simulationstudy. Two empirical equations were presented: one is similar to the dimensionless Vogels model for vertical well; the other

* Guoqing Han currently holds the faculty position at China University of Petroleum Beijing


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2 SPE 152837

one includes the factor for reservoir depletion stage to determine the maximum oil flowrate. The authors also compared the

correlations in the literature including the models of Bendakhlia and Aziz (1989), Cheng (1990), and Retnanto and

Economides (1998). Kamkom and Zhu (2005) used a numerical reservoir simulator to evaluate existing two-phase IPRs forhorizontal wells among the modified Vogels model by Kabir (1992), Bendakhlia and Aziz (1989), Cheng (1990), and

Retnanto and Economides (1998). The Bendakhlia and Aziz correlation yields the closest results at high or low recovery

factors, while the modified Vogel model performs the best at the recovery factors in between. Chase and Steffy (2004)

presented a single-oil-phase IPR curve for a single-stage fractured horizontal well. This IPR model is characterized by the

ratio of the external drainage radius to the fracture half-length, based on Joshis model, which is not appropriate for two-phase systems.

The industry has also devoted great efforts to modeling fracturing jobs in recent years. Valko and Economides (1998)

introduced a concept of proppant number unified fracture design (UFD), which is proportional to fracture permeability and

propped volume, and inversely proportional to reservoir permeability and reservoir volume. The proppant number can be

used to calculate the maximum PI corresponding to the optimum fracture conductivity in pseudosteady-state condition. Later,Daal and Economides (2006) further optimized the hydraulically fractured wells in irregularly shaped drainages by the UFD

method. This work provided a factor labeled optimization function (F-function) and fit it into the dimensionless PI equation

for a square reservoir. The maximum PI can be calculated in differently shaped reservoirs for any proppant number by this

method. However, this works only for single-phase flow. Because of the high cost of hydraulic fracturing operations, industry

used numerical simulations to estimate and optimize the net present value (NPV) and fracture number (Guglielmo et al. 2006;Sadrpanah et al. 2006; Butter et al. 2006). The predictions are even when compared with analytical results. However, it is

difficult to estimate the accurate fracture half-length and infinite permeability. The results are quite different considering theincapacity of the analytical solution for multiphase flow and transient conditions. Yuan and Zhou (2010) presented a new

model to predict the PI of a fractured horizontal well at steady-state condition. This model adds up the inflow from matrix

and fractures, but again it is for a single-phase system.

Here we propose a new method to predict the PI of a multistaged fractured horizontal well for a two-phase solution-gas drivereservoir. We also develop an iterative scheme to calibrate the new IPR model with the network model and obtain an optimal

production system for mature reservoirs.

New IPR Model DescriptionTo develop the general coupling concept of the inflow from rock matrix and fractures under solution-gas drive two-phase

condition, this work starts with the idea proposed by Yuan and Zhou (2010) in Eq. 1. The summation gives the total

contributed production from fracturing stimulation.

... (1)=

+=N

i

ifm qqq1

,

where qis the total flow rate for a multiply fractured horizontal well, qmis the flow rate from matrix without stimulation, qf,iis the flow rate from the i-th fracture, andN is the total number of fractures.

In the first step, the Retnanto and Economides model (1998) is used for demonstration purposes to calculate the inflow frommatrix, qm, in Eq. 2.

=

r

wf

r

wf

mp

p

p

pqq 75.025.00.1max

n

(2)

where n is the exponent on the IPR curve as a function of bubblepoint pressure and reservoir depletion in Eq. 3, qmaxis the

absolute open flow which is calculated in Eq. 4,pwfis the bottomhole flowing pressure, andpris the average reservoir

pressure.

( bb

r

b

r pp

p

p

pn 31066.1496.046.127.0 +

+= )

2

.. (3)

wherepbis the bubblepoint pressure in psi.


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SPE 152837 3

nq r

75.025.0max

+=

Jp...... (4)

whereJis the productivity index which is calculated in Eq. 5.

+

=

totale

D

ec

sL

xpB

J

222.887

xk .... (5)

where kcis the corrected reservoir permeability due to the existence of fractures in Eq. 6,xeis the extent of drainage area in

the x-axis direction (reservoir length),Bis the oil formation volume factor,is the oil viscosity,pDis the calculateddimensionless pressure in Eq. 7 from Economides et al. (1996),Lis the well length, andstotalis all the damage and

pseudoskin factors.

=

=

N

i e

ifi

m

if

mcr

x

L

w

k

kkk

1

,,

121,0max ........... (6)

where kmis the matrix permeability, kf,iis the fracture permeability for the i-th fracture,wiis fracture width,xf,iis half-length

of the fracture, and reis the effective reservoir radius, which can be approximated by/, withyebeing the extent ofdrainage area in y-axis direction (reservoir width). Note that when the summation in the bracket is larger than 1, it indicates

that the inflow is dominated by fractures and the inflow from the rock matrix is ignored.

+=

L

h

r

h

L

x

h

Cxp

w

eHeD

62ln

24 (7)

where CHis the shape factor,h is reservoir thickness, and rwis the wellbore radius. Note that Eq. 7 is a simplified form of theequation from Retnanto and Economides (1998) assuming that the horizontal well is placed vertically in the middle of the

reservoir.

The second step is to calculate the contributed inflow from fractures as proposed by Yuan and Zhou (2010).

+

= 5.0

2ln

100059.0,

h

x

rhB

pwkq

f

w

ff

if

... (8)

where pfis the pressure drawdown from the tip of the fracture to the wellbore, which is calculated in Eq. 9. Note that Eq. 8is derived with the assumption of a linear and a radial flow inside the fracture.

The pressure distribution of the horizontal well needs to be calculated to obtain the pressure drawdown, pf, by Eq. 9.

( )2222

f

f

wfef

xbla

lxppp

+

+=

22

. (9)

wherepeis the pressure at the reservoir outer boundary, l is the distance between a fracture and the center of drainage ellipse,

aandb are respectively the half major axis and minor axis of the drainage ellipse from Eqs. 10 and 11.

( )4/225.05.02

LrL

a e++= ... (10)


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4 SPE 152837

( )222

Lab = ... (11)

Finally, we use Eq. 1 to precompute the grid of the total flowrates as a function of different fracture stages,N.

This model has made several assumptions which are summarized for future field application:

1. Inherited assumptions from general IPRs apply, and include homogenous reservoir, Vogels form IPR, etc.2. The horizontal well is placed vertically in the middle of the reservoir.3. The flow around the wellbore inside the fracture is radial.4. This work considers the effects of the fracture inflow depending on the contacted reservoir volume. However, it

does not take into account the interference among any adjacent fractures.

Assumption 2 can be removed by using the original equations (Economides et al. 1996), with a definition of eccentricityeffects in the vertical direction. Assumption 3 can be modified by replacing Eq. 8 with other flow patterns (Yuan and Zhou

2010). Once the total inflow is known, the fracture stages of a horizontal well can be optimized to maximize the NPV.

Fracture Number Optimization SchemeBecause of the high cost of the stimulation operations, excessive fracturing jobs may not lead to the maximum NPV.

Assuming that a horizontal well is multiply fractured and PI is obtained, we can estimate the cumulative production for a

steady-state period. Total revenue is estimated by using an average oil price. Given a certain confidence interval on the oilprice and a single-fracture job cost, the optimal fracture stage number,N, can be quickly determined by a sensitivity analysis.

The ideal behavior of the NPV versus fracture stage number is shown in Fig. 1.

Fig. 1NPV and PI versus numberof fracturing stages.

However, in a mature field, the allowable flowrate is subject to further network constraints like wellhead pressure

requirement, choke, manifold capacity, sale point condition, etc.The deliverable PI is then used as an input to the surface

network model which takes into account all the engineering constraints from facilities and wellbores. This requires the use ofnetwork simulation methods (Narahara et al. 2004). The network simulation results might predict that a horizontal well will

not produce at the maximum PI. It implies that the network constraints dissipate the contribution from m stage(s)

fractureswhere the value of m can be obtained from the precomputed grid of total flowrates. Then the number of fracture

stages is set toN-m. By rerunning the proposed IPR model, we can update the PI. This new PI is then used in the surfacenetwork model. An iterative optimization scheme in Fig. 2can lead to the final optimal production system.

Case StudyAn example is presented below (most data are adapted from Economides et al. 1996).

A horizontal well ofL= 1,500 ft long is situated vertically in the middle in a square reservoir withxe= 2,000 ft,ye= 4,000 ft.

Later we use drainage radius for calculation. We approximate the horizontal well drainage area, re, by conserving the area of

rectangular regions with the side lengthsxeandye. Consequently, 1595.77/ee == yxer ft. Ash= 20 ft, rw= 0.4 ft,

07.215006

20

4.02

20ln =

=

xs . (12)


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SPE 152837 5

Assuming thatye/xe= 2,L/xe= 1500/2000 = 0.75, the shape factor is determined CH= 2.53. Thus,pDis obtained from Eq. 7,

57.2007.215002204

=

+

=

Dp200053.22000

(13)

Fig. 2 Scheme of the optimization workflow.

If kh(isotropic throughout) = 10 md, kv= 1 md, total skins= 2,Bo= 1.25 res-bbl/STB, ando= 1 cp, PI can be calculated byEq. 5, where the corrected permeability is based on the 3D average,

Dpsi

STB399.0

215002

200057.20125.122.887

20001103 2

=

+

=

J ..... (14)

We assume reservoir pressurepr= 5,000 psia, and bubblepoint pressurepb= 4,600 psia. By Eq. 3, the exponent of IPR, n, is

calculated.

( ) 13.246001066.144600

500096.0

4600

500046.127.0 3

2

=+

+= n ... (15)

By Eq. 4, the maximum flowrate, qmax, is calculated.

D/STB84.107913.275.025.0

5000399.0max =

+

=q ...... (16)

Substituting qmaxand bottomhole pressure = 2,800 psia into Eq. 2,


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D/STB693.125000

280075.0

5000

280025.00.184.1079

13.2

=

=mq

... (17)

Inflow from matrix is obtained, but the corrected permeability is the formation matrix average. Once the fracture effect of

Eq. 6 is considered, the inflow is expected to decrease.

We assume all the fractures have the same width, half-fracture length, and permeability: w= 1 in.,xf= 180 ft, kf= 3,000 md.

The half major axis and minor axis of the drainage ellipse, aand b, can be calculated by Eqs. 10 and 11.

( ) ft18.16861500/77.1595225.05.02

1500 4=++=a ... (18)

( ) ft20.15102

150018.16862

2 ==b ... (19)

The next step is to calculate the pressure drawdown from the pressure at the drainage boundary,pe, to the tip of the fracture.The average reservoir pressure can be obtained from periodic pressure buildup tests5,000 psia in this example. Assuming

pe= 5,630 psia, we plug Eqs. 18 and 19 into Eq. 9 and get

( )

1022

22

10389.718.1686

1802830

+

+=

l

lpf . (20)

Different placements of the multiple fractures lead to different results. This case study investigated two scenarios (Fig. 3).

(A) (B)

Fig. 3 Top view of different placement of the multiple fractures along the horizontal well.The horizontal line represents the 1,500 ft-long well.

Each horizontal line represents the horizontal well with a different fracture pattern.

The blue dotted line indicates the mid-point of the horizontal well.The short perpendicular lines represent the multiple fractures

and their location along the well. Scenario (A) includes five different cases, and scenario (B) includes eight different cases.

In scenario (A), the pressure at each fracture tip is obtained by Eq. 20 with l= 750, 562.5, 375, 187.5, and 0 ft. Each lis

labeled with a numerical case number from top to bottom in Fig. 3(A) and Table 1. The related fracture tip drawdown andinflow at different locations are also summarized. The total inflow from Eq. 1 is summarized in Table 2.


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SPE 152837 7

Table 1FRACTURE PRESSURE ANDINFLOW ANALYSIS

ON DIFFERENT LOCATIONS FOR SCENARIO(A)

Distance of the Fracturefrom the Middle ofHorizontal Well, ft

Pressure AtThe Tip Of

Fractures, psi

Inflow fromFracture,STB/D

750. 4101.54 61.59562.5 3800.46 47.34375. 3511.43 33.67187.5 3258.52 21.70

0. 3137.31 15.96

Table 2INFLOW ANALYSIS FORSCENARIO (A)

CaseNumber

CorrectedPermeabil

ity, md

Inflow fromMatrix,STB/D

Total Inflow fromMulti-Fractures,

STB/D

0 4.642 693.04 0.1 4.623 690.24 61.592 4.604 687.43 108.94

3 4.585 684.62 142.604 4.566 681.82 164.305 4.548 679.01 180.26

Varying the bottomhole pressure from 14.7 psia to the reservoir pressure 5,000 psia, IPR curves for different fracture stages

under two-phase condition are calculated in Fig. 4.

Fig. 4IPR curves of different fracture stages under two-phase condition for scenario (A).The blue dashed line represents the original Economides model (1998).

From Fig. 4, we can see after a fracturing job with 2 to 3 stages, the marginal production enhancement is less. This suggeststhat an economic analysis of the NPV is needed. According to the cost of any single-stage fracture and the projected oil price,

we can optimize the stage number. Let us assume the fracturing cost is USD 1 million per stage and the current oil price is

estimated at USD 100 per barrel for a year. A simple economic analysis based on this 1-year period is given in Table 3andFig. 5. Note that the cost used in this analysis is much simplified compared to the cost in reality.


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Table 3ECONOMIC ANALYSIS FOR SCENARIO (A)

StageNo.

InflowIncreased

Rate1-year

ProductionAdditionalProduction

Revenue Cost ValueNumerical

PI

STB/D STB/D MMSTB MMSTB Million $ Million $ Million $ STB/D/psi

0 693.04 0. 0.25 0. 24.95 0. 24.95 0.315

1 751.83 58.78 0.27 0.02 27.07 1.00 26.07 0.342

2 796.37 103.32 0.29 0.04 28.67 2.00 26.67 0.362

3 827.23 134.18 0.30 0.05 29.78 3.00 26.78 0.376

4 846.12 153.07 0.30 0.06 30.46 4.00 26.46 0.385

5 859.27 166.23 0.31 0.06 30.93 5.00 25.93 0.391

Fig. 5Economic analysis for scenario (A) which meets the expectation of Fig. 1.

Further, the results from scenario (A) indicate an opportunity for the investigation of different patterns of the fracture

placement. In scenario (B), the fractures are evenly placed along the horizontal well. The same procedure as was followed forscenario (A) gives the results in Tables 4 and 5, and Figs. 6 and 7.

Table 4INFLOW ANALYSIS FOR SCENARIO (B)

CaseNumber

CorrectedPermeability,

md

Inflow fromMatrix, STB/D

Total Inflow fromMulti-Fractures,

STB/D0 4.64 693.04 0.001 4.62 690.24 15.962 4.60 687.43 67.333 4.59 684.62 139.154 4.57 681.82 173.865 4.55 679.01 206.48

6 4.53 676.20 240.857 4.51 673.39 275.218 4.49 670.59 292.97


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SPE 152837 9

Fig. 6 IPR curves of different fracture stages under two-phase condition for scenario (B).The blue dashed line represents the original Economides model (1998).

Table 5ECONOMIC ANALYSIS FOR SCENARIO (B)StageNo.

TotalInflow

IncreasedRate

1-yearProduction

AdditionalProduction

Revenue Cost ProfitNumerical

PISTB/D STB/D MMSTB MMSTB Million $ Million $ Million $ STB/D/psi

0 693.04 0. 249.50 0. 24.95 0. 24.95 0.321 706.20 13.16 254.23 4.74 25.42 1.00 24.42 0.322 754.76 61.72 271.71 22.22 27.17 2.00 25.17 0.343 823.77 130.72 296.56 47.06 29.66 3.00 26.66 0.374 855.67 162.63 308.04 58.55 30.80 4.00 26.80 0.395 885.49 192.44 318.78 69.28 31.88 5.00 26.88 0.406 917.05 224.00 330.14 80.64 33.01 6.00 27.01 0.427 948.60 255.56 341.50 92.00 34.15 7.00 27.15 0.438 963.56 270.51 346.88 97.38 34.69 8.00 26.69 0.44

Fig. 7Economic analysis for scenario (B) which meets the expectation of Fig. 1.

DiscussionAs the average distance between the fracture location and the center of the ellipse increases, we see a better enhancement in

production. Fractures at the ends of the horizontal well make more contact with the reservoir. From the above two case

studies, evenly placing the fractures can maximize the NPV and minimize the interference effects of adjacent ones. Different

patterns of the placement of fractures affect the stimulation results. In scenario (A), three stages of fracturing leads to the

optimal NPV; while in the scenario (B), the pattern of seven-stage fracturing gives the best NPV (USD 0.37 million more


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10 SPE 152837

profit).

Field operations do not always allow the maximum flowrate due to the network constraints such as the situated node

pressure, allowed flowrate, placed choke, etc.Incorporating a network simulation to derive deliverable flowrate isrecommended. An example is given in Fig. 8. All the node pressures fromJ1toJ5 are in the range of [2080, 2150] psia. If

we connect a horizontal well into this network, the joint node pressure must be evaluated in advance to avoid backflow and

eventual well shut-in. Fig. 2 shows an iterative process to model a feasible production system and maximize the NPV.

Finally, this work not only presents a further integration of the model of Retnanto and Economides (1998), but also allowsthe use of other two-phase IPRs with the same concept.

Fig. 8An example of network constraints and iterated calculation is normally required.

Future workThe authors would like to validate the model with field data and potentially conduct further investigation on the fracture

patterns.

ConclusionsThis paper presents a new method to predict the PI of a multiply fractured horizontal well for two-phase solution-gas drivereservoir. The study shows that placement of fractures evenly and close to the fracture ends can improve NPV. It also

proposes an iterative scheme between the new IPR model and surface network model to obtain the optimal production system

for mature fields. The proposed approach can be used to quickly determine the number of fracture stages for a horizontal wellin a two-phase solution-gas drive reservoir. This presents an innovative and effective way to assist operators in formulating a

field development plan.

AcknowledgmentThe authors would like to thank Schlumberger for allowing us to publish this paper and Colin Watters, ChukwuemekaOvuworie, and Ghislain Fai-Yengo for the valuable discussions.

Nomenclature

a = half major axis of the drainage ellipse, ft

b = half minor axis of the drainage ellipse, ft

Bo = Formation Volume Factor, res-bbl/STBCH = shape factor, dimensionless

D = time, days

h = reservoir thickness, ftJ = productivity index, STB/d/psi

l = distance between a fracture and the center of drainage ellipse, ft

L = well length, ft

m = the number of unnecessary fracture, dimensionless

N = total number of fractures, dimensionless

n = exponent on IPR curve

p = pressure, psi

q = flow rate, bbl/D


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SPE 152837 11

re = reservoir radius, ft

rw = wellbore radius, ft

s = skin factor, dimensionlesssx = Kuchuk skin factor, dimensionless

w = fracture width, inch

xe = extent of drainage area inx-axis direction, ft

x = half length of the fracture, ft

ye = extent of drainage area in y-axis direction, ft = viscosity, cp

Subscripts

b = bubble-point

c = corrected

f = fracture

h = horizontal

i = i-th fracture

m = matrix

max = maximum

o = oil

r = reservoirtotal = total

v = vertical

wf = bottomhole

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spe-152837 new methods to predict inflow performance of multiply fractured horizontal wells under...

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