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phase Flow. A pressure-buildup test was run in a well suspected of

being in a highly faulted reservoir. In addition to confirming reservoir size, the test also had the objective of confirming a fault quite close to the well. Before shut-in, the well produced oil, gas, and water si- multaneously. Table 2.22 gives the pressure and time data from the buildup test, while other known data are summarized here. Fluid properties were evaluated at p*, used as an approximation to p.

Determine the following: (1) the start and end of the middle-time region on a semilog graph, given that type-curve analysis indicates that wellbore-storage distortion ended at about 0.25 hours and that boundary effects began at about 5.3 hours; (2) effective permeabili- ties ko, kw, and kg; (3) the skin factor, s; (4) the distance, L, to the

boundary believed to be near the well; and (5) the current average drainage-area pressure, p, for this well. Geological evidence sug- gests that the well is completed one-eighth of the distance from the long edge of a 2 1 rectangle.

qo+ 1,100 STB/D qw+ 4,200 STB/D qgt+ 1,800 Mscf/D tp+ 20.5 hours Rs at p*+ 537 scf/STB Rsi+ 705 scf/STB mo+ 0.49 cp mw+ 0.231 cp mg+ 0.01778 cp Bo+ 1.34 RB/STB Bw+ 1.057 RB/STB B_g+ 1.424 RB/Mscf co+ 2.04 x 10–4 psi–1 cw+ 9.79 10–6 psi–1 ÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁ ÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁ ÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁ

TABLE 2.22—PRESSURE-BUILDUP-TEST DATA, EXAMPLE 2.13 ÁÁÁ ÁÁÁ ÁÁÁ Shut-in Time (hr) ÁÁÁÁ ÁÁÁÁ ÁÁÁÁ Pressure (psia) ÁÁÁÁ ÁÁÁÁ ÁÁÁÁ Horner Time Ratio ÁÁÁ ÁÁÁ ÁÁÁ Shut-in Time (hr) ÁÁÁÁ ÁÁÁÁ ÁÁÁÁ Pressure (psia) ÁÁÁ ÁÁÁ ÁÁÁ Horner Time Ratio ÁÁÁ ÁÁÁ 0.0ÁÁÁÁ ÁÁÁÁ 1,658ÁÁÁÁ ÁÁÁÁ — ÁÁÁ ÁÁÁ 3.0 ÁÁÁÁ ÁÁÁÁ 1,963 ÁÁÁ ÁÁÁ 7.83 ÁÁÁ ÁÁÁ 0.1 ÁÁÁÁ ÁÁÁÁ 1,778 ÁÁÁÁ ÁÁÁÁ 206 ÁÁÁ ÁÁÁ 4.0 ÁÁÁÁ ÁÁÁÁ 1,976 ÁÁÁ ÁÁÁ 6.13 ÁÁÁ 0.2 ÁÁÁÁ 1,829 ÁÁÁÁ 103.5 ÁÁÁ 5.0 ÁÁÁÁ 1,984 ÁÁÁ 5.10 ÁÁÁ ÁÁÁ 0.3ÁÁÁÁ ÁÁÁÁ 1,853ÁÁÁÁ ÁÁÁÁ 69.3 ÁÁÁ ÁÁÁ 6.0 ÁÁÁÁ ÁÁÁÁ 1,992 ÁÁÁ ÁÁÁ 4.42 ÁÁÁ ÁÁÁ 0.4ÁÁÁÁ ÁÁÁÁ 1,864ÁÁÁÁ ÁÁÁÁ 52.3 ÁÁÁ ÁÁÁ 7.0 ÁÁÁÁ ÁÁÁÁ 1,997 ÁÁÁ ÁÁÁ 3.93 ÁÁÁ 0.5 ÁÁÁÁ 1,877 ÁÁÁÁ 42.0 ÁÁÁ 8.0 ÁÁÁÁ 2,003 ÁÁÁ 3.56 ÁÁÁ ÁÁÁ 0.75ÁÁÁÁ ÁÁÁÁ 1,898ÁÁÁÁ ÁÁÁÁ 28.3 ÁÁÁ ÁÁÁ 9 ÁÁÁÁ ÁÁÁÁ 2,008 ÁÁÁ ÁÁÁ 3.28 ÁÁÁ ÁÁÁ 1.0ÁÁÁÁ ÁÁÁÁ 1,912ÁÁÁÁ ÁÁÁÁ 21.5 ÁÁÁ ÁÁÁ 10 ÁÁÁÁ ÁÁÁÁ 2,013 ÁÁÁ ÁÁÁ 3.05 ÁÁÁ 1.5 ÁÁÁÁ 1,933 ÁÁÁÁ 14.7 ÁÁÁ 11 ÁÁÁÁ 2,016 ÁÁÁ 2.86 ÁÁÁ ÁÁÁ 2.0 ÁÁÁÁ ÁÁÁÁ 1,944 ÁÁÁÁ ÁÁÁÁ 11.3 ÁÁÁ ÁÁÁ 12 ÁÁÁÁ ÁÁÁÁ 2,021 ÁÁÁ ÁÁÁ 2.71

INTRODUCTION TO FLOW AND BUILDUP TEST ANALYSIS—SLIGHTLY COMPRESSIBLE FLUIDS 53

Fig. 2.35—Horner semilog plot, Example 2.13. cg+ 5.33 10–4 psi–1 Sw+ 0.57 Sg+ 0.10 cf+ 3.9 10–6 psi–1 f+ 0.165 h+ 114 ft r_w+ 0.411 ft gg+ 0.80 (no N2, H2S, or CO2)

oil gravity+ 32°API

T+ 250°F

total dissolved solids+ 10,000 ppm

A+ 23 acres Solution.

1. Prepare a Horner semilog plot of the data. From Fig. 2.35, the middle-time region begins at (tp)Dt)/Dt+69.3 or Dt+0.3 hours

and ends at (tp)Dt)/Dt+3.93 or Dt+7.0 hours. The slope, m, of the

semilog straight line of the middle-time region is

m+116 psi/cycle.

2. Calculate the total flow rate, (qB)_t:

(qB)t+ qoBo) qwBw)

ƪ

qgt*

ǒ

qoRsń1, 000

Ǔƫ

Bg

+ (1, 100)(1.34) ) (4, 200)(1.057) )

ƪ

1, 800*(1, 100)(527)

1, 000

ƫ

(1.424) + 7, 635 RBńD.

3. Calculate the total system mobility and the effective permeabi- lities to each flowing phase. First, the total system mobility is

lt+

162.6(qB)_t

mh +

(162.6)(7, 635)

(116)(114) + 93.9 mdńcp.

The effective permeabilities to oil, water, and gas are, respectively:

ko+ 162.6qoBomo mh + (162.6)(1, 100)(1.34)(0.49) (116)(114) + 8.9 md kw+ 162.6qwBwmw mh + (162.6)(4, 200)(1.057)(0.231) (116)(114) + 12.6 md, and kg+ 162.6

ǒ

qgt*qo Rs_1,000

Ǔ

Bgmg mh +(162.6) ƪ1, 800* (1, 100)(0.537)ƫ(1.424)(0.01778)_{(116) (114)} + 0.38 md.

4. To calculate the skin factor, we need p1hr and ct. From Fig. 2.35,

we find p1hr+1,912 psia at (tp)Dt)/Dt+(20.5)1)/1+21.5. Total

system compressibility is

ct+ coSo) cgSg) cwSw) cf

+

ǒ

2.04 10*4

Ǔ

(0.33))

ǒ

5.33 10*4

Ǔ

(0.1) )

ǒ

9.79 10*6

Ǔ

(0.57))

ǒ

3.9 10*6

Ǔ

+ 1.30 10*4_psi*1_.

The skin factor is computed with the total mobility.

s+ 1.151

ƪǒ

p1hr_m* pwf

Ǔ

* log

ǒ

ltńfctr 2 w

Ǔ

) 3.23

ƫ

+ 1.151

NJ

(1, 912₁₁₆* 1, 658) * log

ƪ

93.9 (0.165)(1.30 10*4_)(0.411)2

ƫ

) 3.23

Nj

+ * 2.3.

5. Although we see the effects of the boundary on the pressure re- sponse, the slope does not double during the test. Thus, we cannot use the methods presented in this chapter to estimate the distance to the boundary. However, we can calculate the radius of investigation at a shut-in time of 7 hours, when boundary effects began to be felt.

ri+

ǒ

l tDt 948fct

Ǔ

½ +

ƪ

_{(948)(0.165)(1.3}(93.9)(7) _10*4)

ƫ

½ + 180 ft. We can estimate the distance to the nearest boundary from geologi- cal considerations. The well is thought to be one-eighth of the width of a 2 1 rectangle from the nearest edge. The length of the short side of a 23-acre, 2 1 rectangle is given by

2L2_{+ 43, 560A;}

therefore, L+

ƪ

(43, 560)(23)

2

ƫ

½

+ 88.5 ft.

We would expect boundary effects to be felt when the radius of in- vestigation reaches the image well; that is, when

ri+ 2L + (2)(88.5) + 177 ft.

Thus, we conclude that the pressure response is consistent with the geological interpretation.

6. To estimate the current average drainage-area pressure, p, we extrapolate the semilog straight line to obtain p*+2,068 psia from Fig. 2.35. The MBH dimensionless time group is

tAD+ 0.0002637lttp fctA + (0.0002637)(93.9)(20.5) (0.165)(1.3 ₁₀*4_{)(1, 001, 880)} + 0.024. From Fig. 2.36, pMBH,D+ 2.303(p ** p) m + * 0.55

Fig. 2.36-MBH chart for Example 2.13. Then -= * - mpMBH,D = 2 068 _ (1 1 6 )( -0.55) P P 2.303 ' 2.303 = 2,096 psia. 2.7 Chapter Summary

This chapter introduces the traditional methods for flow- and build­ up-test analysis for a well producing a single-phase, slightly com­ pressible liquid from a single-layer reservoir. Sec. 2.1 presents an overview of the chapter.

In Sec. 2.2, we discuss flow tests, also known as draw down tests, where the pressure response to a known production rate or rates is measured. These tests can be used to estimate permeability, skin fac­ tor, reservoir PV, and distance to nearby linear boundaries. Flow tests are often conducted when economic considerations prohibit the use of pressure-buildup tests.

Sec. 2.2.1 presents an analysis method for estimating permeabil­ ity and skin factor for constant-rate drawdown tests. This method is based on plotting wellbore pressure vs. producing time on a semilog scale and drawing a best-fit straight line through the data.

It is often difficult to maintain a constant flow rate during a flow test. In fact, the inability to maintain a strictly constant flow rate is the major disadvantage of flow tests compared with buildup tests. Accordingly, in the following two subsections, we present methods for analyzing flow tests where the rate is a function of time.

Sec. 2.2.2 introduces the Winestock and Colpitts3 method of anal­ ysis, which is applicable to flow tests where the flow rate is changing s

�

owly and smoothly. In this technique, we plot the pressure change dIVIded by the production rate vs. the production time, again on sernilog coordinates. The analysis provides estimates of formation permeability and skin factor.

Sec. 2.2.3 presents an analysis method applicable to a well that has had n discrete rate changes during its production history. This method is based on the assumption that the logarithmic approximation to the Ei function is applicable to each production period; i.e., the reservoir

54

is infinite-acting throughout the entire production history. We consid­ ered both the special case of a two-rate test and the more general case of an n-rate test. From the two-rate test, it is possible to solve for the initial pressure, Pi, in addition to permeability and skin factor; for the n-rate test, the initial pressure must be known.

In Sec. 2.3, we turn our attention to pressure-buildup tests. As noted earlier, flow tests are subject to variation in rate, which ad­ versely �ffects our ability to analyze the tests. By shutting in a well, we can Impose a strictly constant surface rate on the well. Thus, pressure-buildup tests traditionally have been used in lieu of flow tests for measuring permeability and skin factor.

Sec. 2.3.1 presents the Horner analysis method, which is applica­ ble to pressure-buildup tests where the well is shut in following a constant-rate flow period. To use this method, we plot the shut-in wellbo�e pressure. vs. the Horner time ratio, defined as (tp + l!,.t)/l!,.t,

on semllog coordmates. The Horner method provides estimates of formation permeability, skin factor, and, for infinite-acting reser­ voirs, original reservoir pressure.

Secs. 2.3.2 and 2.3.3 present analysis techniques for wells where the shut-in period is preceded by two and by n -1 different flow rates, respectively. Both these methods are based on the use of su­ perposition, where the logarithmic approximation to the Ei-function solution describes the contribution to the pressure drop from each producing period. Again, both methods provide estimates of forma­ tion permeability, skin factor, and initial reservoir pressure.

We also presented two alternatives to the superposition method, for wells in which the number of flow-rate changes makes the super­ position analysis inconvenient. The Odeh and Selig method is appli­ cable to tests where the shut-in time is greater than the actual pro­ ducing time. This method is applied by modifying the production time and flow rate with Eqs. 2.50 and 2.51, then applying Horner's method with these modified values.

Horner's approximation is often used when the flow period im­ mediately preceding the shut-in is comparable in length with the shut-in period. To use this approximation, we compute a pseudopro­ ducing time tp as

tpH = 24Np/qlast'··.·· ... (2.111 )

INTRODUCTION TO FLOW AND BUILDUP TEST ANALYSIS—SLIGHTLY COMPRESSIBLE FLUIDS 55 then use this value in place of the actual producing time in the Horn-

er method.

Actual tests invariably are affected by any of a number of factors that cause the pressure response to deviate from the ideal response assumed in developing the methods considered in Secs. 2.2 and 2.3. In Sec. 2.4, we discuss a number of these factors, including reser- voir heterogeneity, reservoir boundaries, wellbore storage, and wellbore damage and stimulation.

Sec. 2.4.1 introduces the concept of radius of investigation. The radius of investigation is given by

r_i+ kt

948fmct

Ǹ

. . . (2.112) for a flow test and by

ri+ kDtń948fmc

Ǹ

t . . . (2.113)

for a buildup test. The radius of investigation provides a framework for understanding the influence of reservoir heterogeneities on the wellbore-pressure response. Heterogeneities outside the radius of investigation have no effect on the wellbore response. As the dura- tion of the test increases, the radius of investigation also increases, as does the likelihood that significant heterogeneities will be en- countered.

In Sec. 2.4.2, we discuss the three time regions (early, middle, and late) into which flow- or buildup-test data are often divided. During the early-time region, the pressure response deviates from the ideal owing to wellbore storage and/or wellbore damage or stimulation. During the middle-time region, the pressure response corresponds to the ideal response on which the analysis methods presented in this chapter are based. Thus, a straight line drawn through the data in this region is often referred to as the “correct semilog straight line.” Dur- ing the late-time region, the pressure response again deviates from the straight-line response, because of the presence of significant res- ervoir heterogeneities, interference from other wells, or reservoir boundaries.

Sec. 2.4.3 discusses wellbore-storage effects, also known as well- bore unloading for flow tests and afterflow for buildup tests. This phenomenon results from the fact that the sandface flow rate lags the surface flow rate when the wellbore has a finite volume. We charac- terize the degree of wellbore storage by the wellbore-storage coeffi- cient, C. We present expressions to estimate the wellbore-storage

coefficient either for a rising or falling gas/liquid interface (Eq. 2.62) or for a wellbore containing a single-phase fluid (Eq. 2.63).

Sec. 2.4.4 discusses various ways of characterizing the degree of damage or stimulation in the near-wellbore region. The skin factor is often used to characterize an alteration in the near-wellbore region, with positive values representing damage and negative values repre- senting stimulation. The simplest skin-effect model, applicable only to damaged skin, consists of an infinitesimally thin zone of reduced permeability adjacent to the wellbore, causing a step change in the pressure profile across the skin zone. Another model involves a two- zone reservoir, with altered permeability ks within a radius rs of the

wellbore. This model can represent either damage or stimulation. An alternative model involves the effective wellbore radius, rwa. For

damaged wells, rwa is smaller than the actual radius rw; for stimulated

wells, it is larger than r_w. For wells with a high-conductivity hydraulic fracture, the fracture half-length may be related to r_wa by L_f+2r_wa. The skin factor may also be characterized by the additional pressure drop associated with the damaged zone,

Dps+

ǒ

141.2qBm

kh

Ǔ

s. . . (2.114)

A final method of characterizing the skin effect is the flow efficien- cy, E, where values less than 1 represent damage, and values greater than 1 represent stimulation.

The next two sections discuss the apparent skin factor owing to the presence of either an incompletely perforated interval or partial penetration (Sec. 2.4.5) or a deviated wellbore (Sec. 2.4.6.)

In Sec. 2.5, we turn our attention to the late-time region, discus- sing methods of estimating drainage-area pressure, distance to boundaries, and reservoir PV.

Sec. 2.5.1 introduces three methods for estimating the drainage- area pressure. If at least one side of the reservoir is still infinite-acting, the Horner straight line may be extrapolated to infinite shut-in time to obtain the initial pressure, pi. If the reservoir is finite-acting and the

drainage-area shape and well placement within the drainage area are known, the MBH method is applicable. The Horner straight line is ex- trapolated to infinite shut-in time as in the case of an infinite acting reservoir, but the extrapolated pressure p* must then be corrected by use of the appropriate MBH function to obtain average drainage-area pressure, p. The modified Muskat method is limited to wells centered in the drainage area. Because it depends on the availability of pres- sure-buildup data within the transition from the middle-time region to the boundary-dominated region, it is restricted in its applicability.

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