well performance equations

Chapter 35

Well Performance Equations R.A. Wattenbarger, Texas A&M U.*

Introduction This chapter summarizes the equations that apply to the performance of a well in a reservoir. The equations are used to calculate the relationship between rate and pressure of a well and the properties of the fluids and formation. These equations apply only in the “drainage area” of the well and do not describe the entire reservoir performance, except for the case of a single-well depletion reservoir. For more complete treatment of the entire reservoir performance, refer to Chap. 37-Solution-Gas-Drive Oil Reservoirs, Chap. 38-Waterdrive Oil Reservoirs, or Chap. 39-Gas Condensate Reservoirs.

There have been several excellent references developed over the past few years on well pressure behavior. I-’ These are much more detailed than this chapter and the reader should be aware of them. This chapter is a brief summary of this technology.

Diffusivity Equation The equations that relate pressure and rates for a well are solutions of the diffusivity equation. This equation can be written as

v2p= 1 ~PC, ap

o.ooo264 k at ) . . . . . .

where p = pressure, psi, 4 = porosity of reservoir rock, fraction, p = fluid viscosity, cp,

C I= total compressibility of system (see Eq. 5), psi-‘,

k = permeability of reservoir rock, md, and f = time, hours.

‘Aulhor of Ihe onginal chapter on ths topic in the 1462 edltmn was Ralph F. Neilsen

The vector notation used on the left side of the equation has the following meaning. In one dimension (lD),

6 1 her ap -= ax2

o.ooo264 k at ) . . . . . . (24

where x is the distance coordinate in a one-dimensional flow system, ft. In two dimensions (2D),

a2p a2p 1 a,:+2=

4wt aP -- ay 0.000264 k at’ .““““.

. (2b)

where x and y are distance coordinates in a 2D flow system, ft. In radial coordinates,

3% 1 ap 1

ar2+--= hc, ap --

r ar 0.000264 k ar ’ “.‘.’ ” ” @cl

where r is the radius in radial flow system, ft. Eq. 2c gives the most useful solution of the diffusivity

equation for reservoir and well performance. The geometry of the reservoir is in cylindrical coor-

dinates with an inner radius, rw, into which the fluid flows at a constant rate and an outer boundary, rc , which is closed and represents the outer boundary of the reservoir. The solutions of this cylindrical coordinate problem have been presented by van Everdingen and Hurst’ and are presented again in Chap. 38.

Eq. 1 is a linear partial differential equation that models how pressure changes with location and time. Theoreti- cally, solutions of Eq. 1 are valid only for reservoirs where the fluid and rock properties are constant. The application of the solutions of Eq. 1, then, are literally applicable for fluids with constant compressibility and

35-2 PETROLEUM ENGINEERING HANDBOOK

Fig. 35.1-Pressure behavior for constant rate in a closed reservoir.

viscosity and for formations with constant permeability. These conditions are very nearly met in the case of aquifer flow or for oil reservoir flow at pressures above the bub- blepoint. The solutions of Eqs. 1 and 2 can be extended to multiphase reservoir flow for most practical cases.

Multiphase Flow When more than one phase exists in the reservoir, it is still possible to write the differential equation in a form similar to Eq. 1. This equation was presented by Martin’ as

v 5 vp= (> P t

o.;2H cpc,;. . . . , . . (3)

This equation shows that the conditions of homogeneity are not necessarily met. The concepts of total mobility, (k/p), , and total compressibility, ct, are introduced.

The total mobility is the sum of the individual phase mobility as follows.

k ko kg kw 0 - =-+-+-, . . . . . . . . . . . . . . . . . (4)
P f PO Pg Pw
TABLE 3&l-ANALOGIES OF SINGLE-PHASE VALUE TO MULTIPHASE EQUIVALENT

Single-Phase Value

w C

98

Multiphase Equivalent

WI4 t Ct

9&3,

. . . . . . . . . . . . . . . . . . . . . . . . . (6)

where -

;; I total reservoir flow rate, STB/D, total formation volume factor, RB/STB,

40 = oil flow rate, STBID, B, = oil formation volume factor, RBISTB, qg = gas flow rate, Mscf/D, R, = solution gas-oil ratio, scf/STB, B, = gas formation volume factor, res cu

where k, = effective permeability to oil, md, k, = effective permeability to gas, md, k, = effective permeability to water, md, PO = oil viscosity, cp,

PLp = gas viscosity, cp, and CL, = water viscosity, cp.

The total compressibility is the volumetrically weighted average of the compressibilities of the fluids and pore space as follows.

ct =cf+s,c, +s,c, fSwC,, . . (5)

where cf = formation compressibility, psi - ’ ,

S, = oil saturation, fraction of pore volume (W,

CO = oil compressibility, psi-’ , S, = gas saturation, fraction of PV,

Cg = gas compressibility, psi - t , S, = water saturation, fraction of PV, and CW = water compressibility, psi - ’ .

The flow rate also must be expressed in terms of the equivalent total flow rate for multiphase flow. The expression for total reservoir flow rate is

q,B,=q,B,+(1,000q,-R,q,)B,/5.615+q,B,,

ftlscf water flow rate, STBID, and water formation volume factor, RB/STB.

Martin’s equation is a nonlinear partial differential equation. Therefore the general case does not have analytical solutions. However, for practical purposes, Eqs. 3 through 6 can be used for most well performance equations if the meaning of the mobility, compressibility, and flow rate are taken in this general three-phase sense. The single-phase solutions of Eq. 1 can be applied to the multiphase case by using the analogies given in Table 35.1.

Oil Well Performance Well Pressure Performance-Closed Reservoir The performance of a constant-rate well in a closed reservoir (of any geometry or heterogeneity) has the general form shown in Fig. 35.1.

The lower curve of Fig. 35.1 shows that the wellbore flowing pressure, p 4, goes through a rapid pressure drop

WELL PERFORMANCE EQUATIONS 35-3

at early (transient) times and then flattens out until it reaches a constant slope. On this coordinate plot, the closed-reservoir, constant-rate case has the properties

aP, --co

at

and

a*Pwf >O at* - .

When p of reaches a straight line on the coordinate plot, the period of pseudosteady state has been reached. Ev- ery pressure point in the reservoir declines at the same constant rate of depletion after that time. Of particular importance is the decline of the average reservoir pressure, j?~, which assumes the pseudosteady-state depletion rate from the very beginning of production.

The constant elope of Fig. 35.1 is valid only for constant-compressibility single-phase fluid. However, the general concept of the transient period and the pseudosteady-state period is the same for a multiphase flow with changing compressibilities. The PR slope would be changing according to the changes in compressibility, and the pR curve after a pseudosteady-state would not be ex- actly parallel to the p,,,f curve. This nonideal behavior would be typical of a solution gas drive reservoir or a dry gas reservoir where the compressibility and mobili- ties are continually changing. The infinite-acting solutions and the pseudosteady-state solutions to follow are still ap-

log t

Fig. 35.2-Typical constant-rate drawdown test graph.

plicable for the multiphase flow case by using the analogies in Table 35.1. The value of pR, however, must be calculated by the material balance method that applies for this case.

Infinite-Acting Solution (MTR) The pressure behavior of constant-rate flow in a closed reservoir goes through several periods: the early-time region (ETR), middle-time region (MTR), and late-time region (LTR). These periods are illustrated on a semilog plot ofp$ vs. log t in Fig. 35.2. The MTR solution is discussed first.

Eq. 1 can be solved for the infinite-reservoir case, which is useful for application at early times. The solution applies to a well producing at constant rate, beginning at t=O, and a homogeneous reservoir of constant thickness.

PO=

10 I IO 102 IOJ IO’

tDr = tD/rD 2

Fig. X.3-Dimensionless pressure for a single well in an infinite system, no wellbore storage, no skin. Exponential-integral solution.


There are two important solutions for the intinite- reservoir case. One solution8 assumes that the wellbore has a finite radius, r,. This solution is used mostly for aquifer behavior with the oil field being the inner radius rather than a wellbore. This solution is given in Chap. 38 for the infinite-aquifer case.

A simpler solution applies for well behavior. This solution, called the “line-source” or “exponential-integral” solution, assumes that the wellbore radius, rw, approaches zero. This solution has the form

where po = kh(pi-p)l(141.2 q&)=dimensionless

pressure, rD = r/r,,, =dimensionless radius, tD = (O.O00264kt)l$+c,r,.’ =dimensionless time,

h = formation thickness, ft, pi = initial pressure, psi, and rw = wellbore radius, ft.

The exponential-integral function, Ei, is a special function that results from the solution of the line-source problem. A more practical solution to the problem is the plot of the dimensionless pD vs. t&rD2, which is shown in Fig. 35.3. The tDr term is the dimensionless time based on external radius, re. Fig. 35.3 can be used to determine the pressure at any time and radius from the producing well. This solution is valid as long as the radius at which the pressure is calculated is greater than 20 r,+ or at the wellbore of the producing well (at r,v) at a value of fo/rD * > 10.

Fig. 35.3 is used mostly to determine the pressure at distances away from the well such as at a nearby well location during an interference test.

The more common solution of the exponential integral solution is the “semilog straight line solution,” which applies after to is greater than 100. After this time, Eq. 8 applies at the wellbore:

pD=% hl t,+0.406. (8) .....................

In customary oilfield units, this equation has the form

kt pKf=pj -In log

+crrw2 -3.23

> , . . (9)

where m equals (162.6qBp)lkh and p,+f is the flowing bottomhole pressure, psi. This equation results in a semilog plot of p,,f vs. log t with a slope of -m psi/cycle (the MTR of Fig. 35.2.)

Eqs. 7 through 9 are used for infinite-acting solutions before the effects of boundaries affect the pressure transient behavior. When the closest boundary begins affect- ing the behavior at the wellbore, this time is the end of the semilog straight line, t,,d . The last column in Table 35.2 shows tend for various drainage shapes (shape factors).

Skin Effect The solutions to Eq. 1 are modified to account for formation damage near the wellbore. The damage near the wellbore can be considered concentrated into a very thin radius around the wellbore such that the thickness of the damage is insignificant but a finite pressure drop results from this damage.

Fig. 35.4 shows a sketch of the physical concept of the damaged region and Fig. 35.5 shows the pressure pro- file resulting from this damage.

The magnitude of the pressure drop caused by the skin effect Ap, is

Ap,=O.87ms, . . . . . . . . . . . . . . . (10)

where s is the skin effect, defined in terms of dimensionless pressure such that it would have the following effect on Eq. 8.

pD=% ln tD+o.@ts+s. . . . . . . . . . . .(ll)

The value of the skin effect is calculated from transient well test data such as a buildup test or a drawdown test. The exact nature of the cause of the skin effect might not bc known but might be caused by a combination of several factors. Some of these factors are (1) mud filtrate or mud damage near the wellbore, (2) the cement bond, (3) limited perforations through the casing and cement bond, and (4) partial penetration (completion). On the other hand, the value of the skin effect, s, might be nega- tive. This would indicate an improved wellbore condition, which might be caused by (1) improved permeability in the vicinity of the wellbore because of acidizing or other well treatments, (2) a vertical or horizontal hydraulic fracture at the wellbore, or (3) a wellbore at an angle rather than normal to the bedding plane.

The determination of the skin effect is important in determining the need for a workover or the benefits of a workover. The effect of the skin can be stated as a modifi- cation to the wellbore radius by calculating an effective wellbore radius, r’,,,, calculated by

r’w=r,e - s ............................. .(12)

This effective wellbore radius, rlw, can be considered the equivalent wellbore radius in an undamaged or un- improved formation, which would have the same flow characteristics as the actual well with the skin effect.

Wellbore Storage Effect (ETR) At very early times the fluid production tends to come from the expansion of the fluid in the wellbore rather than the formation. This tends to delay the production rate from the formation. The relationship between the surface production rate, the expansion of the wellbore fluids, and the formation production rate are shown in Eq. 13:

24C. Lb q$=q+L+

B at . . . (13)


TABLE 35.2-SHAPE FACTORS FOR VARIOUS CLOSED SINGLE-WELL DRAINAGE AREAS

In Bounded Reservoirs

0

0

A

n

q3

c&ID

In Vertically-Fractured Reservoirs*

IO x”xe ,m

,&

,@

,&

>@

In Waterdrive Reservoirs

0 In Reservoirs of Unknown

Production Character

@DA)pss Exact

cA In CA For tDA >

31.62 3.4538 - 1.3224 0.1 0.06 0.10

31.6 3.4532 - 1.3220 0.1 0.06 0.10

27.6 3.3178 - 1.2544 0.2 0.07 0.09

27.1 3.2995 - 1.2452 0.2 0.07 0.09

21.9 3.0865 - 1.1387 0.4 0.12 0.08

0.098 - 2.3227 f 1.5659 0.9 0.60 0.015

30.8828 3.4302 - 1.3106 0.1 0.05 0.09

12.9851 2.5638 - 0.8774 0.7 0.25 0.03

4.5132 1.5070 - 0.3490 0.6 0.30 0.025

3.3351 1.2045 -0.1977 0.7 0.25 0.01

21.8369 3.0836 -1.1373 0.3 0.15 0.025

10.8374 2.3830 - 0.7870 0.4 0.15 0.025

4.5141 1.5072 - 0.3491 1.5 0.50 0.06

2.0769 0.7390 + 0.0391 1.7 0.50 0.02

3.1573 1.1497 -0.1703 0.4 0.15 0.005

0.5813 - 0.5425 + 0.6758 2.0 0.60 0.02

0.1109 -2.1991 + 1.5041 3.0 0.60 0.005

5.3790 1.6825 - 0.4367 0.8 0.30 0.01

2.6896 0.9894 - 0.0902 0.8 0.30 0.01

0.2318 - 1.4619 +I.1355 4.0 2:oo 0.03

0.1155 -2.1585 + 1.4838 4.0 2.00 0.01

2.3806 0.8589 - 0.0249 1 .o 0.40 0.025

2.6541 0.9761 - 0.0835

2.0348 0.7104 + 0.0493

1.9988 0.6924 + 0.0583

1.6620 0.5080 +0.1505

1.3127 0.2721 + 0.2685

0.7887 - 0.2374 + 0.5232

0.08 Cannot use

0.09 Cannot use

0.09 Cannot use

0.09 Cannot use

0.09 Cannot use

0.09 Cannot use

19.1 2.95

25.0 3.22

- 1.07

- 1.20

0.175

0.175

0.175

0.175

0.175

0.175

-

-

-

-

-

-

ftDA)end Use Infinite System

Less Than Solution With Less 1% Error Than 1% Error For t, > For t, <

‘Use (xJx,)’ in place of A#: for fractured systems.


EC

1

REGION OF DAMAGED

PERMEARILITY

SEALED

/ CIRCULAR

,UND+RIES

/

WELL BORE

Fig. 35.4-Radial flow model showing damaged zone.

where C, equals V,,,cd and qsj is the flow rate at the “sandface,” STB/D and C, is the wellbore storage constant, equal to the volume of the wellbore, V, , times the wellbore fluid compressibility, c wf.

The effect of the wellbore storage is to make the very early transient pressure behave as though it were reflect- ing production only from the wellbore fluid expansion. This pressure drop can be calculated from

qB Pi-Pwf’- 24C,t. . . . . . . . ..~..............

Note that this shows a linear relationship between & and time. Consequently, a p vs. t plot will be linear during the wellbore storage period. Also, a plot of log Ap vs. log I is a straight line with a slope of unity. This wellbore storage effect may last for just a few seconds or it may last for many hours-i.e., for a deep, low-permeability gas well that has a large storage volume in the wellbore, a high-compressibility gas, and great resistance to flow from the formation.

After a period of time, this wellbore storage solution gives way to the semilog straight line (for the radial flow case). The period between the linear relationship and the semilog straight line is from one to one and one-half cycles of log t. Fig. 35.6 shows that Eq. 12 applies during ETR, then gives way to Eq. 11 during MTR. lo This log- log dimensionless plot has the same shape as a plot of log (pi-p,,+) vs. log r. This is sometimes called a “type curve.”

Pseudosteady-State Behavior (LTR) After a well produces at constant rate for a period of time, the boundary effects interrupt the infinite-acting pressure behavior. If the well is in an irregularly shaped drainage area, the closest boundary to the well causes the earliest departure from the infinite-acting pressure solution. Af- ter a transition period, the well begins pseudosteady-state behavior. The pseudosteady-state behavior begins after the effects of the farthest boundary have been felt at the wellbore.

When pseudosteady-state behavior begins (see Fig. 35.2) the rate of pressure decline, (a~/&)~~~, is constant

log r

Fig. 35.5--Schematic of pressure distribution near wellbore.

at every point throughout the reservoir. This is a depletion period at which every point of the reservoir drops at a rate according to the pore volume, VP, and compressibility of the drainage area, ct,

ap ( >

-0.234qB --$ pss= . . . . . “~‘pCt (15)

During pseudosteady-state behavior, wellbore pressure is related to the average reservoir pressure, PR, by a productivity index (PI), J, as follows.

q=J(pR -p,j). . . . . . . . . . . . . . . . (16)

This PI equation relates the pressure drawdown to the production rate. For a circular drainage area we can write out the complete expression for the PI equation as

7.08x 10 -3khl(B/t) _ 4=

In r,/r, -0.75s~ 1 (Pi?-Pwj), ‘. . . . . . (17)

where re is the exterior boundary radius, ft. Note that the quantity in brackets is equivalent to J in Eq. 16 for the circular drainage area. J is a constant if the viscosity and formation volume factor of the producing fluid are constant. If these fluid properties are not constant, Eqs. 16 and 17 still apply but the PI value changes with the changing fluid properties. For multiphase flow these equations still can be used by substituting the definition in Ta- ble 35.1 into Eqs. 16 and 17.

Eq. 17 has to be modified if the drainage area is not circular with the well in the center. A general form of the pseudosteady-state equation has been worked out by Dietz l1 and has been cited by other authors. I-5 The gen- eralized pseudosteady-state equation has the form

7.08x 10-3khl(Bp) 4= 2.2458 A 1 (PR -pwf), . . . (18)

‘15 ln-- CA rw

2 +s


Fig. 35.6-Dimensionless pressure for a single well in an infinite reservoir including wellbore storage and a finite skin-composite reservoir.

where A is the drainage area, sq ft. and CA is the shape factor (Table 35.2). This equation can be applied by using the values for CA in Table 35.2 or by moving the terms in the denominator to the form

7.08 x 10 -3khl(&) 9=

2.2458 A I

@R -Pwfh % hl-

CA

+% lnT+S rw

. . . . . . . . . . . (19)

This form is easier to use because the first term of the denominator also is tabulated in Table 35.2.

In Table 35.2, x, is the distance from the well to the side of the square drainage area, and xf is the distance from the well to either end of the vertical fracture. Table 35.2 also shows the dimensionless time, tom, at which the infinite-acting solution ends, and also the time at which pseudosteady state begins, (t~~)~~,r.

Example Problem 1 (Transient and Pseudosteady State). A well is centered in an approximately square drainage area. The following data are given.

A = 1.74~ lo6 sq ft (40 acres), h = 21 ft, s = 1.6,

rw = 0.25 ft, k, = 45 md, PO = 1.5 cp,

$fJ = 0.18, cc7 = 8.5~10~~ psi-‘, CW = 3.2~10~~ psi-‘, cf = 3.0X10p6 psi-‘,

S, = 0.25, B, = 1.12, and pi = 5,100 psi.

Calculate the bottomhole pressure (BHP), pwf, after 12 hours and after 120 days for a constant oil production rate of 80 STB/D.

Solution. From Eq. 5,

Cr=CffS,C, +s,c,

=[3.0+(0.75)(8.5)+(0.25)(3.2)] x 1O-6

=10.2X 10e6 psi-‘.

Calculate the time required to reach pseudosteady state. From Table 35.2,

(tDA)pss=O.l= O.O00264(45)t,,,

(O.18)(1.5)(1O.2x1O-6)(1.74x1O6)’

where tpss is 40.3 hours. So the well is infinite acting after 12 hours. By using Eq. 11,

p~=‘h h tD+o.do&i+s.

By using the definitions of pD and tD in Eq. 8, we have

WKWW~-p,vf)

141.2(80)(1.12)(1.5)

=% In 0.000264(45)( 12)

(O.18)(1.5)(1O.2x1O-6)(O.25)2 +0.4045+1.6;

0.0498(5,100-p,,&=% In (8.28~10~)+0.4045+1.6;

5,100-p,,=(8.82)/(0.0498)= 177; and

p,f=4,923 psi at 12 hours.

35-8

0 tl 12 t3 t4 t N-2 t N-I

FLOW TIME, t, HOURS

Fig. 35.7-Schematic representation of a variable production- rate schedule.

At 120 days, the well is in pseudosteady state (greater than 40.3 hours). First, calculate PR. Using Eq. 15, the rate of pressure decline can be calculated.

aP (-> - -0.234qB

at P==

“pc,

-0.234(80)( 1.12)

= (21)(O.18)(1.74x1O6)(1O.2x1O-6)

= -0.313 psi/hr.

p,=5,100-0.313(120)(24)

=4,199 psi.

Now, using Eq. 19,

7.08x 1O-3 khl(&) 90 = 2.2458 A

% In- +% In-+s 1 CA rM

(80) = 7.08x10-3(45)(21)!(1.12x1.5)

1.74x 10-6 - 1.3224+ % In

(0.25)* 1 +1.6

*(4,199-p++&

3.982 (80) =

-1.3224+8.571+1.6 1 (4,199-p!&

PETROLEUM ENGINEERING HANDBOOK

4,199-p,,f= 178; and

p,,=4,021 psi at 120 days.

Production Rate Variation (Superposition) These solutions have included only the constant-rate case. Of general interest, of course, are the cases where rate changes with time. These cases are best handled by using the principle of superposition.

The principle of superposition amounts to dividing the production history into a sequence of rate changes such as that shown in Fig. 35.7. The total effect of the production on the pressure response, Ap, is the additive effects of each of the rate changes. In Fig. 35.7, rate q1 applies from t=O to the current time. At t, the rate in- creases to q2. The effect of this rate change can be viewed as an incremental rate, q2 -91, which has been in effect for a period of time t-t l . Then q3 would also be seen as a rate change, q3 -92, which has been in effect for a period of time t- 12. The effect of all these rate changes is computed by superposing the solutions that applied to each rate change and its corresponding time that it has been in effect. The equation for computing the total pressure drop, Ap,, is

N

p; -p,#= c (qj -qj-,)f((t-tr-,) , . . (20) i=l

when qieI =0 when i=l. The functionf(t) can be called the unit responsefinc-

tion. The unit response function is the pressure drop, pi -pKf, which occurs at time f for a unit production rate (q= 1). The unit response functions may be quantified by the cases described such as the wellbore storage equation at early times (ETR), the semilog straight line solution at MTR, and finally the pseudosteady-state solution at later times (LTR). For example, if q 1 had been in effect for a time longer than tpss, its contribution to the pressure drop at time t would be calculated from the pseudosteady- state equations, which would comprise the calculation of the reduction in p from Eq. 15 and the pressure drop from p R to pwf in Eq. 16. The effect of the second rate might be still in the transient period, which would call for Eq. 11 to be applied.

Note that the calculation of the pressure decline of p R can be calculated with Eq. 15 only for the constant- compressibility case. For the general case, such as a solution gas drive reservoir, the appropriate material balance equations would be applicable to calculate PR. If the last rate change has been in effect for a time greater than tP,rS and the system has constant compressibility, the following simplification can be made for Eq. 15.

PR’Pi-

5.615 NpB, . . . . . VpCr (21)

The following example problem shows how superposition can be applied for a case where both pseudosteady- state and transient pressure drops are added.

WELL PERFORMANCE EOUATIONS 35-9

the slight increase in cumulative barrels ( 15 STB), which is negligible in this case.

Exynple Problem 2 (Superposition). The well in Ex- ample Problem 1 produces according to the following schedule.

time (hours) (SI%D) Oto2 300 2 to 8 120

thereafter 80

Calculate p,,,, at 12 hours and at 120 days. So&ion. As we observed in Example Problem 1, the

well was infinite acting after 12 hours, so we use Eq. 20.

N

pi-Pwf= C (4i-qi-Of(f-ti-1) i= I

We first needf(t), the unit response function. We can use Eq. 11 to find Ap in terms oft for q=l:

pD=% In tD +0.4@5+3,

141.2(1)(1.12)(1.5)

=% In 0.000264(45)?

(0.18)( 1 .S)( 10.2 x 10 -6)(0.25)2

+0.4045+1.6,

3.98Ap= 1/2 In 6.90x 104t+2.004, and

Ap=O.1256 ln(6.90x lO”t)+0.504,

so

f(t)=Ap=O.1256 ln(6.90x104t)+0.504.

Substituting into Eq. 20,

+@I3 -921f(t--12);

+(120-3OO)f(12-2)

+(80-12O)f(12-8),

so the values off(l2), f(lO), andf(4) are used, giving

5,100-p,,=(300)[0.1256 ln(6.9x lo4 x 12)+0.504]

-(180)[0.1256 ln(6.9x104 x 10)+0.504]

-(40)[0.1256 ln(6.9~10~~4)+0.504]

=(300)(2.22)

-(180)(2.19)

-(40)(2.08)

= 189;

p,,=4,911 psi at 12 hours.

At 120 days, the well has a cumulative production of

N, =300 STB/D x (2/24 days)

+ 120 STBlD x (6/24 days)

t80 STBiDx(l19.5 days)

=9.615 STB.

Using Eq. 21,

pREpi-

5.615NpB,

vpct

=(5,100)- 5.615(9,615)(1.12)

(21)(0.18)(1.74x106)(10.2x10-“)

=5,100-901=4,199.

Using Eq. 19 (the same as Example Problem l), we calculate

and again,

pwf=4,199-178=4,021 psi at 120 days.

The effect of the early rate variation is “forgotten” after the rate is constant for tpss =40.3 hours, except for


Gas Well Performance The performance of gas wells is similar to oil wells (liquid reservoirs) except for two major differences: (1) the fluid properties of gas change dramatically with pressure and (2) flow can become partially turbulent near the wellbore, resulting in a rate-dependent skin factor. These two factors are discussed and alternative forms of gas performance equations are presented.

The application of these principles to gas flow is only slightly more complicated than to liquid flow, but there is often much confusion about gas wells. There are several reasons for this. One reason is that there are many ver- sions of gas flow equations in the literature. Some are in terms of p, some in terms of p2, and some in terms of a real gas pseudopressure, m(p). All these equations can be used and are valid forms. Another reason for confusion is the different coefficients in the equations, which sometimes arise from the assumed temperature and pressure base of a standard cubic foot of gas. The following equations use only the symbols T,, and psC, since the pressure base in different areas does vary significantly.

Still another reason for confusion is that deliverability testing has been customary with gas wells because of government requirements. Deliverability testing, based onalog(pR2 -pwf2) vs. log qg plot, is largely an empirical approach. The deliverability plot approach was developed mainly for low-pressure gas wells and does not work well with the deeper, higher-temperature, and higher-pressure wells that are more common today.

The Effect of Gas Properties In the derivation of the diffusivity equation, the form of Eq. 1 is not achieved because the values of z and p vary with pressure. Consequently the following form occurs in the derivation.

&vp= 1 4 ap --

w o.ooo264 k at , . . . . . . . . (22)

where L is the dimensionless gas-law deviation factor. This equation is a nonlinear partial differential equation and cannot be solved analytically by the methods applied to Eq. 1.

A method for “linearizing” the partial differential equation was developed by Al-Hussainy er al. l2 They introduced a real gaspseudopressure, which may be defined as

m(p)=2;p$p. . . . . . . . .(23)

This pressure-dependent function integrates the variations ofp, Z, and ,U with pressure. When this function is introduced into the derivation of the diffusivity equation, the diffusivity equation for a real gas takes the form

Vm(p)= 1 4cLcg WP)

o.ooo264 k ar . . . .

This equation still is not quite a linear differential equation because p and cR vary significantly with pressure.

The gas compressibility, cg , can be expressed in terms of 2 as

1 Id.2 cg=----.

P zQ

. . . . . . . . . ____ __ (25)

For practical purposes, however, Eq. 23 can be taken as a linear differential equation in terms of m(p). This was confirmed by the result of computer simulations per- formed by Wattenbarger and Ramey. l3 They showed that the pressure transient equations can be used, with very good approximation, in terms of m(p). After pseudosteady-state, PI equations similar to Eqs. 16 through 19 can be used.

The application of the m(p) solutions is not difficult. the values of m( p) vs. p can be determined by graphical integration or can be calculated with computer programs that use built-in correlations to estimate the variation of z and p with pressure.

Since our equations and graphical techniques depend on equations of a straight line of p either on a linear plot or a semilog plot, it is worth analyzing how the slopes of m(p) are related to the slopes of p plots, or p2 plots; we can show that the derivative of m(p) with respect to, for example, log t is as follows.

am(p) --=c$&=<t,&. .C2@ wag 4

These relationships indicate that an m(p) plot, or a p plot, or ap* plot can be used and then the relationships in Eq. 26 applied. The m(p) plot is preferable because it is most likely to have the proper semilog straight line. Thep and p* plots can be used as shortcuts if the proper MTR slope is identified. For example, the slope of a p vs. log r plot can be determined from a plot and then the value of the slope of m(p) vs. log t can be calculated by using Eq. 26 without ever actually plotting values of m(p).

Non-Darcy Flow Darcy’s law applies to gases at lower rates (laminar flow), which are found throughout the reservoir. However, near the wellbore the rates can become extremely high because of the converging flow as the gas approaches the wellbore. At these rates inertial e&ts can become important and Darcy’s law no longer applies. The inertial effects take the form of distorted flow paths and also turbulence in different locations in the pore structure. Although the exact nature of this microscopic flow is not known in the reservoir, the net effect is a higher pressure gradient when these inertial effects become important.

For laminar flow we can rearrange Darcy’s law to the following form.

ap P --.-z--v ax k , ~,............................ . (30)

where apldx is the pressure gradient and v is the macroscopic (Darcy) fluid velocity. At the higher rates, when

WELL PERFORMANCE EQUATIONS

inertial effects become important, the Forchheimer equation is used:

(31)

where p is the fluid density and F, is the turbulence factor. The right side of Eq. 3 1 contains a term for viscous forces and a term for inertial forces, both of which con- tribute to the pressure loss.

Although a number of workers have correlated the value of F, with rock properties, for practial purposes the velocity varies too much in the vertical direction near a wellbore to predict the effect of non-Darcy flow in a particular well. One practical approach is to consider the nonDarcy effect near the wellbore as a rate-dependent part of the skin effect:

s’=s+FDa 1 qg 1 , . . . . . . . . . . .(32)

where FD, is the non-Darcy (turbulence) factor, (lo3 cu ft/D) -’ , 1 qg / is the absolute value of gas rate, lo3 cu ft/D, and s’ is the effective skin effect of a well flowing at a rate qg. Fig. 35.8 shows how s’ varies with rate. The value of FD, varies with pressure but for simplicity can be considered constant. The value of FD, must be evaluated by transient testing of the well at several rates and determining corresponding values of s’

The transient equations (MTR) and pseudosteady-state equations (LTR) are modified for gas wells as shown in the following.

Infinite-Acting Gas Reservoir (MTR) The transient solution for the infinite-acting gas reservoir is analogous to the liquid case shown in Eq. 11. Eq. 11 then must be modified for the effect of non-Darcy flow and fluid property variation with pressure. This results in the following equation.

mD=% In t,+0.4045+s+F, 1 qg ) , . . .(33)

where

and mD = dimensionless m(p),

tD = dimensionless time, T,, = standard condition temperature, “R, PSC = standard condition pressure, psia, TR = reservoir temperature, “R,

m(pi) = m(p) at initial pressure pi, psia2icp, and m(pWf) = m(p) at wellhore flowing pressure pWf,

psia2/cp.

The value of TV is evaluated with &LC evaluated at the initial pressure.

/ / /

*< s’=s

Production rate, q,

Fig. 35.8-Skin factor determination.

Before putting Eq. 33 into a more practical form, consider that the pressure drop term, m( p i) - m( p ,+,f), can be stated as Am(p) and can be related to Ap and Ap2 by the relationships

Am(p)= (z&Q= (;):2. . . (34)

The average values shown in parentheses are the integrat- ed average values over the pressure range. For practical purposes it is accurate to evaluate these average quanti- ties at the midpoint pressure. In other words, 2plzpp is evaluated at j, where ji is equal to (jYR +p,f)/2 and (1l~l.r)~ is evaluated at j?, where jY is equal to (jY~+p,,,f)I2, or ,/(p~+pK,/)/2 for the p* equation. For the infinite-acting reservoir, the average reservoir pressure, PR, is the same as pi.

These relationships are important because they allow us to account for the variation of fluid properties, within engineering accurac

Y> and still express equations simply

in terms of p and p . Eq. 33, when put in more practical form, can be expressed in terms of m(p), p, or p2, as

m(pi)-m(pwf) 1 2.303 0.000264kc

=-log 2 (4P41~W2

+0.4045+~+F~, ( qn / , . . . . . . . (35a)

1.987x 10 -5 (p > t (Pi-P&$)

P

2.303 0.000264kt =-log

2 (4W) i r w 2

+0.4&t5+S+F, ( qe 1 , . . . . . . . . (35b)

35-12

0.08

GAS GRAVITY = 0.7

REDUCED TEMPERATURE= 17(195’F)

0.06

0 2,000 4,000 6,000 8,000 10,000

P, psla

Flg. 35.9-Typical variation of m(p) and zp with pressure.

1.987x 1O-5

2.303 O.ooo264kt =-log

2 (4Pc)irw 2

+0.4045+s+FDa I qg I , . . . . . . (35~)

where (@PC); =&LC evaluated at pi. Eq. 35 can be used to predict p,,f for the infinite-acting period (MTR) between the wellbore storage period and the beginning of pseudosteady state.

Fig. 35.9 shows a typical relationship of zp with pressure. The value of Z,U is almost constant when p is below 2,000 psia. This makes the p2 type of equation fairly accurate below 2,000 psi because Z,U can be taken out of the integral in Eq. 23 if zp is constant. p2 plots and equations tend to work well in reservoir pressures less than 2,000 psia.

Fig. 35.9 also shows that m(p) tends to be linear with pat higher pressures (above 3,000 psia). This means that p plots and equations tend to work well for higher-pressure reservoirs. If there is a doubt about whether these p* or p simplifications apply to a particular reservoir, then m(p) plots and equations should be used.


Pseudosteady-State Solutions (LTR) The pseudosteady-state solutions are analogous to the liquid solutions and can be put in essentially the same form. The only changes are to allow for the changes of fluid properties with pressure and non-Darcy flow. The inclu- sion of these effects is the same as discussed above. The result is the following form of the pseudosteady-state equations, in terms of m(p), p, and p*.

kh

2.2458A % ln----

CAT,' +~+FD, kg I

* [

m(p)-m(p,f) 1 , _. _. _. _. (364

where m@)=m(p) at p R, psia’/cp, and CA =shape factor from Table 35.2.

kh

2.2458A Vi In-

C,4rw2 +s+FDa I qg I

(PR-p,,,,), . . . . . . . . . . . (36b) P

and

kh

2.24584 l/z In-

c*rw2 +~+FDcI I qs I

Eqs. 36 have general application for pseudosteady-state gas flow. Note that these forms of the pseudosteady-state equations are considerably different from the gas deliverability approach that is used extensively. The gas deliver- abili

9 approach is empirical and based on a log-log plot

ofp -p,,,,’ vs. qg. The comparison between Eqs. 36 and the deliverability plot approach is discussed by Lee. 5

WELL PERFORMANCE EQUATIONS

Long-Term Forecast

Long-term forecasting can be accomplished in a fairly straightforward manner using Eqs. 36 along with a p R/z plot. The CR/z plot, of course, is simply a material balance for a closed gas reservoir. Through this plot the value of P.Q can be determined for any value of cumulative production, G,. Given this value of p R, one of the forms of Eqs. 36 then can be used to determine qx.

Note that in deep, high-pressure reservoirs, the influ- ence of formation and water compressibility can become important compared with gas compressibility. At these high pressures, greater than about 6,000 psig, the p R/Z plot should be modified to account for the formation and water compressibilities. A technique for this modified p,& plot is presented by Ramagost and Farshad. tJ

A complete forecast of production rate vs. time can be generated by converting the cumulative production to a time scale. The value ofp%f might be fixed as a condition of the production forecast, or it may be solved si- multaneously with wellbore hydraulic relationships, such as given in Chap. 34.

Example Problem 3. A gas well produces from a drainage area that approximates a 4: 1 rectangle with the well in the center. The following data apply.

A = 6.96x lo6 sq ft (160 acres), h = 34 ft, s = 2.3,

FD, = 0.0052 (lo3 cu MD)-‘, rw = 0.23 ft, k, = 0.52 md,

ZPR = see Fig. 35.9, 4 = 0.11,

TR = 210”F+460=670”R, T,, = 6WF+460=520”R, pSc = 14.7 psia, and j?~ = 4,150 psia.

Calculate the pseudosteady-state rate, qg , if pWf= 1,500 psia.

Solution. Use Eq. 36b-the simplest form of the pseudosteady-state equation.

= (4,150 + 1,500)/2

=2,825 psia.

From Fig. 35.9, we estimate ~~~ at 2,825 psia as

zpR =0.0165

2(2,825) =-=3.42x lo5

0.0165

35-13

From Table 35.2,

CA z5.3790.

Eq. 36b is

kh

2.24584 l/2 In T+S+FDO hi: 1

CArw

2P (-> (PR-Pwf);

z/J p

q,=1.987x10-5 (520)

(14.7)(670)

(0.52)(34)

% In 2.2458(6.96x 106)

(5.379)(0.23)* +2.3+0.0052 1 qK 1

*(3.42x105)(4,150-1,500)

= 1.987x 10 -5(0.0528) 17.68

8.91+2.3+0.0052 1 qR 1

.(3.42x 105)(2,650)

1.68~10~

11.21+0.0052 ( qg 1

(11.21+0.0052 1 qg I)q,=1.68x104.

This equation can be solved as a quadratic equation, or simply by trial and error, by using estimates of I qx I starting with I qg 1 =0:

(11.21+O)q,=1.68x104

4,: = 1,499.

Next, try

(11.21+0.0052x1,499)q,=1.68x104;

qg =884.

Next, try

(11.21+0.0052x884)q,=1.68x 104;

qR = 1,063.


970

.(I, 960 8

; 950 0.

E 2

940

2 E

930

P 920

?I ii 910

FLOW TIME, t, hours

Fig. B&10-Semilog data plot for drawdown test.

Next, try

(11.21+0.0052x1,063)q,=1.68x104;

qg=1m4

until the solution converges at

qg = 1,018 x lo3 cu ft/D.

Transient Well Test Analysis The subject of transient well test analysis can be very complicated and has been covered very thoroughly in the literature. I-5 These references show not only the straightforward cases of transient well test analysis but also go into many exceptions, alternative techniques for analysis, and other complications. It is the intent here to cover only the most straightforward and routine methods for analysis of oil and gas wells.

The most common values to calculate from a transient well test analysis are kh, s, and PR. With these three values plus a knowledge of the drainage area and shape of the drainage area (values of CA and A), the flow rate can be calculated or forecast for a particular BHP, p,,,f, by using the pseudosteady-state equations. The method of analyzing kh and s for a drawdown test and a buildup test are summarized now.

Drawdown Test The drawdown test is accomplished simply by putting a well on a constant production rate after the well has been shut in. Variations of the drawdown test involve analysis of variable rates, but only the constant-rate case is covered here. The analysis is based on the infinite-acting solution (MTR). The data are plotted on a pressure vs. log time semilog plot and the slope of the plot, m, is determined graphically in units of psi/cycle (see Fig. 35.10). The equations for determining w1 for an oil well or a gas well are as follows.

For oil wells,

k,h= - lf=%oBo~o

, . . . . . . . . . . . . . . . . . . . (374 m

and for gas wells,

k,h= -5.792 x 104q,(p,,TR/Ts,)

. . . . . . . . m*

Wb)

where m* is the slope of m(p) plot,

k,h= -5.W!X ~04q,(p,,TR/Ts,)

m’ wb ’

. . . . . . . . . . . . . . . . . . . . . . . . . . . (37c)

where m’ is the slope of p plot, or

k,h= -5.792x 104q,(p,,TR/T,,)

nP (z~~)wb. Wd)

where m” is the slope ofp* plot and subscript wb refers to wellbore. The values of zpl2p in Eq. 37c and zp in Eq. 37d are evaluated at pW, rather than’(pR+p,,)/2, which is used in the pseudosteady-state equations.

The value of the skin effect, s, is determined from one of the following equations for oil and gas wells.

For oil wells,

Pi-P1

I

k x=1.151 -log-

112 ~wtr,2

. . . . . . . . . . . . . . . . . . . . . . . . (384

where p 1 is the pressure at AZ = 1 hour; and for gas wells,

m(pi)-m(pl) k s=1.151

m* -log-

+crr,2

. . . . . . . . . . . . . . . . . . . . . . . . . . . (38b)

I

k -log---

hc,r,2

or

. . . . . . . . . . . . . . . . . . . . . . . . . . . (38~)

. . . . . . . . . . . . . . . . . . . . . . . . . . . (3W

The disadvantage of this equation (compared to buildup testing) is that pi must be known to calculate S.


It is important to evaluate the proper semilog straight line. In many cases it is difficult to tell whether an apparent semilog straight line is in the MTR solution or is still being affected by wellbore effects (ETR) . It is often help- ful to make a log-log plot of Pi -pwf vs. flowing time, t, to analyze when the wellbore effects are finished. A straight line with a slope of unity on this log-log plot in- dicates that the pressure behavior is being totally domi- nated by wellbore storage. The semilog straight line then can be expected to begin at about 1.5 log cycles after the data points leave the log-log straight line of unity slope.

Buildup Testing Buildup testing is more common than drawdown testing. The main reason for this is that the well rate is known when the well is shut in (q=O). The analysis of a buildup test is based on the assumption that a constant flow rate is maintained for a producing time, tp , and then the well is shut in. Variations of the buildup test include analysis of variation in production rate before shut-in, but only the constant-rate production period is covered here. The pressure, p$ (At=O), is measured just before shut-in and then at different shut-in times, A?, after the time of shut-in.

A plot is made of the shut-in pressures, PDF, vs. a time scale based on the shut-in time, At. The time scale is either log At or log (I,, +At)iAt. The first of these plots (Fig. 35.11) is called an “MDH plot” (Miller, Dyes, and Hutchinson 15). The second type of plot (Fig. 35.12) is called a “Homer ~10~“~~ Both plots give the same semilog straight line slope, which is also the same as measured in the drawdown test.

The kh for an oil or gas well can be determined from the slope of this semilog straight line by the following equations (identical to Eqs. 37, except for the sign).

For oil wells,

k 0

h= 162.6qoBofio , . . . . . . . . . . . . . . . . . 094

m

and for gas wells,

k g h= 5.792x 104q,hJ’/dL) . . (39b)

m*

k,h= 5.792x 104q,(p,,WW

(39c) m’ wb ’

or

k g

h= 5.792 x 104q,hJ-dW m"

hg)wb. WW

Note that the signs are reversed for the Homer plot. The skin factor, s, can be determined from one of the

following equations. For oil wells,

-log ko

4ihctr?

. . . . . . . . . . . . . . . . . . . . . . . . . . . (404

3350

= 3317

!i 30000 ,454 6 10-I I IO

SHUT-IN TIME, At, hours

Fig. 35.11 -MDH plot for buildup test.

SHUT-IN -TIME, At, hours

01 ‘E3300

i PI, * 3266 P

P -40 PS/o/CYCL -3250

2 OF STORAGE 3

2 u3200

h

43 2 8 654 3 2 86543 2 a

IO’

(tp +At),A:’ 2 IO’

Fig. 35.1 P-Horner plot of pressure buildup data from Fig. 35.11.

and for gas wells,

s=1.151 (I m(pl)-m(p,f)

m* I kg -log- +3.23

4ClgCt > , . . . . . . . . . . . . . . (40b)

s=l.151 Pl -Pwf mr I -log kg

hsctrw2

. . . . . . . . . . . . . . . . . . . . . . . . . . . (4Oc)


2 2 ,300

z Kcr a, w QIZOO

d ; IQ 8 I=

II00

8

IO00 343 2 86543 2 82.3.1 2 I02 IO

(to + Af)/U

Fig. 35.13-Horner plot of typical pressure buildup data from a well in a finite reservoir.

or

(I P2 I -P2 wf kg

> s=l.l51 -log

CbgCd +3.23 .

m”

. . . . . . . . . . . . . . . . (404

The slope refers to the corresponding semilog straight line. prr,f is the last pI(,f at At=O. These equations are based on the equation of the semilog straight line. Therefore, if p ws does not fall on the extrapolated semilog straight line at At= 1 hour, then p I is read on the semilog straight line rather than at the actual data.

Again, be reminded that transient well test analysis can be very complicated and can depart in many ways from the simple analysis presented here. These equations are presented only for quick reference and to show the proper interpretation of the real gas formulas for the normal cases. The reader should refer to Refs. 1 through 5 for more details and explanation of departures from these simple cases.

Determination of p 8 The value of PR represents the average reservoir pressure in the drainage area of the well. It is important to determine PR from a buildup test so that PR can be used for material balance calculations, history matching in reservoir simulation, or in pseudosteady-state performance equations.

There are several methods for determining Jo from a buildup test but the most general is the MBH (Matthews, Brons, and Hazebioek I’). This method is generally applicable because a number of different reservoir drainage area shapes are available for analysis. These reservoir shapes are the same as those used for evaluating shape factors in Table 35.2.

Fig. 35.13 shows how the method is applied. The buildup test has a semilog straight line, which begins bending at the later shut-in times because of the effect of the boundaries. The data normally will bend down and become flat from this curve, but for unusual cases the data actually can bend up from the semilog straight line before it even- tually becomes horizontal.

Asymptotically, the data approach the correct value of PR as At approaches infinity. Since our shut-in time normally is limited, the MBH method is based on extrapolating the semilog straight line to At= 03, or (fp +At)lAt= 1 .O. This value is called p*. The method then provides a correction to calculate the correct value of j?~ from the extrapolated value of p*.

The MBH method assumes that the well flowed at a constant rate for tp and that the drainage area A is known for the well. The dimensionless producing time, tpDA , is calculated. If tpDA is greater than (tp~A)psJ, the later value can be used as tpDA . In other words, it is not important what the rate history was before pseudosteady state was achieved.

Now that p* has been extrapolated from the data and tpDA has been calculated, then the correction between p* and jYR is made by using the MBH correction curve that best represents the drainage shape. The MBH correction curves are presented in Figs. 35.14 through 35.17. A step- wise procedure to determine p.8 can be summarized as follows.

1. Make a Horner plot. 2. Extrapolate the semilog straight line to the value of

p* at (tP +At)lAt= 1.0. 3. Evaluate m, the slope of the semilog straight line. 4. calculate tpDA =(o.ooo264kt,)/~pcr~. 5. Find the closest approximation to the drainage shape

in Figs. 35 _ 14 through 35.17. Choose a correction curve. 6. Read the value of 2.303(p*-jY~)lrn from the cor-

rection curve at t,~~. 7. Calculate the’ value of 5 R. This procedure gives the value of p R in the drainage

area of one well. If a number of wells are producing from the reservoir, each well can be analyzed separately to give a j?~ for its own drainage area. This is done, assuming that all wells are producing in pseudosteady state, by dividing the reservoir up into drainage areas for each well by constructing no-flow boundaries between the wells. Fig. 35.18 shows an illustration of such a segmentation of a reservoir. These no-flow boundaries represent the “watersheds” of the different drainage areas. The drainage areas are calculated so that each drainage area has the same reservoir flow rate compared to its PV. Thus,

(qr/Vp) 1 =(qr/Vp)2 =(q,lvp)3=(qtlvp)i. . . . C41)

This relationship divides the drainage area (or PV) according to the producing rate of the well. As the well’s rates change, then the drainage area changes for the well. If q=O, for example, then no area would be allocated to that well. This procedure of calculating the drainage area and approximating drainage shape is repeated at the time of each pressure survey. The drainage areas and shapes keep changing as rates change.

There is often confusion about the meaning of p* in the Horner plot. The value of p* has no physical meaning except in the special case of an infinite-acting well (T?=w). This is the case that Horner16 originally ad- dressed in determining the initial pressure, pi ,-in a newly discovered well. In this special infinite-acting case, p*= p R =pi. Otherwise, p* has no physical meaning.

PansH =2.303( p’-fn)lm P meH =2.303(p*-pR)/m

N Y h u

I! ”

0

o- js, E P r


,-

I-

I -

DIMENSIONLESS PRODUCTION TIME. tCD.

Fig. 35.16-MBH dimensionless pressure for different welt locations in a 2: 1 rectangular drainage area.

,o- I DIMENSIONLESS PRODUCTION TIME. tpo4

Fig. 35.17-MBH dimensionless pressure for different well locations in 4: 1 and 5: 1 rectangular drainage area


Fig. 35.18--Reservoir map showing approximate no-flow boundaries.

Example Problem 4 (Pressure Buildup Analysis) (after Earlougher 2 ). Pressure Buildup Test Analysis- Homer Method. Table 35.3 shows pressure buildup data from an oil well with an estimated drainage radius of 2,640 ft. Before shut-in the well had produced at a stabilized rate of 4,900 STBiD for 310 hours. Known reservoir data are

D = 10,476 ft, rw = (4.25112) ft, C - 22.6~10~~ psi-‘,

4; i 4,900 STB/D, h = 482 ft,

pdAt=O) = 2,761 psig, PO = 0.20 cp,

c#l = 0.09, B, = 1.55 RBISTB,

casing di = (6.276/12) ft, and rp = 310 hours.

Solution. The Horner plot is shown as Fig. 35.12. Residual wellbore storage or skin effects at shut-in times of less than 0.75 hour are apparent. The straight line, drawn after At=0.75 hour, has a slope of -40 psigicycle, so m=40 psiglcycle.

Eq. 37a is used to estimate permeability:

k,= 162.6(4,900)(1.55)(0.20)

=12.8 md. WWW

Skin factor is estimated from Eq. 40a using p ,hr = 3.266 psig from Fig. 35.12:

s=1.1513 3,266-2,761

40

-log (12.8)(12)2

(0.09)(0.20)(22.6x 10 -6)(4.25)2 1 +3.2275 =8.6.

I

TABLE 35.3-PRESSURE BUILDUP TEST DATA FOR EXAMPLE 4,i, =310 HOURS

At t, +At

(hours) (hours)

0.0: 0.10 310.10 0.21 310.21 0.31 310.31 0.52 310.52 0.63 310.63 0.73 370.73 0.84 370.84 0.94 37 0.94 1.05 311.05 1.75 371.15 t .36 37 1.36 1.68 311.68 1 .ss 311.99 2.51 312.51 3.04 313.04 3.46 313.46 4.08 314.08 5.03 315.03 5.97 315.97 6.07 316.07 7.01 317.01 8.06 318.06 9.00 319.00

10.05 320.05 13.09 323.09 16.02 326.02 20.00 330.00 26.07 336.07 31.03 341.03 34.98 344.98 37.54 347.54

(At, + At) PW At (Psk3)

Pwn-Pwt (PW

- 2,761 - 3,101 3,057 296 1,477 3,153 392 1,001 3,234 473

597 3,249 480 493 3,256 495 426 3,260 499 370 3,263 502 331 3,266 505 296 3,267 506 271 3,268 507 229 3,271 510 186 3,274 513 157 3,276 515 125 3,200 519 103 3,283 522 SO.6 3,286 525 77.0 3,269 528 62.6 3,293 532 52.9 3,297 536 52.1 3,297 536 45.2 3,300 539 39.5 3,303 542 35.4 3,305 544 31.8 3,306 545 24.7 3,310 549 20.4 3,313 552 16.5 3,317 556 12.9 3,320 559 11.0 3,322 561

9.9 3,323 562 9.3 3,323 562

We can estimate Ap across the skin from Eq. 10:

Ap, =0.87(40)(8.6)=299.

Average Drainage-Region Pressure-MBH. We use the pressure-buildup test data of Table 35.3. Pressure buildup data are plotted in Figs. 35.12. Other data are

A= ?rre2

=a(2,640)2 sq ft.

To see if we should use tp = 310 hours, we estimate tpss using @DA lpss =O.l from Table 35.2.

(0.09)(0.2)(22.6x 10 -6)(7r)(2,640)2(0. 1) tpss =

(0.0002637)(12.8)

=264 hours.

Thus, we could replace tp by 264 hours in the analysis. However, since tp is only about l.l7t,,,, we expect no difference in j?~ from the two methods, so we use t,=310 hours. As a result, Fig. 35.12 applies.

Fig. 35.12 does not show p* since (t,, +At)lAt does not go to 1.0. However, we may compute p* from pws at (tp +At)lAt= 10 by extrapolating one cycle:

p* = 3,325 + (1 cycle)(40 psi/cycle)

=3,365 psig.


Using the definition of tpDA:

(0.0002637)(12.8)(310)

rpDA = (0.09)(0.20)(22.6x 10 -6)(a)(2,640)2

=0.117.

From the curve for the circle in Fig. 35.14, poMnn(~~D,+,=O. 117)= 1.34. Then, from our step-wise procedure,

pR=3,365- 4o p(1.34) 2.303

=3.342 psig.

This is 19 psi higher than the maximum pressure recorded.

Nomenclature A = drainage area of well

cfi = total compressibility evaluated at p; cWf = wellbore fluid compressibility CA = shape factor from Table 35.2 C, = wellbore storage constant

f(t) = unit response function F Da= non-Darcy (turbulence) factor

F, = turbulence factor m = (162.6qBp)lkh

mD = dimensionless m(p)

m(p) = 2jPtdp= real gas pseudopressure

0

m(p) = m(P) atpR m( pi) = m(p) at initial pressure pi

m(p,,,f) = m(p) at wellbore flowing pressure p,,,f m* = slope of m(p) plot m’ = slope of p plot m ” = slope of p* plot p* = MTR pressure trend extrapolated to

infinite shut-in time po = kh(pi -p)/( 141.2qBp) =dimensionless

pressure PDMBH = 2.303(p*-pR)lm, dimensionless

pressure, MBH method Aps = additional pressure drop across

altered zone (qg 1 = absolute value of gas rate qsf = flow rate at the sandface rD = r/rw =dimensionless radius re = external drainage radius rw ’ = effective wellbore radius s’ = effective skin effect

tD = dimensionless time tDA = dimensionless time based on drainage

area, A bDA)Pss = time required to reach pseudosteady

state, dimensionless t end = end of MTR in drawdown test

tpDA = dimensionless producing time

tpss = time required to achieve pseudosteady state

u = macroscopic (Darcy) fluid velocity V, = volume of the wellbore xe = distance from well to side of square

drainage area xf = distance from well to either end of a

vertical fracture

Subscript

wb = wellbore

Key Equations in SI Metric Units

v2p= 1 46 ap -- 3,557x10-9 k at’ .......,,...... (1)

where p is in kPa, 4 is a fraction, p is in Paas, c, is in kPa-t, k is in md, and t is in hours.

4t4 =qoB, +(s, -R,q,)B, +q,B,, (6)

where qo,qr,qw are in std m3/d, B,,BI,B, are in res m3/std m3, qg is in std m3/d, and B, is in res m3/std m3,

, . . . . . . . . . (7)

where PD = [kh(pi -~YW-W%41,

r rD = -,

rw

3.557 x 10 -9kt tD =

4wrrw 2 h,r,rw are in m, k is in md, p,pi are in Pa, q is in m3/d, I3 is in res m3/std m3, p is in Pa*s, t is in hours, 4 is a fraction, and c,is in kPa-*.

kt pwf =pi -m log choir, 2 -8.10

> , . . . . . . . . . (9)

where m=2.149~lO”qB~/(kh). See Eq. 7. for other units.

ap (-> -

-4.168~1O-~qB

at ) . . . . . . . . . . . VpC, . (15)

PSS


where VP is in m3, See Eq. 7 for other units.

5.356x10p1E

4= BP

In T’ -0.75+s >

(PR -Pw& . .

rw where

re =m, s is dimensionless, and p~,p~f are in kPa.

See Eq. 7 for other units.

pR=pi-- vpc, ) . . . . . . . . . . . . . . . . . . . . . .

where Np isinm3, VP is in m3, B, is in res m3/std m3, c, is in Wa-‘, and p~,p; are in kPa.

V2m(p)= 1 4Wg am(p) ~-

3.557x10-9 k at ’ ....I

. .

. .

. (17)

. (21)

. (24)

where m(p) is in kPa2 and cg is in kPa-’ . See Eq. 7 for other units.

,,lj,=% h, t,+o.4@,5+S+FD,IqgI, . . . . . . . (33)

where

mD = 2.708x10-”

3.557x 10-9kt tD =

dw, r,,’ 2 ’

s is dimensionless, FD, is dimensionless, qg is in m3/d, T,,.,TR are in K, prc is in kPa, k is in md,

h is in m, and m(p;),m(p,j) are in kPa2/Pa.s.

See Eq. 7 for other units.

k 0

h= _ 2.149x 106qoB,~o . . (374 .._ m

16. Homer. D.R.: “Pressure Build-Up in Wells,” Proc.. Third World Pet. Gong., The Hague (1951) Sec. II, 503-23.

17. Matthews. C.S., Brons, F., and Hazebroek, P.: “A Method for Determination of Average Pressure in a Bounded Reservoir,” Trans., AIME (1954) 201, 182-91

See Eqs. 7 and 9 for units.

k,h= m*

. . . . . . . . . . WW

where m* is in kPa2/Pa* s-cycle. See Eq. 33 for other units.

s=1.151 (I? ) -log4pc;rw,2 .,.lO),

,......................... (384

where m is in kPa/cycle. See Eq. 7 for other units.

References I. Matthews, C.S. and Russell, D.G.: Pressure Buildup and Fknv Tests

in Wells, Monograph Series, SPE, Richardson, TX (1967) I. 2. Earlougher, R.C. Jr.: Advances in Well Test Analysis, Monograph

Series, SPE, Richardson, TX (1977) 5. 3. Dake, L.P.: Fundmmntals ofReservoir Engineering, Elsevier Scien-

tific Publishing Co., Amsterdam (1978). 4. Gas Well Testing-Theory and Practice, fourth ed., Energy

Resources and Conservation Board, Calgary, AIL, Canada (1979). 5. Lee, John: Well Testing, Textbook Series, SPE, Richardson, TX

(1982). 6. Pressure Analysis Methods, Reprint Series No. 9, SPE, Richard-

son. TX (1967). 7. Pressure Transient Testing Methods, Reprint Series No. 14, SPE,

Richardson, TX (1980). 8. van Everdingen, A.F. and Hurst, W.: “The Application of the

Laplace Transformation of Flow Problems in Reservoirs,” Trans. AIME (1949) 186, 305-24.

9. Martin, J.C.: “Simplified Equations of Flow in Gas Drive Reser- voirs and the Theoretical Foundation of Multiphase Pressure Buildup Analyses,” Trans., AIME (1959) 216, 309-l 1.

10. Wattenbarger, R.A. and Ramey, H.J. Jr.: “An Investigation of Well- bore Storage and Skin Effect in Unsteady Liquid Flow: II. Fimte Difference Treatment,” Sot. Pet. Eng. J. (Sept. 1970) 291-97; Trans., AIME, 249.

11. Die& D.N.: “Determination of Average Reservoir Pressure From Buildup Surveys,” .f. Pet. Tech. (Aug. 1965) 955-59; Trans., AIME. 234.

12. Al-Hussainy, R., Ramey, H.J. Jr., and Crawford, P.B.: “The Flow of Real Gases Through Porous Media,” J. Pet. Tech. (May 1966) 624-36; Trans., AIME, 237.

13. Wattenbarger, R.A. and Ramey, H.J. Jr.: “Gas Well Testing With Turbulence, Damage and Wellbore Storage,” J. Pet. Tech. (Aug. 1968) 877-87; Trans., AIME, 243.

14. Ramagost, B.P. and Farshad, F.F.: “p/z Abnormally Pressured Gas Reservoirs,” paper SPE 10125 presented at the 1981 SPE Annual Technical Conference and ExhibItion, San Antonio. Oct. 4-7.

15. Miller, C.C., Dyes, A B., and Hutchinson, C.A. Jr: “The Esti- mation of Permeability and Reservoir Pressure From Bottom Hole Pressure Build-Up Characteristics,” Trans., AIME (1950) 189, 91-104

well performance equations

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