SPE 10179 SPE Society of Petroleum Engineers of AIME

Transient Pressure Analysis: Finite Conductivity Fracture Case Versus Damaged Fracture Case

by Heber Cinco·Ley* and Fernando Samaniego-V., * PEMEX and UNAM

*Member SPE-AIME

@Copyright 1981, Society of Petroleum Engineers of AIME

This paper was presented at the 56th Annual Fall Technical Conference and Exhibition of the Society of Petrole~m Eng.ineers of AIME, held in San Antonio, Texas, October 5·7, 1981. The material is subject to correction by the author. PermisSion to copy IS restricted to an abstract of not more than 300 words. Write: 6200 N. Central Expressway, Dallas, Texas 75206.

ABSTRACT

This work presents the basic pressure behavior differences between a finite conduc tivity fracture and different types of damaged fractures.

Two kinds of fracture damage conditions are studied: a) a damaged zone around the fracture, and b) a damaged zone within the fracture in the vicinity of the wellbore. The first case is caused by the fracturing fluid loss in the formation and the last case 1S originated by crushing, embedding or loss of propant within the fracture in the vicin ity of the wellbore.

This paper emphasizes that although nite fracture conductivity and fracture damage condition are both flow restrictions, their effectson transient pressure behavior are quite different at early time.

Type curves and both linear flow and bilinear flow graphs can be used to identify different cases when applied properly.

INTRODUCTION

Evaluation of hydraulic fracturing thrQugh transient p~essure analysis has become a common practice today. Initially, the main objective of the application of pressure analysis in fractured wells was to estimate the formation flow parameters and fracture extension. 1-7 These techniques considered an infinity conductivity vertical fracture and inv01ved a trial and error procedure unless prefac information was available. To avoid these limitations Gringarten et a1 8 presented the type curve analysis method which allows the identifi tion of different flow regimes and the est tion of both formation permeability and fra~ ture half length.

References and illustrations at end of paper.

Recent)y,the analysis of pressure data for fractured wells has been directed to­wards the determination of both flow and geometric characteristics of a fracture 9 - 18 •

This has been possible because of the development of new solutions 9

- 18 which can sider a well intercepted by a finite conduc tivity vertical fracture.

Frequently, it is observed that the pressure behavior of a fractured well does not match the infinite conductivity vertical fracture solution; instead these cases exhibit an extra pressure drop caused by a flow restriction somewhere in the system. Several models have been proposed:

a) A damaged zone around the fracture 34 ,lQ

15,19,20,21 (fluid loss damage)

and b) a damaged region within fr ture 1n the vicinity of the wellbore ,18,

(choked fracture); both cases consider an infinite conductivity vertical fracture and are referred as damaged fractures.

The purpose of this work is to show and emphasize that although finite fracture con ductivity and fracture damage conditions are both flow restrictions their pressure tran sient behavior are quite different at ear time.

While the finite conductivity case exhibit the bilinear flow behavior, the fra~ ture damage case is characterized by an extra pressure drop caused by the damaged zone. These differences become evident when pressure data are plotted on a log-log graph.

PRESSURE BEHAVIOR OF FRACTURED WELLS

For the better application of the tran sient data analysis techniques, it is necessary to understand the basic flow equations that describe the flow towards hydraulically fractured wells.

2 PRESSURE TRANSIENT ANALYSIS; FINITE CQNDUCTIVITY FRACTURE CASE VERSUS DAMAGED FRACTURE CASE SPE 10179

Let us consider a well intercepted by a fully penetrating vertical fracture as shown on Fig. 1. The well produces at constant flow rate a slightly compressible fluid from a homogeneous isotropic reservoir whose properties are independent of pressure. Three cases are presented: a) a finite conduc tivity fracture (Fig. 2), b) an infinite con­ductivity fracture with fluid loss damage -(Fig. 5) and c) an infinite conductivity choked fracture (Fig. 8).

FINITE

It has been shown 9,11 that when the dimen

sionless fracture conductivity (kfbf)Dis -greater than 300 the pressure drop along the fracture is negligible; thus the infinite conductivity vertical fracture solution 8 can represent these cases, for practical purposes.

The pressure transient behavior for a well intercepted by a finite conductivity frac ture, (Fig. 2), whose half length, width-and permeability are xf, bf and kf' respectively, is gived byl7;

... (1)

where PwD' (kfbf)D and tDxf represent the dimensionless pressure, dimensionless fr ture conductivity and the dimensionless t respectively. These variables are defined as follows:

Oil

Gas

and

khl1m(p) a q T

g

, .. (2)

Figure 3 shows a log-log graph of PwD versus tD where solutions are presented

xf

for different values of (kfbf)D' Cinco and Samaniego l7 have shown that these cases may exhibit three flow perios: a) bilinear flow period, b) linear flow period and c) pseud~

radial flow period.

The bilinear flow behavior is character ized by one fourth slope straight line as -shown on Fig. 3. The pressure behavior for this case is given byl7:

, .. (3)

That is, the dimensionless pressure (pressure drop) is directly proportional to the fourth root of dimensionless time (real time). This flow period generally «kfbf)D~~ ends when fracture tips affect the wellbore pressure; that occurs when:

... (4)

period exists when is characterized by a

one half slope straight line in a log-log graph, the beginning and the end of this period occur when:

:::: 100 tDblf (k f b f )D2

... (5)

and

tDelf

:::: 0.016 ... (6)

respectively. The pressure behavior for this case is given by:

.. , (7) f

That is, the dimensionless pressure (pressure drop) is directly proportional to the square root of dimensionless time (real

The pressure drop along the fracture s negligible for this case and the flux

distribution is uniform.

The pseudo-radial flow is exhibited by all cases re~ardless the value of fracture conductivity. The pressure behavior can be calculated by:

... (8)

this equation indicates that the dimension less pressure (pressure drop) is proportional to the logarithm of dimensionless time (real time). This type of flow is characterized by both a constant pressure drop and a st lized flux distribution along the fracture.

Figure 4 presents the stabilized flux distribution for different values of dimensio less fracture conductivity. Notice, as cated by Cinco et a1 9 , that for a highly con

SPE 10179 HEBER CINCO-LEY AND FERNANDO ~AMANIEGO V. 3

ductive fracture most of the fluid enters the fracture in the region near the tips; on the otherhand, as conductivity decreases the flow entering the fracture in the vicinity of tl;e wellbore becomes steadily more important.

"Choked Fracture". In this section we consider a vertical fracture with two regions; one of them has a reduced permeability and the remaining part possesses an infinite conductivity (Fig. 5). A "chocked fracture" is originated when propant within the fra~ ture is crushed, embedded or lost in the vicinity of the wellbore. The effect of a "choked fracture" on steady flow, constant pressure transient flow rate and unsteady pressure behavior has been studied by Smith 22 ,

Bennet et al 18 and Narasimhan 15, respectively.

If the zone of reduced permeability has a small length compared to the fracture length, the transient pressure behavior of this system can be expressed as:

+ (Sfs) ch .,. (9)

where p DetD ' (k b )D = ~) is the infinite H x

f f f"

conductivity vertical fracture solution and (S) is a skin factor representing the

fs ch dimensionless extra pressure drop caused by the damaged zone.

By considering steady-state flow in the d zone, the skin factor can be defined as follows:

... (10) is fs

where xs' b fs and s represent the length,

the width and the permeability of the damaged zone, respectively.

zone

Oil:

The pressure drop due to the damaged is given by:

, .. (11)

Figure

versus tD xf

6 is a log-log graph of PwD for a "choked fracture" for

different values of (Sfs)ch' At early time,

the curves show an almost horizontal portion, and all of the cases approach the infinite conductivity vertical fracture solution from

above.

It is obvious from Eq. 9 that if the skin factor is substracted, the pressure behavior of a "choked fracture" is identical to the infinite conductivity solution; hence, a "choked fracture" exhibits linear, transi tion and pseudoradial flow periods, discussed by Gringarten et ale for the infinite conduc tivity undamaged fracture case. Also, the stabilized flux distribution for this case is given by the corresponding to infinite con ductivity fracture. Figure 7 shows that in -the flux distribution for this case, most of the fluid enters the fracture in the tip region, As previously mentioned, the foregcing discussion is valid when is very small compared to x

f'

In some cases, loss of fluid during a fracturing operation into the formation occurs,reducing the permeability in the vicinity of the fracture impairing, as a consequence the well productivity; this case is referred as "fluid loss damaged fracture" Figure 8 shows this type of system for an infinite conductivity fracture with a damaged zone of uniform width b

s and permeability k

s'

The effect of this type of fracture damage on well behavior has been studied by van Poollen 19

, Prats 20, Raghavan 21 , Cinco and

Samaniego 10, Narashimhan 15

, Wattenbarger and Ramey3, Ramey23 and Millheim and Cichowicz 4 •

Cinco and Samaniego 10 showed that the pressure behavior for an infinite conductiviry fluid loss damaged fracture may be expressed as:

f(t D ,Sf) x

f s

... (12)

where Sfs is a skin damage factor defined by:

'11 b - 1) ... (13)

Figure 9 presents a log-log graph of versus tD for this situation. It has PwD x

f been shown 10

, that this case exhibits three flow periods: a) a linear flow, b) a tran tion flow and c) a pseudoradial flow.

The solution for the linear flow period is given by:

... (14)

The end of the linear flow is indicated by the dashed line on Fig. 9. Notice that the larger the fluid loss damage skin factor the longer the linear flow period.

4 PRESSURE TRANSIENT ANALYSIS: FINITE CONDUCTIVITY FRACTURE CASE VERSUS DAMAGED FRACTURE CASE SPE 10179

For the transition and the pseudoradial flow periods 9 :

00)

... (15)

where 6PUs is the additional dimensionless

pressure drop caused by the skin damage. When the pseudo-radial flow prevails the extra pressure drop becomes a function of Sfs only.

Figure 10 presents the stabilized flux distribution for different values of the fluid loss damage skin factor. The curves for small values of S approach the infinite conductivity fract case; however, as Sfs increase the flux distribution curves become more uniform approaching the uniform flux case.

Figure 11 shows a semilog graph of PwD - S versus t for Sf equal to 0.2 fs DX f s

and 1. This figure clearly indicates, that the curves for fluid loss damaged fractures fall in between the infinite conductivity solution and the uniform flux solution; in such a way that the curves become closer to the uniform flux solution as S increases.

fs

COMPARISON OF SOLUTIONS

It is evident from the previous section that there are significant differences when the solutions presented are compared. First of all, at early time, the finite conduct! vity solution exhibits the bilinear flow behavior while t~e_other twD cas:s show the linear flow cond1t10n. The solut10n for the fluid loss damaged fracture case is identical to the choked fracture solution when linear flow prevails in both cases.

However, at large values of time, when pseudoradial flow occurs, the behavior for all cases is given by a logarithmic function of time plus a term depending on Sfs' (Sfs)ch

or ( Figure 12 presents a graph

of the difference between dimensionless pressure for the three cases and the infinit conductivity solution versus the skin factors and the reciprocal of fracture conductivity, at the pseudoradial flow period. The curves for both fluid loss damage and choked fractures yield different values for extra pressure drop for a given value of Sf and (Sf ) l' sse 1

The pressure drop for the former is higher than the pressure drop for the latter

These differences can be explained by considering that the damaged zones are located in different p_artq of the system and by examining the flux distributions (Figs. 7 and 10). The flux distribution for a fluid loss damaged fracture is more uniform than the correspondin~ one for choked fractures. Gringarten et al showed that the uniform flux solution shows a higher pressure drop then the infinite conductivity case.

It should be emphasized that the skin factor (Sfs)ch is additive for any flow

period to the undamaged case when choked fractures are considered; however this is not case for fluid loss damaged fractures when transition and pseudoradial flow pr~ vail.

FRACTURE DAMAGE DETECTION

It is of prime importance for fracture design improvement to determine the charac teristics of fractures in situ, such as lenght, conductivity and nature of flow restrictions (damage).

Several techniques has been presented to estimate fracture and formation properti these method lude the linear flow graph (p versus , the bilinear flow graph17

wF

(pw versus v-i) and the type curve matching f

graph 8 ,9,11,17 (log 6p versus log t).

Identification of the cases discussed in this work can be achieved through the use of the different graphical techniques because there are important differences in pressure behavior exhibited at early time. Next,a discussion is presented on the application of the various methods of sis.

F 13 shows a squemat c gr for differ ent cases. The behavior for both the choked fracture and the fluiu loss damaged fracture cases is given by a single curve exhibiting a straight line portion at early time. This straight line has a slope m

lf and intercept

equal to (6p)s' The curve· for the finite conductivity fracture is concave downwards intercepting the origin.

The fracture half length for the two damaged cases can be estimated as indicated by Millheim and Cichowicz 4 and Clark', from the slope of the straight line; and the damage factor Sfs of (Sfs)ch is calculated by

using the following equations:

Oil:

Sfs or (Sfs) ch

Gas: Sfs or (Sfs) ch

kh(6p)

ex q a

... (16)

SPE 10179 HEBER CINCO-LEY AND FERNANDO SAMANIEGO V. 5

Notice that from a linear flow graph there is no way to differentiate between the choked fracture case and the fluid loss damaged case.

'::"':~-=-::::"':::-=::":=--'="'::::""::"''':':''''''-='='''::=..L.:'::- • Fig u r e 1 4 s how s a versus ~ for difer

ent cases. can be observed that, for the finite conductivity case, the pressure curve exhibits a straight line portion at early time corresponding to the bilinear flow period. This straight line has a slope m

bf and passes through the origin when the flow restriction is only caused by the conducti~ ity of the fracture itself. The fracture conductivity kfbf can be estimated, as indicated by Cinco and Samaniego 17 , from the slope of the bilinear flow straight line.

Notice in Fig. 14, that the curve for both the fluid loss damage fracture and the choked fracture solutions is concave upwards and intercepts the L1p axis at (L1p) . At large values of time all cases behave id tically, as discussed in previous sections.

The type curve mat seen extensively used to analyze pressure data for fractured wells. For this method the pressure data are plotted as a function of log L1p versus log t as shown on Fig. 15, and then overlayed on different published type curves to estimate both formation and fracture parameters.

The log-log graph is an excellent tool for diagnosis, because there are chara£ teristic features for different situations. This become evident when Fig. 15 is analyzed.

In a log-log graph a finite conductivity fracture yields a one quarter slope straight line while an infinite conductivity undam aged fracture exhibits a one half slope -straight line corresponding to bilinear flow and linear flow, respectively. Both the choked fracture and the fluid loss damaged fracture show a flat almost horizontal curve in this type of graph; the pressure curves are concave upwards as shown on Fig. 15, 6 and 9.

There are several type curves published in literature for fractured wells~,lO,11 16, ,21 Some of the type curves present the uniqueness problem, that iS,curves for dif ferent cases have the same shape making difficult to erform a unique match; such is the case of F ures 3, 6 and 9. Gringarten et a1 24 discussed this problem and pointed out that a better analysis is possible when type curves emerge from or converge to a single line.

Cinco and Samaniego l7 presented two type curves for finite conductivity vertical fractures. Figure 16 show one of them where a graph of log PWD(kfbf)D versus log t

Dxf'

(k f bf

)D2 is presented. This type curve has

the advantage that both the bilinear and linear flows are represented by a single line and the curves for different values of (kfbf)D emerge from this line. These curves

are particularly useful when pressure data match part of the transition curve located in between the bilinear (one quarter slope) and the linear (one half slope) straight lines.

Figure 17 presents the second type curve for fractured wells; here a graph of log P

wD versus log t

Dr, is presen ted. t

Dr, is the

w w dimensionless time based on the effective wellbore radius of the fracture. These curves converge to a single line representing the pseudo-radial flow behavior. Figure 17 becomes particularly useful when part of the pressure data fallon the pseudo-radial flow period.

Figure 16 and 17 present the same information graphed on Fig. 3.

Figure 18 presents the relationship between (kfbf)D and the dimensionless effe£

tive wellbore radius for a finite conduct ity vertical fracture. This graph is us to estimate once both r~ and Ckfbf)D are

known.

Figures 16, 17 and 18 can be used to estimate k, kfb f , x f ' r~, (kfbf)D' the end

of the bilinear flow and the beginning of both the linear and pseudo radial flows, as illustrated by Cinco and Samaniego 17 ,

The use of Figs. 6 and 9 for choked and fluid loss damaged fractures leads to the uniqueness problems because in some regions curves for different cases have the same shape. This problem can be avoided if the curves are plotted on graphs similar to those presented in Figures 16 and 17.

in or

log

the Appendix that a graph log (PwD!(Sfs)ch) versus

tD /(S£) h yields a xf

s c

single line for both the flat portion of the curves and the half slope straight line, as presented on Fig. 19. Different curves for various values of Sfs or (Sfs)ch emerge from

a single line. This type curve is useful when pressure data match part or all of the transition curve between the flat portion and the one half slope straight line.

When pressure data fallon the ps radial flow period,Figures 6 and 9 do not provide a unique answer unless type curves are plotted as a function of log p D versus log t

Dr, (Fig. 20), where t

Dr, isw the

w w

6 PRESSURE TRANSIENT ANALYSIS: FINITE CONDUCTIVITY FRACTURE CASE VERSUS DAMAGED FRACTURE CASE SPE 10179

dimensionless time based on the effective wellbore radius. Figure 21 shows the relationship between the dimensionless ef fective wellbore radius (r~/xf) and the skin factor Sfs and (Sfs)ch'

Application of Figures 19, 20 and 21 allows the estimation of Sfs' k, r~, x f ' and

both the end of linear flow and the beginning of the pseudoradial flow.

The pressure behavior differences between the three cases discussed in this work are without any question evident when Figure 16 is compared to Figure 19.

Notice that a comparison of Figure 17 to Figure 20 does not show any significant differences between the solutions at large values of time since all cases exhibit a pseudoradial flow.

The discussion on the three methods of analysis presented indicates that to have a reliable interpretation of pressure data it is necessary to examine the data on different type of graphs. At early time; the log-log graph provides an excellent tool to identify either the finite conductivity case or the damaged fracture cases. Since the solution for choked fracture is similar to the solution for fluid loss damaged case, it will be extremely difficult to differenti ate one case from the other If all of­the pressure data fallon the pseudoradial flow, there is no way to identify any case because all of them behave identically.

CONCLUSIONS

From the material presented in this work the following remarks can be made;

1.

2.

3.

4.

5.

There are significant differences in transient pressure behavior for damaged fractures and finite conductivity fra£ tures.

At early time, a finite conductivity fracture exhibits the bilinear flow, while the damaged fracture cases show the linear flow.

The differences between the damaged fracture behavior and finite conducti vity fracture case become evident when a log-log graph is used.

All type of graphs (bilinear, linear, logarithmic and log-log graphs) must be combined to identify different cases and calculate both fracture and formation parameters.

Suitable type curves should be used for the finite conductivity fractures to avoid the uniqueness problem.

6.

7 •

B

b s

c

New type curves are presented to analyze early time and long time pr sure data for both choked and fluid damaged fractures.

Identification of flow restriction conditions is not possible if all pressure data are taken on the ps radial flow period.

formation volume factor

fracture width

damage zone width

compressibility

h formation thickness

k permeability

kf

fracture permeability

m(p) ~ real gas pseudo-pressure

permeability of damaged zone

p pressure

qf fracture flux density

q well flow rate

t time

Sfs~ fracture damage skin factor

T reservoir temperature

x,y

x s

ch

ebf

f

fs

space coordinates

half fracture length

choked fracture length

viscosity

porosity

choked

end of bilinear flow

fracture

fracture skin

* Units are defined in Table 1.

oss

SPE 10179 HEBER CINCO-LEY AND FERNANDO SAMANIEGO V. 7

i initial

based on xf

S damaged zone

t total

w wellbore

The authors are rateful to the administration of Pet leas Mexicanos and the University of Mexico for providing the time and permission to present this paper.

1.

2.

3.

4.

5.

6.

7.

8.

9.

Russell, D.G. and Truitt, N.E.: "Tr sient Pressure Behavior in Vertical Fractured Reservoirs", (Oct. 1964); "

Raghavan, R., Cady, G.V. and Ramey, H. J. Jr.: "Well Test Analysis for Ver cally Fractured Wells", (Aug. 1972) ;-=--=...=..:~_

Wattenbarger, R.A. and Ramey, H.J., Jr.: "Well Test Interpretation of Vertically Fractured Gas Wells", (May. 1969).

Millheim, K.K. and Cichowicz, L.: "Testing and Analyzing Low-Permeability Fractured Gas Wells", . (Feb. 1968) .

Matthews, C.S. and Russell, D.G.: "Pres sure Buildup and Flow Test in Wells", Monograph Series, SPE of AIME, Dallas (967) •

Earlougher, R.C.: "Advances in Well Test Analysis", Monograph Series, SPE of AIME, Dallas, (1977)

Clark, K.: "Transient Pressure Testing of Fractured Water Injection Wells".

(June 1968).

Gringarten, A.C., Ramey. H.J. Jr. and Raghavan, R.: "Applied Pressure Analysis for Fractured Wells", _J __ .. ___ --:::-

(July 1975); Trans., A 9.

Cinco-Ley, H., Samaniego-V .• F. and Do­minguez, A., N.: "Transient Pressure Behavior for a Well with a Finite-Conduc tivity Vertical Fracture",

(Aug. 1978).

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

CincO-Ley, H. and Samaniego-V. F. "Effect of Wellbore Storage and Damage on the Transient Pressure Behavior of Vertically Fractured Wells", Paper SPE-6752 presented at the 52nd Annual Fall Technical Conference and Exhibition of the SPE of AIME. Denver. Colorado, Oct. 9-12, 1977.

Agarwal, R.G., Carter, R.D. and Pollock, C. B.: "Evaluation and Performance Predic tion of LOW-Permeability Gas Wells Stimulated hy Massive Hydraulic Fractur ing", P ., (March, 1979).-

Barker, B.J. and Ramey, H • .1., Jr.: "Transient Flow to Finite Conductivity Vertical Fractures", Paper SPE 7489 presented at the 53rd Annual Fall Techni cal Conference and Exhibition of SPE of AIME, Houston,Tex., Oct. 1-3,1978.

Scott, J.~. :"A New Method for Determining Flow Characteristics of Fractured Wells-Application to Gas Wells in Tight Formations", Paper presented at the American Gas Association Transmission Conference, Montreal, Canada, May. 8-10, 1978.

Ban:JYoJ)ihl\"/;lY . r. and Hanley, E.J.: "An Improved Pressure Transient Method for Evaluating Hydraulic Fracture Effectiveness in Low Permeability Reservoirs l1

, Paper presented at the Congreso Panamericano del Petr6leo~ Mexico City, March 19-23, 1979.

Narasimhan, T.N. and Palen, W.A.: "A Purely Numerical Approach for Fluid Flow to a Well Intercepting a Vertical Fracture", Paper SPE 7983 presented at the California

of SPE of AIME, Ventura, , 1979.

Guppy, K.H., Cinco-Ley, H., Ramey, H.J., Jr. and Samaniego V. F.: "Non-Darcy Flow in Gas Wells with Finite Conductivity Vertical Fract1.lres", Paper SPE 8281 presented at the 54th Annual Fall Technical Conference and Exhibition of SPE of AIME, Las Vegas, Nevada, Sep. 23-26,1979.

Cinco-Ley. H. find Samaniego V., F.: "Transient Pressure Analysis for Frnctured Wells", to be published in (Sep. 1981).

Bennett, C.O., Rosato, N.D., Reynolds, A.C., Jr. and Raghavan, R.: "Influence of Fracture Heterogeneity and Wing Length on the Response of Vertically Fractured Wells", Paper SPE 9886 presented at the SPE-DOE Low Permeability Sympos ll1m, Denver, Colorado, May. 27-29, 1981.

van Poollen, H.K.: vs Permeability Damage in Hydraulically Produced Fractures", Drill and Prod. API (1957) 103.

Prats, N.: "Effect of Vertical Fractures on Reservoir Behavior-Incompressible Fluid Case",

(June 1961)

8 PRESSURE TRANSIENT ANALYSIS: FINITE CONDUCTIVITY FRACTURE CASE VERSUS DAMAGED fRACTURE CASE SPE 10179

21. Raghavan, R.: "Some Practical Consideder ations in the Analysis of Pressure Data IT

,

(Oct. 1976).

22. Smith, J. E.: "Effect of Incomplete Fracture Fill Up at the Wellbore on Productivity Ratio", Paper SPE 4677 presented at the 48th Annual Fall Meeting of SPE of AIME, Las Vegas, Nevada, Sep.30 - Oct. 3, 1973.

23. Ramey, H.J., Jr.: "Short-Time Well Test Data Interpretation in the Presence of Skin Effect and Wellbore Storage", J.Pet. Tech. (Jan. 1970) 97 104.

24. Gringarten, A.C., Bourdet, D., Landel, P. A. and Kniazeff, W.: "A Comparison Between Different Skin and Wellbore Storage Type Curves for Early Time Tran sient Analysis", paper SPE 8205 presented at the 54th Annual Fall Technical Conference and Exhibition of SPE of AIME, Las Vegas, Nevada, Sep. 23-26, 1979.

DIMENSIONLESS VARIABLE GROUPS FOR DAMAGED

FRACTURE TYPE CURVES.

From Figures 6 and 9 it can be seen that curves for ~ifferent damaged fracture cases have the s~me shape at small values of tDx

f originating a uniqueness problem when a type curve match is performed. To avoid this problem this graph should be modified in such a way that curves having the same shape become a single line.

At early time, the pressure behavior for damaged fractures is given by:

+ (A-1 )

(A-2)

If both equations are divided by Sfs and

(Sfs)ch and terms are rearranged:

(A-3)

+ 1 (A-4)

Equations (A-3) and (A-4) indicate that graphs of PwD/Sfs versus /(Sfs)2 and

PwD/(Sfs)Ch versus tDxf

/(SfS)Ch 2 provide a

single curve for all values of Sfs and

TABLE 1.-

Parameter or Variable

k

k f

k s

h

qo

qg

II

B

¢

c t

P

m(p)

t

bf

b S

a 0

a g

B

£

T

SI PREFERRED UNITS, PRACTICAL UNITS AND UNIT

CONVERSION CONSTANTS USED IN THESE SYSTEMS

SI preferred units Practical units

wrn 2 rnd

Wm 2 rnd

Wm 2 md

m ft

m 3 / d STB/D

m 3 / d MSCF/D

Pa.s cp

m 3 / m3 RB/STB

fraction fraction

Pa- 1 pSi- 1

kPa psi

kPa 2 /Pa.s psi 2 /cp

h h

m ft

m ft

1842 141 . 2

1293 1424

3.6 x 10- 9 2.637 X 10- 4

Tra Tra

oK oR

IMPERMEABLE BOU N DARI ES

WELLBORE

FRACTURE

Fig. 1 - Vertical fracture in an infinite slab reservoir

k FRACTURE

.' . ' .. b'" '.' k'" ... . ,... ',... . f ' .. ' .... . f' .' ~ -'. : . : '." ..

I~

Fig. 2 - Finite conductivity vertical fracture

h

IO~--------~--------~----------~--------~----------'

- - - END OF BILINEAR FLOW

/

END OF LINEAR FLOW

I

Fig. 3 - Pressure behavior for a finite conductivity vertical fracture

II Q -d

o~ __ ~ __ ~ __ ~ __ ~ __ ~ o .2 .4 .6

_ X XD --xl

.8

Fig. 4 - Stabilized flux distributions for finite conductive fractures

DA.MAGED ZONE

k FRACTURE

Fig. 5 - Infinite conductivity choked vertical fracture

1o----------~--------~--------~----------~--------~----------

0.5

PWDI------~0~.2~-- I

10J END OF LINEAR FLOW

_3L-______ ~--------~--------~------~--------~------~ 10

10-5

Fig. 6 - Pressure behavior for a choked fracture

II

Q

<r;;1>

3

2

'0=_1-If

Fig. 7 - Stabilized flux distribution for choked fractures

10

DAMAGED ZONE WELL FRACTURE

Fig. 8 -Infinite conductivity vertical fracture with fluid loss damage

10

\

Srs /\ I

/ BEGINNING OF

PWD

THESEMILOG 0.2 STRAIGHT LINE

10-1

Fig. 9 - Pressure behavior for a fluid loss damaged fracture

J

2

00-

X

II

3

2

OL-__ -L ____ ~ __ _L ____ L_ __ ~

o .2 .4 .6 .8 x X --0- Xf

Fig. 10 - Stabilized flux distribution for a fluid loss damaged fracture

UNIFORM FLU

FLUID LOSS DA~!JAGE

Sf's I 0.2

" INFINITE

CONDUCTIVITY

/0

Fig. 11 - Comparison of fluid loss damaged fracture solutions and undamaged fracture solutions

o

/ /

,.,p/

-

/' r

h /­

/.f;' ,/.7":

"y

;t

// .~'/

.//

/;/ -- INFINITE CONDUCTIVITY o

/ /

/ /

/

/

/ /

/

// CHOKED FRACTURE {Sfs)ch

FINITE CONDUCTIVITY FRACTURE I (kf bf}O

-'-'- flUID LOSS DAMAGED FRACTURE, Sfs

_2 _I 10 10

(SfS)Ch' SfS 0 r (kf bf)~1 Fig, 12 Stabilized skin factors for finite conductivity and damaged fractures

INFINITE CONDUCTIVITY CHOKED FRACTURE, Sfs

FLUID LOSS DAMAGED FRACTURE, Sfs

END OF LINEAR FLOW

FINITE CONDUCTIVITY fRACTURE, (kfbf}D

Fig, 13 - Linear flow graph

0.... <J

END OF CHOKED LINEAR

II P FRACTURE FLOW

AND

END OF BILINEAR FLOW

FINITE CONDUCTIVITY

o

CHOKED FRACTURE

AND

FLUID LOSS DAMAGE

Fig. 14 - Bilinear flow graph

END OF 81 LI NEAR FLOW

~ L-------------~~--o ....J

FINITE CONDUCTIVITY

FRACTUR I NFINITE CONDUCTIVITY UNDAMAGED FRACTURE

LOG t

Fig. 15 - Log-log graph

p wD

1

'+-.0

- 2 ~ 1u

a 0.3:

_2 10

APPROXIMATE START OF SEMI~LOG STRAIGHT LI N E

Fig. 16 - Type curve for finite conductivity vertical fracture

--- End of Bilinear Flow

End of Linear Flow

t 1

Dr' w

Beginning of Semilog Straight Line

Fig. 17 - Type curve for finite conductivity vertical fracture

I i 8 7 6 r I 5

W .4, --Xf

10-1 1 9 8 7 6

5 .4,

jI

Z

10 3

PWD

Sfs 102

OR

P 10 WD

(Sf'S)Ch I

10 I

A 6 7 891 2 .

I I lI:N 'l'IY , , ':t I'

! I'

i IIHIIIIII

"

-:

! 1

I~TI I I i

II II I

I I

I I !

I' , :r

111:i1tH ! i 'i ,

If I I, I ;

,I'

I :~I~. I,

I

11

11, ,I 1'1,

!I' II I ,

I: "i, I I I ,I I! .

I I I

II I', 1'1

I I! I, I' I " ,

Fig. 18 - Effective wellbore radius for a finite conductivity vertical fracture

FLUID LOSS DAMAGE

---- CHOKED FRACTURE

o END OF LINEAR FLOW

Fig. 19 - Type curve for damaged vertical fractures

4 ~ 9

F "

k I II~ I '.,

I!I;I" I'"

I~;:'" I"'

~:I I~ [:::,

!'r< ~"'

, i II':]I:!: 'iIW!

."1,:;,

!!:.I'"

Fig. 20 - Type curve for damaged vertical fractures

Fig. 21 - Effective wellbore radius for damaged vertical fractures