Matthews_ c. s. and Russel_ d. g. - Pressure Buildup and Flow Tests in Wells

!

~

'¥'p)

2

,

.,

,

.,

,

.(2,4)

0 ok oY oY + -F--!. az oz

~.I. ,,'/'

,,(2.7)

If c is small, if the permeability is constant and isotropic, if the porosity is constant and if it is assumed that the pressure gradients involved are small so that th: gradient squared terms may be neglected, the foregOIngreduces to ~!!:!-~= ~~ ax2 + oy2 + OZ2 k

(28) .

ot .,

F

d ' I fl b" or ra Ia ow, com Ination 0f Eqs. 2,6 and 2. 5 yields (viscosity constant) ~~ +~~~+C 2

(~ )

( r~ )

r or

or

k.. or or

or

at

Eq. 2.4 represents a general form for the combination of the continuity equation and Darcy's law. The final differential equation which will result from this equa~on depends on the fluid and the equation of state of Interest. For the radial flow case we obtain in similar manner: -~

~-~-

) *

"

at

-o(pu..)/or

~(rpu..) = -~ (cpp). or ot

uX

) [ ( )

( )

~t

uX

OP 2 + k" ( ay + k,,~ OP ) 2 a2p + kz""'J"Z2 02P + C k" ax

=cpp.c-+J.I.-

.ot d(pUr)/dr~ and SInce

=~

.

If c is constant then the above relationship can be

0 + k z -Foz

[ - ( ) ] --~ ~ pu..dr rd pUr -A rdr ~ r

ap

~

integratedto yield

This reduces to

~ ox

OP)

-;-"aT

FLOW TESTS

I

~his IS S!~p!y a dIrect application of the cOntin.UItypnncIple. DIVIding the equation by dX dY dZ dt YIelds dx

( rpk..

AND

dY]

= i/>pdXdY dZ -i/>pdX dy dZ I ..An

- [~~

BUILDUP

cpp.c~ + ~ ~ = T ot k.. ct ' *To establishthis relationship we have made use of a op o.p -a-i" (.pp)= .pat + Pat op o.p = .ppCat+ Pat .

MATHEMATICAL

BASIS

FOR PRESSURE

ANALYSIS

METHODS

If we assume constant permeability and porosity, cond h ap 2. ' b' li stant and small compressl I ty, an t at -IS ar negligibly small, the above equation becomes

( )

1 0 --rr or This

(

op ) -o2p --+ or ar2

equation

is

one

of

1 op -I/>p.c ap r or

the

most

used

and

a fluid

of

small

and

d

=

-a'

an

h

were

b

th

0

th

'

e

porosl

ty

an

d

in

a2p

petroleum

constant

a2p

a2p -I/>p.

~+az+T2-T(c+Cf)a' x y

op

z

t

.(2.12)

If Eq. 2.11 is expressed for radial flow it becomes a2 1 a 1 ak a 2 a -& +r-!r+ ( c +Ta)( fr) = ~ (c + Cf)-ft.

com-

p

pressibility must be assumed to obtain this equation from the original nonlinear equation with which we began. The reader should keep these assumptions in mind since solutions to this particular equation form the foundation of pressure~~ ~hniques. --Gas Eq. 2.8 and Eq. 2,9 are called diffusivity equations ~~ ,.. and the constant,~, IS called the hydraulIc dlffu-

(2.13) 2,4 Single-Phase Gas Flow An important class of single-fluid flow equations is that describing flow of gas through a porous medium. flow equations are different than those for liquid flow in that the equations of state which are used are quite different in functional form from those for liquids. The equation of state for an ideal gas is given by the ideal gas law as m pV = MRT,

~

si~y~iStOrlCaTiy~-th1sequation first arose in the study of heat conduction. Lord Kelvin called a corresponding constant in the heat-conduction equation the thermal diffusivity. Equations similar to Eq. 2.8 also arise in the study of diffusion and electrical potential distribution. Equations of this general type are known as the diffusivity equation.

where V is the volume occupied by the mass m of gas of molecular weight M, R is the gas law constant and

!

If we wish to obtain the differential equation for flow of a fluid of small and constant compressibility,

T is the absolute temperature. Since the density, p = ~, in this case is

but for the case of pressure-dependentporosity and permeability, we can further refine Eq. 2.7. If we assume constant viscosity, isotropic permeability and neglect gravity, we obtain

V

.then ~.

[( ap a2p a2p a2p n + 32 + T2 + C a x y z x

-+

~ k

)

2

+

( op )

~

2

ay

( ap) 2 + -:e-- ] Z

[ ~ax ~ox + ~ay ~ay + ~oz ~az ] = ~k ,

~= ox

~~ cp

(and

similarly

,

gas viscosity and constant rock properties, and neglecting gravity, Eq, 2,4 becomes

.(2.10)

for

y

and

(

( ~

( ~)

3ax p ~ax) +~oy p oy) + 3cz p oz =~~. k at

z),

.,

(2,14)

ax

This equation can be rewritten as 01/> -~

02p2 c2p2 a::t2 + ~+

op

at-a-p-~' 02p F2-+n+~ x y

) (

a2p + z

[ CP] 2 ) -I/>p. + -a-z -T(C

1 ak c+Ta p

op + Cf) at

) ([ ap] a

"

x

2

02p2 21/>p. ap --aZ2 = k~"

(2,15)

r ap ] 2 + ay

In the case of radial flow Eq. 2,15 becomes 02p2 1 cp2 21/>p. op I/>p.op2 -+ --= --=~.(2.16) ar2 r or k ot kp ot Either of the two right-hand forms is often used. This

(2,11)

equation is nonlinear and has been solved mainly by numerical methods.

If we rearrange Eq. 2.10 it now becomes

( 02P

for isothermal variations in pressure, op -M ap ot RT ot . From kinetic theory, the viscosity of an ideal gas depends only upon temperature. Thus, for constant

and

x

M p = liT p ,

~ + !:-.-~. at k at

This equation can be simplified somewhat by noting that

~\,

I

.1 al/> I/> P permeability are pressure-dependent.In cases in which the gradient squared terms can be neglected, Eq. 2.11 Cf

can be reduced to

engineering-the equation for radial flow of a fluid of small and constant compressibility. It is quite important to not~ that small p~essure gradients, constant rock properties,

h

were

(29)

k at'..

often

7

8

PRESSURE BUILDUP AND FLOW TESTS IN WELLS

In the case of flow of a non-ideal gas, the gas deviation factor z is introduced into the equation of state to give p =:r

~. z

of gas liberated from a volume of oil to the oil volum~ (all referred to standard conditions) is the gas solubility factor Rs. Similarly, a gas solubility factor for water can be defined and representedby Rsw. The use of a formation volume factor to allow for the changesin volume which occur in each phase upon

If we assume laminar flow, neglect gravity and assume constant rock properties, then Eq. 2.4 becomes, for isothermal conditions,

transition from reservoir to standard surface conditions of temperature and pressureis a well known procedure. These volume factors are defined for each phase as

(~ ) ~ox(~ILZ~ox )+ ~oy(!!.-ILZ~oY) + ow ~ (~ILZ~oW) = ..!.k ~ot Z

B = oil and dissolvedgasvolume (reservoirconditions), 0 oil volume (standardconditions)

(2.17) In Eq. 2.17 we have used the symbol W for the Z co-

gasvolume (reservoirconditions) Bg = gasvolume (standardconditions)'

ordinate to avoid confusion with the gas deviation factor z.

Bto --. watervolume (standardconditions)

For radial flow Eq. 2.17 can be expressedas 1

-T r

C r

(p

aP

-r /LZ

-:ar

)

cf> '0 kat

=

(

P z

) ...(2.18)

A version of Eq. 2.18 in which higher-order terms are neglected can also be derived. This equation is ~

+ ~

or2

~

r

or

= ~ k

~ P

~ at

(~

,

-water anddissolvedgasvolume(reservoirconditions) In addition to these quantities, the concept of relative pe~meability m.ust be introduced. When th~ee immiscible 'flUIds (e.g., 011, gas and water) flow sImultaneously through a porous medium, the permeability of the rock

to each flowing phase depends on the interfacial tensions betweenthe fluids and the contact angles between the rock and the fluids. It has been found

)

..(2.19)

Z

Russe11 et a1.ave 4 h

commonly the rock

encountered conditions the to each phase is independent

that for

permeability of bulk

of fluid

19 shown that use 0f Eq. 2.as a ..nction substitute for the more rIgorous Eq. 2.18 can lead to serIous errors...permea m gas-well performance predictions for low-permeability gas reservoIrs.

properties and of flow rate (for laminar flow), and is a fu f h fl d h 0 t e UI saturations only. T e relative perb l ti t h h d fin d mea 11 es 0 eac p ase are e e as the ratIo 0f t he b1lit y t 0 a phase at preVaI .1mg saturation conditions to the single-phase permeability of the rock.

The equations for flow of a single fluid which are essentialto this Monograph have now been developed. In reality, of course, the pore space of a reservoir is occupied by more than one fluid, and any or all of these fluids may occur at saturation levels such that simultaneous flow will take place. It is essentialto an

Thus, for oil, gas and water,

. ... .

.

..

. .

krtO= kiD(So,Sto), k ko (So,SiD) kro = k '

understanding of pressure analysis methods that some basic facts about multiphase flow be developed. The brief section which follows is devoted to this.

.

.

k rg

= kg (So,SiD) k '

M I h FI 2.5 u tip ase ow A completely rigorous formulation of the equations

where

for multiphase flow should consider the spatial distribution of each component in the hydrocarbon-water systemas a function of time.5.18The approach which we take in this section will be much less rigorous. All hydrocarbon liquid which is present at atmospheric conditions, as obtained by differential vaporization, we refer to as oil. The gas phase we refer to simply as gas, without regard to its composition, and we consider the solubility of gas in the oil and water phases. Our derivation will be for radial flow only. At any instant an element of the reservoir will contain certain volumes of oil, gas and water which, when reduced to standard conditions, will be modified as a result of gas solubility in the oil and water and the compressibility of each phase. The ratio of the volume

It is beyond the scope of this Monograph to present a definitive discussionof two- or three-phaserelative permeability. For the purposes of our derivations, we shall consider simply that these are physically meaningful quantities which can be measured on a rock sample in the laboratory. Consider a unit volume of the reservoir. In this volume there is a mass of oil given by

S + S + S = 1 . 0 to g

cf> So ~Pos , 0 and a mass of water given by cf> SiD B;;; Ptos ,

MATHEMATICAL

BASIS

FOR

PR,ESSURE ANALYSIS

METHODS

wherepo.and pw.are oil and water densitiesat standard conditions. In the same reservoir unit there is a massof free gas S ~ PUB Bu and a massof dissolvedgas .l. R

'I'

P .U'

S

B

0

+

-loR 'I'.w,.

P

S

~

PU'

B

porous mediumunder conditionsof neglectof gravity forcesandcapillarypressuredifferencesbetweenphases. They representa simultaneousset of four nonlinear equationsdescribingfour unknowns,So,Su,Swand p. This complexsystemcan be solvedonly by numerical means. Martin6 has shown that in the casewhere higherorder

tD

B' 0 tD so that the total massof gas per unit volumeof reservoir is + f/>R. PU'SO + cJ>R-p,. StD B

9

..

terms

-ko po uro ---

.-!.-~ r or

neglected

(r

~

Bo op

+~~ ~

+

~

~

Bo op

-~~

BtD

fJpw ' pw.ar

expansion

of

) = !!:!!-+ ~~ = ~ or2 r or (-k )

~ or

ce =, -~

B0 po.a r '

ktD

the

the

p.

~ ot' e

where Ct is the total systemcompressibilitygiven by

and for water PtDUroo = -~

in

"",,(2.24)

opo

p.o

op

-~

~

BtD op

+ Cf'

(2.25)**

Bu op

and the quantity (klp.)t is the sum of the mobilities

For gas,

(kip.) of the fluids;i.e., k

--u Pu uru ---;;ij;

R k

op, .0 PU' ar -PUB B:- ~

t

f 0

ti' equa

f ons

II 0

1 0

[r

ko OP] -0 -at

( ~k

k

k

+ -!- + ~ ) ,11.0 p.u p.w

e

.(2.26)

~th a pressure-dep~ndent diffus~vitycoefficient..This Impo~ant fact proVl~esthe. baSISfor pressu:e.mt~rpretati~npro~e~uresm multiphasecases.This IS discussed

ows.

m

detail

m

later

chapters

of

the

Monograph.

For the sake of completeness, the simplified forms of the precedingequationsin the case of two-phase,

Oil: Tar

p.

Comparisonof Eqs. 2,24 and 2.9 showsthat under the assumed multiphase flow in aequation porous medium can conditions, be described by the diffusivity

If we neglect capillary pressuredifferences * in the systemand neglectgravity, then a continuity equation for eachphasecanbe written as in Eq. 2.3. se

( -=k )

apo --a;;-

-PUB~Bw ~ jJ.w~or '

e

be

B'

u 0 tD By use of Darcy's law we can expressthe radial massflux of oil as

Th

can

quantitiesin Eqs. 2.20, 2.21 and 2.22, theseequations canbe combinedmathematicallyto yield

(

~ar

So ) cJ>~.

(2.20)

Gas:

gas-oil flow are ~resented. The differential become the folloWIng.

equations

Oil:

(~+~+~ )~ ] -.!-.!.r or [ r p.oBo IJ.wBw jJ.,B, or [ ( R.So R.wStD Su)] =-cJ>-++ ot Bo Bw Bu 0

~~ r

(2 21)

or

-.!-.~ r or r ~+~ p.oBo

]

~ ~ [r~ ~ =~ r or IJ.wBtD or ot

( cJ>~BtD)

,

(2.22)

So+S,+Sw=1

, , , .,

, ..(2.23)

Eqs. 2.20 through 2.23 constitutethe equationsfor simultaneousflow of oil, gas and water through a *Capillary forces are not completelyneglectedbecause effectivepermeabilityterms are affectedby capillarity.

--

)~] =~ot [ (~+~ Bo

,II.,Bu or

cJ>

, , , , ., and

where

.(2.27)

Gas..

[ (

Water:

[ r ~p.oBo ~or ]=~ot ( cJ>~Bo ) ,

,

Bu

)]

(2.28)

So+ s, = 1 ,

This set of equationshas beenstudied extensivelyby Perrine,7Wellersand West et aV4 by means of numerical solutionsobtainedon digital computers.L **The term c, was added to Martin's equationsto ac-.the count for formation compressibility.

10

PRESSURE BUILDUP AND FLOW TESTS IN WELLS

2.6

Solutions for Radial Flow of Fluid of Small and Constant Compressibility Thus far in the presentation of the math:matical basis for pressure analysis methods, we have discussed the physical laws which govern fluid flow in a porous medium and the combination of these laws into differential equations which describe the various flow ti f Eq W h th regimes which may occur. It e excep on o. 2.8 and its form for radial flow, Eq. 2.9, all the equa-

.

.

..

Fluid of small and constant compressibility; Constant fluid viscosity; Small pressuregradients; and ... Negligible graVIty forces. Again, the equation is 02 1 0P .I. P 'f'JJ.COp ¥ + -3 = ~at . r r

tions which were developed are nonlinear and not easily solved. Eqs. 2.8 and 2.9, however, are linear and can be solved analyticany for boundary conditions of interest, as we shan see presently. Not only can they be solved, but application of these solutions to reservoir conditions has, over the years, demonstrated their practical value. Because of this utility and simp~icity, these equations have become the fundamental basIs for the commonly used pressure analysis techniques. For the development of the pressure analysis theories discussed in this Monograph, three basic solutions of Eq. 2.9 are needed. These are presented in the section which fonows. Others may be found in Carslaw and Jaeger9or in the paper by Rowan and Clegg.15 The assumptions made in the development of Eq. 2.9 are summarized as fonows: Radial flow into wen opened over entire thickness of formation; Homogeneous and isotropic porous medium; Uniform thickness of the medium; Porosity and permeability constant (independent of pressure);

Ghe solutions of this equation of interest to us in the developmentof pressure analysis methods are those for the case of flow into a centrally located wen at a constant volumetric rate of production, q) As win be mentioned later in this chapter, the basic solutions for constant rate can be combined by the principle of superposition to yield solutions for arbitrary rate histories. (Three basic casesare of interest: (1) Infinite Reservoir -the case in which the wen is assumed to be situated in a porous medium of infinite radial extent; (2) Bounded Cylindrical Reservoir-the case in which the wen is assumed to be located in the center of a cylindrical reservoir with no flow across the exterior boundary; and (3) Constant Pressure Outer Boundary -the case in which the wen is situated in the center of a cylindrical area with constant pressure along the outer boundary. The specific application of each of these caseswin become apparent in the later sections of this Monograph.) The geometry and boundary conditions for these three casesare indicated schematically on Fig. 2.2. To INFINITE

RESERVOIR

P- PiASrCONSTANT

PRESSURE

BOUNDARY P = p.

CASE

co

OUTER

CASE

AT r = r

Ie""

/

/'

0

'" BOUNDED

CIRCULAR

RESERVOIR

CASE

~or Ire

=0

"" -.J

I

I

re I-r

I

~

W

; =:j : I:=-

I

I I

Fig. 2.2 Schematicdrawingof geometryand boundaryconditionsfor radial flow, constant-ratecases.

MATHEMATICAL

BASIS

FOR PRESSURE

ANALYSIS

METHODS

11

expressthe condition for constant flow rate at the wellbore (i.e., at r = rw), we may write from Darcy's law ( r ~ .Thus, q = ~ p. or r~ Thus, if we require a constant rate at the well, then we

The symbol Y is Euler's constant and is equal to 1.78. 4kt for ~ > 100,

impose the following condition on the pressure gradient at the well: ( ~ ) = --.!!!!:-~.. (2.29)

or

)

(

qp. p( r ,)t = P. + -In 41Tkh

[ 1n~

p(r, t) = P. -41Tkhqp.

)

yf/Jp.cr2

4kt'

kt

+ 0.80907 ] .

or r~ 21Tkhrw (For no flow across an exterior boundary, r = re, we must have zero flow velocity; therefore, the pressure gradient must be zero.) OP -Pwf a -0 (2.30) r r. (In all caseswe require that at t = 0 (i.e., initially) ~h~.reservoi.r.isuniformly pressured.at a value Pi)~he ffiitial condition could also be specified as a function of radius from the well; however, for our purposes the

...(2.32) The expression for pressure at the wellbore (i.e., at r = rw) is

assumptionof initial uniform pressureis adequate. The mathematical statement of the boundary conditions and development of the mathematical solutions for each of these cases is presented in Appendix A. These solutions are, of course, quite well known and have been incorporated into this Appendix solely for the sake of completeness. As is usually the case, the exact form of the mathematical expressions for the solutions of the foregoing

The solution we have presented for the infinite reservoir case is an approximation to the actual finitewellbore infinite reservoir case, and is based on the assumption of a vanishingly small wellbore radius. However, when it is evaluated at practical values of radius and time (including normal wellbore radius values), it yields almost identical results with the lesstractable finite-wellbore solution. More information on this approximation can be found in Appendix A.

problems depends on the approach taken in the analytical treatment. In this regard, several slightly different

Bounded Circular Reservoir

( )

solutions of th~mproblems in w~ch we are interested have ap'peared the petroleum literature.. Rather than

qp. In = P. + 4:;;kh

(

yf/Jp.crw2 4kt'

)

or k

Pwf = P. -~

[ln~f/Jp.crw 41Tkh

+ 0.80907

].

(2.33)

p(r, t) = P. -2;khqp.

{ reD22-1 (4rD2 +

)

tDw -

attempting to present all of these solutions and an accompanying critique, we have chosen to utilize in each of the three casesthat solution most convenient to

reD2In rD (3reD'-4reD' In reD-2reD2-1) 00 ~-=T4(reD2 -1)2 + 1Tn=l

the needs of this Monograph. The reader who is interested in a variety of these solutions is referred to Muskat,lO van Everdingen and Hurst,S Homer,l1 or CarslawandJaeger.9 The mathematical solutions for each case are listed

e-a."tD~112(anreD)[11(an)Yo(anrD)-Y1(a,,)lo(anrD)] a,,[112(anreD)-112(an)] } (2.34) where

in the section of the text which follows. Infinite Reservoir, Line Source Well

I

qp. 21Tkh

p(r,t)=pi

1

.'t' 2 E, ( -

.I.

r rD = -,rw 2

p.C r 4 kt )

re reD= -tDw rw '

=

kt f/JpocrfD2 '

and the an values are the roots of

~

,

J1 (anreD)Y 1 (an) -J1

(an) Y 1 (anreD) = O.

(2.31) where

(2.35) For the pressure at the wellbore, Pwf, for the casewhere re > > rw, Eq. 2.34 can be written

-E,

. (-x)

For x < 0.01,

=

00

f e-U udu.

Pwf = P. -2;kh"qp.

z

( 1) -E.

(-

x) ~ -y -In ( x) = In

1nreD-4 3

e-a."tD. J 2 ( ar ~ 2 2 1 2"'eD) } .(2.36) n=l an [J1 (anreD)-J1 (an)] The an values in Eq. 2.36 take on monotonically

+ 2

--05772 x'. ~

~

00

{~+2tow

12

PRESSURE BUILDUP AND FLOW TESTS IN WELLS

increasing values as n increases; i.e., a1 < a2 < as "". Thus, for a given value of tDw,the exponentials decrease monotonically (e-at'tD~ > e-a22tD~ > ). Also, the Bessel Function portion of the terms of the seriesbecomesless as n increases.Thus, as tDwbecomes large, the terms for large n become progressivelysmaller. For tDwsufficiently large, all the terms of the summation become negligibly small. Prior to this tDwvalue, however, there is a period of time in which all terms but the first in the summation can be neglected. This period will be referred to later in the Monograph as the "late transient" period. Thus, for sufficiently large tvw, the summation term in the solution approaches zero and Eq. 2.36 becomes 2t [~+ 11' reD

PtD/= Pi -~

In reD-4]

3

of the limited drainage area manifests itself. Until the time of boundary interference in the solutions the pressure behaviors for both casesare identical. This may be seen from the behavior of the reD = 10 curve which breaks away from infinite reservoir behavior at tvw= 16. The curves for reD= 6 and reD= 8 had broken away earlier. A comparison of the dimensionlesspressure drop for an infinite reservoir with that for the constant-pressure outer-boundary case is shown on Fig. 2.4. Here, again, the solutions are identical until the effect of the boundary is felt. Since in this casea constant pressureis being maintained at the outer boundary, the flow system reaches an equilibrium condition (steady state) and the pressure at the well becomes constant. This is in contrast to the bounded reservoir solution. In that case no fluid enters the flow system and the pressure in the

.(2.37)

A form of Eq. 2.36 which is convenient for use in pressure buildup analysis and determination of average reservoir pressure is obtained by adding and subtracting a term In( ycf>p,crw2 /4kt) to obtain 2 Pw/ = Pi +~[ln~Y (t)], .(2.36a) 11' t

well and throughout the reservoir declines with time as a result of the depletion of mass from the system. Note in Fig. 2.4 that the larger-size reservoirs follow infinite reservoir behavior for a longer time than the smaller ones. Further, the steady-statevalue of Pi -Pw/ is proportional to log reD'The specific pressure behavior during the various flow regimes will be discussedin great-

where

er detail in the section which follows.

ycf>lJ.Crw272+ 4tvw 2 (In reD -4)3 Y(t) = In 4~+ eD 00 e-a.8tD~112(a r D) + 4 ~ 2[1 2 ( ) 1 "2e ( )] n=l

a"

1

a"reD

1

2.7 Concepts of Transient, Semi-Steady State and Steady-State Flow Behavior .If

we

consider

a

hypothetical

example

in

which

the

an

2.30 Constant Pressure Outer Boundary

In this case we present only the solution for the pressure behavior at the well. This expressionis Pw/ = Pi --

2 kh qp.

{ In reD-2

11'

2.20

00 ~ n=l

e-P.'tD~102(,8"reD) ,8,,2[112(,8,,) -102 (,8"reD)]

} ,.

where

re reD = ~'

(2.38)

2.1

~

a. I ::Ll.c ~

kt tvw = cf>p.CrtD2 ,

ci- C" C\J II

and ,8" is a root of 11(,8,,) YO(,8"reD)-Y

~ OE ,ill:'!, ({;?(Q{'t :~.. j, a ~{'r'..4'"'

~.".,..;

Pressure Buildup Analysis

,"!,"'.'\"~~,'"

i'.\~;'~;;;!~':j':'~~::~ ...'

.e... i~'r

3.1 Basic Method In the previous chapter we developed the basic equations for describing the pressure behavior in an oil reservoir. In this chapter we will show how these are applied in analyzing pressure buildup curves. We begin with the "line source" solution (Eq. 2.31) for one well in an infinite reservoir. This equation indicates that after a well has produced at rate q for time t, the bottom-hole flowing well pressure P"'f will be given by 2 ( -~ ) , (3.1) Pwf = P. + ~Ei 41Tkh 4kt

well pressure after shut-in and Prof to designate the pressure during the production period before closing in. Eq. 3.4, which was presented by Homer1 in 1951, will be our basic equation for pressure buildup analysis. As discussedin the previous chapter, it is a solution for an infinite, homogeneous, one-well reservoir containing a fluid of small and constant compressibility. As might be expected, the equation applies quite well without modification to newly completed wells in oil reservoirs above the bubble point. Modifications necessary for application of this equation to other cases will be discussedlater in this chapter.

which at times of interest reduces to

When we express Eq. 3.4 in practical oilfield units of psi, BID, cp, md and ft, it becomes**

-q}l. Pwf -P.

(

+ _4 kh 1T

so that the pressure

In

)

yCP}l.Crro2

4k

* ,..

t

(3.2) Pro. = P. -162.6-log

drop is

( Ycf>,ucrw2)

= (pressure drop caused by rate q for time t + ~t) + (pressure drop caused by rate change -q for time ~t);

or P. -Pw.

-4;;jiJi --q,u1

2

(

)

2

) ( ycf>p.crw 4k~t

+ qp.1 ycf>p.crw n 4k(t + ~t) 4;kJi n

(3.3) and

(

(I+

kh

P. ---q,u Pwf -4;k"h 1n 4kt' ..(t If ",:,enow ~lose m our w~ll for a time ~t, after ~roducmg for ti~e ~, we obtam the. ~ressu.redrop ~t time ~t by the pnncIple of superposItion discussed m Section 2.8, as P. -Pw.

q,uB

)

~t

) ..(3.4a)

~I

This equatio~ tells us tha~ if we plot the pressure Pw-; o~~,:a:Y!!~~os~perioa vs the iOgarithni ~f + ~t) I ~t, we should obtain a straight line. Pig. 3.1 shows a plot of data from a new well in an oil iservoir. As may be seen, the theory and practice ~gree very well in this case. T 0 ana1yze the curve m PIg. 3 .,1 note th at the

. .

absolute value of the slope of the curve m is equal to the coefficient of the logarithm term in Eq. 3.4a. Therefore* * * kh =

162.6qp.B

(3.5)

m

Extrapolation of the straight-line section to an infinite shut-in time, [(t + ~t)1 ~t] ~~y~s a pressure we will call p* throughout this Monograph. In this case

Pro. = P. -4kii qp. 1n. I + ~I .(3.4). 1T ~I In these equations we have used Pw. to designatethe

**See Section2.10 for derivation of the factor 162.6; this factor is also discussedin the Nomenclature.

*Throughout this Monograph, "In" will refer to the natur.al logarithm, while "log" will refer to the base 10 logarIthm.

slope should be used in this equation. This is true in all uses of the slope throughout the Monograph. Note that the slope is also given by m = 2.303 qp/4trkh, as may be seen f1lom Eq. 3.4.

***Only the magnitude (not the + or -sign)

of the

PRESSURE BUILDUP ANALYSIS

19

ii* = pi,the-itiiti-alpressure. Determination of kh and p* in this manner forms two of the basic steps in pressure buildup analysis. The quantity p* is the pressure which would be obtained at infinite shut-in time. In the case of one well in an infinite reservoir, p* is also the initial reservoir pressure. In finite reservoirs and even in infinite reservoirs containing more than one well, p* is less than the original pressure after some depletion occurs. The difference between Pi and p* is a reflection of this depletion. As will be discussedlater, p* is approximately equal to, but usually slightly greater than, the average pressure in the drainage area around the well. Note that for values of ~t small compared with t (the usual case during a.buil~up), a plot of Pro.vs log ~t should also be a straight line, as may be seen from Eq. 3.4a. The slope of the curve will be the same (though reversed in sign) whether Pro. is plotted vs log ~t or log [(t+~t)/ ~t]. However; the plot orplO, vs ~!E=~o~ be ex_tra~la!ed to p *in a ~im~ manner so that it is usually s~. over-~11,tn.p1at Pro. vs 102 r CL-!:~bt;)-I-M.To account for the fact that the production rate of a well may vary considerably over its life, one should, theoretically, use the principle of superposition discussed in Section 2.8 to approximate the true rate history (see also, Section 3.8). However, an acceptable approXimation, as discussedbelow, is to take the rate q as the last rate before closing in and to compute the flowing time from t =

.een cumulative well production smce completion production

wellbore damage. Additional pressure buildups will usually be made to obtain values for kh and wellbore damageafter a well is completed and "cleans up". Thus, the drillstem test values usually need only be approximate. Nisle8 has shown that if the production time subsequentto a short term shut-in is at least 10 times the duration of the shut-in, the error in kh arising from use of Eq. 3.6 will be less than 10 percent. For all these reasons, Eq. 3.6 will be used throughout this Monograph. For interpreting short produ~tion tests and for obtaining accurate kh values from drillstem tests, the Odeh and Seli~- method should be used. A method similar to this has also been suggestedby Trebin and Shcherbakov.41 3.2 Skin Effect, Skin Factor, and Flow Efficiency Skin Effect In many cases it has been found that the permeability of the formation near the wellbore is reduced as a result of drilling and completion practices. Invasion by drilling fluids, dispersion of clays, presence of a mud cake and of cement, presence of a high gas saturation around the wellbore, partial well penetration, limited perforation, and plugging of perforations are some of the factors responsible for this reduction in permeability. Since the effect is close to the well, transients caused by it are of small duration and may be neglected. Hence, the effect of a reduction in permeability near the well can be taken into account as an additional pressure drop ~p proportional to the rate of production q. The zone of reduced permeability has b

rate just before closing in

ca 11ed a " s ki n "4 .an 5

d th e resu lti. ng e ff e ct a " skin

effect".

6)

Skin F'attor

Another approximation for t and q has been discussed bY~_~d._S~_:!~a~oximation-I~a better ~e for o~~~ng kh from short production tests ana-drlllstem tests. Even for these cases, the approximation of Eq. 3.6 leads to correct extrapolated pressures and to reasonably accurate values for kh and

... .Quantitatively, after van Ev~rdingen,4we define the skin ~actor as. a constan~ s w~ch relates the pressure drop m the skin to the dimensionlessrate of flow.

(

)

qp. ~P.kln = S 2:;;:kji""

(3.7)

1320

Here-s is called the skin IaC"fOf.After introducing Eq. 3.7 into Eq. 3.2, we find for the well pressure after a

130

production time t

-~ ~1280

PIO!= Pi + ~ qp.

:)

[ In ( rc/>JiCrIO2 4kt ) -2s ]

.(3.8)

~ 1260 ~

This flowing pressure Prof is lower by an amount sqp./21Tkhthan the pressure in the absence of a skin

~1240 m :: 1220 ~ 120

(see Eq. 3.2). The skin effect is illustrated in Fig. 3.2 (from Hurst5). In the idealized case shown there, the pressure should rise by an amount ~P.kln immediately after shutin. In practice, the order of magnitude of the skin effect can be estimated from the difference between the pressure before shut-in and that shortly after.

118 100

10 It+ 6t) /6t

Fig. 3.1 Pressurebuildup in a nearly ideal reservoir.

I

To calculate the skin factor, it is necessaryto measure -

20

PRESSURE

the well pressure both before and after closing in. By co~bini~g ~q. 3.8 which .give~ the pressure before cloSIng m, WIth Eq. 3.4 which gives the pressure after

PIC' -PIc! --qp. -4;kh"

,"«

SK 0 ZON DA

]

(3.9)

m ;", ,~,

m

(-~

)+

JLCrw2

FROM

(3.10)

STATIC PRESSURE

""

skin

ACROSS

SKIN

HURST5

Fig. 3.2 Pressuredistributionin a reservoirwith a skin.

In this equation we have replaced the factor qp./41Tkh by its equivalent based on Eq. 3.4, m/2.303. The pressure Pw! is that measured before closing in; the pressure PI hr is obtained from the straight-line portion of the pressure buildup curve 1 hour after closing in. The italicized statement is most important. If the pressure buildup curve is not straight at 1 hour, it is necessary to extrapolate the curve backward as shown on I

IN WELLS

"~;; f1 1m!: II JJ !i~~ FLOWING PRESSURE

]

3.23

FLOW TESTS

»illji PRESSURE IN !~ FORMATION !: ::; 11A P ..PRESSURE DROP 1;

For ~t small compared with t, we can approximate (t + ~t)/t as 1. Rearranging this equation, choosing ~t = 1 hour so that P",. = PI hr, and introducing practical oilfield units, we get for s

s = 1.151[PI hr -PIc! -log

AND

WELL BORE ,;.,.(;, W *1

closing in, we find

[ In ( 'YJLCrw2 (t + ~t» ) 4kt(~t) -2s

BUILDUP

6tt

Fig. 3.3. This is necessary because Eq. 3.4 is only applicable to the straight-line portion of the curve. Usually, at early times, the curve deviates from a straight line becauseof flow into the wellbore after the well is closed in at the surface. The basic theory does not take this into account. To compensatefor this well fillup effect, it is necessaryto extrapolate the straightline portion of the curve backward to early times. hours

10

100

~ ~ .

4600

r"I

4400

4200

4000

3

3

34

100 I

(t+6t)/6t

Fig. 3.3 Pressurebuildup showing effect of wellbore damageand afterproduction. --

~' PRESSURE BUILDUP ANALYSIS fJ -t.i:? IP

21

It would have been possible to choose any time besides 1 hour in developing Eq. 3.10. T~ wo~erel'y ~ choos~

~t ch~~e_the == 10 !!o~s~v!!!~e the of constant the cQns~~ would

(~~-==19.H.-lQl

nr2-21.-

other so

~.23.~ become

The

V

factor

8 by6

( ~-

3 = Thus,

) In 2.

1

k.

(3.11)

the permeability

in the skin is greater

if if

than that in the

formation, as from fracturing or acidizing, 8 -will be negative, Hydraulically fractured wells often show values of 8 ranging from -3 to -5. te that even I f k d k t t No , 8 an rw are nown, I IS no possible to obtain both the radius of the skin and its P ermea

b i li t y

from

Eq

3

11

0

...ne

t

.

ff

ti

may

ge

lIb

d aroun

di

thi

1n!!r ' w

=

1n!-!r

riD' = If 8 is positive,

s

+ 8

Flow

from

Eq.

.,

3.7 which,

...(3.7a)

of about

2,0 may be obtained

may reach 5.0 after a fracture

Pressure

e

ffi

buildup

...

B,

treatment.

Calculation

calculations

clency. are convemently

for

kh,

summanze~

Example

.

radius riD' is small-

..

1).

8 and

flow

on a form

sheet

are

the

Calculations

h b ld pressure UI up curve sown m FIg. ... reservoIr IS above the bubble pomt.

3.3,

for

where

or p* total

= 4,585

psig.

compressibility

for Example

1-is-obtained

laboratory

measurements

(see also TrubeIO).

pressibilities are of the order of 10 X 10-6 psi-l. compressibility of non-gas saturated water varies

If 8 is negative,

the effective

fr~2

than r w. For example,

8 values

of

-3

and

-5

corre-

spond to effective well radii of 5 and 37 ft, respectively, for rw = 3 in. This effective wellbore radius concept is especially useful in discussing results of hydraulic fracturing. FI A better

relative

ow

Eff " Ic/ency

index

than

3 X 10-6 psi-l is usually

skin effect for deciding

.

= ~

quired to obtain

J

'P. Ii! p"..,etuc/ttti(t( -7'-C"-u4

-q actual -.

33

-Pwf

Rock

compressi-

th elr . .m enter

only into a logva I ues IS not re-

.

accurate values for skin and

B d dR . Gun e eservolrs

Thus far in this chapter, we have presented tibns for only one well in an infinite reservoir. approximations

for bounded

equaThese reser-

voirs if production time is not long but become poor with additional production.'In this section we will discuss modifications of previously presented theory to enable

p*

satisfactory.

reasonably

are good

.;;

and use of a value of

damage.

-equations ~;W-~ncy .Jldw Smce

The. only

bilities may be obtained from Hallll (Fig. G.5). They vary from 3 X 10-6 to 10 X 10-6 psi-!. The compressibility of gas-saturated water varies from 15 X 10-6 psi-l at 1,000 psi to 5 X 10-6 psi-l at 5,000 psi (see Ramey12). Since compressibilities an. thmi c t erm, high accuracy

upon the efficiency with which a well has been drilled andd completed fin d this providedf by a "flow efficiency". This IS e e as e ratio 0 actual productiVIty mdex of a well to its productivity index if there were no skin (8 = 0).

.

X 10-6 to 4 X 10-6 psi-l

form from

Oil com-

must theoretically travel through addito give the required pressure drop). is larger

the

To obtain p* in Fig. 3.3, we must extrapolate from P = 4,445 psig (at right ordinate) two cycles to the right at a slope of 70 psi/cycle. Thus, p* = 4,445 +

er than r,o (fluids tiona1 formation

radius

after

of moderately high formations, the flow

using the equation shown at the bottom of the sheet. It is best to obtain the oil compressibility

wellbore

A

its

usage

solution

in bounded

for

reservoirs,

the pressure

behavior

of a well

in

and J Ideal =

q p*

-Pwf

~

we obtain

~P.kln

'

*Strictly speaking, one should use p, the average pres~ure, rather than p* in this _equation. However, since p* IS a good approximation for p, and since this quantity oc-

curs in both. numerator

caused by USIng p*.~ff~

v'F;;:;

I-tp.£

..-

--

Example

The

wellbore

*"

(3.12)

qp./41Tkh, is

efficiencies

efficiency

2(70)

w riDe-B.

the effective

IS obtained

3(;;;)1. ':~

(Appendix

"

difficulty by defin mg ." an e ec ve we ore ra us , r ' W d fin thi di th t hi h k th w. e e e s ra us as a w c ma es e calculated pressure drop in an ideal reservoir equal to that in an actual reservoir with skin. Thus,

or

m = 2.303

-~P.kln -Pwf

hydraulic fracturing in formations permeability; in low-permeability

. .

.

~P.kin

Vlrnp~.87 I~DA'U

in the skin zone is less than

that in the rest of the formation, 8 will be positive; the permeabilities are equal, 8 will be zero. Finally,

-Pwf

-p*

-Ihe flow efficiency has also_been called the~ ~ctivity ratio, the condition ratio,7 and the com~~~ tacror.8 When subtracted from unity it gives the damage factor. 9 7J.,C'

rw

if the permeability

quantity

using

The radius r. of the "skin" zone around the well and the permeability k. in this zone are related to the skin

.-p* efficIency

Flow

and denominator,

little

error

is

22

PRESSURE BUILDUP AND FLOW TESTS IN WELLS

a bounded cylindrical reservoir was obtained in Chapter 2. Eq. 2.36a of that chapter gives the following relation for flowing pressure Prof. qp. P"" = P. +"4;kji""

)

[(In

Yc/>JLCr",2 4kt -Y(t)

] .(3.13)

13"

~j 12 PROBABLE

The factor (q/L/41Tkh) Y(t) may be thought of as a pressure drop additional to that in an infinite reservoir, caused by the fact that no fluid can flow in across the

p=1190psiQ II

outer cylindrical boundary. To obtain the shut-in pressure in this bounded reservoir, we superposethe pressure drop given by Eq. 3.13 at time (t + L;:.t) on the change at time L;:.t,obtaining

[ (t+

P",. = P. --In q/L 41Tkh

L;:.t

)

L;:.t

+ Y(t + L;:.t) -Y(L;:.t)

] .(3.14)

p* = P. --.!!!!:- Y(t). 41Tkh

...(3.15)

For a cylindrical, bounded reservoir, Y(t) is a positive function which increases with time. Thus, p* will be less than Pi and the difference will increase with increasing production time. If we substitute Eq. 3.15 into Eq. 3.13, we obtain -*

q/L

+ "4;kji""ln

.1015.!.i!!'

Qoo61 .. Fig. 3.4Observedpressurebuildupcurve In well In finite reservoir. If on Fig. 3.4 we extrapolate the straight-line portion

From Eq. 2.36a we find that for small L;:.t,Y(L;:.t) ~ 0* and Y(t+ L;:.t)~ Y(t). Then when Pro. is plotted vs In [(t+ L;:.t)/ L;:.t] and extrapolated to [(t+ L;:.t)/ L;:.t] = 1, we find the extrapolated value, p*, from Eq. 3.14 as

p",,-p

10

( Yc/>JLCr",2 ) 4kt'

.(3.13a)

of the buildup curve to ~nfinite closed-in .tim~, [(I + L;:.t)/ L;:.t] = 1, we obtain a value p* which IS

greater than the average pressure p, as shown. Usually a well will not be closed ~ long enough to obtain the flattening and to observe p. However, it is possible to estimate p from the extrapolated value of p*. This is done by using the Ei-function and other functions developed in Chapter 2 (see Ref. 13) to developequations for (p* -p) vs time for drainage areas of various shapes. For a circular drainage area, a graph of (p* -P>/(q/L/41Tkh) in oilfield units is given in Fig. 3.5 as a function of kt/c/>/LcA,also in practical oilfield units (see Nomenclature). The quantity A is the drainage area of the well; for one well in a bounded reservoir, it is the reservoir area. Values of p* -p for drainage areas of other shapes will be found in Chapter 4. Discussion of the use of graphs such as Fig. 3.5 toobtainpfromp*willbedeferredtoChapter4.

On comparing this equation for a bounded reservoir with Eq. 3.2 for an infinite reservoir, we see that p* has the same meaning in an equation written for a bounded reservoir as does Pi in an equation written for an infinite reservoir. This has an important corollary. An equation written for pressure behavior in an infinite reservoir may be immediately rewritten for the finite reservoir case by substituting p* for Pi. We will use this corollary later in the Monograph. Returning now to Eq. 3.14, we see that it differs from the case of one well in an infinite reservoir by the two Y(t) terms. On evaluating Y(t) from Eq. 2.36a, we find that the e~ect of the two Y(t) terms is to cause the pressure buIldup curve to bend over at large ..-e time, as shown by Fig. 3.4. The flattened curve ~ll ~pproach, asymptoti~ally, the average pressure p m this bounded reservoIr.

If there are other wells in a reservoir, the effect of predfJ(;tion at the other wells is to cause a well to be surrounded by a drainage boundary, as shown in Fig. 3.6. On one side of this boundary fluid flows toward that well, and on the other side toward another well. For some time after a well is closed in it can be treated as if its drainage boundary still exi~ts. Thus, a well surrounded by other wells will have a buildup curve qualitatively similar to that in Fig. 3.4. For very long closed-in times this is not true, as will be discussed later under Interference Tests. A more extended discussion of average well and reservoir pressuresis presented in Chapter 4. 3.4 Pressure Buildup For Two- or Three-Phase Flow B 1ow t he bubble pomt ' 0f the 01 .1 .m the reservoIr, . gas flow will begin. At this time the pressure buildup behavior is governed by the more complicated nonlin-

* T0 Sh ow th IS we must repIace th e term In [( Y"'I'C'F.")/ 4kt] in Y(t) by the equivalentEi-function.

ear differential equations given in Chapter 2; and since the. equations are nonlinear, strictly speaking, the foregomg methods cannot be used.

O

--

PRESSURE BUILDUP ANALYSIS

23

Practical experience has shown, however, that with modifications the above methods also apply quite well below the bubble point. To arrive at the modifications, one should first note that the pressure in the oil phase in a given pore in the reservoir will be almost the same as that in the gas phase in tQe same or an adjacent pore. The two pressures will differ by the oil-gas capillary pressure, which for most situations of interest will be less than a few pounds per square inch. Thus, for practical purposes the buildup will be identical in

each phase. If we concentrate our attention on the oil phase, we can liken the buildup in this phase to buildup in a single-phasesituation. Two differences will arise. First, the compressibility will be higher in any set of pores because of the presence of gas. Secondly, the change in pressure with distance and time will be caused by the simultaneous flow of both oil aJld gas. We might expect that we could apply, at least approximately, the single-fluid methods if we use total compressibility and

7

6

5

.c

Ia.

~ 4 m

*1

:l

a.

0lD

~ "3

2

1

)

00.01

0.1

.10 0.000264

kt

9>JLCA

Fig. 3.5 Pressurefunction for one well in centerof cylindricalreservoir. DRAINAGE

BOUNDARY

1 ~ ~ :)

z

i

1000

0

..J Lo.

800

60010

100

1000 TIME

IN MINUTES

Fig 5.4 Flowing pressurevs logarithm of flowing time, extendedpressuredrawdowntest, Denver Basin Muddy Sandstonewell.

10,000

PRESSURE

DRAWDOWN

ANALYSIS

53

000 bbl and a water saturation of 35 percent, this amounts to a recovery efficiency of about 30 percent. A recovery of this magnitude is quite reasonable for Denver Basin depletion-type reservoirs. The example demonstrates the economic value of a transient pressure analysis. For roughly $1,000, information was obtained which prevented an obvious loss of much greater magnitude. However, all the separate facets of the analyses are not in complete agreement. In particular, why is it that the kh and s values determined from the transient and late transient methods do not agree more closely? In this case the explanation lies in the fact that the well was fractured on completion. In fractured wells the kh and s values will depend on the flowin,g time range of the pressure data used in their calculation. At early time, flow into the well is the result of essentially linear flow into the fracture, and the pressure drop per unit of production is less than with radial flow. Thus, the calculated kh value will be too high. As flowing time increases, the more radial flow away from the fracture controls the pressure behavior and truer estimates of kh result. In both instances negative skin factors are obtained. This is discussedmore completely in Section 10.5. Because of this effect and the fact that the reservoir volumes determined from the late transient and reservoir limit analyses are in near-perfect agreement, we believe the

kh and s values calculated from the late transient method to be preferable. This example has served to illustrate the need for considering all available information when analyzing pressure data. In this case, the additional knowledge of pressure behavior in fractured wells gained from theoretical studies provided a basis for choosing between answers. There are cases in which a unique interpretation of reservoir characteristics may not be possible from pressure analyses and the available geological and petrophysical data. In such casesthe engineer must acknowledge the existence of more than one possible interpretation. If he must make recommendations concerning future exploitation of a reservoir, he should be especially aware of the economic implications of each alternative solution. ..." 5.5 Operational Considerations with Pressure Drawdown Tests The properly run pressure drawdown test can yield valuable information about the reservoir. This type of test is harder to run and control, however, becausethe well is flowing during the test. The analysis methods are based on the assumption of a constant flow rate from the well. If the well will not flow at constant rate, then the pressure drawdown behavior can sometimes be analyzed by recourse to the multiple-rate

2000

1800

".-0 ~

1600

1&1

~ ~ II) II) 1&1

f

1400

1&1 -J 0

i

~

1200

I0

m ~ z

i

0 -J

1000

I&.

800

600 0

600

1200

1800 TIME

2400

IN MINUTES

Fig. 5.5 Flowing pressurevs time, extendedpressuredrawdown test, Denver Basin Muddy Sandstonewell. :

3000

54

PRESSURE

test methods which are presented in the next chapter. In the case where the well surges or "heads" due to slug flow through the tubing string, the resulting pressure data generally will not be usable. It is advisable to have some idea of the flowing characteristics of a well before committing funds and equipment for a drawdown test. The flow rate of the well should be great enough to cause easily discernible pressurechanges on all phasesof the test.

BUILDUP

AND

FLOW TESTS

IN WELLS

If it is available, special test equipment with which the oil, gas and water produced during the test can be metered as a function of time is highly desirable. Pressure measuring equipment should be especially calibrated for the range of pressuresto be encountered on the test. If an internally recording pressure bomb is used, then an effort should be made to eliminate as much as possible the necessity for retrieving the bomb to rewind the clock. Especially in those cases in which

1000

b='320 ~"-

~

,,'-..,...

/

",.0~ 0.

0/~=1'300

-0 """

" 0'0, 100

.

0

'"

'

0\",

""""'-0 ~ = 1400

'"

\

~ -0 U)

\ \

a.. ~a.

1

\

~= slope = ~= \

,

-\

a.-

0.1'35

\ \

\ \\

10

A

p = 1460

\

\

\

\0 \ A p=1490

1

0

2

4

6

8

flowing

time -hours

Fig. 5.6 Late transient analysisplot, extendedpressuredrawdown test, Denver Basin Muddy Sandstonewell. ~

-

PRESSURE DRAWDOWN ANALYSIS

55

small changes in pressure are being observed, the disturbance in the tubing created by pulling and re-running a pressure bomb when combined with gauge hysteresis effects can render significant portions of the pressure data unusable. Surface recording bottom-hole pressure gauges8are very helpful in drawdown tesung. When properly run, the pressure drawdown test affords a method for establishing the formation permeability and skin effect which is equally as reliable as the pressure buildup procedure. From the late transient portion of a pressure buildup or drawdown, the contributory drainage volume of the well can be estimated.

The long-term pressure drawdown test offers the engineer an additional means for estimating reservoir size (reservoir limit test). The pressure buildup is operationally simpler than the drawdown test, however, because it requires no measurementof production rates during the test. We believe it absolutely necessaryto devote some discussion to precautions concerning reservoir limit tests. These tests are probably the easiestof the pressure analysis techniques to misapply and obtain erroneous results. Invariably, the question which is asked concerning reservoir limit test data is: "Did

300

15, 000

..J m m

...

-&..

0

~

m

u

-

-200 I&J

10

000 ,

0 -

~

c

U)

J

I&J

0

~

~

-

...I

U)

~ ~

100

5,000

~ "

,~ ',\" 5000 ';1

10,000 CUMU~:~:

IlL

20,000

25,000

Fig. 5.7 Productionperformance,DenverBasinMuddy Sandstonewell. -~--

---

30,000

8

...

56

PRESSURE BUILDUP AND FLOW TESTS IN WELLS

the well really reach semi-steady state drawdown behavior?" The longer the test, the greater the certainty as to whether semi-steady state was reached. If the test data are misinterpreted and the slope of the pressure vs time plot is derived from transient or late transient pressure data, then the resulting calculated drainage volume will always be conservative. This is true because the slope of the pressure-time plot (dp/dt) decreases monotonically until semi-steady state flow is reached, and calculated reservoir size is inversely proportional to the slope value. Some simple results from theoretical fluid flow studies will help to answer whether the test time was sufficient. Transient flow endsll at a flow time of t ~

2 q"ucre (practical units), 0.00264 k

and flow at semi-steadystate begins9 at a time of t ~ 2 -cJ>JLcre -or three times the first value. (See also Odeh 0.00088 k and Nabor,12 who give approximately these same values.) Thus, if we can pick the end of the transient period from the pressurevs log t plot, then the semi-steady state beginning can be estimated at three times this value. This factor of three will increase as the drainage shape departs from circular and ~s the well location .is shifted from the center of the draInage area. In certaIn

viously, difficulties in maintaining proper test conditions and the cost of the test almost preclude reservoir limit testing in such cases. The pressure measurement costs will usually be $200 to $300 per day. In planning for such tests, the engineer should estimate the permeability and other parameters and make rough calculations of the time required to reach semisteady state for various drainage radii. In this way he can help decide on the practicality of attempting a reservoir limit test. Another problem may arise in economic handling of products produced from the well. In the case of oil this usually is not too much of a problem. However, with a gas well that has no pipeline connection, considerable volumes of gas may have to be produced and flared. This adds heavily to the cost of the test. In areas where production is prorated, permission must be obtained from the regulatory body to undertake any extended drawdown tests that would violate allowable restrictions. . ProVided t he p1anmng and economIC .. requirements indicate it to be feasible, a considerable amount of useful reservoir information can usually be obtained from a well executed reservoir limit test. Surface recording bottom-hole pressure gaugesprovide a ready check on pressure behavior as the test progresses. 5.6 Behavior in Non-Ideal Cases

.

10.3.

Section

In

eory

-

C

th

pra

d

.

UI

e-

1

SIng

fl

some

. 0

t

on

e

en

e

a

e

ca

presen

an

ave

ng

we

.

es

own

much

IS

factor

the

reservoIrs dIscussed

as

than

...or

heterogeneous

of

types larger

three

(

f d

ar

tubing or casing is proportional to the rate of change of the difference between the tubing- or casing-head pressuresand bottom-hole pressures.Thus, if one plots ~(BHP-THP) vs flow time, then stable conditions prevail in the well for times beyond the point at which ~(BHP-THP) becomes essentially a constant. Generally, it is possible to recognize this point on field data from the linearity of the p vs log t plot. Ramey5 has studied wellbore storage effects in pressure buildup and drawdown of oil and gas wells. Fortunately, the casing-tubing annulus usually is isolated by a packer in a gas well so that the only gas volume available is in the tubing string. Ramey found that the time required

raw

Extended drawdown tests have been valuable to many companies in making post-discovery decisions as to further developmentdrilling. As can be seenfrom the formula for the time at which semi-steady state 2 flow begins t ~ -~~ ), however, the time re0.00088 k quired to obtain valid reservoir limit test data is directly proportional to the compressibility and inversely proportional to the permeability. In the case of a lowpermeability gas reservoir, for example, the time required for semi-steady state flow to occur for a 640acre area may be in the order of several months. Ob-

..us

Another check on the reservoir limit information can be obtained from data derived from a pressure buildup test run immediately after the drawdown test. For a fluid of small and constant compressibility, the time required to reach semi-steadystate pressure decline is the same as the shut-in time required for the well to completely build up to the average reservoir pressure. Therefore, if a post-drawdown pressure buildup is run for a shut-in time at least as great as the length of the drawdown test, a check on whether semi-steady state was reached can be made. This is done by noting whether the buildup plot (PW8vs log [(t+ ~t) / ~t]) deviates from a straight line at large time and flattens out toward the averagepressure. Unless some flattening occurs, the flow test of equal time was too short.

.

Th d

. .

h t d th t ti d 11d tt ti ti I ca aspect s 0f the meth0ds. I n this secti on We shall point out the modifications to the basic theory which must be effected to handle gas flow, multiphase flow, etc. If the well has been closed in prior to a drawdown test, there is some fluid stored in the wellbore which unloads when the well is opened. Van Everdingen and HursPo originally studied this effect in transient production analysis. Until mass equilibrium is restored in the wellbore, the surface-measuredproduction rate will be the sum of the bottom-hole inflow and wellbore storage depletion rates. On short drawdown tests in wells with extended periods of wellbore depletion, the calculated results may be in error becauseof the use of erroneous rates. Stegemeierand Matthews4 show that the net rate of fluid accumulation or depletion into the f

PRESSURE DRAWDOWN ANALYSIS

effects

are

likely

less

than

1

of

wellbore

storage

In

case

the

li mi

reservoir point of

day.

will

case tests

must

substitute

the

and

system

fluid replace of the

total

pressures

th

of the

single-fluid

In

tests

in gas wells

formulas is for two P hase

analysIs

as dis-

drawdown

by using a modification that for pressure buildup

usually

of the analysis

can

,ugBg

product

f

mean

0

sures,

and

should

..

h

t e static the

reservoir

be

evaluated

reservoir

an

CtJ.tg product

pressure.6

If

the

the

of

stant

for

studies

all

to

the

. owmg be

pressure

is not greater CtJ.tg products the

d fl

should

being analyzed the J.tgBg and mid-point

theory analoin gas wells.

range

rigorously

the

h

we

ore

pres-

on

static

of

the

data

than, say, 400 to 500 can both be evaluated and

considered

to

there

substantiate

1

.

ate

1

transient

our

.

been

d

0

we

manner for gas wells. that pressure changes

say,

psi

to

500

are

encountered

h

of

during

wells,

the

should

of the

be

taken

account.

1

for

properties plotting

including

in

gas

gas propthe

In

transient

pressures

directly,

drawdown they

of

analysIs. first

the

Instead

transform

larg-

the

gas of pres-

sures into a pseudo-pressure variable by means of an integral involving the pressures and associated gas properti ' e Thi . ful d f .of s.

s

IS

a

use

proce

ure

or

gas

a test

to

at stat-

to

cases, next

if the

possible

difficult

the

of Pressure

Tech.

(Dec.,

main-

the multichapter

are

Build-Up

An-

1966)

F.: Tables of New York.

1624-1636.

Functions,

Fourth

213, 44-50.

Pet.

Tech.

(Feb.,

J.

Pet.

Tech.

R. and

(Jan.,

Ramey,

8. Kolb..

~.

AIME

H.:

"Two

(1960)

223-233.

G ..an E d B rus Gas Well Per-

1966)

99-108. "Application

of

Testing and Deliver(May, 1966) 637-642.

~ottom-Hole

Automatic 219,

1965)

H. J., Jr.:

Re,a.l Gas Flo~ T,~eory to Well ability Forecasting, J. Pet. Tech.

M

h

.att

C

ews,

Pressur.e

Surface

Instruments

Recordmg",

Trans.,

346-349.

for

Bounded 191. 10.

,:an

reservoirs

S

B

rons,

Reservoir",

in Reservoirs", C.

F.

Laplace

C.,

H

b

aze

of

Trans.,

A.

the

d

.an

Determination

Everdingen,

11. Miller,

F

..,

Method

than,

tests

pressure-dependence

is In these of

J. Pet.

J.

formance",

b

est pressure changes are encountered during transient tests and the smallest for the semi-steady state tests AI-Hussainy and Ramey7 have described a unique pro~ cedure

frequently

"Extensions

Wells",

7. AI-Hussainy,

no

more

Usually,

to begin

6 R 11 D G G d . P erry, .usse, .., 00 nc, h J ..,H kotter, J. F.: "Methods for Predicting

a so I e

pressure-dependent

into

it is not

H. J., !r.: "Non-Darcy Flow and Wellbore Effects m Pressure Build-Up and Drawdown

of

vanation

time

methods

G.:

(1958)

5. Ramey, Storage

..tlon

erties

cases

rate.

volume. tests

believe

Id

s ou

drainage

of flowing

e.

D.

AIME

psi, at

be con-

have

view,

th

ana YSlS me

modified in this In the event 400

it

Methods",

Gas

anthmetic

lIb

based

range

.Provldmg Although

analysIs.

at

should

3. Jones, P. an~ M.ca;hee, E,',: "a;ulf Coast Wildcat Verities Reservoir Limit Test, 011 and Gas J. (June 18, 1956) 184. 4 St . G L d M tth C S ." Anomalous Pressure Build-Up Behavior", Trans.,

9

h

t at t e

Also,

2. Jahnke, E. and Emde, Ed., Dover Publications,

be

..of The

sufficient

analysIs bl

1. Russell, alysis

the total of these

many

pro~ucing

app Ica

work

References

reservoir

apter.

Pressure handled gous to

buildup

test r

In

and

simplest

for

a constant

0 ten

compressibil-

mobility with The calculation

pressure

conditions.

Their

drawdown tests are an operasound means f or evauating 1 .

are the

are right.

in a well

drop.

parameters

tests

.egemeler,

of

pressure details.

reservoir

pIe-rate f

u

of We

the

oil

3

case

bble

e

the equations for analysis must be modified slightly.

the single-fluid flowing fluids.

Ch

the .

In d

flow

b

ow

gas

compressibility

close

tain

and

for

critical

ic (including

of

flowing to for

conditions

tests,

1 e

flow

a large

Drawdown

drawdown

tests

quantities and their use in the analysis p letel y analo gous to the modifications com

cusse

in

not be important.

b

a

string wellbore

longer-duration

two-phase

rock

flow

for

t run

with

be referred

In summary, pressure tiona 11y an d th eoretically . .

cases

drawdown

) s

reservoir,

will result. In this pressure drawdown

the

some

For

pressure

test

of

in

by

)

umts.

to be important

probably of

t

the

ity and mobility

length that

.

at

wells

f

tests

is the

gas

is given

0

Lt

in

y

storage

.

J.tcgrw2Lt ( kh

concluded

out effect,

d

formula

Ramey

die skin

St

this

to no

A

) 4 785 -practic,

ys

having

...u

In feet.

effects

and

ews,

(da

t

storage

packers,

a

wellbore

without

..an

for

57

and

AIME Hur~t,

Trans.,

AIME A.

B.

W.: to

(1949) and

P

"

.:

Pressure

(1954)

Transformation

Dyes,

k

roe,

Average

201,

"The

A

in

a

182-

Applica-

Flow

Problems

186, 305-324.

Hutchinson,

C.

A.,

Jr.: "Estimation of Permeability and Reservoir Pressure from Bottom Hole Pressure Build-up Characteristics", Trans., AIME (1950) 189, 91-104. 12. Ode~,

A.

duction acteristics

.S. and History From

Nabor,

~.

"!'.:

"The

Effec.t

on Determmatlon of Formation Flow Tests", J. Pet. Tech. (Oct.,

of

ProChar1966)

1343-1350.

13. Root, P. J., Warren, J. E. and Hartsock, J. H.: "Implications of Transient Flow Theory: The Estimation Gas 141.

Reserves",

Drill.

and

Prod.

Prac.,

API

(1965)

Chapter 6

Multiple-Rate Flow Test Analysis

The methods for analyzing flowing well behavior discussedthus far have been based on the assumption of a constant producing rate. In some cases, however, the rate will vary with time. In other cases, regulatory bodies require flow tests...0 made at a series of different ..pressure rates. Gas-well tests fall Into thIScategory. ThIS chapter IS devoted

to development

of pressure

analysIs

methods

..W for handlIng both of these cases.The methods of thIS ...era chapter are partIcularly useful In the case of a floWIng well which produces at constant rate where it is not operationally or economically feasible to shut in the well for a pressure buildup or to allow the pressure to equalize prior to a pressure drawdown test. In these

Chapter 3) can be used to forecast future deliverability. 6.1

General Equations for Analysis of Flowing Well Tests with Variable Rate T deve1op the genera1 equatIons, we dIVIde the

.

d raw d own

.

.

based

on

case

wIth

any

measurements

...esire pressure

obtaIned

...rate.

analysIs

whIle

the

well

on the obtaining of made in the discussion

good of

measurements which were pressure drawdown analysis

are equally valid here. It should be realized, of course, that the measurement of production rates is more critical in the case of multiple-rate tests than with ordinary pressure drawdown at constant rate. We shall discuss this point further in the presentation of the various multiple-rate analysis methods. As will be seen, the multiple-rate methods to be presented are applicable to gas wells as well as oil wells. The objective of these methods is to determine permeability, skin effect and reservoir pressure. Determination of gas well deliverability is not a primary objective. However, once the basic parameters of kh and skin factor are determined, the methods of Swift and Kiel (Ref. 22, Chapter 3) 0; Russel[et al. (Ref. 27,

0f

.. .. ..

.

intervals

f

rom

q = q"

,

t"-l ~ t .

may

as

small

be

d t 0 app I y t 0 th e case

Th

IS

producIng, care must be exercIsed to obtaIn good production rate and pressure measurements. The remarks

eac h

q = ql , 0 ~ t ~ t1 , =., t < t< t q q- , 1 --2 , q = qs , t2 ~ t ~ t3 ,

d

method

.

.

for estimating the kh product, the skin factor and the reservoir pressure. ...The IS the

..

t erva 1s d unng .

d t. t b h h th IC e pro uc Ion ra e can e consldered const ant. Th t ti hd I e- me sc e u e IS as f 011 ows:

instances dependable transient pressure data can often be obtained by measuring the pressure responsecaused by a change in flow rate. Analysis of these data by the interpretation methods of this chapter affords a means

As

t es t .mom t .

E

p,

.

and

as

numerous

as

0 f a con ti. nuous I y c h a n gt. ng

. ..

.

d d th ' . ti I t d e pressure rop unng e mi a Ime peno IS,

q. .-=

5 2 .,

P"'f

162.6

kh

qlJL B

[I

og

t +

-] s,

(61) .

where s = log ---~~ -3.23 + O.87s .(6.2) cf>p.cr", Applying the principle of superposition (as in Section 2.8), we find the pressure drop during the second time period to be 162.6 q1/A. B P. -P"'f = kh [log t + s] +

162.6(q2 -ql)JL B [log (t -t1) kh

+ s] . (6.3)

For the third time period,

MULTIPLE-RATE FLOW TEST ANALYSIS

59

_162.6 q1JA.B -162.6 P. -Poo! -kh [log t+s] + [log (t-t1)

-162.6 + s] +

[log (t-t2)

-rate + s] ..(6.4)

(q2 -q1) JLB kh (q -q) B ~h 2/L

down tests which are begun at stabilized pressure conditions in the reservoir and in which the production is non-constant. The basic plot requires that the initial pressure value Pi be known. If it is not known,

Thus, dunng tIme penod n the pressuredrop IS gIvenby .-= p,

162.6

q,/L

P,C!

B

[log

t +

s]

162.6(qa-q2)JL kh

...+

-on [log

kh

+

(t-t1)

+

B [log (t -t2)

This equation can also be written 162.6/LB Pi -Poo! = kh[q110g t + (q2 -q1)

Odeh

-rate + s] +

by

trial

and

be

that

which

error.

That

yields

is,

the

the

best

correct straight

Pi line

and

Jlmes!

presented

the

.. foregoIng

vana bl e-

~nalysis technique a?d also some examples of applYIng the method to ollwell and gas-well pressure

drawdowns. The basic data for the oilwell example + s] .presented in their paper are depicted graphically on Fig. 6.1. Fig. 6.2 shows the variable-rate analysis method plot for these data. Further details of the example calculation can be found in the Odeh and Jones paper.

log (t -t1)

+ + ( qa-q) + 2 log (t-t.) -I ...(q" -q"-1) og (t -t"-1)] + 162.6q"p.B -

kh

Pi should

..

the basic plot.

s]

162.6(qn -qn-1)/L B[log (t -t"-1) kh

In a laterofsection of this chapter will present the application a modification of thewe variable-rate analysis theory to the determination of the kh product and

s ,

500

or P. -Poo! -162.6/LB :: [ Aqj -"" -og I q" kh j=l q"

]

( t -t j-1)

]

rlog~- k 162.6JLB kh

+

...

value

162.6(q2 -q1)JL B "

then the utility of this type of test and analysis becomes h t d somew a ImtffiShed .ISIt pOSSI ble, 0f course, t 0 determine

kh

+

The foregoing equations and plotting technique comprise the basic method for analyzing pressure draw-

3.23 + 0.87s ,

~ m

400

> a: l1J ~

300

~

(6.5) where

a:

A.qj = qj -qj-1

.;

A.q1= q1

20

to = O. From Eq, 6,5 we see during the nth period of constant rate, i,e., t"-1 $: t, if we plot p, -P

!

to

vs

q..

n -logA.qj

~ j=1

(I

-Ij-J

,

Pi = 3000

]

p.cr 00

kh = .!~~~,. and

~r

Co

.:. ~ 15

..(6.6)

m

[ bl. k -,-log-:;:--2+ m

200 "OJ

From these values we can determine the kh product and skin factor from

S -1.151

psi

q..

we should obtain a straight line of slopem'=-~~~~and intercept bl= 1626 k'h /LB [log~- k 3.23 + 0.87s

,!

100 2500

] 3.23 ...(6.7)

100

DATA

V-

MISSING

'l'JLCroo

Damage ratio or flow efficiency can be calculated for flow tests exactly as for pressure buildups (see Chap3).

0

TIME,MINUTES Fig. 6,1 Multiple-ratetestpressureand productiondata. (Data from Odehand Jones.')

-

200

60

PRESSURE BUILDUP AND FLOW TESTS IN WELLS

skin effect from gas-well open-flow potential test data. The developmentof an analysis theory has led to the use of multiple-rate flow tests as a means of generating transient pressure data. One of the most popular of these techniques is the two-rate flow test described in the section which follows. v .and --ssuInlng 6.2 Two-Rate Flow Test AnalysIs Method The two-rate flow test, developed by Russell,2offers a means for determining the kh product, skin effect and extrapolated pressure which overcomes many of the wellbore problems experienced with pressure buildups, and also eliminates the need for shutting in the well. The required pressure data are obtained by observation of the transient bottom-hole pressure behavior V after the stabilized production rate of the well is changed

is reduced. The period of transient pressure change caused by the change in rate (labeled A on Fig. 6.3) is followed by a period during which boundary and interference effects are felt (B), and finally the well returns to a stabilized pressure decline (C). It is necessaryto measure only the pressuresjust prior to the rate change during a portion of the transient responseinterval. A that the ra te 0f pro duct Ion q2 becomes operative immediately after the change in rate, we may combine Eq. 5.2 and the principle of superposition to yield the following expression for the flowing bottemhole pressure after the rate change.

.

.

p"" = Pi -kh

to another, higher or lower, rate. In preparation for a two-rate flow "test, a well is usually stabilized for several days at a constant producing rate, and both oil and gas production rates are measured on a daily basis. Three to four hours prior to the rate change, the bottom-hole pressure bomb is lowered into the well, and pressure measurement is

[log~-

3.23 + O.87s

[log~+~t'

~log ql

162.6 q2JLB

-162.6

kh

qIPB

k p.C'"

]

~t' ] ' (6.8)

where . ql = rate pnor to the rate change,

begun. This is necessaryin order to obtain a dependable value for the flowing pressure prior to the test. The producing rate is then changed by adjustment of the

q2 = rate after the rate change, ... t = producIng time pnor to the rate change,

choke at the wellhead, and, after a usually short period of transition, the rate stabilizes at its new value. Meanwhile, the transient pressure response caused by the. rate change is being measured. The flow test pressure and production rate behavior with time are shown schematically on Fig. 6.3 for the case in which the rate

~t' = producing time measured from the instant of rate change. It IS assumed that the well has produced at constant rate ql for time t prior to the test. (This is the same assumptionmade in pressure buildup analysis theory.) We see from Eq. 6.8 that a plot of p"" vs

/

6.0

-/

/0 /

/

[IOg¥+~log~t']

0

0

~INITIAL

~

0

..I-

I.

~

-I~~

~

U)

:

@

PORTION OF PRESSURE HISTORY USED IN FLOW TEST ANALYSIS

@

BOUNDARY

-'

Co

I

PREssuRE

@II'\

... ~

'in

Co

@

:0 '"

'-CD

-0

will yield a straight line.

0

.0 50

"f

/

2

lc

PAST PRESSuRE INOT NEEDED

~

0"

ANALYSISI

HISTORY -, FOR

i

I

~

WELl. RETURNS TO STABILIZED

z:

~

PRESSURE

'.

1

AND INTERFERENCE

EFFECTS ARE FELT

DECLINE

0;-:-6"-TIME-

'.'j£~1 ,~)t

... I-

SHORT TIME LAGUSUALLYREOUIRED BEFORE NEW STABLE RATE IS REACHED

:

~

4.

q,

-

:0 0

3.0

..6.0

7.0

t

n qj-qj-l

I

J=I

)

~

1- ~

8.0

0

f

q.

I I

, I

I

log (t-tj-l)

1--1

qn

.1-

6t'-TIME-

Fig. 6.2 Multiple-ratetestbasicplot. (Data from Odeh and Jones.') 'I

-I

IU

~l.J)

Fig. 6.3 Schematicplot of productionrate and bottom-hole pressureperformancefor two-rate flow test (q, < qJ.

MULTIPLE-RATE FLOW TEST ANALYSIS

61

From the slope m (in psi/cycle) of this plot the kh product can be determined by y -162.6 qlJLB V I__~ -m'. (6.9)

cussion of this example and the details of the calculations employed in the analysis can be found in Appendix E. 6.3 Two-Rate Flow Test Analysis

...In In a manner sImIlar to that used to denve the pressure buildup theory it can be established that the skin factor is given by

Non.ldeal Cases ... As IS the case WIth pressure buIldup and pressure drawdown analysis theories, the two-rate flow test analysis theory is based on flow of fluid of small and constant compressibility. If the pressures on a tworate flow test are below the bubble point, it is necessary to substitute the total mobility and compressibility of the system into the analysis formulas, as in Example

3 = 1.151

I

[(

ql

ql -q2

)(

PI

) -log~

hr -PID

m

f/llJ-CrtD2 (6.10)

3.23]

...+ "

where p~~win2 pressure at the time of the rate change an~ is the pressure at 1 hour after the rate change on the straight-line section of the flow test Plot. Damage

ratio

or

flow

efficiency

can

be

calculated

2A, Appendix B.

for

q2 < q I

I.J

-

a:

0000

two-rate flow tests exactly as for pressurebuildups (see Chapter 3).

~ ~

cfI\

The value of Pi (equivalent to p. in pressurebuildup: theory; see Chapter 3) is given by

RETURN TO 0' SEMI STEADY:r STATEFLOW

/ EARLY DEVIATION FROM

I

[ P.

=

PtD

+

m

kt log

] -3.23

+

2 ~

0.873

f/lp.crtD2

LINEARITY DURING RESTABILIZATION

b

(6.11)

CD ~

The pressure drop across the skin zone is b" aiven by ~p (skin) = 0.87 ms (at rate ql) or q ~p (skin) = 0.87 ~ ms (at rate q2).

0

~ ~ ~

4--INCREASINGFLOW TIME

( ) ql

The interpretation

i ,

'

L-

1+61' q + -!61 ql

log ~

theory is based on the infinite

homogeneous reservoir.

3250

.0

caseof a well producIng from a bounded homogeneous dramage voIume .IS shown on P.Ig. 6 ..n 4 I t his case,I.e., q2 < ql, the effect of a boundary is to cause the points to bend over and deviate from the straight line. This d b .k eVlation ecomes progressIvely greater as the well fina11 y reaches a semI-steady state pressure decline at rate q2. PIg. 6.sows 5 h

.

.0

.

.

a typIcal fi eld example of a two-rate

flow test from the paper by RUsseiI.2This flow test was run .m a Iow-permeab1li ty li mestone reservoIr m t he

.

..

Permian Basin region of West Texas. A complete dis-

ql * 107STB/D P. * 3118 pliO q2*46STB/D h*59fl c,*9.32.'0-5pI'-1 rw*0.2fl IJ.*0.6Cp B*I.5 ~*0.06 Np*26.400 STB 1*~ '24* 5922hr 107 BASIC DATA-WELL A

324

[

PtD!vs log t + ~t'+ !l!.log ~t' ] ~t' ql will be linear for a period of time after the rate change and will then start to deviate from linearity as boundary and (or) interference effects are reflected at the well. The times at which such effects will be felt will be of the same magnitude as would be the case if a conventional buildup were being run in the well. The general appearanceof a two-rate flow test curve in the

..

log 61'

Fig. 6.4 Appearance of two-rate flow test curve in bounded

reservoir solution of the radial flow equation for a slightly compressible fluid. In practice, however, the method will generally be applied to wells which produce from bounded drainage volumes. It may be expected h th h ten at t e plot of

.

RATE

323 0-

.~ 3220 ,.j ~ 3210 ~ ~ 3200 ~ i 3190 :: m

SLOPE *90pII

31BO

317

RESUL TS =30md

316

1*-3.6 . P* *3548psI9

3.0 3.

3150

...3.5

3.6

109~'

.38

+ .9..z. 10961'

61

ql

Fig. 6.5 Two-rate flow test plot, Permian Basinwell.

,..,,-

,£"c~

3.9

62

PRESSURE

BUILDUP

AND

FLOW TESTS

IN WELLS

For the case of a gas well, the technique of employing "average" gas properties as outlined in the discussionsof pressure buildup and pressure drawdown analysis is applicable for two-rate flow tests. If the pressure range encountered on the test is quite great then the methods suggestedby Al-Hussainy and RameyS can be applied to account for pressure dependenceof the gas properties. The effects of partial well penetration, perforations, etc., are quantitatively the same on two-rate flow tests

in a well. Although the last few points appear to lie on a straight line, in reality the true straight-line section has not yet been reached. In this case it was necessary to extend the shut-in period of the well to 6 or 7 days to overcome compression and afterproduction effects and obtain a straight-line section on the pressure buildup plot. The plot of a two-rate flow test from this same well is shown on Fig. 6.7. The flow test of 22 hours' duration enabled g?od estimate~of .kh, sand p* without loss of production from closing In the well.

as on pressure buildups. Another point pertinent to practical use of two-rate flow tests concerns the assumption of instantaneous change in flow rate in developmentof the interpretation theory. Such an instantaneous change in flow rate is never fulfilled in practice because the adjustment in bottom-hole inflow rate results from a change in choke size at the wellhead. Thus, the flow test performance of a well after a change in choke si~e is directly related to the vertical lift performance characteristics of the well. Field experiencewith the flow test method indicates that rapid stabilization of flow conditions usually occurs within the flow string and, therefore, surface producing rate measurements are not greatly distorted. Rapid restabilization appears to be directly related to the fact that the flow string is continuously resupplied with mass during a two-rate flow test. In general it has been found from field tests that the

Fig. 6.8 shows a humping buildup curve obtained from a Wilcox Sand gas-condensateproducer in South Texas. The two-rate flow test run in this well is shown on Fig. 6.9. This is a very striking example of the utility of two-rate flow tests.

restabilization period is shorter in the case of a rate reduction than in the case of a rate increase. The reason for this seemsto be as follows. In the case of a decrease in rate, an additional amount of fluid is needed to establish new flow equilibrium in the flow string. This can be provided quickly by the entering fluid. In the case of an increase in rate, there must be a net decrease in the mass content of the flow string for stabilization. This decreaseoccurs more slowly because fluid is now entering the flow string from the formation at a higher rate than before. The method of Stegemeier and Matthews4 can be used to detect when flow through the tubing has stabilized. This technique was discussedin Section 5.6. 6.4

Elimination of Wellbore Effects with Two-Rate Flow Tests Two-rate flow tests have found their primary area of application in wells in which wellbore effects obviate the pressure analysis technique. Two principal types of conditions are involved. One is the case in which compression of gas in the wellbore and a long, low-rate afterproduction period combine to render the normal pressure buildup plot non-linear. The other is the instance in which phase redistribution in the tubing string subsequent to shut-in causes "humping" buildup behaVlor. ..30

Some

examples

from

Ref.

2

are

Included

to

illustrate graphically the cases referred to above. Fig. 6.6 shows a 72-hour pressure buildup obtained

As a final note in our discussion, we would like to emphasizethat in planning and executing two-rate flow tests, one needs to have a general idea of the flow characteristics of the well. If field personnel are not familiar with the behavior of the well, it is advisable to observethe flowing behavior of the well at two or three different flow rates to obtain a general impression of its performance characteristics. By obtaining such observations in advance, one is able to make a better choice of the flow rates to be used during the flow test. A basic requirement of the two-rate procedure is that the well flow without surging or heading at each rate. 6.5

.. Transient AnalysIs of Gas-Well Multi-Point Open-Flow Potential Tests

An important and plentiful source of multiple-rate transient pressure data, especially in low-permeability reservoirs, is the flow-after-flow type of open-flow potential test run in gas wells. The general multiple-rate analysisprocedure set forth by Odeh and Jonesl can be applied to determine the kh product and skin factor from such data. 4800 440 ~ 4000 ~ a 3600 g: ~0 3200 T ~ 2800 ~ 2

72-hr PRESSURE BUILDUP Np=4145 STB

:J

.J:f Q-"o.P' -~

2000 ,.000

10 000

1+61

At Fig. 6.6 Pressurebuildupcurve,problemwell.

10

MULTIPLE.RATE

FLOW TEST ANALYSIS

3600 ~t'=22hr ~ ~ Q Q 3500\ .~ ~ w

63

°'b

~

'

(/) ~ 3400-.0 U! SLOPE =670 pslg ~ "7

At'=5.5hr 00 0 ° °

~ BASICDATA 0' 6 UI3300-I qo.=70STB/D P-0'0.34cp q =40STB/D t=8151hr h02=8 ft Ct=2.77 x10-5psi-I

3200

3.3

In several areas of the United States and Canada, the stat.e regulat?ry bodies that control oil. and gas production requIre that open-flow potential tests (OFPT) be obtained in gas wells. Results of these tests are often used in conjunction with other parameters to determine the allowable flow rate of a well. The OFPT consists of a series (commonly from one to four) of measurementsof flowing bottom-hole pressures made at various flow rates. Generally, the well is allowed to flow several hours at each rate and then the pressure is measured. In the flow-after-flow type of OFPT, the pressures are measured at the end of a flow period at a given rate, after which the rate is changed immediately to a new value without closing in the well. The pressure-rate data from an OFPT are usually analyzed by the familiar steady-state

RESULTS k .1.1md s =-4.84 p*. 5770psig

r w2.0.13 f t 2 ~ '0.12

(~) 29.8 mdIcp II =t320~ s.

Bo' 1.~2 I

:w .3186PP:. Ihr

flow method of Rawlins and Schellhardt5to determine the open-flow potential of the well, i.e., the theoretical rate at which the well would flow if the sand-facepres. sure were reduced to atmosphenc. In permeable reser-

tt21 hr 0 0

voirs each pressure measurement usually represents essentially semi-steady state flow conditions at the re-

spective rates; however, in tight reservoirs the flowing pressures measured are usually still in the transient

I

3.4

3.~ 3.6 3.7 3.8 ~9 4.0 1°9~ + ~ 1°9 At' Fig. 6.7 Two-rate flow test plot, ~rOducingbelow bubblepoint.

stage. The idealized pressure-rate-time behavior during an OFPT of a new gas well in a tight reservoir is depicted on Fig. 6.10. The "tight reservoir" restriction is in-

3600

~ 3500 '" 0-

w

FLOWINGPRESSURE = 3255psig

a: :> In In

U! a: a.

U! .J 0 :I:

3400

I

~ 0

ff0 UI

/

3300

I

.0

6t, MINUTES Fig. 6.8 Pressurebuildup curve,Wilcox well.

64

PRESSURE BUILDUP AND FLOW TESTS IN WELLS 3200

.,.-6 t/: 22 hr 3190 SLOPE: 35 psiQ

.~ 3180

RESULTS

U)

Co

~

w

a: =>

~ 3170

k

:

5.7

s

:

5.47

md

in

p* : 3560 psig

w

a: Q.

BASIC DATA-WELL C

w

0

6 3160

ql : 8781 Mcf/O

BQ: 0.0056

:I:

q2 : 6002 Mc f /0

Ct :2.24xI0

Np : 32,254 MMcf

fLQ : 0.02 cp

h

p : 3084psiQ w Plhr:3180pSiQV

I

~ 0 I-

b m 3150

: 142 ft

rw: 0.25ft ~ : 0.15

-4

psi-I

Sw : 03 t : 88,157 hr

314 ",

1

t6t: 3130 4.5

...4.9

5.0

I

3 mln 4

5.5

,

, .I.

t +6t 6t'

log

+

q2 -q;-

' log

6t

Fig. 6.9 Two-rate flow test plot, Wilco-x well.

I

serted to insure that the method will be applied only in those cases in which the measured pressures are within the transient portion of the pressure history. In Fig. 6.1 0 a four-point OFPT is shown in which the rates increase. The analysis procedure is independent of whether the rates increase or decrease during the test.

~ ~ ~ ~

By modifying the general equations (Eqs. 6.1 through 6.7) presented earlier in the chapter for flow of gas, we find that at point n(n = 1, 2, ...) of an OFPT the following expression can be written:

~ i 3 II.. 0

p.

0 Pi -PtDf..=28,958IJ.,Bg q.. k,h

[10g-.!~-

c!>IJ.,clrtDl

I

I

I

tl

t2

t3

] 3.23 + O.87S

p wf4 1 t4

TIME w

+ 28,958IJ.,Bg ~ k,h

( q/

i=1

I-

-q/-l

)

log

(t..

-t/-1)

,

q"

(6.12)

I

:

:

Z 0

I I

t

q2

:

=>

where qo and to are identically zero. The factor 28,958 is used in this equation rather than 162.6, because throughout Section 6.5, q is expressed in units of Mcf per day.

g ql ~ 0 0

:.: tl

t2

:

: :

t3

t4

TIME

Thus

,

a plot of

Pi -PtDf" .q"

vs

Fig. 6.10. Idealized pressure-fate-time. relationship for fourpomt gas-well open-flow potential test (OFPl).

MULTIPLE.RATE

FLOW TEST ANALYSIS

) IOg (I"

.~ (~ 1=1

q"

65

-li-1)

to yield a linear relationship with slope m' = ~. 0'"

., , -28,958 should be linear with slope m -koh

/LgBg. andmter-

The .kgh product.can be determined fro~ the slope m', and If slope and mtercept values from this plot are used

cept

with Eq. 6.7, the skin factor can be calculated. 3.23 + 0.87s 1 .sho~n

b' = 28,958 /LgBg[log--~koh

~/Lgctrw2

This procedure is depicted schematically on Fig. 6.11. The kh product and skin factor are determined from kgh = 28,958 /LoBg , m' and [ b' k s = 1.151 -, -log ~ g 2 + 3.23 m 'j'/LgCtrw

.

Th e gas OFPT

th e

ana thod

d

properties I . tli

YSiS

an th me

d

.

0 th

ate h

s

ld

di

pressure. However, OFPT pressure measurements are (6.14) frequently made at the wellhead with a dead-weight d .tester. In this case one should exercise care in con-

parameters b I

ou

e .

eva

d uate

f

-4.7 determined from a subsequent pressure buildup test. The use of this technique depends heavily on the accuracy of the pressure data. If possible, it is desirable to obtain direct measurements of flowing bottom-hole

]

d

I re

d

(6.13)

use b

10 h y

t

... vertmg

these

pressures

back

and

are

to

sand-face

e

II

me ou ne m e SCUSSion0 gas-we pressure B d bUl1dup and d raw d own ana IYSiS. Th e JJ.g g pro uct should be evaluated at the mean between the static reservoir and flowing wellbore pressures, and the CI,url product should be based on static reservoir pressure. If the pressure range on the OFPT is only 400 to 500 psi, all gas properties can be evaluated at the mid-point

.

.

A field example of the application of this method is on Fig. 6.12: The pressure data are from OFPT obtamed at completion on a Morrow-Chester sandstone well in the Anadarko basin of Oklahoma. The basic data and calculations lor this analysis are presented in Appendix E. The calculated results are kg = 3.5 md and s = -4.7, as compared with k" = 3.8 md and s =

conditions.

.' If

condensate

water

present

m

the

flow

string,

... this converSion can be difficult. .' . If the pressure is not fully built up to the static reservoir pressure during the shut-in period prior to an OF~T, allowanc~ mus~ be made for the effects of the preVious produ~tion. history of the. well upon the ~esuIts. The modifications to the basic procedure which

of this range and considered to be constant for all the analysis. If the total pressure range becomes greater,

0.017

Eq. 6.12 can be divided by JJ.gBg so that

0.016

P. -Pw/" q" /Lg"Bg"

0.015

may be plotted against

~

( qi

j= 1

0.014

) log (I"

-qi-1

-li-1)

,

0.013 u ~

q"

m'=0.02904 0.012

In

a.

0

-0.011 .E ~~ cf 0.010

-::::u

a

n=I,2,'.'

~ 0 .-SLOPE "' ':"

'

=m =

k g = 3 . 5 md

28958 8

0.009

kh g

0.008

,

p.g g

RESULTS kgh =140md.fl 5 =-4.7

c

ic~ 'i'" ii: 0"

~

c., 28 958p. B INTERCEPT =b'=~~'~~-fg -g log g

k9 2 -3.23 +0.875 V ~Ctp.grw

0.007, (.., / 0

00

n

.L

)=1

( qj-qj-l ) qn

b =0.00625

0.006 0.1 n

log (In-lj-,)

Fig. 6.11 Illustration of type of plot used to determine k.h and s from OFPT data.

.}:;

jcl

(

0.2 qj-qj-l

q

n

0.3

) log (tn-lj-l)

Fig. 6.12 Calculation of k.h and s from OFPT data, Anadarko Basin well.

0.4

66

PRESSURE BUILDUP AND FLOW TESTS IN WELLS

are necessaryin this case can be established by use of the principle of superposition. In

this

chapter

we

have

presented

the

techniques

which are necessaryto the analysis of transient pressure data from variable-rate flow tests in wells, The types of

multip le-rate

tests

which

ha

ve

been

d'scussed 1

do

d n

d ee

h ,

t

., ere

are

many

d

vanations

on

these

prace

1. Odeh, A. S. and Jones, L. G.: "Pressure Analysis, Variable-Rate Case", J. Pet. Tech.

IStlCS 1963)

which the enterprising engineer may employ to accomplish a given objective. The general approach used in the theoretical presentation in this chapter may be helpful to the engineer who wishes to analyze other multiple-rate test combinations including those involv...' 109 closed-In penods.

From Two-Rate 1347-1355,

.

3 Al H '

ures

-ussamy,

Real

Gas

Flow

R

,an

Flow

d R

Theory

arney,

Tests,

J,

Well

r.,

Testing

;

"_._"-~c

If!)

..,_'::3:;~~Jr~: ,+"

~";;,; '" ~-~" ..... '", "$,.;"..n.~~.. i [;H

a..~ (J~1-

"'!tlI;'J

"'

Tech.

A

pp and

.

Ilca

(Dec.,

t'

Ion Deliver-

0

f

ability Forecasting",J. Pet. Tech. (May, 1966)637-642. 4, Stegemeier,G. L, and Matthews,C. S.: "A Study of AnomalousPressureBuild-Up Behavior", Trans.,AI ME (1958)213, 44-50, 5. Rawlins, E, L. and Schellhardt, M, A:: "Bac~-Pr,essure Data on Natural Gas Wells and Their ApplicatIon to Production Practices",Monograph 7, USBM.

,

\

Pet.

H J J ..,

to

,~

"

Drawdown (Aug., 1965)

960-964, 2. ~~ssell,D. G.: "Determination of"F°rmation Character-

not

represent a complete library of these types of tests. I

References

Chapter 7

V

Analysis of Well Interference Tests

7.1 ,Reasons for Interference Tests W en one well is closed in and its pressure is

-Pw. = p' -162,6

[

~~~th..r~ ia. tbe~ ~ir are prooriced, the test is termed an interference test. The comes from the fact that the pressure drop causedname by the producing wells at the closed-in observation well "interferes with" the pressure at the observation well, This type

of

test

can

give

information

on

reservoir

I

which cannot be obtained from ordinary pressurebuildup or drawdown tests. First- of all, one can determine y' Is the Portion of the reservoir at reservoocoonectivit , this dlwell locationf being drained d by other wells? ? A hi 1 3How A rapl y. n Inter erence test can eterInlne t s. -n-~ ot~er I~~nt use of Interference tests ISto deterInlne dire~arrese.rvoir now patter~s. This is ,done by selectively opening wells surrounding the shut-In well. In addition to this qualitative information, it is possible to obtain a quantitative estimation of connected porosity from such a test. Porosity cannot be d f b Id I Elk' 4 estimate rom a pressure UI up test a one. ms

.

.

.

.

.

has also

used

..e Interference

tests

to determine

the

nature

and magnitude of an anisotropic directional reservoir permeability. Groundwater hydrologists5.6have made much more use of interference tests than have petroleum engineers. Muskat7 shows an example of their work. .If 7.2 Equations for Pressure Interference The mathematical basis for interference tests was first presentedby Theis" in 1935,* The following method usesthe same basic equations but differs in treatment and method of analysis. This method is based on su rpo..:'!!tion ?--~~~~~~~h of.the prod~cing v:ells at tile Slffit:m oDservl1tiunwt;il. USIng the El-function solution of Chapter 2 (Eq. 2,31), we find that the pressure at the closed-in observation well due to continued production at Wells 1, 2, 3, etc., is given by:

*In this same paper Theis also developed a pressure

buildup equation similar buildup methods never groundwater hydrologists.

N~Wq;

i=lq

-E'

properties

to Eq. 3.4, However, pressure gained great popularity with

9~log kh

(

(~ ~I ) +

70,6 ~ kh

{ E' I ( O.OO105k -~l1.ca;2 (t; +

.6.t;) )

-~I1.Cai2 )}]

O.OO105kt;

..(7.1)**

h were -h .q -t

d '. . e p.ro uctlon rat.eat our observation well before It was closed m t W II ' qi = the rate of Prod Uct'on 1 a e 1 ..i ' te rf enng we 11 prIor t 0 -1 = P roducing time of l'th m shut-in of the observation well ~ .6./; = producing time interval of the jth interfering well subsequentto shut-in of the observation well,. I NW :: n~mber of mterfe~~n~wells. ' a; -distance. of the 1 mterfenng well from the observationwell. All th t't " Ifi Id ' E 71

.

quan

lies

m

q.

.are

..m 01

e

.

Units,

The log term in Eq. 7.1 gives the effect of producing and shutting in the observation well itself, The Ei terms give the pressure drop at the observation well causedby production at Wells 1,2,3, ...at distancesat, a2,a3 ' , , a reservoir bou~~.J!:.-Sf1°se by. it may be taken into account by the method of images to be discussedin Chapter 10. The "image terms" are exactly like the Eifunction terms in Eq. 7.1, there being one such term for each "image" well. The distance a; in this case is the distance from the image well to the observation well. For use in Eq. 7,1 we obtain times t, tt, .6.4, etc., by the same type of equation as in Chapter 3. 1 = cumulative production at observation well rate of production just before closing in '

**Eqs. 7.1 through 7,3 apply to a more general case than the corresponding equations in the original monograph. The authors are indebted to Raj K. Prasad for the derivation.

68

PRESSURE BUILDUP AND FLOW TESTS IN WELLS

cumulative l?roduction at :WeIll prior to shut-In of observatlo~ well averagerate of productIQnq1 during interference test

11 =

well, best represents these quantities in the reservoir between the interfering wells. '" . An example Interference test IS shown In FIg. 7.1. The dotted line in this figure, called "Extrapolated Buildup Pressure", was obtained by extrapolating the linear portion of the log plot a~ shown on Fig. 7.~. From Eq. 7.1 we see that the dIfference between thIs extrapolated curve and the observed pressure is the sum of the Ei functions, or

incremental production at Well 1 subsequent to shut-in of observatio~ well averagerate of production q1 during interference test

1::.11=

and similarly for 12, 1::.12, etc. These equations for 4, 1::.1],etc., apply best when the rates of production at these wells are reasonably constant during the interfer-

( P*

ence test. We have already implied that this is the case by u~ing only one Ei term to represent ~ach producing well In Eq. 7.1. If the rate at a producIng well varies considerably during the test, a series of Ei functions should be used in Eq. 7.1 to represent the rate at that well; that is, the method of superposition (see Chapter 2, Section ~.8) should be employed rat~er t.~an the a?ove equations for 4. ~nd 64. Eq. 7.1 IS written for sIngle-phaseflow condItIons above the bubble point. For two-phase flow below bubble point, total mobilities and compressibilities should be used as in pressure buildup examples 2A and 2. Using Eq. 7.1, it is possible to determine the quantity (I/>JLc/k). The value of this quantity which, by trial and error, gives the best fit between observed and calculated values of the pressure drop at the observation

-162.6

~Iog kh

= -70.6~ ., kh .~ q -Ei(-

~ ~I

)

-P""

'

[ i=1 N!

.-1/>fJ.Cai2 { Ei ( O.OOI05k (I; +

)

1::.1;)

-1/>,uca;2 --: )}. ] O.OOI05kl}

(7.2)

Since the first two terms on the left side of the equation representthe straight-line extrapolation on Fig. 7.2 and the third term, P,V,',represents the observed pressure, this may be rewritten as

(/)

0.200 W

~ 1900 (f)

~

DUP

Ct: 180

EXTRAPOLA~E'py~l~

a..

w ~ 1700

\

7

I 1600 v

v

I-

I()

I()

~

w.!.~.!...~.!...

(II

~

58 psi

~ 0

PRESSURE

150

I

I

.'1; v I

I()

I

I

I

I

r--

a>

a>

m

I

143 psi ~

I

I

I

I

I

v

v

I()

I()

I()

I()

I()

I()

.!.

.!...

I

I() I

m Q

1:

I

.'1; v I

r\ ?

I

I

=

~

I.!... -C\I

..!

!..

140

1

~1

1300

",,"'! 0

40

80 120 CLOSED-IN TIME,

160 days

Fig. 7.1 Interference test in a low-permeability reservoir.

200

,olt_! -.Ic,rIU;;

,,:ii'"

ANALYSIS OF WELL INTERFERENCE TESTS

Pext -Pcb.

[

-m 2.303

(

-Ei

I '

[

69

=

incremental production at WeIll

NW ~ -El qj

{ .(

i=l q -cpp.caj2

)

-cpp.caj2 0.00105k

(4 +

)} ]

!:1t -subsequent 1 -avera

!:1tj)

23050 = -iso

..(7.3)

0.00105ktj

Example Calculation,

Calculate

the

pressure

Interference drop

at

the

and assume tha

-270 well

~P

~aused by production at Well 1 at the time when Incremental production at Well 1 subsequent to shut-in is 23,050 bbl. There is no production prior to shut-in and thus t1 = O. 0 a ta q = 140 ~/D,

~p.C/k = 10"6.

=

2.303

[ 180

.

(

-10-6

T4(fEl

(1835)2

0.00105

(3070)-

= -117.2

[1.285 Ei (-

= -117.2 = 32 psi.

[1.285 ( -~.21)]

)J

,

1.042)], ,

one can calculate the effect of production at Similarly, other wells-Wells 2, 3, 4-as a function of time and for several values of cpp.c/k. Fig. 7.3 shows results

rate at observation well prior to

shut-m, m = 270 psi/cycle, for buildup curve in observation well (from Fig. 7.2), B = 1.1, formation volume factor,

calculated rounding producing that their

C = 6.9 X 10-6 psi-1, compressibility, qt = 180 B/D,

~

cu ation The calculated pressure drop at the observation well caused by production at Well 1 is, from Eq. 7.3,

Test observation

= 128 days = 3,070 hours,

a1 = 1,835 ft,

The terms on the nght-hand sIde of thIs equatIon represent the calculated pressure drop at the observation well due to production at Wells 1, 2, 3, This will be made clear by the following example calculation. 7.3

to shut-in ge rate at WeIll

as above for the effect of four wells surthe observation well. Wells 1 and 2 began at the time the observer was closed in. Note influence was not appreciably felt at rates of

125 to 180 B/D until 60 days had passe