analysis of pressure falloff tests following cold water injection

5/10/2018 Analysis of Pressure Falloff Tests Following Cold Water Injection - slidepd...

http://slidepdf.com/reader/full/analysis-of-pressure-falloff-tests-following-cold-water-in

Analysis of Pressure-Falloff TestsFollowing Cold-Water InjectionReldar B. Bratvold, * SPE, and Roland N. Home, SPE, Stanford U.

~umma~. This p~per presen~s generaliz~~ procedures. t~ interpret pressure i~jection and falloff data following cold-water injection

Into a hot~OlIrese': '?lr. Th~ relative permeability ch~ractensl:lcs of the porous medium are accounted for, as is the temperature dependence

of t~e.flu~d mobilities. It IS .shown that the saturation and temperature gradients have significant effects on the pressure data for both

the Injection and falloff penods. The matching of field data to type curves generated from analytical solutions provides estimates ofthe t~mperature.-dependent mobilities of the flooded and uninvaded regions. The solutions also may be used to provide estimates of

the size of the Invaded region, the distance to the temperature discontinuity, heat capacities, and wellbore-storage and skin effects.

Introduction

Numerous full-field waterflooding projects are currently under way

throughout the world to improve recovery. In many large oilfields,

water injection is initiated during the early stages of reservoir de-

velopment. Exploratory wells are tested for injectivity, and injectors

are tested during field operation. If properly interpreted, these tests

can give information about the progress of the flood (i.e., frontal

advance), residual oil saturation, the flow characteristics of the virgin

formation, and near-wellbore damage.

In a water-injection well test , the injected fluid usually has a tem-

perature different from the initial reservoir temperature. During

injection, both a saturation and a temperature front propagate intothe reservoir, Furthermore, because of differences in oil and water

properties, a saturation gradient is established in the reservoir. The

water saturation is highest close to the well and continuously

decreases with distance from the well. Ahead of this invaded region

is the unflooded oil bank at initial water saturation.

For the interpretation of well-test data, the most important temper-

ature-dependent fluid property is the viscosity. The viscosity of both

oil and water may change by an order of magnitude between 50

and 572°F, with the major change occuring between 50 and

212 of. 1 This temperature effect strongly influences the fluid mo-

bilities, and hence, the saturation gradient and the transient pressure

response. The total fluid mobili ty changes continuously in the in-

vaded region and has to be accounted for in reservoir modeling

and data interpretation.

Many different models have been introduced for the analysis ofwater-injection and falloff tests. Typically, these models neglect

the temperature effects, the saturation gradient, or both. Refs. 2

and 3 provide reasonably complete reviews of previous works.

For this paper, the most important reference is Fayers'f exten-

sion of the fractional flow theory of Buckley and Leverett> to ac-

count for a radial temperature gradient in the reservoir. Fayers'

work was put into a mathematical framework by Karakas et al.>

and Hovdan.? Hovdan also used this incompressible-fluids solu-

tion to derive a pressure-transient solution for the late stages of a

cold-water-injection test.

Recently, Abbaszadeh and Kamal? presented procedures to ana-

lyze falloff data from water-injection wells. Their procedures are

based on analytical solutions not presented in their original paper

and include the effect of the saturation gradient inthe invaded region.

Nonisothermal effects were not considered.

In summary, a number of studies pertaining to well-test analysisof injection and falloff tests have been presented. However, none of

these account for both of the two most important effects in a typical

waterflood: the saturation gradient and the temperature effect.

The principal objectives of this paper are (I) to derive analytical

solutions that include the most important effects in a nonisothermal

water-injection/fal loff test, (2) to examine the parameters that in-

fluence the well injectivity, and (3) to present procedures to obtain

detailed and accurate information about the important reservoir and

fluid properties in a waterflood. Specifically, we consider the pres-

sure behavior at the well resulting from the simultaneous flow of

'Now a t IBM European Pet roleum App li cat ion Center .

Copyr igh t 1990 Society of Pe troleum Enginee rs

SPE Formation Evaluation, September 1990

oil and water in a reservoir with a radial temperature gradient. Ana-

lytical solutions that account for the effects of temperature and satu-

ration gradients are derived and discussed. Consequences of

neglecting the temperature and saturation effects are illustrated.

Solutions for linear systems, including the effects of linear bound-

aries in cylindrical reservoirs, were presented by Bratvold and

Larsen. 8

Mathematical Model

Fig. 1 presents a schematic of the reservoir configuration consid-

ered. The reservoir is assumed to be cylindrical with the well atthe center. The well penetrates the entire formation thickness, and

fluid is injected at a constant rate. The reservoir is assumed to be

a uniform, homogeneous porous medium, completely saturated with

oil and water, Liquid compressibilities are assumed to be constant,

while the viscosities are assumed to be functions of temperature

only. Neglecting effects of gravity, as well as heat transfer to the

surrounding formation, permits the use of a ID radial model.

Injection Period. The transient, nonisothermal two-phase flow of oil

and water requires that saturations, pressures, and temperatures be

determined simultaneously at any time. Furthermore, because cold-

water injection into a hot-oil reservoir is a moving-boundary prob-

lem, it cannot be solved with standard linear techniques, such as

eigenfunction expansion, integral transforms, or Green's function

methods.

To circumvent the problem of simultaneously solving the coupled

second-order conservation equations, we derive an alternative

approximate solution to the injection problem using a two-step

procedure.

Step 1. Assume incompressible fluids. Then use fractional flow

theory to solve the resulting first-order coupled energy- and mass-

conservation equations.4,6,7 This essentially amounts to decoupling

the equations for saturation and temperature from the pressure equa-

t ion. The saturation profile obtained is a Buckley-Leverett> pro-

file including (convective) temperature effects.

Step 2. With the saturation and temperature proftles and the mobil-

i ties and diffusivities known from Step I, solve the diffusion equa-

tion for pressure by assuming that the fluid compressibilities are

small and constant . Hence, the pressure distribution in the system

is obtained by superimposing pressure-transient effects on a satu-

ration profile known a p r io r i.Fig, 2 shows an example of a saturation and temperature distri-

butions as functions of the similarity transform t=7rcphr2/qt, as

calculated from the Buckley-Leverett model and including temper-

ature effects. 4,6,7 Note that the profile exhibits two saturation dis-

continuities. In addition to the discontinuity depicted by the standard

Buckley-Leverett theory, the saturation distribution shows a sec-

ond discontinuity caused by the step-change in temperature. The

magnitude of the saturation change at the temperature discontinuity

is related to the ratio between the mobility ratios in the hot and

cold zones. The saturation distribution obtained from a numerical

simulator is superimposed on the analytical saturation profile. The

simulations were performed with a two-phase, 2D, black-oil simula-

tor developed by Nyhus? that is described later in the paper.

293



Region 1

Region 2

Region 1

Fig. 1- Two-region, radial moving-boundary problem.

As Fig. 2 shows, the saturation distribution obtained numerically

isvery close to the analytical solution for this data set, which isgiven

in Table 1and which is typical for water-injection tests. On the basis

of Fig. 2 and other examples.I it is reasonable to conclude that

the nonisothermal Buckley-Leverett solution is a good approximation

to the actual saturation distribution (neglecting gravity) for typical

water-injection tests where the compressibilities Co and Cw are of

the order 10 - 5 to 10 - 6 psi - l and the mobility ratio M < ! 1 : 100.

Step 2 consists of solving the diffusion equation for pressure in the

composite system schematically illustrated in Fig. 1. With the satura-

tion distribution known as a function of time and space from Step 1,

the total fluid mobility and total system compressibility become func-

tions of time and space. The locations of the saturation and tempera-

ture fronts are time-dependent, and hence this is a moving-boundary

problem. In general, moving-boundary problems are nonlinear and

cannot be solved by standard techniques based on superposit ion.

The moving-boundary problem can be linearized by introducing

the Boltzmann variable (similarity transform) y=rl)4tD' As

shown in Appendix A, the Boltzmann variable is constant at the

moving boundary by virtue of the integrated frontal-advance equa-

tion (Eq. A-lO). Hence, the problem can be transformed to a com-

posite problem with a fixed interface where the independent variable

is y. The mathematical formulation and solution to the injection

problem is presented in Appendix A. To obtain the wellbore pres-

sure in an infinite system we evaluate Eqs. A-22 and A-23 at

rD=1.

PwD =(MI2)E, (1I4tD)' to s, 1I4YBL (1)

294

1.0I

I- - Temperature

I-- Buckley-Leverett00000 Numencal simulator

I

I~I

f-O .S I.

0

0

~ I

(f) I Set 2 0

I

I M = 5

I fj = 21.6

-I

0.00.0 1.0 2.0 3.0

~

Fig. 2-Comparison of simulated and analytical saturat ion and

temperature distributions.

TABLE 1-SIMULATOR INPUT DATA

USED IN VERIFICATION

Reservoir Properties Fluid Properties

TR , OF 180 SWj , fraction 0.2

Tj , OF 60.7 SO" fraction 0.14 > , fraction 0.2 co, psi " ' 1 x 10-5

h, ft 25 cw, psi-1 1 x 10-6

r w , ft 0.25 (pC)o' Btu/(ft3_0F) 23.00

r e' ft 1,000 (pC)w, Btu/(ft3_OF) 62.35

k, md 20 (pC) s' Btu/(ft3_OF) 42.45

P R , psi 1,000 krw(Sw) Sw3

q, BID 250 kro(Sw) (1-Sw) 3

Viscosity, cp

Data Set 1

2.00

8.00

0.25

0.40

Data Set 2

2.40

7.20

0.401.20

flon

floc

flwh

flwc

and PwD='/2E, (l/41)tD) - V2E1 (YBL/1)+ (MI2)E1 (YBL)

+ '/2IBL (f"/f')[(lIF).) -1]dSw, to ~ 1/4YBL' ..... (2)

sw /

Note that YBL is constant and that the saturation at the wellbore

sandface, Swf' is a function of time because of the line-source

inner-boundary condition. Pressure is scaled with respect to the

water propert ies at injection conditions, while time is scaled with

respect to the oil properties in the uninvaded oil zone.

For late times, the above solution approaches

pwD='h(ln tD+O.80907)+sa' (3)

where the apparent skin factor, sa' is given by

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

The late-time solution is equivalent to the solutions derived by Hov-

dan," Benson et al., 10 and Barkve. * These authors assumed the

inner region to be incompressible, not only in describing the satu-

ration profile." but also in solving for the pressure distribution.

This implies that the fluids in the invaded region essentially be-

have as incompressible at late times in the injection period.

'Pers~nal communication with T. Barkve, U. of Bergen. Norway (March 1987).




1.0 -,--------------------,

-_ Injection doto

00000 Folloff doto

"f) 0.5

Set 1

M 5f 7 = 21.6

0.0 +---r----,----.---..,.---,.----,---~o 200 400 600

Fig. 3-Saturatlon distribution during falloff.

Following the approach used by Verigin 1 1 and Ramey, 1 2 we can

derive approximate solutions to the finite outer-boundary problems

by solving a three-region problem with moving boundaries. The

wellbore pressure for the closed system is

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

The constant-pressure outer-boundary solution is given by Eqs. 5

and 6 without the exponential (last) term.

Finally, note that the above solutions apply equally well to thecase where temperature effects are neglected, provided that the ap-

propriate fractional-flow curve is used. As demonstrated in Appen-

dix A, these solutions reduce to the two-region solutions derived

by Woodward and Thambynayagam 13 and Barkve, 14 if the flow

is piston-like.

Falloff Period. The solid line in Fig. 3 shows the saturation distri-

bution obtained from the numerical model after 100 days of injection

with the data in Table 1. The saturation profile obtained from the

same simulator after the well is shut in for 100 days after the 100

days of injection is also shown. The saturation profiles are virtually

identical, implying that the saturation distribution during falloff re-

mains stationary. This is consistent with use of th e Buckley-Leverett

theory for the injection solution because the assumption of incom-

pressible fluids will result in the immediate stoppage of the mov-ing saturation distribution and temperature discontinuity. The falloff

problem can then be approximated by a linear problem (fixed in-

terface) where the mobilities and diffusivities are given by their

value at the end of the injection period. The solution to this variable-

coefficient problem can be obtained by dividing the invaded region

into several regions where the saturation is approximated by its aver-

age value within each region. The pressure distribution at the end

of the injection period is the appropriate initial value for the falloff

problem. The formulation and solution to the problem are given

in Appendix B. Because we do not rely on the use of a similarity

transform in solving the falloff problem, we can implement a finite

wellbore radius with storage, skin, and a finite outer boundary. The


30~-------------------,-- Similarity solut ion00000 Numerical solution

- - Piston-like displacement

20

10

Set 2

10'

t o10 • 10 •

Fig.4-Comparison of analytical andnumerical solutions for

the Injection period in an infinite reservoir.

wellbore-pressure solution for the SD =0 case is obtained by in-

verting Eq. B-14 and evaluating the pressure at rD =I:

+ £-I[X llK O (-.j~ )]+ £ -1[X I2 Io(~ )], (7)

while the solution including wellbore storage is obtained from the

inner-boundary condition (Eq. B-6).

As demonstrated by Bratvold ' and Bratvold and Horne, 15 the

falloff solution can also be obtained by superposing the solutions

to the stationary saturation-distribution problem.

Solution Verification

In this section, the analytical solutions for the injection and falloff

periods are compared with numerically simulated results. The simu-

lator is a single-well thermal model, which numerically solves a

more-detailed model of the physical situation." The appropriate

partial-differential equations are solved with a finite-difference tech-

nique, and the solution procedure is fully implicit with respect topressure, saturation, and temperature. Effects of gravity, capillary

pressure, and variations in relative permeabilities are accounted for

in the simulator. The simulator can include heat convection and

conduction in two dimensions in the reservoir, as well as conduction

between the reservoir and the cap- and bedrock. The heat capacities

and conductivit ies are assumed to be independent of temperature

and pressure. The densities are assumed to be linear functions of

pressure and temperature, while the viscosities are entered in table

form as functions of pressure and temperature.

Gravity was neglected in all the simulations conducted. A total

of 1,000 gridblocks was used in the simulator to permit an accurate

determination of the saturation, temperature, and pressure through-

out the reservoir.

Injection Period. Fig. 4 presents data from two injection tests in

an infinite reservoir. Table I gives the data used in the numerical

reservoir simulator. The dimensionless pressures are plotted vs.

injection time. The two cases shown differ in the values of the vis-

cosities at injection and reservoir temperature, and hence, differ

in mobility ratios. The numerical results compare very well with

the analytical solution for both cases.

Fig. 4 also shows the results from a three-bank analytical solution,

where the displacement is assumed to be piston-l ike and the fluid

discontinuities move according to mass and energy balance. 13,14

Obviously, significant errors are introduced by neglecting the vari-

able saturation profile. In particular, application of the three-region

solution results in a large underestimation of the apparent skin fac-

tor, and hence, an overestimation of the formation damage.

295



2 5 ~ - - - - - - - - - - - - - - - - - - - - - - - - - - ~ - - - - - - - ,

15

Set 2

20

-- Similarity solution00000 Numerical solution

08

o

•.10

Fig. 5-Comperlson of analytical and numerical solutions for

the Injection period In a finite reservoir.

Fig. S shows injection results for closed and constant-pressure

outer boundaries. Again, we get a very good match between the

numerical and analytical solutions for the closed outer-boundary

case. The numerical simulator used lacked the capability of im-

plementing a constant-pressure outer boundary. Unlike the constant-

pressure case, however, the closed outer-boundary solution actu-

ally violates a boundary condition.? and hence we expect the

constant-pressure outer-boundary solution to give a better approx-

imation to the actual pressure behavior.

Falloff Period. Fig. , presents falloff results for the two injection

cases shown in Fig. 4. Fig. 7 shows falloff results following injec-

tion into a finite reservoir. The injection time was 100days in both

figures.

As discussed previously, the falloff problem is a variable-

coefficient problem that is solved by dividing the -invaded region

into several regions where the saturations and the coefficients in

the governing equation are approximated by their average values.

The accuracy of this approach increases with the number of regions

and is also a function of the saturation gradient in the flooded region.

For the examples presented in Figs. 6 and 7, we found eight regions

to be sufficient. As with the injection results, the analytical solu-

t ions compare excellently with the results obtained numerically.

In addition to the examples presented here, numerous other cases

were investigated. 3 In all cases where typical water-injection data

were used, the agreement between the numerical and analytical so-

lutions was excellent. Hence, the validity of the mathematical model

2 5 , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,Set 2

-- Analytical solution00000 Numerical solution

20

15

o•.10

5

10'

6t O

Fig. 7-Comparl80n o f .... ytlc.l.nd numerical solutions for

the f.lloff period In a finite reaervolr.

296

10 •

3 0 ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,-- Analytical solution00000 Numerical solution- - Piston-like displacement

20

o

•.10

10 •10 •

llto10 • 10 •

Fig. 6-Comparison of analytical and numerical solutions for

the falloff period in an infinite reservoir.

is established and the use of the analytical solutions for well-test

interpretation is justified.

Results

Inthis section we illustrate the general behavior and the applicability

of the solutions derived in the previous sections. All examples use

the data in Table 1.

Injection Period. For both cases shown in Fig. 4, two well-defined

straight lines are evident. From the solution (Eqs. 1 and 2), we

see that the first straight line corresponds to the mobility of the virgin

reservoir fluid and has slope M12. The second line has slope 1/2

and corresponds to the mobility in the completely flooded region

where Sw = I-Sor·

Fig. 8 is a plot of the solution in real variables with Data Set

2. The calculated wellbore pressures for isothermal injection tests

are also shown. The uppermost curve is generated with the cold

viscosities, 1 - ' 0 = 7.2 cp and I-'w = 1.2 cp, while the lowermost curve

depicts the result when only hot viscosities are used, 1 - ' 0 =2.4 cp

and I - 'w= 0.4 cp. It is obvious that any estimate of the permeability-

thickness product, kr1kh, 1=0, w , will be greatly in error if the vis-

cosities are not adjusted for temperature. Furthermore, the appar-

ent skin factor caused by the injected fluid bank will be significantly

overestimated, resulting in an equally large underestimate of the

formation damage at the wellbore.

The solid lines in Fig. 9 show the injection profiles including

temperature effects for Data Sets I and 2. The isothermal solution

3500~----------------------------------__,

-- - Nonisothermol solut ion- - Isothermal, cold v iscosities------ Isothermal. hot viscosit ies

3000

/

Set 2

/

/

/

0:,2000

/

1500 /

l0001-~mr~~_n~rT~~~mr~~-n~rT~10" 10--' 10-. 10 10•

t, hours

Fig. 8-Ef fects of assuming temperature-independent viscosi-

t ies during inject ion.




3 0 , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,--- Nonisothermal

25 00000 Isothermal, p.." I J . o > ,

20

o~15

Q.

oo~

oo~Set 2

10

5

10•

to10• 10•0•

Fig. 9-Comparlson of nonlsothermal and Isothermal solu-tions with temperature-adjusted viscosities during Injection.

where the water viscosities are taken at their cold values (l'w=OA

and 1.2 cp) while the hot values of the oil viscosities are used

(1'0=2.0 and 2.4 cp) is also included. From this graph, we see that

the isothermal solution-the solution where the temperature dis-

continuity in saturation is neglected but the temperature-dependent

values of the viscosities are used-will yield both the correct krlkhand a good estimate of the formation damage.

Ifwe take a closer look at Fig. 9, which compares the noniso-

thermal and isothermal solutions, we see that the nonisothermal solu-

tion has a discontinuous time derivative. This discontinuity does

not occur at the point where the solution switches from Eq. 1 to

Eq. 2 but at some later time corresponding to the location of the

step-change in temperature. For Data Sets 1 and 2, this occurs at

the dimensionless t imes 1,535 and 1,280 (about 4 minutes). At this

time, the viscosities used in the solution switch from hot to cold

values, and we get a discontinuous time derivative. This abrupt

change in the mobilities has the same immediate effect as a perme-

ability boundary (increasing permeability in this example). Short-

ly thereafter, the invaded region starts behaving like a positive skin

zone, and the pressure starts to increase more rapidly. This sud-

den change in the mobilities is also apparent in the solution ob-

tained numerically (Fig. 4). The temperature change, however, is

gradual in the simulator because of the finite grid, and the slope

iscontinuous although different from the isothermal case. The major

effect of this peculiarity for the injection case is the shortening of

the two semilog straight lines as seen in Fig. 9. Because nonlinear

regression techniques typically rely on accurate calculations of the

pressure derivatives with respect to the parameters being investigat-

3 0 , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,o 0

00

o 0

00

-- Nonisothermol solution

- - lsotherrnol, / L w o . Jl.ot,

0000 0 lsotberrnot, Ji..-h. / J . o n

. A6 6.6 Isothermal. J . l . w c . J . ' o c

25

.. .20

..o~15

0...

oo

10

oo

Set 2

• 0

• 0

o5

10• 10•

6tO10•

Fig. 11-Comparison of nonlsothermal and Isothermal solu-tions during falloff.


25

15

o(f)

Corey rei. perm.

n = 3 - 8

5

o 6 8 104

M

Fig. 10-Apparent skin factor.

ed, this difference in the rate of change may lead to errors if the

isothermal solution is used. This obviously is also a concern if type-

curve-matching techniques are used to estimate reservoir and fluid

parameters.

A closer inspection of the equation for the apparent skin factor

(Eq. 4) shows that it is a function of the saturation-dependent pa-rameters. In particular, i t will be a function of the relative permea-

bility data, which usually are not known in advance. In Fig. 10

we have plotted the apparent skin factor as calculated from Eq. 4 vs.

the mobility ratio, M. The graphs correspond to different values

of the exponent in Corey's 1 6 relative permeability equations. Note

that the functional form of the relative permeability data will de-

pend on the exponent, while the endpoint values are independent

of the exponent. The mobility ratio, M, on the abscissa, however,

is a function of the endpoint values and not of the functional form

of the relative permeability data. From Fig. 10, we see that the

data for the various exponents are very close, in particular for the

lower range of mobility ratios. The interpretation is that if the mo-

bil ity ratio is known, the apparent skin factor can be estimated from

the graph in Fig. 10. Similar graphs can be generated from the

analytical solution for any chosen set of relative permeability equa-tions. Furthermore, because the apparent skin factor does not de-

pend strongly on the functional form of the relative permeability

data, but rather on the endpoint values that can be obtained from

the analysis of the injection and falloff data, we can estimate saand obtain the formation damage (or stimulation), s, as s=St -sa'where SI is the total skin factor obtained by standard analysis. 1 7

10,------------------------------------,

(l)

0...o(f)

--- Nonisothermol solution00000 Numerical solution- - Isothermal solution. JJ-. J .loo,

~(l)

C~oI

Set 2

10' 10• 10•0 •

M o

Fig. 12- Temperature and saturation effects on pressure-de-rivative type curves.

2 9 7



The injection solution for the constant-pressure, outer-boundary

case (Fig. 5) has a sign change in the slope as the outer boundary

is "felt" in the wellbore pressure. This slope change is described

mathematically by the late-time solution to the constant-pressure

outer-boundary case and the late-time slope- is

1/2(I-M). . (8)

Hence, the slope will change sign at approximately tD =teD' where

teD is given by Eq. A-28, if M> 1. Physically, this can be ex-

plained by comparing the mobility of the reservoir oil with the mo-

bili ty of the injected water. If the cold water is more mobile than

the hot oil, the overall resistance to flow (or injection) will decreaseas water occupies a larger fraction of the reservoir. Eventually,

the pressure must be given by the steady-state solution

PwD= lilreD , (9)

which requires a drop in the wellbore pressure if M> 1.

Finally, to apply the above results to field data, note that the well-

bore storage effect may mask the first straight line. Furthermore,

in a relatively small reservoir, the outer-boundary effects may mask

the second straight line; i.e., the line corresponding to the com-

pletely flooded region at the residual oil saturation. As shown in

Ref. 3, however, the injection data will eventually plot as a straight

l ine with slope 2M7r on a dimensionless Cartesian graph for the

closed-outer-boundary case. If the outer boundary is a constant-

pressure type, the dimensionless late-time data will plot as a semi-

log straight line with slope 'h(l-M). Hence, if the test is longenough, the injection data will give an estimate ofthe water mobil-

ity for the completely flooded region if the system is infinite-acting,

or it will give M if outer-boundary effects mask the infinite-acting

semilog straight line.

Falloff Period. Let us now return to Fig. 6 to investigate the char-

acteristics of the pressure/time curve during the falloff period. Fig.

11presents the analytical solution from Data Set 2. The graph also

includes the solutions with cold and hot viscosity values. As in the

injection case, we see that neglecting the temperature effects on

the viscosity results in gross errors in the calculation of both krlkh

and s. The dashed line in Fig. 11 represents an isothermal analyti-

cal solution of the cold-water and hot-oil viscosit ies. The isother-

mal solution is shifted above the nonisothermal at early times but

will give .the correct values at late times.

A cursory inspection of the falloff data in Fig. 6 suggests that

the pressure/time data will yield two straight lines separated by a

transition period. Intuitively, and from other work, 13,14, 18·21 we

would expect the slope of these straight lines to correspond to the

completely flooded and uninvaded regions, respectively. Fig. 12is a log-log plot of the pressure derivative, d pwD /[ d I n( tD . + atD) /

atD], vs. time. The analytical solution (the solid line) indi~ates that

the falloff data will not plot as a semilog straight line correspond-

ing to Sw =I-So" This is explained by Fig. 3, the saturation pro-

fi le in the reservoir during the falloff period, which shows that the

only place where S w =I-So, is at the sandface. From the Buckley-

Leverett theory, we know that

ro drD I o c r I s (10)

dtD Sw w

or rD21s ocrls to. (11)w w

When the analytical relative permeabili ty curves are used, /'(1-

So,) =0 and hence rDl1-sor =0; i.e ., the only location where the

saturation can be at its endpoint value, I-Sop is at the line source.

This may be an artifact of the analytical solution, but even if the

injected fluid was slightly mobile at the sandface, the fully flooded

region would be very small, and the corresponding straight line

would be too short to be identifiable. If this is the case, why do

the injection data yield the properties of the completely flooded

region? The difference in the wellbore-pressure behavior between

the injection and falloff cases can be explained with the qualitative

concept of a radius of investigation. 22 Remember that the injection

29 8

problem is a moving-boundary problem where the saturation front

propagates into the reservoir with time. Inthe falloff problem, how-

ever, the saturation distribution is approximately stationary for all

times. The wellbore-pressure response reflects the average reser-

voir and fluid properties within the radius of investigation. In the

injection case, the location of the saturation front is proportional

to the radius of investigation and, after a short time, the wellbore

pressure will be dominated by the mobility near the sandface. In

the falloff case, however, the saturation reflected by the wellbore

pressure will be I-So, at atD =0, and then will continuously

decrease to SW ; , as the saturation distribution in Fig. 3 and the

pressure derivative in Fig. 12 show. Hence, if the well is shut inlong enough, the falloff data will plot as a straight line representing

the mobility of the uninvaded region with saturation S w =Sw; '

A second interesting feature of the analytical falloff solution is

the decreasing slope about halfway through the time period investi-

gated in Fig. 12. The solid line corresponds to the nonisothermal

analytical solution, the dashed line is the isothermal solution with

temperature-corrected viscosities, and the circles reflect the numer-

ical solution. The oscillating slope, which is caused by the same

phenomenon, is related to the discontinuous slope in the injection

solution. At early times (small atD) , the radius of investigation en-

compasses the cold region and the mobilities are low. Later, the

average fluid properties reflected by the wellbore pressure will shift

gradually toward their hot or high values in the invaded region,

resulting in a decreasing slope. As the radius of investigation in-

creases further, the propert ies within the radius of investigationeventually will be dominated by the properties in the uninvaded

region-i.e., the hot oil properties. This is also reflected by the

nonisothermal solution obtained numerically. The oscillatory slope

effect, however, is more diffuse in the numerical solution because

of the finite grid system. The resolution in the slope obtained nu-

merically is also sensitive to the timestep size used. Fig. 12 also

shows the isothermal solution for cold-water and hot-oil viscosities.

Although the isothermal solution will represent the init ial and the

straight-line portions of the falloff data correctly, its slope character-

istics are significantly different from the nonisothermal solution.

Hence, in analysis of the falloff data by type-curve methods or non-

linear regression techniques, the temperature effects will be im-

portant and must be accounted for to obtain a good representation

of the solutions. This is particularly true if the falloff test is not

long enough to obtain the semilog straight line.

For the examples presented here, the straight-line portion repre-

senting the uninvaded region will be reached almost one full log

cycle earl ier in the nonisothermal case than in the isothermal case

because the mobili ties used in generating the isothermal solution

give a larger invaded region-i.e., a larger rDBL than the noniso-

thermal solution used in the match.

Finally, the shut-in time required to reach the semilog straight

line of the uninvaded region may be too long to be practically attain-

able. In the example discussed here, the well must be shut in for

about 30 days or about one-third ofthe injection time to get within

5% of the correct slope. This suggests that type-curve-matching

techniques, preferably automated, will be essential for the analysis

of the falloff data.

Conclusions

1. Both temperature and saturation effects must be accounted for

to analyze injection-falloff tests associated with cold-water injection

into hot-oil reservoirs accurately.

2. A combined semilog and type-curve (nonlinear regression)

analysis of field data can provide estimates of the mobilit ies of the

completely flooded and the uninvaded regions; the apparent skin

factor caused by saturation and temperature gradients, and from

this, the formation damage (or stimulation); the size of the uninvaded

region and the distance to the temperature discontinuity; and the

product of the density and heat capacity of the reservoir rock or

one of the fluid phases.

3. Changes in slope caused by nonisothermal pressure transients

may be incorrectly interpreted as reservoir boundaries.




4. If saturation gradients are ignored, the formation damage can

be grossly overestimated.

5. The solution technique presented is applicable to a wide range

of injection-falloff problems.

Nomenclature

ai = distance to Interface iin multizone problem

an = distance to oil /water interface in multizone problem

C = compressibility, psi - 1

ct = total compressibility, ct=ct(Sw), psi r !

C

to = total compressibili ty of oil region, psi=!C = specific heat capacity, Btu/(ft3 -OF)

d= right-side vector

E1(x) = exponential integral, 1 00

e-U/udu

f = fractional flow of water

f' = df/dSw

I" = d2f/dS~

FA = total mobility, (A o +Aw)/~w' dimensionless

h = formation thickness, ft

Io(x) = modified Bessel function, first kind, order zero

IIx) = modified Bessel function, first kind, order one

k = permeability, md

t:= kro(Swi)

krw = krw(l-Sor)

Ko = modified Bessel function, second kind, order zero

K, = modified Bessel function, second kind, order one.£ -I(u) = inverse Laplace transform of u

M = mobility ratio, ~wc/~oh

P = pressure

PD = dimensionless pressure, 27r~w(P-PR)/q

P R = ini tial reservoir pressure, psi

PwD = dimensionless wellbore pressure, 27r ~w(Pwf -PR)!q

Pwf = wellbore pressure, psi

q = injection rate, BID

r = radius, ft

fe = exterior reservoir boundary radius, ft

rt = radius to temperature discontinuity, ft

rw = wellbore radius, ft

s = skin factor

SBL=

Buckley-Leverett saturation, fractionSD = dimensionless wellbore storage coefficient based on

injected water properties

Sor = residual oil saturation, fraction

Sw = water saturation, fraction

Swf = water saturation at the wellbore sandface, fraction

Swi = initial water saturation, fraction

t = time, hours

to = dimensionless time, ~ot!</>ctor~

teD = dimensionless time at which outer boundary is felt

in wellbore pressure

ilt = shut-in time, hours

T = temperature, OF

TD = dimensionless temperature, (T-T;)I(TR-T[)

T, = injection temperature, OF

TR = initial reservoir temperature, OFu = Laplace transform parameter

X i = unknown constants in multizone problem

y = Boltzmann variable, rB/4tD

YBL = Boltzmann variable evaluated at rBL, r8SL/4tD

'Y = Euler constant , 0.57722 ...

r = matrix

E = qCto!27r~oh

.I= similarity transform, 7rf2h</>!qt

1] = diffusivity ratio, M(ctolct)

~ = endpoint diffusivity ratio, M(ctolctw)

K = 1 I F A 1 1

A = mobility


~o = endpoint oil mobility, kkro(Swi)/ J.loh, md/cp

~w = endpoint water mobility, kkrw(l-Sor)/J.lwc, md/cp

J. I = viscosity, cp

~= variable of integration, dimensionless

p = density, lbm/ft '

< /> = porosity, fraction

if i = pressure distribution at shut-in time, dimensionless

Subscripts

a = apparent

BL = Buckley-Leverett

D = dimensionless

f = reservoir rock

i=Region iin falloff solution or injection

1= 0 or w

n =number of regions used in falloff solution

0= oil

oc = oil at injection temperature (cold)

oh = oil at reservoir temperature (hot)

P = production time

r = relative

t = total

w = water

wc = water at injection temperature (cold)

wh = water at reservoir temperature (hot)

1 = invaded zone2 =uninvaded zone

Superscripts

= average

= vector

r = first derivative

" = second derivative

Acknowledgments

This work was supported financially by SUPRI-D, the Stanford U.

Research Consortium for Innovation in Well Test Analysis. Finan-

cial aid to one of the authors from the Norway-America Assn., The

Royal Norwegian Council for Scientific & Industrial Research

(NTNF), Rogaland Research Inst., and Statoil are gratefully ac-

knowledged. Many thanks also to Svein M. Skjaeveland at RogalandU. Center for suggesting the problem and to Takao Nanba for

providing us with the nonlinear regression program he developed

as a student at Stanford U.

References

I. Prats, M.: Thermal Recovery, Monograph Series, SPE, Richardson,

TX (1982) 7.

2. Abbaszadeh, M. and Kamal, M.M.: "Pressure-Transient Test ing of

Water-Injection Wells," SPERE (Feb. 1989) 115-24.

3. Bratvold, R.B.: "An Analytical Study of Reservoir Pressure Response

Following Cold Water Injection, " PhD dissertation, Stanford V., Stan-

ford, CA (March 1989) .

4. Fayers, F.J.: "Some Theoretical Results Concerning the Displacement

of a Viscous Oil by a Hot Fluid in a Porous Medium, Part 1," 1. Fluid

Mech. (1962) 1365-76.

5. Buckley, S.E. and Leverett , M.C.: "Mechanism of Fluid Displacementin Sands," Trans., AIME (1942) 146, 107-16.

6. Karakas, M., Saneie, S., and Yortsos, Y.: "Displacement ofa Viscous

Oil by the Combined Injection of Hot Water and Chemical Additive,"

SPERE (July 1986) 391-402; Trans., AIME, 282.

7. Hovdan, M.: "Water Inject ion-Incompressible Analytical Solution

With Temperature Effects," technical report MH-I/86, Statoil,

Stavanger, Norway (Sept. 1986) (in Norwegian).

8. Bratvold, R.B. and Larsen, L.: "Effects of Linear Boundaries on

Pressure-Transient Injection and Falloff Data," paper SPE 19830

presented at the 1989 SPE Annual Technical Conference and Exhibition,

San Antonio, Oct . 8-11.

9. Nyhus, E.: "Modelling of Thermal Injection and Falloff Tests ," techni-

cal report , Rogaland Research Inst., Stavanger, Norway (Oct . 1987)

(in Norwegian).

299



10. Benson, S.M. et al.: "Analysis of Thermally Induced Permeability En-

hancement in Geothermal Injection Wells," Proc., Workshop on Geo-

thermal Reservoir Engineering, Stanford U., Stanford, CA (1987).

11. Verigin, N.N.: "On the Pressurized Forcing of Binder Solutions Into

Rocks in Order to Increase the Strength and Imperviousness to Water

of the Foundations of Hydrotechnical Installations," lsvestia Akademii

Nauk SSSR Odt. Tehn. Nauk (1952) 5, 674-87 (in Russian).

12. Ramey, H.J. Jr.: "Approximate Solut ions for Unsteady State Liquid

Flow in Composite Reservoir," J. Cdn. Pet. Tech. (Jan.-March 1970)

32-37.

13. Woodward, O.K. and Thambynayagam, R.K.M.: "Pressure Buildup

and Falloff Analysis of Water-Injection Tests," paper SPE 12344 avail-

able from SPE Book Order Dept., Richardson, TX.

14. Barkve, T.: "Nonisothermal Effects in Water-Injection Well Tests,"SPEFE (June 1989) 281-86.

15. Bratvold, R.B. and Horne, R.N.: "An Analytical Solution to a Multiple-

Region Moving Boundary Problem-Nonisothennal Water Injection Into

Oil Reservoirs," Proc., Second Annual Joint IMA/SPE European Con-

ference on th e Mathematics of Oil Recovery, Cambridge U. (July 1989).

16. Corey, A.T.: "The Interrelat ion Between Gas and Oi l Relative Per-

meabilities," Producers Monthly (Nov. 1954) 19, No. 11,34-41.

17. Earlougher, R.C. Jr.: Advances in Well Test Analysis, Monograph Ser-

ies, SPE, Richardson, TX (1977) 5.18. Benson, S.M. and Bodvarsson, G.S. : "Nonisothermal Effects During

Injection and Falloff Tests," SPEFE (Feb. 1986) 53-63.

19. Weinstein , H.G.: "Cold Waterflooding a Warm Reservoir ," paper SPE

5083 presented at the 1974 SPE Annual Meeting, Houston, Sept. 30-

Oct. 3.

20. Sosa, A., Raghavan, R., and Limon, T.J.: "Effect of Relat ive Perme-

abili ty and Mobili ty Ratio on Pressure Falloff Behavior," JPT (June

1981) 1125-35.

21. Kazemi, H., Merrill , L.S. , an d Jargon, J.R.: "Problems in Interpretation

of Pressure Fall Off Tests in Reservoirs With an d Without Fluid Banks,"

JP T (Sept. 1972) 1147-56.

22. Streltsova, T.D.: Well Testing in Heterogeneous Formations, John Wiley

& Sons, New York Ci ty, Exxon Monograph (1988) .

23. Barkve, T. : "A Study of the Verigin Problem with Application to Analy-

sis of Water Injection Wells ," PhD dissertation, U. of Bergen, Bergen,

Norway (March 1985).

24. Stehfest, H.: "Numerical Inversion of Laplace Transforms," Communi-

cations of the A CM (Jan. 1970) 13, No.1, Algori thm 368.

Appendix A-Injection Solution

We wiJI derive the injection solution in an infinite reservoir with

the similarity-transform approach. The disadvantage of this approach

is that, although it is exact for a given saturation distribution, it

is limited to a line-source well, and hence, wellbore-storage effects

cannot be included. To include wellbore-storage effects, the solu-tions obtained by the quasistationary approach3,15 should be used.

We limit our discussion to cold-water injection into a hot-oil reser-

voir. The procedure, however, applies to any system for which the

saturation profile can be described a priori.

Infinite Reservoir. Assuming that the reservoir consists of two

different regions separated by a moving discontinuity in fluid satura-

tion, as outlined in Fig. I, we obtain the following mathematical

model for an infinite system with a line-source well.

Governing equations:

10 ( OPD2) OPD2

and -- ro=rr: =--, ro <rD<C:X> ' (A-2)ro orD orD otD BL

where rDBL

=rDBL

(tD)'

1 ) [ Sw(rD ,tD ) ] =1 ) (rD ,tD ) =M(ctol ct),

c t=Swcw+(I-Sw)co +cJ '

and FX[Sw( rD , tD ) ] =F } . .( rD , tD )=(Ao +Aw) /~w'

Initial conditions:

P Dl = PD2=0, tD=O, (A-3)

and rDBL

=0, to =0. . (A-4)

300

Boundary conditions:

lim rD (o PD lo rD ) =-I (A-5)'D-O 1

and lim PD =0. . (A-6)'D -co 2

Moving-boundary conditions:

PD1=PD

2' rD=rD

BL.••.•••.••••••••••••....•••• (A-7)

OPDl 1 OPD2and F x--= ---, rD=rD (A-8)

orD M orDBL

All variables and parameters are dimensionless. PD and PD are

the pressures in the invaded and uninvaded regions,'respecti~ely.

rDBL is the position of the moving interface between the two re-

gions. A second moving discontinuity-the temperature discontinu-

ity-exists in the reservoir. In the above model, this is accounted

for through the time- and space-dependent total mobili ty, F } . . .

From the Buckley-Leverett- theory, we know that the satura-

tion profile is defined by the frontal-advance (mass-conservation)

equation

[rD(drDldtD)]s =ff'ls, (A-9)w w

where f= [(qc to)/(21f ~oh)] , and where f' =dfldSw denotes the

slope of the fractional-flow curve. By integrating Eq. A-9, we obtain

(rD2ItD)S =2ff'ls =constant. (A-IO)

w w

Hence, by transforming the problem from the independent vari-

ables ro and to to the Boltzmann variable y=rl j14tD' we fix the

moving interface and transform the moving-boundary problem into

a composite problem in one variable.

d ( dPD ,) Y dpD l- F xY-- +---=O,O< Y< YBL, (A-II)dy dy 1) dy

d ( dPD2) dpD2- Y-- +Y-- =0, YBL< y <00 ....•...... (A-12)dy dy dy

lim y(dpD,/dy)=-'/z, (A-l3)y-O

lim PD2=0, (A-14)

y-co

PD , = PD2, Y= YBL' (A-IS)

dPDl 1 dpD2and F ) ..--= ---, Y=YBL' (A-16)

dy M dy

where YBL= lh.ffBL=constant. . (A-l7)

The problem now consists of two ordinary linear differential equa-

t ions with variable coefficients. The moving boundary is fixed in

terms of Y, and the solution is obtained by integration.

YB L dy' ( Y' dy" )PD1(y)=V:zj --,exp -j --

Y F xY 0 FX1)

+ M exp(YBL _ j YB L dy )E1(YBL), O ::; Y::; YB L ..... (A-IS)

2 0 FX 1)

M ( J YB L dy )andpD

2(y)=-exp YBL -- E1( y) , YB L: :; Y :: ;C : X> '

2 0 Fx 1)

.................................. (A-19)

Eq. A-IS is singular atthe line source y=O. This singularity is re-

moved by adding and subtracting:

I' f Y BL( - 'liIl ')d ., ( 2012 J e Y Y y. A- )

Y

Furthermore, for a typical water-injection test, the reservoir and

fluid parameters are such that e is small and YBL= ff'12 is of order




10 - 2 to 10 - 3, while FA 7 J > 1. This makes the exponential terms

in the solution close to one after the first few seconds of injection,

and the fluids in the invaded region essentially behave as if they

were incompressible. Because

y=rj)4tD =c/>cor2/4i..ot= 'hE/', (A-21)

we can change variables to obtain the solution in terms of radius

and time (note that YBL is constant):

PD2 (rD,tD) =(MI2)E I (rji4tD)' to S, rji4YBL (A-22)

and PDI (rD,tD)= '12EIrlJl4~tD) -'12EI(YBL/~)

S r ( 1 ) M+1/2J BL - --I dSw+-EI(YBL)

I-Sor/' FA 2

I-S f"(I) r2+ 'l 2 J or - - -I dSw' tD? __!2_. . (A-23)

Sw(rD.tD) I' FA 4YBL

Evaluating the pressure at the wellbore, we note that for late times

Sw(l,tD)-I-Sor> and the last integral in Eq. A-23 becomes

negligible. At these times, we can also use the approximation

-EI(x)",dnx+" and obtain the wellbore pressure as

pwD=1/2(ln tD+0.80907)+sa+s, (A-24)

where sa is given by

Sa= '12 SBL /': ( _ 1 _ - I )dSw + '12(lnYBL +,,)(l-M). . . (A-25)

I-Sor f FA

Therefore, the wellbore pressure is given by the familiar Theis

solution 1 7 plus an apparent skin factor caused by the saturation

gradient and the propagating temperature and phase discontinuities.

In this expression for the wellbore pressure, we have added a me-

chanical skin factor, s, to account for formation damage or stimu-

lation near the wellbore.

Ifthe relative permeabilities and viscosities of the fluids are such

that the injection results in piston-like displacement, FA =1, 7 J =~,

and removal of the singularity from Eq. A-18 gives the Veriginsolution 1 1,23 :

PDI = '12EIrlJl4~tD) -'12E IYBL/~)

+(MI2)eYBL(l-I/~) EIYBL)' o s ro S, rDBL

(A-26)

and PD2 =(MI2)eYBL(l-I/~) E, (Y), rDBL

S, ro <00 . (A-27)

Finite Outer Boundary. During injection into an infinite reservoir,

only the compressibilit ies in the uninvaded zone are significant. 3

Consequently, it is likely that the outer boundary of a finite cylindri-

cal reservoir will start influencing the wellbore pressure at the time

given by the radius-of-investigation concept->:

t-o = 'Ar~D' (A-28)

The Buckley-Leverett phase-front position at time teD is given by

rbBL =2EfBLteD=(EfBL/2)r~D' (A-29)

Because E is of order 10 -4 to 10 - 5 while filL is of order I, the in-

vaded region is still occupying only a small part of the total reservoir

at the end of the infinite-acting period. Hence, it is expected that

the compressibility in the large uninvaded region also dominates

after the outer boundary is felt in the wellbore pressure.

Verigin II and Ramey 12 discussed approximate solutions for

moving-boundary problems in finite domains. Applying their ideas,

we can add an outer region to obtain an additional moving boundary

in an infinite reservoir. The two outer regions have constant, but

different, mobilities, while the inner region has a saturation gradient

and a temperature discontinuity as before. The interfaces move ac-

cording to mass and energy balances and hence the saturation and

temperature distributions remain constant at any constant value of

the Boltzmann variable. An approximation to a closed outer bound-

ary can be obtained by taking the limit as the mobility in the outer-


most region approaches zero while i ts interface is kept at r.o- Theapproximate closed-outer-boundary solution is given by Eqs. 5

and 6.

The constant-pressure outer-boundary approximation is obtained

by taking the limit as the mobility in the outermost region approaches

infinity. The solution to this problem is given by Eqs. 5 and 6

without the exponential terms.

Appendix B-Falloff Solution

Inthe paper we show that the falloff problem can be approximated

by a linear, variable-coefficient problem. This variable-coefficient

problem can be solved by dividing the invaded region into (n+ 1)

regions where the coefficients are evaluated at the average satura-

tion of each region. The governing equations and boundary condi-

tions in dimensionless form for the (n+ lj-region composite

reservoir follow.

Governing equations:

................................... (B-1)

................................... (B-2)

Initial conditions:

PDJj

=v"I(rD)' tD=O (B-3)

and PD2=h(rD), to =0, (B-4)

where 1fi(rD) is the injection solution for the appropriate outer-

boundary condition evaluated at to =i»;

Inner-boundary conditions:

(SDIM)(dPwDldtD)-[rD(iJPD/iJrD)]rD=1 =0 (B-5)

and PwD =[PD1 -srD(iJPD/iJrD)]rD=I' (B-6)

Outer-boundary conditions:

Infinite: lim PD =0. . (B-7)'D-OO 2

Closed: iJPD/iJrD I reD=0. . (B-8)

Constant pressure: PD I r =0. . (B-9)2 eD

Interface conditions:

PDIi

=PD1(i+I)' rD=ai' i=I n-l, (B-IO)

iJPDli iJPDI(i+1) •Mi--= ,rD=ai' 1=1 n-I, (B-ll)

iJrD iJrD

PD1n

=PD2' rD=an, ••••. .. •• .. .. •• •. .. .. •. .. .• . (B-12)

- iJPDln iJPD2andFA i1 M--=--, rD=an, (B-13)

iJrD iJrD

where Mi=F).,iIFAi+I'

We now use the average values for F)"i and 7 J i in each of the n re-

gions. The problem is then a linear, constant-coefficient problem

and can be solved by Laplace transformation. Note that the nonzero

init ial condit ion results in a nonhomogeneous problem in Laplace

space. The two linearly independent, homogeneous solutions in

~place space are KoCJ~ rD) and lo(~ rD), where Ki=(l1

FA illi'The general solution in Laplace space to the nonhomogeneous

problem is then obtained by variation of parameters.

301



Authors

Reldar Brumer

Bratvold Is asenior scientistwith IBM EuropeanPetroleum Applica-tion Center InStavanger, wherehe works with com-puter applicationsIn reservoir engi-neering. Heholds a

Bratvold Horne DH. Ing degree Inpetroleum engi-

neering from Rogaland Regional C., an MSdegree In petrole-um engineering from the U. of Tulsa, and an MS degree inmathematics and PhD degree In petroleum engineering fromStanford U. Bratvold currently serves on the Editorial ReviewCommittee. RolandN. Horne Is an associate professor ofpetroleum engineering at Stanford U. He holds BE and PhDdegrees In theoretical and applied mechanics, and a DScdegree in engineering, all from the U. of Auckland, NewZealand. His research Interests are computerized well-test anal-ysis, tracer testing, flow through fractures, geothermal reser-voir engineering, simulation, optimization, and microcomputerapplications. Horne, the 1982 recipient of the SPE Distin-guished Achievement Award for Petroleum EngineeringFaculty, has served on various committees and Is currentlyserving on the Textbook Committee and a Technical Program

Committee for the 1990 Annual Meeting.

P D1i

= K i K o ( ~ r D )I» ~ 1 / t l ( D ! o ( - . r ; ; ; ; ~)d~ai_l

+ K i 1 o ( - . r ; ; ; ; r D )I aj

~ 1 / t 2 m K o ( - . r ; ; ; ; ~)d~r»

30 2

+Ii'ru r D ) I reD ~ 1 / t 2 ( ~ ) K O ( . J ; ~)d~ro

+ X ( n + 1) 1 K O ( . J ; r D ) + X ( n + l ) 2 I O ( . J ; r D ) ' (B-I5)

The constants are determined by applying the boundary and inter-

face conditions. In doing this, we need to solve for the following

(n+ 1) x (n+ 1) block tridiagonal system of equations for the con-

stants Xii and x i 2 , i=1 n+ 1:

rx=d. (B-I6)

The coefficients of matrix r and the right-side vector d depend

on the auxiliary conditions given in Ref. 3.

The well bore pressure in Laplace space now can be obtained by

substituting Eq. B-I4 and its gradient into Eq. B-6 with ro =1. The

inverted wellbore pressure for the case where SD=O is given by

Eq. 7. The last two terms of this solution are inverted numerically

with the Srehfest-" inversion algori thm.

51Metric Conversion Factors

bbl x 1.589873 E-Ol m3

Btu x 1.055056 E+OO kJ

cp x 1.0* E-03 Pa-sft x 3.048* E-Ol m

ft3 x 2.831 685 E-02 m3

OF (OF-32)/1.8 °C

md x 9.869233 E-04 p.m2

psi x 6.894757 E+OO kPa

psi-1 x 1.450377 E-Ol kPa-1

'Conversion factor is exact. SPEFE

Original SPEmanuscript received for reviewOct. 2. 1988. Paper accepted for publication

May 21,1990. Revised manuscript received March 19, 1990. Paper (SPE 18111) first

presented at the 1988 SPEAnnual Technical Conference and Exhibition held in Houston,

Oct. 2-5.

SPE Format ion Evaluat ion, September 1990

analysis of pressure falloff tests following cold water injection

Documents