Proofs 1st_ Radius of Investigation

UESO #200271 (EXP) [ESO/06/066] Received: ? 2006 (November 26, 2006)


Propagation of Radius of Investigation fromProducing Well

B.-Z. HSIEHG. V. CHILINGARZ.-S. LIN

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Energy Sources, Part A, 29:000–000, 2007Copyright © Taylor & Francis Group, LLCISSN: 1556-7036 print/1556-7230 onlineDOI: 10.1080/15567030601003759

Propagation of Radius of Investigation fromProducing Well Q1, Q2, Q3

B.-Z. HSIEH

Department of Resources EngineeringNational Cheng Kung University 5Tainan, Taiwan

G. V. CHILINGAR

Environmental Engineering DepartmentUniversity of Southern CaliforniaLos Angeles, CA, USA 10

Z.-S. LIN

Department of Resources EngineeringNational Cheng Kung UniversityTainan, Taiwan

Abstract The purpose of this study is to estimate the pressure disturbance area, or 15the propagation of the radius of investigation, from a producing well by both analyt-ical and numerical methods. A linear coefficient in the relation between the squareof the dimensionless radius of investigation and the dimensionless time is studiedand derived. The coefficient in the equation is a constant, and varied with differentcriterions of radius of investigation defined, i.e., the amount of pressure change from 20the initial formation pressure at the pressure front of the pressure disturbance area.For the dimensionless pressure defined at the pressure front changing from 0.1095 to10−9, the coefficient varied from 4 to 71.15, respectively. The coefficient of radius ofinvestigation is independent of the level of the flow rate for a well producing at aconstant flow rate. For a well producing with variable flow rates, the coefficient is not 25a constant for the case of larger pressure drops defined at the pressure front. The skinfactor does not affect the result of the calculated radius of investigation. The wellborestorage volume will affect the propagation of the radius of investigation only at anearly time, depending on the wellbore storage volume.

Keywords Q430

1. Introduction

As fluid is produced from the reservoir by a producing well, the pressure disturbancearea expands outward from the wellbore and increases as time increases. The pressure

Address correspondence to Zsay-Shing Lin, Dept. of Resources Engineering, National ChengKung University, No. 1, University Rd., Tainan City, 701, Taiwan. E-mail: [email protected]

1

2 B.-Z. Hsieh et al.

disturbance area created by a producing well is the area enclosed by the pressure frontwhere the disturbed pressure (or the pressure drop from the original pressure) being 35defined an extremely small value (a value very close to zero). The radius of the pressuredisturbance area is called the radius of investigation (ri), or the radius of drainage.

The radius of investigation created by a producing well is a function of time, such asa linear relationship between the square of the dimensionless radius of investigation (r2

iD)and the dimensionless time (tD) was found in the literature (Muskat, 1934; Tek et al., 401957; Jones, 1962; Van Poolen, 1964; Lee, 1982). The radius of investigation equationwith a dimensionless form is r2

iD = αtD , where α is a coefficient.The radius of investigation with the coefficient (α) of 4 is used very often in well

test analysis (Muskat, 1934; Van Poolen, 1964; Matthews and Russell, 1967; Lee, 1982;Chaudhry, 2004). Some other studies obtained the coefficient of 16 and 18.4 (Jones, 451962; Tek et al., 1957). These radius of investigation equations are derived for the caseof constant flow rate. And no mention in the literature is made on the condition at thepressure front or the boundary of the disturbed area. It is infrequent that the coefficient(α) in the equation is obtained for the case of variable flow rates. Also, skin factorand wellbore storage are not considered in the studies of radius of investigation in the 50literature.

The purpose of this study is to estimate the pressure disturbance area (i.e., thepropagation of the radius of investigation) from a producing well using both analyticaland numerical methods. Also, this study is going to establish the radius of investigationequation in terms of dimensionless radius and dimensionless time for different criterion 55of dimensionless pressure defined at the pressure front. The effects of skin and wellborestorage to the linear coefficient are also included in this study.

2. Basic Theory

2.1. Analytical Solution for Estimation of Radius of Investigation

For an isotropic porous medium that is isothermal and homogeneous, with uniform thick- 60ness, constant porosity and constant permeability, the dimensionless equation describingsingle phase fluid flow in a circular reservoir is (Lee, 1982):

∂2pD

∂r2D

+ 1

rD

∂pD

∂rD= ∂pD

∂tD(1)

where

pD = kh(pi − p)

141.2qµB(2)

tD = 0.000264kt

µcφr2w

(3)

rD = r

rw(4)

For a well producing at a constant production rate with zero wellbore radius inan infinite cylindrical reservoir with uniform initial pressure before production begins, 65

Radius of Investigation from Producing Well 3

the analytical solution of the diffusivity equation for an infinite cylindrical reservoir is(Earlougher, 1977):

pD = −1

2Ei

(− r2

D

4tD

)(5)

where

−Ei(−x) = E1(x) =∫ ∞

x

e−u

udu = −0.5772 − ln x +

n∑k=1

(−1)k+1xk

k(k!) (6)

In a well producing at variable flow rates, the pressure drop in the formation at the“nth” flow rate (n > 2) can be calculated by using the superposition of Eq. (5) as follows 70(Earlougher, 1977):

(�p)total = −70.6µB

kh

{q1Ei

(− r2

D

4tD

)+

n∑i=2

(qi − qi−1)Ei

(− r2

D

4tD[t − ti−1]

)}(7)

or

pD1 = −1

2

{Ei

(− r2

D

4tD

)+

n∑i=2

(qi − qi−1

q1

)Ei

(− r2

D

4tD[t − ti−1]

)}(8)

where

pD1 = kh(�p)total

141.2q1µB

and

tD[t − ti−1] = 0.000264k(t − ti−1)

µcφr2w

.

When a well is producing with a constant or variable flow rates, the pressure distur-bance area gradually extends outward in the formation as producing time increases. Theradius of investigation (ri) is the distance from the center of wellbore to the pressure 75front where the pressure difference between the initial pressure and formation pressure,or pressure drop (�p) at the pressure front, is less than the defined small value.

For the case of constant flow rate, the value of r2D/4tD in Eq. (5) can be estimated

by defining a small dimensionless pressure (pD) value, which is proportional to thepressure drop (�p) at the pressure front for the defined small value. Thus, a linear 80relationship between the square of the dimensionless radius of investigation (r2

iD) and thedimensionless time (tD) can be obtained from the analytical Ei solution (Eq. (5)). Notethat the coefficient (α) of the radius of investigation equation is dependent on a criterionof pD value chosen.

For the case of variable flow rates, the relationship between the dimensionless radius 85of investigation (riD) and the dimensionless time (tD) can be estimated from Eq. (7) orEq. (8) by specifying (�p)total or pD1 value.


2.2. Numerical Solution for Estimation of Radius of Investigation

In this study, the radius of investigation equation is derived not only from the analyticalsolution, but also from a numerical solution. The IMEX simulator (CMG, 2004) used in 90this study is basically a three-phase black-oil simulator with a Cartesian or cylindricalgrid system. The simulator is also capable of modeling two-phase (oil-water or gas-water)fluid flow. One phase of oil flow can be simulated by using two-phase oil-water flowwith zero water relative permeability.

The numerical simulation is started with dividing a reservoir into grids. After rock 95and fluid properties are assigned to each grid block, the numerical simulation can be runto model a well to produce at specific flow rates, either at constant rate or variable rates.Formation pressures for each grid block at each time step from the result of the simulationrun can be used to track the pressure front changes as function of time. Note that thepressure front is a point or a line where the pressure drop in the formation is equal to a 100certain small value. The radius of investigation is the distance from the wellbore to thepressure front.

3. Results

An oil reservoir used in this study has a uniform thickness of 60 ft, constant porosity of0.2, and the constant permeability of 150 md. The initial reservoir pressure is 3,000 psi 105(Table 1). The oil PVT data and rock fluid properties shown in Table 1 are used in the Table 1study of the radius of investigation for both the analytical solution and the numericalsolution.

3.1. Radius of Investigation Studies from Analytical Solution

In this study, the pressure front is defined as a line or a curve where the pressure drop 110from the original pressure is equal to a certain small value. The following criterionsare used to define the pressure front: (i) piD = 0.1095 (criterion I), (ii) piD = 0.01095(criterion II), and (iii) piD = 0.001095 (criterion III). The pressure front in the reservoir ischanging as function of time due to a well is produced. Thus, the radius of investigation,or pressure front, is a function of time, and is derived from the solution of the diffusivity 115equation.

Applying criterion I of the radius of investigation (or piD defined at pressure frontof 0.1095) to Eq. (5), we can obtain the results of r2

iD/4tD . In other words, the rela-tionship between the dimensionless radius of investigation and the dimensionless time

Table 1Basic reservoir parameters used in this study

Parameters, unit Values Parameters, unit Values

pi , psi 3,000 kh, md 150q, stb/day 100 kv , md 150µo, cp 13.2 φ, fraction 0.20Bo, rb/stb 1.06 h, ft 60ct , psi−1 2.01 ∗ 10−6 rw, ft 0.35


is r2iD = 4tD . The value of the linear coefficient (α) is 4. This equation is the same

as the radius of investigation equation derived by Lee (1982) and can be expressed as 120ri = (kt/948µcφ)1/2.

When criterion II of the radius of investigation is applied, i.e., piD defined at pressurefront is 0.01095, the relationship between radius of investigation and time is r2

iD =10.39tD . The radius of investigation equation is r2

iD = 17.82tD for criterion III (piD atpressure front is 0.001095). 125

By applying different criterions of dimensionless pressure (piD) at the pressure front,we obtain equations of the radius of investigation with different coefficients (α) (Table 2).The coefficient (α) increases with decreasing dimensionless pressure (piD) defined at Table 2the pressure front. In other words, at the same producing time, the estimated pressuredisturbance area is larger for the criterion of dimensionless pressure (piD) defined as the 130pressure front becomes smaller. The coefficient (α) is varied from 4.00 to 71.15 whenthe criterion value of the dimensionless pressure (piD) defined at the pressure front ischanged from 0.1095 to 10−9 (Table 2). The above results are obtained for the wellproducing at constant flow rate.

To study the propagation of the radius of investigation for the well producing at 135variable flow rates, the result of superposition, Eq. (8), based on constant flow rate ofEq. (5) is used. For the flow rate increasing from an initial flow rate (q1 = 100 stb/day)to a higher flow rate (q2 = 150 stb/day) (Figure 1), the relationship between the radius Figure 1of investigation and time is obtained for the dimensionless pressures at the pressure frontof 0.1095, 0.01095, 0.001095, and 1.0 ∗ 10−9. 140

The coefficients (α) obtained are 4.101, 10.411, 17.819, and 71.312 for different piD

defined at the pressure front of 0.1095, 0.01095, 0.001095, and 1.0 ∗ 10−9, respectively(Figure 2). The coefficients (α) of the increasing flow rates test are slightly larger than Figure 2those obtained from the constant flow rate test (Table 3). Table 3

For the flow rate decreasing from an initial flow rate (q1 = 100 stb/day) to a lower 145flow rate (q2 = 50 stb/day) (Figure 3), the coefficients (α) obtained are 3.887, 10.374, Figure 317.814, and 71.312 for the criterion piD defined at the pressure front of 0.1095, 0.01095,

Table 2The linear coefficient (α) values derived from

different criterion piD defined at pressure front fora constant flow rate test by using analytical solution

piD = −1

2Ei

(− r2

iD

4tD

)= y α =

(α = r2

iD

tD

)

0.1095 4.000.01095 10.390.001095 17.82

10−4 26.0610−5 34.2810−6 42.6910−7 51.2210−8 59.8410−9 71.15


Figure 1. The designed variable flow rate test (an increasing flow rate test) and the calculatedbottom-hole pressure.

0.001095, and 1.0 ∗ 10−9, respectively. The coefficients (α) for these cases are slightlysmaller than those obtained from the constant flow rate test (Table 3).

In addition to the two-rates studied above, the radius of investigation equation for a 150three-rate test is also analyzed. The first three-rate test was designed to increase the flowrate from the initial rate (q1 = 100 stb/day) to a higher flow rate (q2 = 150 stb/day), thendecrease that to q3 = 100 stb/day (i.e., a middle flow rate increasing test) (Figure 4).The coefficients (α) obtained for the middle flow rate increasing test are 4.246, 10.494, Figure 417.849, and 71.316 for the criterion piD defined at the pressure front of 0.1095, 0.01095, 1550.001095, and 1.0 ∗ 10−9, respectively. The coefficients (α) of the middle flow rateincreasing test are larger than those obtained from the constant flow rate test (Table 3).

Figure 2. The integrate results of the dimensionless radius of investigation for the increasing flowrate test by using analytical solution.


Table 3The radius of investigation equations (r2

iD = αtD) from different riD analysis criteriaby using analytical solution

Flow ratesriD criteria I

for piD = 0.1095riD criteria II

for piD = 0.01095riD criteria III

for piD = 0.001095

q = 100 stb/day r2iD = 4.00tD r2

iD = 10.39tD r2iD = 17.82tD

(Constant rate) R2 = 1a R2 = 1 R2 = 1

q1 = 100 stb/day r2iD = 4.101tD r2

iD = 10.411tD r2iD = 17.819tD

q2 = 150 stb/day R2 = 0.9991 R2 = 0.9999 R2 = 0.9999

q1 = 100 stb/day r2iD = 3.887tD r2

iD = 10.374tD r2iD = 17.814tD

q2 = 50 stb/day R2 = 0.9983 R2 = 0.9999 R2 = 0.9999

q1 = 100 stb/day r2iD = 4.246tD r2

iD = 10.494tD r2iD = 17.849tD

q2 = 150 stb/day R2 = 0.9981 R2 = 0.9999 R2 = 0.9999

q3 = 100 stb/day

q1 = 100 stb/day r2iD = 3.698tD r2

iD = 10.280tD r2iD = 17.782tD

q2 = 150 stb/day R2 = 0.9956 R2 = 0.9998 R2 = 0.9999

q3 = 100 stb/day

aR2 = the coefficient of determination.

The second three-rate test was designed to decrease the flow rate from the initialrate (q1 = 100 stb/day) to a lower flow rate (q2 = 50 stb/day), then to increase that toq3 = 100 stb/day (i.e., a middle flow rate decreasing test) (Figure 5). The coefficients (α) Figure 5160obtained for the middle flow rate decreasing test are 3.698, 10.280, 17.782, and 71.316for the criterion piD defined at the pressure front of 0.1095, 0.01095, 0.001095, and1.0 ∗ 10−9, respectively. The coefficients (α) of the middle flow rate decreasing test aresmaller than those obtained from the constant flow rate test (Table 3).

Figure 3. The designed variable flow rate test (a decreasing flow rate test) and the calculatedbottom-hole pressure.


Figure 4. The designed triple flow rates test (a middle flow rate increasing test) and the calculatedbottom-hole pressure.

3.2. Radius of Investigation Studies from Numerical Solution 165

To verify the results of the radius of investigation from the analytical solution, a numericalsimulation study is also used. A cylindrical oil reservoir is simulated with 5,000 grids inr direction (radial direction), 1 grid (i.e., 360 degree) in θ direction (tangent direction),and 1 single layer in k direction (vertical direction). The grid size in radial directionis small at the vicinity of the wellbore and increases gradually as the distance outward 170from the wellbore increases. The formation parameters used in the numerical model isthe same as that used in analytical model (Table 1).

Numerical simulation studies conducted to investigate the propagation of the radiusof investigation include production well producing at constant flow rate and at variableflow rates. The results from numerical simulation studies for both constant and variable 175flow rates cases are the same as these from analytical solution (Tables 3 and 4). Table 4

Figure 5. The designed triple flow rates test (a middle flow rate decreasing test) and the calculatedbottom-hole pressure.


Table 4The radius of investigation equations (r2

iD = αtD) from different riD analysis criteriaby using numerical simulation

Flow ratesriD criteria I

for piD = 0.1095riD criteria II

for piD = 0.01095riD criteria III

for piD = 0.001095

q = 100 stb/day r2iD = 3.986tD r2

iD = 10.363tD r2iD = 17.799tD

(Constant rate) R2 = 0.9999 R2 = 0.9999 R2 = 0.9999

q1 = 100 stb/day r2iD = 4.082tD r2

iD = 10.381tD r2iD = 17.804tD

q2 = 150 stb/day R2 = 0.9991 R2 = 0.9999 R2 = 0.9999

q1 = 100 stb/day r2iD = 3.868tD r2

iD = 10.344tD r2iD = 17.799tD

q2 = 50 stb/day R2 = 0.9983 R2 = 0.9999 R2 = 0.9999

q1 = 100 stb/day r2iD = 4.227tD r2

iD = 10.464tD r2iD = 17.833tD

q2 = 150 stb/day R2 = 0.9981 R2 = 0.9999 R2 = 0.9999

q3 = 100 stb/day

q1 = 100 stb/day r2iD = 3.679tD r2

iD = 10.249tD r2iD = 17.767tD

q2 = 150 stb/day R2 = 0.9956 R2 = 0.9998 R2 = 0.9999

q3 = 100 stb/day

3.3. Radius of Investigation Affected by Skin Factor

We investigated the effect of skin factor (s) to the propagation of the radius of investi-gation in simulation studies by varying skin factors. The results show that the radius ofinvestigation (riD) is independent of skin factor (s). For the different skin factors (s = 2, 1805, 8, and 10), the entire linear coefficients (α) are 3.986, 10.363, and 17.799 for thecriterion piD defined at the pressure front of 0.1095, 0.01095, and 0.001095, respectively(Figure 6). The radius of investigation equations for different skin factors are the same Figure 6as result from the equations of no-skin factor (s = 0).

3.4. Radius of Investigation Affected by Wellbore Storage 185

The effect of wellbore storage volume, in terms of dimensionless wellbore storage volume(CD), on the propagation is also investigated in this study. By using numerical simulationstudies, the radius of investigation equation for different wellbore storage volume (CD =102, CD = 103, CD = 104, and CD = 105) are obtained and plotted the results ofr2

iD versus tD on linear coordinates (Figure 7) for different criterion piD defined at the Figure 7190pressure front. The results are very close to these with no wellbore volume. The resultsshow that the coefficients (α), except at an early time (or small tD), are 3.98, 10.36, and17.80 for the criterion piD defined at the pressure front of 0.1095, 0.01095, and 0.001095,respectively, for all wellbore storage volumes studied (Figure 7).

From the result of dimensionless radius of investigation (riD) versus dimensionless 195time (tD) plotted on log-log plot axis for different wellbore storage volumes (CD = 0,CD = 102, CD = 103, CD = 104, and CD = 105) with criterion of piD = 0.1095,the propagation of the radius of investigation influenced by the wellbore storage effect


Figure 6. The integrate results of the dimensionless radius of investigation for different skin factor(S = 0, 2, 5, 8, 10) by using numerical simulation.

only in the early dimensionless time is observed (Figure 8). A linear relationship exists Figure 8between the dimensionless radius of investigation (riD) and dimensionless time (tD) inlog-log plot when the wellbore storage effect is ended (CD = 0) (Figure 8). The curves 200of dimensionless radius of investigation (riD) versus dimensionless time (tD) for differentwellbore storage volumes (CD = 102, CD = 103, CD = 104, and CD = 105) divergefrom the straight line of CD = 0. The degree of deviation is increased as the wellborestorage volume (CD) increases (Figure 8). In other words, the propagation time requiredto reach the specific boundary or distance is longer for the large wellbore storage volume 205than for the small wellbore storage volume.

Figure 7. The integrate results of the dimensionless radius of investigation for different wellborestorage volumes (CD = 0, 102, 103, 104, 105) by using numerical simulation.


Figure 8. Dimensionless radius of investigation vs. dimensionless time log-log plot for differentwellbore storage volumes (CD = 0, 102, 103, 104, 105) at the criterion pD = 0.1095.

4. Discussion

A linear relationship between the square of the dimensionless radius of investigation anddimensionless time (r2

iD = αtD) was studied in this study for both constant flow rate andvariable flow rate tests. The coefficient (α) of 4, or r2

iD = 4tD , was obtained from the 210results of all tests with constant flow rate for the criterion piD defined at the pressure frontof 0.1095. In other words, the widely-used radius of investigation equation r2

iD = 4tDis derived by the assumption of the dimensionless pressure defined at the pressure frontof 0.1095.

Using different criterion piD defined at the pressure front, we obtained different 215coefficients (α), which vary from 4 to 71.15 for the pressure front varied from 0.1095 to10−9, respectively.

The results from our study shows that radius of investigation is independent ofany constant flow rate. This result is the same as from Lee (1982), mentioning that“in principle, any flow would suffice—time required to achieve a particular radius of 220investigation is independent of flow rate.” However, the radius of investigation affectedby a rate change in a well for the criterion piD defined at the pressure front of 0.1095is observed (Figure 9). When flow rate increases from previous constant flow rate for Figure 9two-rate cases, the coefficient (α) increases and vice versa in the latter time (Figure 9).For the pressure front defined as piD = 0.01095, the slope in the plot of the square of 225dimensionless radius of investigation verses dimensionless time is not affected much bythe rate changes (Figure 10). The results are very close to the case of constant flow rate, Figure 10in which the coefficient is 10.39. When the criterion piD defined at the pressure frontdecreases, such as piD = 0.001095, the rate change affecting the slope in the plot of thesquare of dimensionless radius of investigation versus dimensionless time also decreases 230(Figure 11). Figure 11


Figure 9. The integrate results of the dimensionless radius of investigation for entire constant flowrate and variable flow rate tests at the criterion pD = 0.1095.

In the radius of investigation studies by both analytical solution and numerical so-lution for injecting well, the results of the dimensionless radius of investigation variedwith dimensionless time are the same as those from producing well (Figure 12). Figure 12

5. Conclusions

The propagation of the radius of investigation from a production well has been studied 235by both analytical and numerical methods. The radius of investigation equations in di-mensionless terms are derived with different criterions of radius of investigation defined.The conclusions of this study are as follows:

Figure 10. The integrate results of the dimensionless radius of investigation for entire constantflow rate and variable flow rate tests at the criterion pD = 0.01095.


Figure 11. The integrate results of the dimensionless radius of investigation for entire constantflow rate and variable flow rate tests at the criterion pD = 0.001095.

1. The relationship between the square of the dimensionless radius of investigationand the dimensionless time (r2

iD = αtD) is linear for the case of constant flow 240rate and may not be linear for the case of variable flow rates.

2. The constant coefficient (α) in constant flow rate cases varies from 4 to 71.15when the defined dimensionless pressure at the pressure front of the radius ofinvestigation is changed from 0.1095 to 10−9.

3. The radius of investigation is affected by a rate change in a well for the case of 245dimensionless pressure defined at the pressure front less than or equal to 0.1095.The widely-used radius of investigation equation, r2

iD = 400tD , independent offlow rate is valid only for constant flow rate cases.

Figure 12. The integrate results of the dimensionless radius of investigation for the injecting welland the producing well at a constant flow rate test.


4. The radius of investigation equation for an injecting well is the same as that froma producing well. The radius of investigation is independent of skin factor. The 250propagation of the radius of investigation is affected by wellbore storage effectonly at an early time, depending on the size of wellbore storage volume.

References

Chaudhry, A. U. 2004. Oil Well Testing Handbook. Amsterdam: Elsevier Inc.CMG. 2004. User’s Guide IMEX Advanced Oil/Gas Reservoir Simulator. Calgary, Alberta: Com- 255

puter Modelling Group Ltd.Earlougher, R. C., Jr. 1977. Advances in Well Test Analysis. Dallas, TX: Society of Petroleum

Engineers of the AIME.Jones, P. 1962. Reservoir limit test on gas wells. J. Petrol. Technol. June:613–618.Lee, J. 1982. Well Testing. Dallas, TX: Society of Petroleum Engineers of the AIME. 260Matthews, C. S., and Russell, D. G. 1967. Pressure Buildup and Flow Tests in Wells. Dallas, TX:

Society of Petroleum Engineers of the AIME.Muskat, M. 1934. The flow of compressible fluids through porous media and some problems in

heat conduction. Physicals 71.Tek, M. R., Grove, M. L., and Poettmann, F. H. 1957. Method for predicting the back-pressure 265

behavior of low-permeability natural gas wells. Trans. AIME 210–302.Van Poolen, H. K. 1964. Radius-of-drainage and stabilization-time equations. Oil Gas J. September

14:138–146.

Nomenclature

c fluid compressibility, psi−1

CD wellbore storage effect, dimensionlessB formation volume factor, rb/stbh formation thickness, ftk permeability, mdq flow (production) rate, stb/dayp pressure, psipD dimensionless pressure, dimensionlesspD1 dimensionless pressure of variable flow rates, dimensionlesspi initial formation pressure, psipiD dimensionless pressure defined at the pressure front, dimensionlessr radius, ftrD dimensionless radius, dimensionlessri radius of investigation, ftriD dimensionless radius of investigation, dimensionlessrw wellbore radius, fts skin factor, dimensionlesst time, hourstD dimensionless time, dimensionlessα the linear coefficient in the radius of investigation equation, dimensionless�p pressure drop, psi(�p)total pressure drop in variable flow rates, psiφ porosity, fractionµ viscosity, cp 270

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Proofs 1st_ Radius of Investigation