Dake L.P. Fundamentals of reservoir engineering

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REAL GAS FLOW: GAS WELL TESTING 291

8.13 SUMMARY OF PRESSURE ANALYSIS TECHNIQUES

For those who have struggled through this and the previous three chapters, it is

worthwhile presenting a brief summary of their contents in an effort to simplify and

generalise the theory of well testing.

To start with, the application of the principle of mass conservation, together with

Darcy's law and the definition of isothermal compressibility, lead to the non-linear,

radial, second order, partial differential equation for the flow of a single phase fluid (in

Darcy units) as

1k p p

rr r t

φρ

æö

∂∂ ∂

ç÷

∂∂ ∂

èø

(5.1)

Prior to obtaining useful solutions of this equation it must first be linearized (or partially

linearized) and the method by which this can be achieved depends on the nature of the

fluid under consideration, as follows.

Undersaturated oil

Linearization by deletion of terms, assuming that

(p/ r) 0;

∂∂≈ ≈constant and

cp << 1.

Real gas

Partial linearization using the integral transformation

pdp

m(p) 2

(8.7)

Gas-oil

Partial linearization using the integral transformation

ro o

k(S)

m(p) dp

′

(8.62)

or, strictly speaking, in Darcy units, the correct form of this transformation should be

ro o o

k(S)

m(p) dp

′

Application of any of the above methods leads to the re-formulation of equ. (5.1) as

rr r k t

βφµβ

∂∂ ∂

æö

ç÷

∂∂ ∂

èø

(8.66)

which has the form of the radial diffusivity equation and in which:

for undersaturated oil β = p

REAL GAS FLOW: GAS WELL TESTING 292

for a real gas β = m(p)

and for gas-oil (two phase) β =

m(p)

′

Although the term linearization has been applied to the conversion of equ. (5.1) to

equ. (8.66), it should be remembered that linearization is only achieved for the case of

liquid flow (undersaturated oil) for which the coefficient k/

φµ

c is a constant. For both

real gas and two phase (gas-oil) systems, the

c product is pressure dependent,

meaning that equ. (8.66) is still non-linear.

The basic building block in well test analysis is the constant terminal rate solution of

equ. (8.66), which predicts the pressure or pseudo pressure response at the wellbore,

resulting from the production of a well at constant rate from a state of equilibrium

pressure. Expressing equ. (8.66) in dimensionless form

DD D D

rr r t

ββ

æö

∂∂

∂

ç÷

∂∂∂

èø

(8.67)

where r

= r/r

and t

= kt/

φµ

r (= 0.000264 kt/

φµ

r in field units- t in hours), the

general constant terminal rate solution, for r

= 1, can be expressed as

f(p) (t ) S

æö

ç÷

èø

(8.68)

In this equation the various component parts are as listed in table 8.14, (in field units),

again, dependent on the nature of the fluid.

To interpret the majority of practical well tests requires the superposition of constant

terminal rate solutions, for different constant production rates acting for different

periods of time, to give the value of f(p)

at time t

during the n

flow period, as

nj1

njDDDn

f(p) q (t t ) q S

αβ

−

=∆ − +

(8.69)

in which

Undersaturated

oil

Real

gas

Two phase

gas-oil

(fieldunits)

ooo

7.08 10

−

1422QT

7.08 10

−

f(p) p

−

m (p

) −

m(p

)

iwf

m(p) m(p )

′′

−

)m´

)

SSS + DQS

TABLE 8.14

nj1

oD D DD DA D(MBH)DA

(t t ) (t ) 2 t ln (t )

ββπ β

−

′

′′ ′

−= = + −

(8.70)

REAL GAS FLOW: GAS WELL TESTING 293

where t

= 0.000264kt /

φµ

cA (field units), and β

D(MBH)

) is the ordinate of the MBH

charts, corresponding to the value of t

, and includes allowance for the geometry of

the drainage area and the degree of well asymmetry with respect to the boundary.

Well test analysis techniques presented in the literature invariably require the

superposition of transient constant terminal rate solutions of equ. (8.67) which, in

comparison with equ. (8.70), have the form

nj1

DD D DD

(t t ) (t ) ln

ββ

−

′

−= =

(8.71)

Superposition of such solutions, however, automatically assumes the infinite boundary

condition and is therefore only appropriate for tests of short duration, whereas

superposition of the total solution, equ. (8.70), is theoretically correct for any value of

the flowing time and for any boundary condition.

Examples in this and the previous chapter (exercises 7.8 and 8.2), have shown that the

ad-hoc assumption of transient flow can be particularly dangerous in the interpretation

of multi-rate flow tests. Furthermore, conventional multi-rate test analysis techniques

do not permit the attempted matching of the drainage boundary, even using the correct

function, equ. (8.70), in the superposed test analysis equation, (8.69), as illustrated

in exercise 7.8. Multi-rate tests should only be planned when the engineer is quite

confident that transient conditions apply for the duration of the entire test, which is

difficult to ascertain in advance of the test itself.

In this author's opinion, the safest and most useful type of test, for any fluid system, is

the pressure buildup. Definition of a linear trend, for small values of the closed in time

on a Horner buildup plot, can lead to the unambiguous determination of the effective

permeability and the skin factor. Use of this early linear trend, in fact, imposes

transience on the buildup analysis. In addition, the test can be interpreted to determine

the average pressure within the drainage boundary of the well and also to gain some

impression of the shape, area and well position within the boundary, as illustrated in

exercise 7.7.

It must be admitted that the use of equ. (8.70) instead of equ. (8.71), in the basic well

test equation, (8.69), only serves to complicate well test analysis. Nevertheless, in

using the correct solution one is simply recognising the basic fact that second order

differential equations require the specification of both initial and boundary conditions to

obtain meaningful solutions.

As illustrated in the various exercises, it is not difficult to evaluate equ. (8.70) for use in

test analysis. These days when engineers have access to computers or, at least,

sophisticated electronic desk calculators, the evaluation is much less tedious than in

the past.

For instance, a perfectly general program for evaluating pressure buildup tests, for any

fluid, could be structured as shown in fig. 8.18.

REAL GAS FLOW: GAS WELL TESTING 294

BASIC

INPUT

PVT

PRELIMINARY

ANALYSIS

KAZEMI

ITERATION

SPECIAL

ANALYSES

DIGITIZED

MBH CHARTS

Fig. 8.18 Schematic of a general analysis program applicable to pressure buildup tests

for any fluid system

The various component parts have all been detailed in the text and will only be briefly

summarised here.

Basic input

This consists of the basic reservoir parameters, as detailed at the beginning of

exercises 7.6 and 7.7, together with the specification of the flowing time to be used in

the analysis and the bottom hole flowing pressure at the time of closure. A table of

recorded wellbore pressures as a function of closed in time is also required and, for a

gas well test, wellbore pressures as a function of time during the flowing periods before

and after the buildup (exercise 8.3).

PVT

For a buildup in an undersaturated oil reservoir it is only necessary to specify the

current oil formation volume factor, viscosity and total compressibility. For a gas well

test, however, the program should be capable of generating real gas pseudo pressures

as a function of the actual pressure, as shown in table 8.1. Similarly, for a test in a gas

saturated reservoir the pseudo pressure function described in sec. 8.12, must be

generated.

Preliminary analysis

The program should print a Horner plot, using closed-in pressures or pseudo

pressures, as appropriate. The engineer should inspect this plot and decide which, if

any, of the plotted points constitute an early, linear buildup trend. It is inadvisable to let

the computer perform this task. Having defined the points on the linear trend, the

computer can determine the equation of the straight line which best matches the points

and subsequently determine the extrapolated pressure at infinite closed-in time, the

permeability and the skin factor.

REAL GAS FLOW: GAS WELL TESTING 295

For a gas well test, this part of the program should also analyse the transient pressure

fall-off during the flowing periods, giving a re-determination of the permeability and two

separate skin factors, at different flow rates, which facilitate the calculation of the rate

dependent skin (exercise 8.3).

For an initial test in either an oil or gas reservoir (exercises 7.6 and 8.3, respectively)

the analysis would cease at this point. In the more common instance of routine

pressure surveys made throughout the producing lifetime of the well, the following

steps are necessary which incorporate the boundary condition in the analysis.

Digitized MBH charts

This set of data is the main-spring of the special analyses part of the program. As

already mentioned, the MBH charts (figs. 7.11-15), which are equally appropriate for

dimensionless pressure or pseudo pressures, have al ready been digitized by

Earlougher

, for dimensionless time in the range 0.001 ≤ t

≤ 10. From this data bank,

values of the constant terminal rate solution

, equ. (8.70), can be extracted for any

fluid type, geometry of the drainage area and degree of well asymmetry.

Special analyses

This part of the program should be quite flexible with the main aims being to determine

the average pressure within the drainage boundary at the time of the survey and to

gain some information about the magnitude and shape of the drainage area and well

position with respect to the boundary. For instance, the program could be structured,

as described in exercise 7.7, to match the equation of the observed linear buildup,

obtained in the preliminary analysis, with a theoretical straight line for various assumed

area sizes and shapes. Alternatively, an attempt could be made to match the entire

buildup by evaluating both

+ ∆t

) and β

(∆t

), and plotting the difference

between the two, also as described in exercise 7.7.

Kazemi iteration

Iteration over the entire test analysis may be required for surveys conducted in gas, or

in gas saturated oil reservoirs, as described in sec. 8.11. This measure is necessary to

refine the estimate of the average pressure within the drainage area at the time of the

survey.

REFERENCES

1) Russell, D.G., Goodrich, J.H., Perry, G.E. and Bruskotter, J.F., 1966. Methods of

Predicting Gas Well Performance. J.Pet.Tech., January: 99-108.Trans. AIME.

2) Al-Hussainy, R., Ramey, H.J.,Jr. and Crawford, P.B.,1966. The Flow of Real

Gases Through Porous Media. J.Pet.Tech., May: 624-636. Trans. AIME.

3) Al-Hussainy, R. and Ramey, H.J., Jr., 1966. Application of Real Gas Flow Theory

to Well Testing and Deliverability Forecasting. J.Pet.Tech., May: 637-642.

Trans. AIME.

REAL GAS FLOW: GAS WELL TESTING 296

4) Ramey, H.J., Jr. and Wattenbarger, R.A., 1968. Gas Well Testing with

Turbulence, Damage and Wellbore Storage. J.Pet.Tech., August: 877-887.

Trans. AIME.

5) Aziz, K., Mathar, L., Ko, S. and Brar, G.S.,1976. Use of Pressure, Pressure-

Squared or Pseudo-Pressure in the Analysis of Transient Pressure Drawdown

Data from Gas Wells. J.Can.Pet.Tech., April-June: 58−65.

6) Geertsma, J., 1974. Estimating the Coefficient of Inertial Resistance in Fluid Flow

Through Porous Media. Soc.Pet.Eng.J., October: 445-450.

7) Gewers, C.W.W. and Nichol, L.R., 1969. Gas Turbulence Factor in a

Microvugular Carbonate. J.Can.Pet.Tech., April.

8) Wong, S.W., 1970. Effects of Liquid Saturation on Turbulence Factors for Gas

Liquid Systems. J.Can.Pet.Tech., October.

9) Katz, D.L., et.al., 1959. Handbook of Natural Gas Engineering. McGraw-Hill Book

Company, Inc. 47−50.

10) Kazemi, H., 1974. Determining Average Reservoir Pressure from Pressure

Buildup Test. Soc.Pet.Eng.J., February: 55−62. Trans. AIME.

11) Rawlins, E.L. and Schelihardt, M.A., Back Pressure Tests on Natural Gas

Wellsand Their Application to Production Practices. Monograph 7, USBM.

12) Carter, R.D., Millers, S.C., Jr. and Riley, H.G., 1963. Determination of Stabilized

Gas Well Performance from Short Flow Tests. J.Pet.Tech., June: 651−658.

Trans. AIME.

13) Odeh, A.S. and Jones, L.G., 1965. Pressure Drawdown Analysis, Variable Rate

Case. J.Pet.Tech., August: 960−964. Trans. AIME.

14) Essis, A.E. and Thomas, G.W., 1971. The Use of Open Flow Potential Test Data

in Determining Formation Capacity and Skin Factor. J.Pet.Tech., July: 879−887.

Trans. AIME.

15) Earlougher, R.C., Ramey, H.J., Jr., Miller, F.G. and Mueller, T.D., 1968. Pressure

Distribution in Rectangular Reservoirs. J.Pet.Tech., February: 199−208.

Trans. AIME.

16) Odeh, A.S., Moreland, E.E., and Schueler, S., 1975. Characterization of a Gas

Well from One Flow-Test Sequence. J.Pet.Tech., December: 1500−1505.

Trans. AIME.

17) Raghavan, R., 1976. Well Test Analysis: Wells Producing by Solution Gas Drive.

Soc.Pet.Eng.J., August: 196−208.

18) Ramey, H.J., Jr., 1964. Rapid Method for Estimating Reservoir Compressibilities.

J.Pet.Tech., April: 447−454.

CHAPTER 9

NATURAL WATER INFLUX

9.1 INTRODUCTION

Natural water influx into both gas and oil reservoirs has been described previously

(Chapter 1, sec. 7, and Chapter 3, sec. 7) using the simple, time independent aquifer

material balance

= c W

- p)

for calculating the amount of influx. In this equation

= initial volume of water in the aquifer and is therefore dependent upon

aquifer geometry

= initial aquifer/reservoir pressure

p = current reservoir pressure, which in this chapter will always be assumed to

be equal to the pressure at the original oil (or gas) water contact

c = total aquifer compressibility = c

+ c

The equation is simply a re-statement of the basic definition of compressibility and is

only applicable to very small aquifers. For large aquifers a mathematical model is

required which includes time dependence, to cater for the fact that it takes a finite time

for the aquifer to respond fully to a pressure change in the reservoir. Two such models

will be studied in this chapter: firstly, that of Hurst and van Everdingen

and secondly,

the more recent, approximate method of Fetkovitch

. These techniques will be applied

in the classical reservoir engineering manner of first building a model which will

adequately match the reservoir's production and pressure history when included in the

material balance equation (this is sometimes referred to as aquifer fitting) then, once a

satisfactory model has been built, using it to predict the future performance of the

reservoir, say, for a given offtake policy.

It should be appreciated that there are more uncertainties attached to this subject, in

reservoir engineering, than to any other. This is simply because one seldom drills wells

into an aquifer to gain the necessary information about the porosity, permeability,

thickness and fluid properties. Instead, these properties have frequently to be inferred

from what has been observed in the reservoir. Even more uncertain, however, is the

geometry and areal continuity of the aquifer itself. The reservoir engineer should

therefore consult both production and exploration geologists, concerning the latter,

rather than relying entirely upon his own judgement. Due to these inherent

uncertainties the aquifer fit obtained from history matching is seldom unique and the

aquifer model may require frequent updating as more production and pressure data

become available, as illustrated in exercise 9.2.

NATURAL WATER INFLUX 298

9.2 THE UNSTEADY STATE WATER INFLUX THEORY OF HURST AND VAN

EVERDINGEN

The flow equations for oil into a wellbore are identical in form to the equations

describing flow from an aquifer into a cylindrical reservoir; only the radial scale is

different. In the former case, when an oil well is opened up on production at a constant

rate q, the pressure response at the wellbore can first of all be described under

transient flow conditions, before the reservoir boundary effects are felt, followed

possibly by a period of late transient flow and finally by stabilized, semi-steady state

flow. Irrespective of the flow condition the general equation for calculating the pressure

in the wellbore at any time was presented in Chapter 7, sec. 6, as

()

D D DA DA

DMBH

pt 2t ln p t

=+ −

(7.42)

where

) =

2kh

– p

)

which is the dimensionless pressure function describing the constant terminal rate

case. That is, it determines the pressure drop at r = r

due to a rate change from zero

to q applied at the inner boundary at time t = 0.

In the description of water influx from an aquifer into a reservoir there is greater interest

in calculating the influx rate rather than the pressure drop. This leads to the

determination of the influx as a function of a given pressure drop at the inner boundary

of the system. In this respect Hurst and van Everdingen

solved the radial diffusivity

equation for the aquifer-reservoir system by applying the Laplace transformation to the

equation, expressed in terms of dimensionless variables as

DD D D

rr t t

æö

∂∂

∂

ç÷

∂∂∂

èø

(7.18)

where in this case

(9.1)

and

φµ

= (9.2)

in which r

is the outer radius of the reservoir and all the other parameters in both

equs. (9.1) and (9.2) refer to aquifer rather than reservoir properties, which will be the

case for all the equations in this chapter unless specifically stated otherwise.

Instead of obtaining constant terminal rate solutions of equ. (7.18), Hurst and van

Everdingen derived constant terminal pressure solutions of the form

NATURAL WATER INFLUX 299

()

2khp

∆

(9.3)

where q

) is the dimensionless influx rate evaluated at r

= 1 and describes the

change in rate from zero to q due to a pressure drop ∆p applied at the outer reservoir

boundary r

at time t = 0. These functions can be generated from constant terminal rate

solutions and vice-versa. It is generally more convenient to express this solution in

terms of cumulative water influx rather than rate of influx. Thus integrating equ. (9.3)

with respect to time

DD D

qdt q (t ) dt

2khp dt

∆

òò

which gives

()

Wcr

2khp k

µφµ

∆

and therefore

()

e0DD

W2hcrpWt

πφ

=∆ (9.4)

in which, since Darcy units are being employed

= cumulative water influx (ccs) due to a pressure drop ∆p (atm)

imposed at r

at t = 0

and W

) = dimensionless, cumulative water influx function giving the

dimensionless influx per unit pressure drop imposed at the reservoir

aquifer boundary at t=0.

Equation (9.4) is frequently expressed as

eDD

WUpW(t)=∆ (9.5)

where

U2fhcr

πφ

= (9.6)

which is the aquifer constant for radial geometry

and

(encroachment angle)

360

which is to be used for aquifers which subtend angles of less than 360° at the centre of

the reservoir-aquifer system.

The dimensionless water influx W

) is frequently presented in tabular form or as a

set of polynomial expressions giving W

as a function of t

for a range of ratios of the

NATURAL WATER INFLUX 300

aquifer to reservoir radius r

= r

, for radial aquifers. In contrast, plots of W

versus

for both radial and linear geometry and, in the former case, for select values of r

are included as figs. 9.3 - 9.7. The plots are taken from the published solutions of

equ. (7.18) by Hurst and van Everdingen

. Each chart has a different resolution of the

dimensionless time scale. It should be noted that the graphs are valid for all values of

and hence are applicable for calculating both the early, unstable influx (infinite

aquifer case) and for the influx occurring once the aquifer boundary effects have been

felt. There are differences in the way in which the dimensionless time and aquifer

constant are calculated, dependent on the geometry. These are summarised in

figs. 9.1 and 9.2.

Water

Oil

360

t =

Radian

Fig. 9.1 Radial aquifer geometry

Darcy Units Field Units

φµ

t = constant

φµ

× (9.7)

the constant = .000264 (t-hours)

= .00634 (t-days)

= 2.309 (t-years)

U = 2π f

cr (cc/atm) U = 1.119 f

cr (bbl/psi) (9.8)

Water

Oil

Fig. 9.2 Linear aquifer geometry