SPE 159687

Productivity Index and Inflow performance of Hydraulically Fractured Formations S. Al Rbeawi, SPE, D. Tiab, SPE, University of Oklahoma

Copyright 2012, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 8-10 October 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 been reviewed 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, its officers, 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 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 SPE copyright.

Abstract

Productivity index and inflow performance of horizontal wells intersecting multiple hydraulic fractures are of great importance. This importance comes from the fact that the fracturing process has become a common stimulation technique in the petroleum industry. However, few models for the productivity index and inflow performance have been presented in the literatures due to the complexity governing this topic.

This paper introduces a new technique for estimating the pseudo-steady state productivity index of horizontal wells intersecting multiple hydraulic fractures. Based on the instantaneous source solutions for the diffusivity equation, seven analytical models have been derived for different source solutions. Four of them represent the effect of the formation height and fracture height (the vertical direction), while the other three represent the solution for the horizontal plane. For vertical hydraulic fractures, the four solutions of the vertical direction, representing the pseudo-skin factor, are almost neglected. The three horizontal plane solutions are the main parameters that control the productivity index and inflow performance of the fractured formations. In this technique, the horizontal wells are acting in finite reservoirs where the pseudo-steady state flow is expected to develop. Reservoir geometry, reservoir properties, and fracture dimensions were considered in this technique. The number of fractures and the spacing between them were also investigated in this study. A new analytical model for estimating the required number of hydraulic fractures has been introduced in this study based on the reservoir drainage area and the surface area of fractures.

The models have been used to establish several plots to estimate the shape factor group based on the number of fractures and the half fracture length. This group is one of the main terms in the productivity index model. Several plots for the shape factor of fractured formations have been introduced in this study. The results obtained from the new technique have been compared with the results from previous models. Several numerical examples will be included in the paper. Introduction

Hydraulic fracturing is an important stimulation technique that has been widely used in conventional and unconventional oil and gas reservoirs all over the world. The technique involves the creation of a fracture or fracture system in porous medium to 1) increase the contact area between reservoir matrix and the wellbore, 2) overcome wellbore damage, 3) improve oil and gas productivity in low permeability and tight gas reservoirs, 4) enhance the connectivity of naturally fractured reservoirs, and 5) facilitate the production from shale gas reservoirs. During the last two decades, horizontal wells with multi-stage hydraulic fractures have become a common applied completion technology in the petroleum industry. Because of the large reservoir contact area connected to the wellbore, hydraulic fractures can greatly improve a well’s productivity.

The productivity index is a critical parameter in the oil and gas production process and its management. Regardless of the type of formation and the type of wellbore, the index is defined as the amount or volume of reservoir fluids that can be produced daily by one psi pressure drop at the sand face. For a horizontal well with multiple hydraulic fractures, the productivity index is influenced by several factors such as the number of fracture, the spacing between them and the fracture dimensions. Reservoir permeability and reservoir fluid properties have great influence on the productivity index as well as the geometry of the drainage area. Several models have been introduced during the last two decades for the productivity index of fractured formations. The high cost of the fracturing process and the serious need for a technique that can help in evaluating the benefits of the continuously increasing number of fractures are the two motivating factors to develop the productivity index models for fractured formations.

Raghavan and Joshi (1993) presented a method to evaluate the productivity of wells with hydraulic fractures. A general

2 SPE 159687

and rigorous model was provided in their study for the productivity index as a function of reservoir variables and numbers of radials of fractures. Guo and Evans (1993) proposed a new analytical model for the productivity index of fractured formations. They stated that the transient flow period can’t be used to estimate the productivity index since the drainage area is variable during oil and gas production. Therefore the dynamic productivity index based on the uniform fracture flow rate distribution would be used for this purpose. Their most important finding was the negligible impact of the fracture half length on the productivity index during the pseudo-steady state. Li et al (1996) developed a model to predict the performance of fractured formations and examined it for two case studies. They investigated the effects of the number of fractures and fracture half length on the productivity of the wells.

Valko and Economides (1998) suggested the use “proppant number,” proportional to the fracture permeability and propped volume and conversely proportional to the reservoir permeability and reservoir volume. This number can be used to estimate the productivity index corresponding to the fracture conductivity in pseudo-steady state flow. Larsen (1998, 2001) introduced several analytical models for the productivity of fractured and non-fractured deviated wells in commingled reservoirs. Fokker et al (2003) presented a novel approach to determine the productivity of complex wells. They stated that their model is applicable for the finite-conductivity wells, well interference, non-homogenous reservoirs, and hydraulically fractured formations. Sadrpanah et al (2006) studied the impact of the fracturing process on the deliverability of a well. Their study has been conducted on a pilot plant where two fracture systems and four fractures were used to enhance the production.

The performance of the horizontal wells associated with multiple hydraulic fractures depends on the penetration ratio in two directions. The first one is the vertical direction where the height of fracture does not need to be equal to the height of the formation. The second is the horizontal direction where the fracture wings are designed to propagate; however it is not common that the fracture tips reach the boundaries. Demarchos et al (2006) investigated the effect of the drainage shape and flow regimes in transversely fractured wells on the productivity. The concept of “proppant number,” developed by Valko and Economides, has been also used by Demarchoes et al. Guo et al (2006) presented a new model to predict fractured horizontal wells production. The model combined the shape of formed fractures and the fluid peculation mechanism together.

The productivity and the drainage area of fractured horizontal wells in tight gas reservoir and shale plays are of great importance in the petroleum industry. The importance comes from the high cost of both the completion techniques for these wells and the fracturing process that may include more than twenty stages. Medeiros et al (2008) discussed the performance of the fractured horizontal wells in heterogeneous and tight gas formations. They documented in their study the production characteristics and flow regimes in which the long transient periods may govern the productivity. Guo and Yu (2008) stated that the horizontal wells drilled in the direction of the minimum horizontal stress allow multiple transverse hydraulic fractures to enhance well productivity in low permeability oil and gas reservoirs. Two case studies have been investigated in their argument using three analytical models for three expected flow regimes. Grieser et al (2009) used a 3-D four phase nonisothermal multiwall black oil and pseudo compositional simulator that allows placement of multiple transverse fractures to predict production outcome from horizontal completions in the Barnett Shale. Zhang et al (2012) approached a method to predict the total inflow performance and associated productivity index under two-phase flow conditions. The approach can be used to determine the number of fracture stages required for a horizontal wells extending in a two-phase solution-gas drive. Mathematical Modeling

The hydraulic fracturing process aims to greatly increase the contact area between the wellbores and the pay zones. The production rate from fractured formations is a function of the total surface area of the hydraulic fractures. For a constant production rate, the pressure drop at any point in fractured formations depends on several parameters: permeability, homogeneity, isotropy, formation drainage area configuration, reservoir fluid properties, and fracture dimensions. Total production rate from all hydraulic fractures and total pressure drop at the wellbore are the two items required for estimating the productivity index. The worthy and successful fracturing process has to significantly improve the productivity index of the formation. Therefore, the productivity index is a critical parameter in oil and gas production from fractured formation because of the high cost of the single stage fracturing process. Generally, the simple model for the productivity index can be written as:

P

qJ (1)

For constant sandface production rate, the pressure drop is the main parameter that affects the productivity index. Pressure drop can be modeled using the instantaneous source solutions as:

t

fffmmmxyzfffmmm dhxhztzyxSc

qhxhztzyxP

0

),,,,,,,(),,,,,,,( (2)

In dimensionless form, the pressure drop of multi-stages hydraulically fractured finite formation can be written as (Al Rbeawi and Tiab 2012):

SPE 159687 3

Dl

wDwDfD

xfD

D

fD

DwDDDwDD

m N

fDD

t

n

DwDfDfDD

fD

fDfDD

dzlzlh

lh

l

lh

Nsyym

Nsyym

ym

N

xnxn

xnxn

nx

yxP

D

12

22

1 0

222

0 1

222

)cos()cos()2

sin()exp(14

1

)))2

2(cos())

2

2((cos()

4exp(

11

)2

cos()2

cos()2

sin()4

exp(14

12

(3)

The above model represents the sum of the pressure drop due to transient period and the pressure drop due to pseudo-

steady state. It can be written as:

DaDiDDDDDDDDDDDfDfD

DAD PPdZYXZXYXZYXyx

tPD

022 (4)

Ac

kt

t

xDA (5)

ee yxA 4 (6)

In equation (4), DiP is the dimensionless pressure drop from initial reservoir pressure to average reservoir pressure that

represents the transient period. It can be written as:

DADi tP 2 (7)

DaP is the dimensionless pressure drop between average reservoir pressure and reservoir pressure at any point and any

time. This pressure drop represents the pseudo-steady state. It can be written as:

DDDDDDDDDDDDDfDfD

Da dZYXZYZXYXZYXyx

PD

02 (8)

The model given in equation (8) can be solved for long time pseudo-steady state. It can be written as follows:

pHFDa SCP (9)

HFC is defined as the shape factor group.

pS is defined as the pseudo-skin factor.

Shape Factor group for fractured formations

The dimensionless pressure of multi-stage fractured formations can be approximated for long time to the following model:

2

4ln

2

12

fADAD rC

AtP (10)

The term 2

4ln

2

1

fArC

A in equation (10) is the shape factor group because it contains the shape factor for fractured

formations )( AC . From equation (8), it can be recognized that seven instantaneous solutions exist. Four of these solutions

),,,( DDDDDDDD ZYXZYZXZ represent the effect of the vertical penetration of fractures. While the three solutions

),,( DDDD YXYX represent the effect of the reservoir drainage area in the horizontal plane where the shape factor of

fractured formations is determined by this area. Therefore, the shape factor group from equation (8) can be written as:

DDDDDfDfD

HF dYXYXyx

CD

02 (11)

For long time approximation (pseudo-steady state), the group can be defined as:

4 SPE 159687

1 1 122223

1 1222

1333

)2

cos()2

cos()2

sin(

)))2

2(cos())

2

2((cos(

)44

(

14

)))2

2(cos())

2

2((cos(

14

)2

cos()2

cos()2

sin(116

2

n m N DwDfD

DwDDDwDD

fDfDfD

N

DwDDDwDD

mfD

n

DwDfD

fD

fDfDHF

xnxn

xn

Nsyym

Nsyym

ymxnn

Nx

Nsyym

Nsyym

mNy

xnxn

xn

nx

yxC

(12)

The results of the shape factor group given in equation (12) are plotted in Fig.(1) and Fig.(2) for two half fracture length

)0.1( fDx and )5.0( fDx . These plots are established for different reservoir configurations and different numbers of

vertical hydraulic fractures. They are designed for fully penetrating ratio )0.1( fDh . The spacing between fractures depends

on the number of fractures that intersect the horizontal wellbore )2//( nyyss DeD .

It can be seen that the shape factor group has a linear relationship with )( fDy for almost all the values of fracture half

length when reservoir configuration is close to rectangular with extremely long length )5.0( fDy . The slope of the lines on

log-log plot is (-1). Another linear relationship can be seen between the shape factor group and the length of drainage area for

all half fracture length when the reservoir length is extremely short )5.0( fDy . The slope of the lines on log-log plot in this

case is (1).

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-02 1.0E-01 1.0E+00

Fra

ctu

re s

hap

e f

ac

tor

gro

up

, C

HF

yfD

xfD=1.0, hfD=1.0

Single fracture

Two fractures

Four fractures

Eight fractures

Sixteen fractures

Figure 1: Shape factor group for different vertical hydraulic fractures.

SPE 159687 5

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-02 1.0E-01 1.0E+00

Fra

ctu

re s

ha

pe

fac

tor

gro

up

, C

HF

yfD

xfD=0.5, hfD=1.0

Single fracture

Two fractures

Four fractures

Eight fractures

Sixteen fractures

Figure 2: Shape factor group for different vertical hydraulic fractures.

Number of fractures

Hydraulic fracturing is an effective technique to increase the productivity of formations. The number of fractures is among several parameters that have great influence on the deliverability of the fractured formations. The productivity index of these formations highly depends on the number of hydraulic fractures. Therefore, determining the required number of hydraulic fractures is of great importance because of the high cost of the fracturing process.

By the multi-regression analysis for the results of the shape factor groups that consider different reservoir configurations and different numbers of fractures with different fracture dimensions, the number of fractures can be calculated as:

fDfDHF ynx

C42.1

(13)

Equation (12) can be solved for real parameters:

yf

ee

khx

yxJBn 2

200 (14)

Shape factor

Shape factor for fractured formation can be calculated from the shape factor group given in equation (10):

2

4ln

2

1

fAHF rC

AC (15)

In equation (15), the term )( fr represents the effective wellbore radius of the fractured formation. It has been found that

the estimation of this parameter is difficult for a wide range of configurations especially for reservoirs with high ratio between the length and width of the horizontal boundaries. However, the following simple model can be used to calculate the effective wellbore radius:

snxr f

f

42 (16)

The calculated value of the shape factor group given in equation (12) can be used to calculate the shape factor for

fractured formation using equation (15). The results for several vertical hydraulic fractures are shown in Fig.(3) and Fig.(4). It is easy to recognize that the shape factor value increases as the reservoir length parallel to the wellbore

decreases )( ey . The shape factor value for a small number of fractures is smaller than the shape factor value for a large

number of fractures. These two facts indicate the impact of the reservoir geometry on the value of shape factor for different

6 SPE 159687

numbers of fractures. For square shape reservoirs, the shape factor value is expected to be in its maximum value. Therefore, the productivity index for square configuration reservoir is the maximum productivity index.

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-02 1.0E-01 1.0E+00

CA

yfD

xfD=1.0, hfD=1.0

Single fracture

Two fractures

Four fractures

Eight fractures

Sixteen fractures

Figure 3: Shape factor for different vertical hydraulic fractures.

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-02 1.0E-01 1.0E+00

CA

yfD

xfD=0.5, hfD=1.0

Single fracture

Two fractures

Four fractures

Eight fractures

Sixteen fractures

Figure 4: Shape factor for different vertical hydraulic fractures.

Pseudo-Skin factor

The pseudo-skin factor for multiple hydraulic fractures can be found from the instantaneous solutions that include the vertical direction. It represents the effect of vertical penetration of fractures on pressure behavior of fractured formations. Therefore, the pseudo-skin factor can be written as:

DDDDDDDDDfDfD

p dZYXZXZZYyx

SD

02 (17)

SPE 159687 7

In dimensionless form, the pseudo-skin factor is:

1 1 1 1

2

222224

1 1 1

2

222

3

1 1

2

2224

1323

)cos()cos()2

sin(

)))2

(cos())2

((cos(

)2

cos()2

cos()2

sin(

)44

(

116

)cos()cos()2

sin(

)))2

(cos())2

((cos(

)4

(

1

4

)cos()cos()2

sin()2

cos()2

cos()2

sin(

)4

(

116

)cos()cos()2

sin(14

2

n m l N

DwDfD

DwDDDwDD

DwDfD

xfD

fDfDfDfD

m l N

DwDfD

DwDDDwDD

xfD

fD

fD

l nDwD

fDDwDfD

xfD

fDfDfD

lDwD

fD

xfDfD

fDfDP

zlzlhl

Nsyym

Nsyym

xnxn

xn

h

lymxnnl

hNx

zlzlhl

Nsyym

Nsyym

h

lyml

Nh

zlzlhlxnx

nxn

h

lxnnl

hx

zlzlhl

lhh

yxS

(18)

The pseudo-skin factor in the above model represents the effect of the partial penetration in the vertical direction and the

location of the horizontal wellbore. For full penetration )1( fDh , the instantaneous source solution in the vertical direction

converges to zero when the wellbore extends at the mid-point of formation height. Therefore, the pseudo-skin factor for fully penetrating fractures is:

0.0pS (19)

1.0E-03

1.0E-02

1.0E-01

1.0E-02 1.0E-01 1.0E+00

Ps

eu

do

-skin

fa

cto

r, S

p

yfD

Four vertical fractures, xfD=1.0

hfD=0.75

hfD=0.5

hfD=0.25

Figure 5: Pseudo-skin factor for vertical penetration ratio.

8 SPE 159687

The partial penetrating ratio in the vertical direction may increase the value of the pseudo-skin factor rather than leave the value equals to zero. However, this value is small, so that it can be neglected. The effect of the partial penetration ratio in the vertical direction on the value of the pseudo-skin factor can be seen in Fig.(5). The pseudo-skin factor significantly

increases as the penetration ratio )( fDh decreases. Therefore, the productivity index is expected to be affected by the vertical

penetration ratio. It increases to the maximum value when the vertical penetration ratio is )1( fDh and decreases slightly

when the fractures do not fully penetrate the formation. The full penetration fractures provide the maximum surface area for the reservoir fluid to flow toward the wellbore.

Fig.(6) and Fig.(7) show there is no impact for the vertical penetration ratio on the shape factor group and the shape factor respectively. This fact can be explained as the shape factor group and the shape factor are considering reservoir drainage area configuration rather than fracture geometry.

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-02 1.0E-01 1.0E+00

Fra

ctu

re sh

ap

e

fac

tor

gro

up

, C

HF

yfD

Four vertical fractures, xfD=1.0

hfD=1.0, 0.75, 0.5, 0.25

Figure 6: Shape factor group for vertical penetration ratio.

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-02 1.0E-01 1.0E+00

CA

yfD

Four vertical fractures, xfD=1.0

hfD=1.0, 0.75, 0.5, 0.25

Figure 7: Shape factor for vertical penetration ratio.

SPE 159687 9

The asymetricity of the wellbore location with respect to the formation height may increase the value of the pseudo-skin

factor. The pseudo-skin factor for symmetrical wellbore )5.0( wDz has inconsiderable value so that )0.0( pS .

However, the pseudo-skin factor increases as the horizontal wellbore extends asymmetrically )5.0( wDz in the formation

as shown in Fig.(8). Therefore, it is reasonable to say that the maximum value of productivity index comes from the symmetrical wellbore rather than the asymmetric ones.

Fig.(9) and Fig.(10) show the impact of the asymetricity on the shape factor group and the shape factor respectively. It can be seen that the asymetricity has no influence on the shape factor group and the shape factor. This fact can be explained as the shape factor is a function of the reservoir drainage area that is not affected by the location of the wellbore.

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E-02 1.0E-01 1.0E+00

Ps

eu

do

-sk

in f

ac

tor,

S

p

yfD

Four vertical fractures, xfD=1.0, hfD=1.0

zwD=0.5

zwD=0.25 or 0.75

Figure 8: Pseudo-skin factor for asymetricity.

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-02 1.0E-01 1.0E+00

Fra

ctu

re s

ha

pe

fac

tor

gro

up

, C

HF

yfD

Four vertical fractures, xfD=1.0, hfD=1.0

zwD=0.75 or 0.25

zwD=0.5

Figure 9: Shape factor group for asymetricity.

10 SPE 159687

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-02 1.0E-01 1.0E+00

CA

yfD

Four vertical fractures, xfD=1.0, hfD=1.0

zwD=0.75, 0.5, 0.25

Figure 10: Shape factor for asymetricity.

Half Fracture Length

Fracture half length has two different impacts on the productivity index of fractured formations:

1- Formation with long Length )1.0( fDy : The productivity index is conversely proportional with the fracture half

length. When the fractures are fully penetrating the formation, i.e. fractures’ tips reach the horizontal boundary parallel to the wellbore (reservoir length), reservoir fluid will not be able to flow toward the inner fractures and the flow of fluid will only be to the outermost fractures.

2- Formation with long Length )1.0( fDy : The productivity index is proportional with the fracture half length since

the most flow of fluid will be from the drainage areas between fractures.

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-02 1.0E-01 1.0E+00

Pro

du

cti

vit

y i

nd

ex

, J

p

yfD

Two fractures, Sp=0.0, Sm=0.0

xfD=0.1 xfD=0.2

xfD=0.3 xfD=0.4

xfD=0.5 xfD=0.6

xfD=0.7 xfD=0.8

xfD=0.9 xfD=1.0

Figure 11: Pseudo-productivity index for different half fracture length.

Fig.(11) and Fig.(12) show the pseudo-productivity index for two and sixteen hydraulic fractures respectively. The

pseudo-skin factor can be defined as:

SPE 159687 11

mpHFp SSC

J1

(20)

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-02 1.0E-01 1.0E+00

Pro

du

cti

vity

in

de

x,

Jp

yfD

Sixteen vertical fractures, Sp=0.0, Sm=0.0

xfD=0.1 xfD=0.2

xfD=0.3 xfD=0.4

xfD=0.5 xfD=0.6

xfD=0.7 xfD=0.8

xfD=0.9 xfD=1.0

Figure 12: Pseudo-productivity index for different half fracture length.

Spacing between fractures

The spacing between fractures has a considerable effect in the productivity of the fractured formations. The uniform spacing between fractures along the length of the wellbore is determined by the number of fractures and the length of the reservoir. The reduction in the spacing between fractures causes increasing in the shape factor group, which means reduction in the productivity index. Fig.(13) shows the effect of the spacing between fractures in the fracture shape factor group for ten vertical fractures. Fig.(14) shows the shape factor for different spacing between ten vertical fractures.

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-02 1.0E-01 1.0E+00

Fra

ctu

re s

ha

pe

fac

tor

gro

up

, CH

F

yfD

Ten vertical fractures, xfD=0.5, hfD=1.0

SD=0.005

SD=0.01

SD=0.02

SD=0.04

SD=0.05

Figure 13: Shape factor group for different spacing between fractures.

12 SPE 159687

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-02 1.0E-01 1.0E+00

Fra

ctu

re s

ha

pe

fa

cto

r, C

A

yfD

Ten vertical fractures, xfD=0.5, hfD=1.0

SD=0.005

SD=0.01

SD=0.02

SD=0.04

sD=0.05

Figure 14: Shape factor for different spacing between fractures.

Productivity Index

The productivity index can be written as:

mpHFp SSC

CCJJ (21)

where

B

hkkC

yx

2.141 (22)

mS is the mechanical skin factor.

1.0E-02

1.0E-01

1.0E+00

1.0E-02 1.0E-01 1.0E+00

Pro

du

cti

vit

y in

de

x,

Jp

yfD

Two fractures, Sp=0.0, Sm=4.0

xfD=0.1 xfD=0.2

xfD=0.3 xfD=0.4

xfD=0.5 xfD=0.6

xfD=0.7 xfD=0.8

xfD=0.9 xfD=1.0

Figure 15: Mechanical skin factor for two vertical fractures.

SPE 159687 13

The mechanical skin factor negatively affects the productivity index. It increases the resistance of the formation to the flow of fluid from the drainage area close to the fractures. Fig.(15) and Fig. (16) show the effect of the mechanical skin factor on the pseudo-productivity index for two and sixteen hydraulic fractures.

1.0E-01

1.0E+00

1.0E-02 1.0E-01 1.0E+00

Pro

du

cti

vity

in

de

x,

Jp

yfD

Sixteen vertical fractures, Sp=0.0, Sm=4.0

xfD=0.1 xfD=0.2

xfD=0.3 xfD=0.4

xfD=0.5 xfD=0.6

xfD=0.7 xfD=0.8

xfD=0.9 xfD=1.0

Figure 16: Mechanical skin factor for sixteen vertical fractures.

Application

Table-1 includes the required data of the formation, reservoir fluid properties, and the fracture dimensions. The measured production rate of the well was 128 STB/D for pressure drawdown of 1323 psi (Li et al 1996 and Guo and Yu 2008).

Table-1: Formation, fluid and fractures data.

Pay zone thickness 39 ft Reservoir permeability 7.5 md Average reservoir pressure 2602 psi Oil viscosity 4.8 cp Horizontal wellbore length 1820 ft Wellbore radius 0.19 ft Skin factor 0 Fracture length Fracture No. 1 246 ft Fracture No. 2 246 ft Fracture No. 3 246 ft Fracture No. 4 246 ft Estimated fracture width 0.9 in Fracture permeability 30000 md Drainage Area 16 acres Bottom hole flowing pressure 1279 psi Measured flow rate 128 STB/D Formation volume factor 1.084 res-bbl/STB

135.0fDy

64.0fDx

2.0Ds

25.0wDD yy

14 SPE 159687

0.0wDD xx

5.0wDD zz

135.0xfDh

0.1fDh

Fig.(17) shows the plot of CHF that has been generated for the above conditions. It can be found that:

1.4HFC

Using equation (22): 398.0C

Since the formation is fully penetrated by the fractures )0.0( pS and the mechanical skin factor is found to

be )0.0( mS :

Using equation (21):

097.0J STB/D/psi

The measured productivity index is (0.0967). Therefore, the error percent is less than (0.5%).To check the number of fractures, equation (15) can be used:

4n fractures

The calculated number of fractures is identical with the existed fractures in the field.

Figure 17: Shape factor group for the case study.

Conclusions

1- Shape factor for fractured formations can be calculated from the pseudo-steady state pressure response when the effect of the pressure pulse reaches the boundaries.

2- The productivity index for fractured formation can also be calculated from the pseudo-steady state pressure model using long time approximation.

3- An analytical model for the number of the required hydraulic fractures has been proposed. 4- A shape factor group has been developed in this study for the shape factor and productivity index calculations.

SPE 159687 15

5- The pseudo-skin factor for fractured formations can be neglected since it has inconsiderable value. 6- The asymmetricity - the location of the wellbore with respect to the formation height - and the vertical penetration

ratio have no impact on the shape factor of fractured formations. However both of them affect the value of the pseudo-skin factor.

7- The productivity index increases as the number of fractures and the spacing between them increases. Nomenclatures B formation volume factor, res-bbl/STB CA shape factor, dimensionless CHF shape factor group, dimensionless h formation height, ft hf fracture height, ft ct total compressibility, psi-1

J productivity index, dimensionless Jp Pseudo-productivity index, dimensionless kx permeability in the X-direction, md ky permeability in the Y-direction, md kz permeability in the Z-direction, md n number of fractures

P pressure difference, psi Q total flow rate, STB/D q fracture flow rate, STB/D rf equivellent wellbore radius s spacing between fractures, ft Sp pseudo-skin factor, dimensionless Sm mechanical skin factor, dimensionless t time, hrs xe reservoir half width, ft xf fracture half length, ft ye reservoir half length, ft Greek Symbols

porosity

viscosity, cp

diffusivity

References Al Rbeawi, S. and Tiab, D., 2012. Effect of penetrating Ratio on Pressure Behavior of Horizontal Wells with Multiple-

Inclined Hydraulic Fractures. SPE 153788 presented at the SPE Western Regional Meeting held in Bakersfield, California, USA, 21-23 March, doi:10.2118/153788-PA.

Demarchos, A., Porcu, M., and Economides, M., 2006. Transversely Multi-Fractured Wells: A Recipe for Success. SPE 102263 presented at the 2006 SPE Russian Oil and Gas Technical Conference and Exhibition held in Moscow, Russia, 3-6 October. doi: 10.2118/102263-MS.

Fokker, P., Verga, F., Egberts and P., 2005. New Semi analytical Technique to Determine Horizontal Well Productivity Index in Fractured Reservoirs. SPE Reservoir Evaluation & Engineering, 123-131, SPE 84597, doi:102118/84597-PA.

Grieser, B., Shelley, B. and Soliman, M., 2009. Predicting Production Outcome From Multi-Stage, Horizontal Barnett Completions. SPE 120271 presented at the 2009 SPE Production and Operation Symposium held in Oklahoma City, Oklahoma, USA, 4-8 April. doi:10.2118/120271-MS.

Guo, B. and Yu, X., 2008. A Simple and Accurate Mathematical Model for Predicting Productivity of Multifractured Horizontal Wells. SPE 114452 presented at the CIPC/SPE Gas Technology Symposium 2008 Joint Conference held in Calgary, Canada, 16-19 June. doi: 10.2118/114452-MS.

Guo, J., Zeng, F. and Xu, Y., 2006. A New Model to Predict Fractured Horizontal Well Production. Paper 20006-076 presented at the 57th Annual Technical Meeting of Canadian Petroleum Society held in Calgary, Alberta, Canada, 13-15 June. doi: 10.2118/2006-076.

Guo, G., Evans, R., 1993. Pressure-Transient Behavior and Inflow Performance of Horizontal Wells Intersecting Discrete Fractures. SPE 26446 presented at the 66th SPE Annual Technical Conference and Exhibition held in Houston, Texas, USA, 3-8 October. doi: 10.2118/26446-MS.

Larsen, L., 1998. Productivity of Fractured and Nonfractured Deviated Wells in Commingled Layered Reservoir. SPE Journal, 191-1998, SPE 38912. doi:10.2118/38912-PA.

16 SPE 159687

Larsen, L., 2001. General Productivity Model for Wells in Homogenous and Layered Reservoirs. SPE 71613 presented at the 2001 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, USA, 30 September-3 October. doi:10.2118/71613-MS.

Li, H., Jia, Z. and Wei, Z., 1996. A new Method to predict Performance of Fractured Horizontal Wells. SPE 37051 presented at the 1996 SPE International Conference on Horizontal Well Technology held in Calgary, Canada, 18-20 November. doi:10.2118/37051-MS.

Medeiros, F., Ozkan, E. and Kazemi, H., 2008. Productivity and Drainage Area of Fractured Horizontal Wells in Tight Gas Reservoirs. SPE Reservoir Evaluation & Engineering, 902-911, SPE 108110. doi:10.2118/108110-PA.

Raghavan, R., Joshi, S., 1993. Productivity of Multiple Drain holes or Fractured Horizontal Wells. SPE Formation Evaluation, 11- 16, SPE 21263, 3 October-2 November. doi: 10.2118/21263-PA.

Sadrpanah, H., Charles, T. and Fulton, J., 2006. Explicit Simulation of Multiple Hydraulic Fractures in Horizontal Wells. SPE 99575 presented at the SPE Annual Conference and Exhibition held in Vienna, Austria, 12-15 June. doi:10.2118/99575-MS.

Valko, P. and Economides, M., 1998. Heavy Crude Production from Shallow Formations: Long Horizontal Wells Versus Horizontal Fractures. SPE 50421 presented at the SPE International Conference on Horizontal Wells Technology held in Calgary, Alberta, Canada, 1-4 November, doi: 10.2118/50421-MS.

Zhang, H., Han, G., Houeto, F., Lessard, R., Wang, W. and Li, J., 2012. New Methods to Predict Inflow Performance of Multiply Fractured Horizontal Wells under Two-Phase Condition and Optimize Number of Fracture Stages. SPE 152837 presented at the North Africa Technical Conference and Exhibition held in Cairo, Egypt, 20-22 February, doi: 10.2118/152837-MS.

Appendix - Models Derivation

Consider a horizontal well penetrates several vertical transverse hydraulic fractures in a finite reservoir as shown in Fig.

A-1. Each fracture is considered as a single plane of length ( fx2 ), width ( w ), and height ( fh ). The spacing between

fractures is (s). If we assume that all fluid withdrawal will be through the fractures, the unsteady state pressure drop created

by production from fractures at any point in the reservoir ),,( mmm zyx is: t

fffmmmxyzfffmmm dhxhztzyxSc

qhxhztzyxP

0

),,,,,,,(),,,,,,,( (A-1)

where;

),,,,,,( hxhtzyxS ffmmmxyz is the instantaneous source function that can be determined as follows:

tzStyStxShxhztzyxS zyxfffmmmxyz ,),(),(),,,,,,,( (A-2)

q is the fluid withdrawal per unit fracture surface area per unit time can be determined as follows:

ff hnx

Qq

2 (A-3)

Figure A-1: Horizontal well intersected by multiple hydraulic fractures.

),( txSx can be written as follows:

SPE 159687 17

12

22

)2

cos()2

cos()2

sin()4

exp(14

1),(n ee

w

e

f

e

x

f

e

e

f

x

xn

x

xn

x

xn

x

tn

nx

x

x

xtxS (A-4)

),( tySy can be written as follows:

1 02

22

))2

2cos()

2

2)(cos(

4exp(

11

2),(

m N e

f

e

f

e

y

e y

Nsyy

y

Nsyy

y

tm

Ny

NtyS (A-5)

),( tzSz can be written as follows:

1

2

22

)cos()cos()2

sin()exp(14

1),(l

ffz

f

f

h

zl

h

zl

h

hl

h

tl

lh

h

h

htzS (A-6)

Substituting Eqs. (A-4), (A-5), and (A-6) in Eq. (A-2) first and then substitute Eqs. (A-2) and (A-3) in Eq. (A-1) gives

the pressure drop in dimensionless form as:

Dl

wDwDfD

xfD

D

fD

DwDDDwDD

m N

fDD

t

n

DwDfDfDD

fD

fDfDD

dzlzlh

lh

l

lh

Nsyym

Nsyym

ym

N

xnxn

xnxn

nx

yxP

D

12

22

1 0

222

0 1

222

)cos()cos()2

sin()exp(14

1

)))2

2(cos())

2

2((cos()

4exp(

11

)2

cos()2

cos()2

sin()4

exp(14

12

(A-7)

where:

eD x

xx (A-8)

e

ffD x

xx (A-9)

e

wwD x

xx (A-10)

e

D y

yy (A-11)

x

y

e

ffD k

k

y

xy (A-12)

e

fwD y

yy (A-13)

z

x

fxfD k

k

x

hh (A-14)

h

hh f

fD (A-15)

h

zz w

wD (A-16)

h

zzD (A-17)

18 SPE 159687

22f

x

ft

xD x

t

xc

tkt , where

t

xx c

k (A-18)

q

hhxtzyxPhkkP

ffmmmyx

D

),,,,,,(2 (A-19)

e

D ys

s (A-20)

To solve the above model given in Eq. (A-7):

DaDiDDDDDDDDDDDfDfD

DfDfD

D PPdZYXZXYXZYXyx

dyx

PDD

00 22 (A-18)

In this equation, DiP is the dimensionless pressure drop from initial reservoir pressure to average reservoir pressure. It

can be given as:

DfDfD

Di tyx

P2

(A-19)

while DaP is the dimensionless pressure droop between average reservoir pressure and reservoir pressure at any point and any

time. It is given by:

DDDDDDDDDDDfDfD

Da dZYXZXYXZYXyx

PD

02 (A-20)

where:

Dn

DwDfDfDD

fDD d

xnxn

xnxn

nxX

1

222

)2

cos()2

cos()2

sin()4

exp(14

(A-21)

1 0

222

)))2

2(cos())

2

2()(cos(

4exp(

1

mD

DwDD

N

DwDDfDDD d

Nsyym

Nsyym

ym

NY (A-22)

Dl

DwDfD

xfD

D

fDD dzlzl

hl

h

l

lhZ

12

22

)cos()cos()2

sin()exp(14

(A-23)

1 1 1

222

222

)2

cos()2

cos()2

sin()4

exp(1

)))2

2(cos())

2

2()(cos(

4exp(

4

m n ND

DwDfDfDD

DwDDDwDDfDD

fDDD d

xnxn

xnxn

n

Nsyym

Nsyym

ym

NxYX

(A-24)

1 12

22

222

2

)cos()cos()2

sin()exp(1

)2

cos()2

cos()2

sin()4

exp(1

16

l nD

DwDfD

xfD

D

DwDfDfDD

fDfDDD d

zlzlhl

h

l

l

xnxn

xnxn

n

hxZX (A-25)

1 1 12

22

222

)cos()cos()2

sin()exp(1

)))2

2(cos())

2

2()(cos(

4exp(

4

l m ND

DwDfD

xfD

D

DwDDDwDDfDD

fDDD d

zlzlhl

h

l

l

Nsyym

Nsyym

ym

NhZY

(A-26)

SPE 159687 19

1 1 1 1

2

22

222

222

2

)cos()cos()2

sin()exp(1

)))2

2(cos())

2

2()(cos(

4exp(

)2

cos()2

cos()2

sin()4

exp(1

16

n m l ND

DwDfD

xfD

D

DwDDDwDDfDD

DwDfDfDD

fDfDDDD d

zlzlhl

h

l

l

Nsyym

Nsyym

ym

xnxn

xnxn

n

hNxZYX

(A-27) Integrating and approximating (for long time):

1333

)2

cos()2

cos()2

sin(116

n

DwDfD

fDD

xnxn

xn

nxX (A-28)

)))2

2(cos())

2

2((cos(

14

1222

DwDDDwDD

mfDD

Nsyym

Nsyym

mNyY (A-29)

133

2

)cos()cos()2

sin(14

lDwD

fD

fD

xfDD zlzl

hl

lh

hZ (A-30)

1 1 122223

)2

cos()2

cos()2

sin(

)))2

2(cos())

2

2((cos(

)(

14

n m N DwDfD

DwDDDwDD

fDfDfDDD

xnxn

xn

Nsyym

Nsyym

ymxnnNxYX (A-31)

1 1

2

2224)cos()cos()

2sin()

2cos()

2cos()

2sin(

)4

(

116

l nDwD

fDDwDfD

xfD

fDfDfDDD zlzl

hlxnxn

xn

h

lxnnl

hxZX

(A-32)

1 1 1

2

222

3

)cos()cos()2

sin(

)))2

2(cos())

2

2((cos(

)4

(

1

4

m l N

DwDfD

DwDDDwDD

xfD

fD

fDDD

zlzlhl

Nsyym

Nsyym

h

lyml

NhZY (A-33)

1 1 1 1

2

222224

)cos()cos()2

sin(

)))2

2(cos())

2

2((cos(

)2

cos()2

cos()2

sin(

)44

(

116

n m l N

DwDfD

DwDDDwDD

DwDfD

xfD

fDfDfDfDDDD

zlzlhl

Nsyym

Nsyym

xnxn

xn

h

lymxnnl

hNxZYX

(A-34)