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f+ I: "

.'

q6r

PG E O( -

242

I

A,

G(\

IL/zg

t

wi ~ ~

GAS RESERVOIR ENGINEERING

I

:.'

, I

TABLE 10.10-AQUIFER

I. Type of Outer Aquifer Boundary

j for Radial Flow

j for Linear Flow

(STBID-psi)

(STB/D-psi)

'.:

,I "

Finite, no flow

J""

Finite, constant pressure

J""

0.00708kh(/J/360)

J""

p.L 0.001127kwh

0.00708kh(/J/360)

j""

p. In(r.Ir,) j""

Infinite

p.L

0.00708kh(/J/360)

[

B.6.PI-WeOPO(tDl) PD(tDl)-tDOPO(tDl)

W"l =W.o+(tDl-tDO)

iI,

=0+(15.1)

]

dW" dt

454.3(5)-0

[

1.83-0

tr~

]

.

[

B.6.P2- W"lPo(tm) PD(tm)...,.tDlPo(tm)

]

454.3(19)-18,743(0.0155)

= 18,743(15.1)

j!

1~i

[

2.15 -15.1(0.0155)

..:::::::t>

FetkovichI6

(~ ).

.

(10.68)

W",

geometries. Note that we must use the aquifer properties to calcu- "

J

~~

Method.

From Eq. 10.67, we can derive an expression for (Paq -p,),

and , following substitution into Eq. 10.68 and rearranging, we have

To simplify

water influx

calculations

dW" qw=-=J(Paq-p,)n, (10.62) dt where n=exponent for inflow equation (for flow obeying Darcy's law, n=l; for fully turbulent flow, n=0.5). Assuming that the aquifer flow behavior obeys Darcy's law and is at pseudosteady-state conditions, n = 1. Based on an aquifer material balance, the cumulative water influx resulting from aquifer expansiolkjs We=CtWi(Paq,i-Paq)'

"""""""""""'"

.(10.63)

Eq. 10.63 can be rearranged to yield an expression for the average aquifer pressure,

W,,

Paq=Paq.i 1-

) (

=Paq.i 1--

CtPaq.;W;

where We;=CtPaq.iWi

""""""""""""""

We

)

,

dW !... =J(Paq.; -p,)exp dt

fur-

dicted by the previous two models. Similar to fluid flow from a reservoir to a well, Fetkovich used an inflow equation to model water influx from the aquifer to the reservoir. Assuming constantpressure at the original reservoir/aquifer boundary, the rate of water influx is

(

(10.67)

0 W,,;

Table 10.10 summarizes the equations for calculating the aquifer PI for various reservoir/aquifer boundary conditions and aquifer

]

ther, Fetkovich proposed a model that uses a pseudosteady-state aquifer PI and an aquifer material balance to represent the system compressibility. Like the Carter-Tracy method, Fetkovich's model eliminates the use of superposition and therefore is much simpler than van the Everdingen-Hurst method. However, because Fetkovich neglects the early transient time period in these calculations, the calculated water influx will always be less than the values pre-

-

=- J t Jpi dt' -Jp.t

6. Table' 10.9 gives the final results.

"

. . . . . . . . . . . . . . . . . . . . . . . . . . . (10.66)

or Paq-P,=(Paq.;-p,)exp

=84,482 RB.

!~ J ~i !:i

We; dPaq dt Pi

-

.'IP, t' " .. I ii '! .\i

=-

J faq dPaq Paq.i Paq-P,

For n=2, 1'1

W,,2= W"I +(tm -tDl)

t

Combining Eqs. 10.62 and 10.66 and integrating yields

=18,743 RB.

~,.i Iii

1,OOOp.JO.0633kt/q,p.c

is defined as the initial amount of encroachable water and repre-~ sents the maximum possible aquifer expansion. After differentiat- J ing Eq. 10.64 with respect to time and rearranging, we have'

For example, at n=l, '.

kwh

j""

p. InJo.0142kt/q,p.c

i,:i

3(O.001127)kwh

p.[ln(r .Ir,) - 0.75]

.! ;i

PI's

(10.64)

W,,;

(10.65)

-JP.t ~

( ) W,,;

,

(10.69)

which is integrated to obtain the cumulative water influx, We: ~

.

W,,;

W,,=~(Paq.i-P') Paq,l

[ (--

JPaq.;,

l-exp

W"i

(10.70)

)]

Recall that we derived Eq. 10.70 for constant pressure at the reservoir/aquifer boundary. In reality, this boundary pressure changes as gas is produced from the reservoir. Rather than using superpo- , sition, Fetkovich assumed that, if the reservoir/aquifer boundary pressure history is divided into a finite number of time intervals, the incremental water influx during the nth interval is '

Wei -

-

.6.W"n=~(Paq,n-l-Pm)

P~~

[ (

JPaq,i.6.tn

l-exp

~, .'

)]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.71) TABLE 10.11-PRESSURE HISTORY AT THE RESERVOIR! AQUIFER BOUNDARY, EXAMPLE 10.8

t (days) 0 91.5 183.0 274.5 366.0 457.5 549.0

p, (psia) 3,793 3,788 3,774 3,748 3,709 3,680 3,643

GAS VOLUMES

.

AND MATERIAL-BALANCE

-

(

wherepaq.n-l=Paq,i .

- , and P m-

We.n-l

1--

Wei

CALCULATIONS

243 From Eq. 10.65,

)

(10.72)

We; =CIPaq.i Wi =(6 x 10 -6)(3,793)(7.744

Pm-l +p .m 2

.(10.73)

2. CalculateJ. For radial flow in an aquifer with a finite no-flow outer boundary, from Table 10.10, J=

Althoughit was developed for finite aquifers, Fetkovich's method canbe extended to infinite-actingaquifers. For infinite-actingaquifers,the method requires the ratio of water influx rate to pressure

, dropto be approximately --

constant throughout the productive

life

Th:: fc:"h'.-ing calculation pwc'-= m'oir. Aoo\.e the dewpoim. the \.apor phase Ll'nsisrs l'f nl'[ ,'nly Jrocarbon and inert gases but also water vapor. As the reservoir ssure declines, the water in the liquid phase continues to vapo~

to remain in equilibrium with the existing water vapor, thus

reasing the saturation of the liquid water in the reservoir and 'easing the PV occupied by the vapor phases. As the reservoir ;sure declines further, the amount of water vapor present in the phase may increase significantly. However, as the reservoir .sure decreases below the dewpoint. the fraction of PV avail. for the vapor phases decreases as liquids condense from the :ocarbon vapor phase. J develop a material-balance equation that considers the effects 1Scondensation and water vaporization requires that we include ;hanges in reservoir PV resulting from these phenomena. We n witn a material-balance equation for gas-condensate resers. We then extend this equation to include the effects of conwater vaporization. In addition, because changes in fonnation pressibilities often are significant in these deep, high-pressure reservoirs, we include geopressured effects. u-Condensate Reservoirs. We derived the material-balance .tions in previous sectioBS for dry gases with the inherent as)tion that no changes in hydrocarbon phases occurred during ;ure depletion. Unlike dry-gas reservoirs, gas-condensate reser; are characteristically rich with intennediate and heavier ocarbon molecules. At pressures above the dewpoint, gas conates exist as a single-phase gas; however, as the reservoir presdecreases below the dewpoint, the gas condenses and forms lid hydrocarbon phase. Often, a significant volume of this COllate is immobile and remains in the reservoir. Therefore, corapplication of material-balance concepts requires that we ider the liquid volume remaining in the reservoir and any liqproduced at the surface. ,suming that the initial reservoir pressure is above the dew, the reservoir PV is O£cupied initially by hydrocarbons in the JUSphase (Fig. 10.16), or i=Vhi'

(10.92)

e reservoir PV occupied by hydrocarbons in the gaseous phase can be written as vi=GTBgi,

(10.93)

e GT includes gas and the gaseous equivalent of produced conttes and Bgi is defined by Eq. 10.7. later conditions following a pressure reduction below the dew-

.the reservoir PV is now occupied by both gas and liquid

!Carbon phases, or

=Vhv+VhL

.(10.94)

~ Vp=reservoir PV at later conditions, RB; Vhv=reservoir 'lCoccupiedby gaseous hydrocarbons at later conditions,RB; 'hL= reservoir PV occupied by liquid hydrocarbons at later tions,RB. 10.94assumes that rock expansion and water vaporization ~gligibl-e. In terms of the condensate saturation, So' we can =(I-SoWp

(10.95)

'hL=SoVp' """""""""""""""'" (10.96) .ddition, the hydrocarbon vapor phase at later conditions is =(GT-GpT)Bg,

","""""""""""'"

Bg is evaluated at later conditions. ating Eqs. 10.95 and 10.97, the reservoir PV is

(10.97)

:\"c'W. Ll'mbmm:: Eqs. Il).9': anJ material-balance equation:

ll). IN yidJs

[he folllming

S,,(GT-GpT)Bg GTBgi=(GT-GpT)Bg+

I-So

.(10.100)

or, if we substitute Bg/Bg =(pz;)/( PiZ) into Eq. 10.100 and rearrange, P

Pi

Z

Zi

(I-So)-=-

(

1--

GpT

GT

)

,

f I

(10.101)

which suggests that a plot of (I-So)(Plz) vs. GpT will be a straight line from which GT can be estimated. Correct application of Eq. 10.101, however, requires estimates of the liquid hydrocarbon volumes formed as a function of pressure below the dewpoint. The most accurate source of these estimates is a laboratory analysis of the reservoir fluid samples. Unfortunately, laboratory analyses of fluid samples often are not available. An alternative material-balance technique is

i i, i

!1 !

GTBzgi=(GT-GpT)BZg'

""""""""""'"

(10.102)

where GTBzgi=reservoir PV occupied by the total gas, which includes gas and the gaseous equivalent of the produced condensates, at the initial reservoir pressure above the dewpoint, RB; (GT-GpT)BZg=reservoir PV occupied by hydrocarbon vapor phase and the vapor equivalent of liquid phase after some production at a pressure below the initial reservoir pressure and dewpoint pressure, RB; and Bz i and Bzg =gas FVF's based on two-phase Z factors at initial anJ later conditions, respectively, RB/Mscf. If we substitute Bzg/BZg =(pzzi)/(PiZZ) into Eq. 10.102 and rearrange, we have

~=~(iZz ZZi

GpT). GT

j

j

;

:A

(10.103)

where ZZiand Zz=two-phase gas deviation factors evaluated at in-

itial reservoir pressure and at a later pressure, respectively. The form of Eq. 10.103suggests that a plot of p/zz vs. GpTwill be a straight line for a volumetric gas-condensate reservoir when two-phase gas deviation factors are used. Two-phase gas deviation factors account for both gas and liquid phases in the reservoir. Fig. 10.17 is an example of the relationship between the equilibrium gas (Le., single-phase gas) and twophase deviation factors for a gas-condensatereservoir. At pressures above the dewpoint, the single- and two-phase Zfactors are equal; at pressures below the dewpoint, however, the two-phase Zfactors are lower than those for the single-phase gas. Ideally, two-phase gas deviation factors are determined from a laboratory analysis of reservoir fluid samples. Specifically, these two-phase Z factors are measured from a constant-volume depletion study.27-Z9However, in the absence of a laboratory study, correiationsZ9are availablefor estimatingtwo-phase Zfactors from properties of the well-stream fluids. Gas-Condensate Reservoirs With Water VaporiZiltion. In this section, we developa material-balanceequation for gas-condensate reservoirs in which both phase changes and water vaporization occur. Similar to Humphreys'30 work, we include the effects of rock and watercompressibilities,which are often significantin deep, high-pressure reservoirs. The reservoir PV is occupied initially by hydrocarbon and water vapor phases as well as the connate liquid phase, or Vpi= V\'i+ V...i

(10.104)

where V'.i= initial reservoir PV occupied by hydrocarbon and , . (10.98) water vapors, RB, and V...i=initial reservoir PV occupied by the (1-So) liquid water, RB. If the reservoir pressure is above the dewpoint. connate water itituting Eq. 10.98 into Eq. 10.95 and combining with Eq. is the only liquid phase present. From the definition of water satuyields an expression for the reservoir PV at later conditions: ration, we can write the initial reservoir PV occupied by the liquid -G )B + So(GT-GpT)Bg . .... . . . . . . . . . . . . . (10.99) phase as pT g I-SQ (10.105) Vw;=Sw;Vp;' (GT-GpT)Bg

.. , ~ i, ~ ;

i

!

j! ii ;:. ji "

]\.

250 Gn.N

1.0

0.9

Gas-Plus