Dake L.P. Fundamentals of reservoir engineering

Подождите немного. Документ загружается.

NATURAL WATER INFLUX 331

− Insert this initial estimate of p

, which is admittedly too small, in the water influx

equation thus calculating a new value of W

, which will now be too large.

− Rework the material balance with this new value of

W, giving

p which must

now be too large.

− Iterate finding

(W ,p ),

(W ,p ) . . . etc. until the difference between

successive values of p

is less than some tolerance level, at which point the

procedure is repeated for the next time step.

b) Fetkovitch

The iterative solution technique can again be applied but in this case the equations

used are (9.28) and (9.29).

Aquifer material balance

n-1

pp1

−

æö

=−

ç÷

èø

Water influx

()

n1n

in ei

en1n

Jp t / W

Wppp(1e )

−

−∆

∆= − + −

njn

eee

WWW

−

=∆ +∆

Reservoir material balance

pei

GWE

ZZ G G

æö

ç÷

=− −

ç÷

èø

A flow chart showing the sequence of steps is included as fig. 9.20. An example

illustrating this technique is presented in Fetkovitch's paper.

NATURAL WATER INFLUX 332

Time step = n

pp1

−

ç÷

=−

ç÷

k = 1

nn1

pei

GWE

ZZ G G

−

æö

=− −

ç÷ç ÷

ç÷

ç÷ç ÷

èø

en1n

Jp t

iei

W(p(pp))1e

−

−∆

∆= − + −

ç÷

nn1 n

ee e

WW W

−

=+∆

= p

k-1

n=n 1 +

kk=1 +

k = 1

pei

GWE

ZZ G G

æö

ç÷

=− −

ç÷

èø

-1

pp TOL

−−

k = iteration counter

TOL = tolerance pressure

difference

(

psi

)

Fig. 9.20 Prediction of gas reservoir pressures resulting from fluid withdrawal and

water influx (Fetkovitch)

NATURAL WATER INFLUX 333

9.6 APPLICATION OF INFLUX CALCULATION TECHNIQUES TO STEAM SOAKING

The method of predicting aquifer performance, using the unsteady state theory of Hurst

and van Everdingen presented in the previous section, is not necessarily restricted in

use to the description of reservoir-aquifer systems. The same technique can be used to

predict fluid influx in any system which has the same geometry as the reservoir-aquifer

model described in this chapter. As an example, the method will be used to determine

the oil producing rate during the early, transient phase of a steam soak cycle. This

subject was discussed in Chapter 6, sec. 4, in which an expression was derived for the

PI ratio increase during the later stabilized flow part of the cycle. Immediately upon

opening the well on production, however, there will be a period when transient flow

conditions prevail, that is, before the effect of the boundary of the drainage volume has

been felt.

The initial situation is shown in fig. 9.21, in which, after injecting several thousand tons

of steam, a hot zone of radius r

is created around the well in which the temperature is

, the condensing steam temperature at the prevailing reservoir pressure.

cold

hot

T ,

Fig. 9.21 Conditions prior to production in a steam soak cycle

The radius of the hot zone can be calculated using the technique of Marx and

Langenheim

, which allows for heat losses to both cap and base rock during the steam

injection. This simplified description of the temperature distribution is justified by the

experimental findings of Niko and Troost

, which indicate that the dominant factor in the

production cycle is the total amount of heat injected into the formation, the additional oil

production being largely independent of the way in which the temperature is

distributed.

The overall geometry of this system is the same as for an aquifer producing into a

reservoir, only on a smaller scale. During the transient flow period there will be an

influx of cold oil into the heated region which can be described using the water influx

calculation procedures al ready developed in this chapter.

If the transient period is divided into small, equal time steps of length ∆t, then it is

required to calculate the average oil rate

q during the n

time step and, assuming

steady state flow at all times within the small volume of the hot zone close to the well,

then

NATURAL WATER INFLUX 334

()

nwf,n

2khp p

−

(9.38)

which is a re-formulation of the steady state inflow equation presented in table 6.1. The

oil viscosity,

is a function of the average temperature in the hot zone during the n

time interval. The temperature will decline continuously during production as heat is

lost by conduction to the cap and base rock and by convection through the produced

fluids. A simple method for predicting the temperature decline allowing for both effects

has been presented by Boburg and Lantz

. The pressure

p in equ. (9.38) is the

average pressure during the time step at r

, the outer boundary of the hot zone, and

wf,n

p the average wellbore pressure during the same period. If it is assumed that the

pressure declines at r

and r

can be approximated by a series of discrete pressure

steps then, in accordance with equ. (9.15),

p and

wf,n

p can be expressed as

()

nn1n

ppp

−

=+ (9.39)

and

()

wf,n

wf,n 1 wf,n

ppp

−

=+ (9.40)

where p

and p

wf,n

are the pressures at r

and r

respectively at the end of the n

time

step. The cumulative oil influx across the boundary r

by the end of the time step will be

p,n p,n 1

NN q.t

−

=+∆

in which N

p,n-l

is the known influx at the end of the (n - 1)

time step. Using equs. (9.39)

and (9.40), the influx can be expressed as

()

p,n p,n 1 n 1 n wf ,n 1 wf,n

NN p pp p

−− −

=+ +−− (9.41)

where

2kht

∆

An expression for the cumulative oil influx can also be obtained using the unsteady

state influx theory of Hurst and van Everdingen in the manner similar to that of

equ. (9.34)

()

jn1

p,n j D D D n 1 D D D

NUpWTt UpWTt

−

=∆ −+∆ −

(9.42)

in which T = n. ∆t and W

is the dimensionless influx function of Hurst and van

Everdingen, figs. 9.3-9.7. U is the aquifer constant defined in equ. (9.6), only in this

case applied to the oil reservoir for which f = 1, and the radius r

is replaced by r

NATURAL WATER INFLUX 335

Values of the pressure drop ∆p

at the hot zone boundary can be evaluated using

equ. (9.16) as

()

ij1j1

ppp

−+

∆= − (9.16)

and in particular

()

n1 n2 n

ppp

−−

∆= −

therefore, equ. (9.42) can be written as

()

p,n j D D D n 2 n

NUpWTt p p

−

=∆ −+ −

(9.43)

where

= U W

−t

n -1

)

Equations (9.41) and (9.43) can now be equated and solved explicitly for the boundary

pressure p

at r

at the end of the n

time step, i.e.

()

n j D D D p,n 1 n 2 wf ,n 1 wf ,n n 1

p2UpWTt2Npppp

βα

αβ

−

=−−−−

∆−−+++−

(9.44)

Providing that the manner in which the bottom hole flowing pressure will be allowed to

decline is specified, then everything on the right hand side of equ. (9.44) is determined,

thus p

can be calculated and subsequently

p using equ. (9.39). Finally, substituting

this value of

p in equ.(9.38) will lead to the determination of

q . In fact, it is necessary

to iterate on this oil rate to correctly model the heat loss due to convection, since, as

mentioned already, the temperature in the hot zone, which determines the oil viscosity

in equ. (9.38), is reduced due to the removal of heat by the produced oil. This

convective heat loss is directly proportional to the production rate and therefore, the oil

rate and viscosity are dependent upon one another necessitating the iterative solution

technique.

Bentsen and Donohue

have reported the use of the above technique incorporated in

a dynamic programming model for optimising steam soak operations.

REFERENCES

1) van Everdingen, A.F. and Hurst, W., 1949.TheApplication of the Laplace

Transformation to Flow Problems in Reservoirs. Trans. AIME. 186: 305-324.

2) Fetkovitch, M.J., 1971. A Simplified Approach to Water Influx Calculations-Finite

Aquifer Systems. J.Pet.Tech., July: 814-828.

3) van Everdingen, A.F., Timmerman, E.H. and McMahon, J.J., 1953. Application of

the Material Balance Equation to a Partial Water-Drive Reservoir. Trans. AIME.

198:51.

4) Havlena, D. and Odeh, A.S., 1963. The Material Balance as an Equation of a

Straight Line. J.Pet.Techn, August: 896-900. Trans. AIME.

NATURAL WATER INFLUX 336

5) Havlena, D. and Odeh, A.S.,1964. The Material Balance as an Equation of a

Straight Line-Part II, Field Cases. J.Pet.Tech., July: 815-822. Trans. AIME.

6) Schilthuis, R.J., 1936. Active Oil and Reservoir Energy. Trans. AIME. 118: 37.

7) Marx, J.W. and Langenheim, R.H., 1959. Reservoir Heating by Hot Fluid

Injection. Trans. AIME: 118:37.

8) Niko, H. and Troost, P.J.P.M., 1971. Experimental Investigation of Steam

Soaking in a Depletion-Type Reservoir. J.Pet.Tech., August: 1006-1014. Trans.

AIME.

9) Boberg, T.C. and Lantz, R.B., Jr., 1966. Calculation of the Production Rate of a

Thermally Stimulated Well. J.Pet.Tech., December: 1613-1623.

10) Bentsen, R.G. and Donohue, D.A.T., 1969. A Dynamic Programming Model of

the Cyclic Steam Injection Process. J.Pet.Tech., December: 1582-1596.

CHAPTER 10

IMMISCIBLE DISPLACEMENT

10.1 INTRODUCTION

This chapter describes how to calculate the oil recovery resulting from displacement by

an immiscible (non-mixing) fluid which is primarily taken to be water. After considering

several basic assumptions, the subject is introduced in the conventional manner by

describing fractional flow and the Buckley-Leverett equation. Since the latter is one

dimensional its direct application, in calculating oil recovery, is restricted to cases in

which the water saturation distribution is uniform with respect to thickness. In more

practical cases where there is a non-uniform saturation distribution, defined, for

instance, by the assumption of vertical equilibrium, then it is necessary to generate

relative permeabilities, which are functions of the thickness averaged water saturation,

for use in conjunction with the Buckley-Leverett theory. This has the effect of reducing

two dimensional problems to one dimension. The remainder of the chapter

concentrates on the theme of generating these averaged functions, for various

assumed flow conditions in homogeneous and layered reservoirs, and describes their

application in numerical reservoir simulation.

10.2 PHYSICAL ASSUMPTIONS AND THEIR IMPLICATIONS

Before undertaking to describe the mechanics of displacement, it is first necessary to

consider some of the basic physical assumptions which will later be incorporated in the

simple mathematical description of the process. The implications of each assumption

are described in detail.

a) Water is displacing oil in a water wet reservoir

When two immiscible fluids, such as oil and water, are together in contact with a rock

face the situation is as depicted in fig. 10.1. The angle Θ, measured through the water,

is called the contact angle. If Θ < 90° the reservoir rock is described as being water

wet, whereas if Θ > 90° it is oil wet. The wettability, as defined by the angle Θ, is a

measure of which fluid preferentially adheres to the rock.

The two dynamic situations shown in fig. 10.1 (a) and (b) are described as (a)

Imbibition; in which the wetting phase saturation is Increasing and (b) Drainage; in

which the wetting phase saturation is Decreasing. It has been determined

experimentally that the contact angle is larger when the wetting phase is advancing

over the rock face than when retreating and this difference is described as the

hysteresis of the contact angle.

IMMISCIBLE DISPLACEMENT 338

OIL

WATER

(

)

WATER

(

)

Fig. 10.1 Hysteresis in contact angle in a water wet reservoir, (a) wetting phase

increasing (imbibition); (b) wetting phase decreasing (drainage)

Whether the majority of reservoirs are water wet, oil wet, or of intermediate wettability

(Θ ≈ 90°) is still very much a matter of debate and research. It is argued by many that

since all sands were initially saturated with water (water wet), prior to the migration of

hydrocarbons into the reservoir traps, then this initial wettability should be retained. In

this chapter it will also be assumed that the reservoirs described are water wet. This is

not necessarily because this author is convinced by the above argument but more for

the sake of uniformity.

The displacement of oil by water in a water wet reservoir is, therefore, an imbibition

process. As such, the capillary pressure curve and relative permeabilities used in the

description of the displacement must be measured under imbibition conditions.

Conversely, the displacement of oil by water in an oil wet reservoir would be a drainage

process and require capillary pressures and relative permeabilities measured under

drainage conditions. There is a basic difference between the two due to the hysteresis

of contact angle

1,2

The fact that the oil and water are immiscible is also important. When such fluids are in

contact a clearly defined interface exists between them. The molecules near the

interface are unevenly attracted by their neighbours and this gives rise to a free surface

energy per unit area or interfacial tension. If the interface is curved the pressure on the

concave side exceeds that on the convex and this difference is known as the capillary

pressure. The general expression for calculating the capillary pressure at any point on

an interface between oil and water is given by the Laplace equation

cow

Ppp

æö

=−= +

ç÷

èø

(10.1)

where

= the capillary pressure (absolute units)

= the interfacial tension

and r

= the principal radii of curvature at any point on the interface where the

pressures in the oil and water are p

and p

, respectively.

There is also a sign convention that the radii are positive if measured in the oil and

negative if measured in the water, in a water wet system.

IMMISCIBLE DISPLACEMENT 339

Consider a volume of water contained between two spherical sand grains, in a water

wet reservoir, as shown in fig. 10.2.

ROCK

WATE

OIL

Fig. 10.2 Water entrapment between two spherical sand grains in a water wet reservoir

In applying equ. (10.1) to calculate the capillary or phase pressure difference at point X

on the interface, one of the principal radii of curvature, say, r

, is positive, since it is

measured through the oil, while the other, r

, which is measured through the water, is

negative. Since r

< r

, however, the capillary pressure is positive. What is also evident

from fig. 10.2 is the fact that as the volume of water (water saturation) decreases, the

radii decrease, and therefore there must be some inverse relationship between P

and S

This relationship is called the capillary pressure curve (function) and is measured

routinely in the laboratory. Typically, such experiments are conducted, for convenience,

using air-brine or air-mercury fluid combinations and the resulting capillary pressure

curve converted for the oil-water system in the reservoir

3,4

. For the sake of consistency,

however, a hypothetical experiment will be considered, firstly for the (non-commercial)

case of oil displacing water, in a water wet core sample, and then the displacement

process will be reversed. The results of such an experiment are shown in fig. 10.3.

DRAINAGE

IMBIBTION

1 - S

100%

S (% OF PORE VOLUME)

Fig. 10.3 Drainage and imbibition capillary pressure functions

IMMISCIBLE DISPLACEMENT 340

Starting at point A, with the core sample 100% saturated with water, the water is

displaced by the oil, which is a drainage process. If the difference in phase pressures

(imposed pressure differential) is plotted as a function of the decreasing water

saturation the result would be the dashed line shown in fig. 10.3, the capillary pressure

drainage curve. At the connate water saturation (point B) there is an apparent

discontinuity at which the water saturation cannot be reduced further, irrespective of

the imposed difference in phase (capillary) pressure. If the experiment is reversed, by

displacing the oil with water, the result would be the imbibition curve shown as the solid

line in fig. 10.3. The drainage and imbibition plots differ due to the hysteresis in contact

angle, the latter being the one required for the displacement described in this chapter.

When the water saturation has risen to its maximum value S

= 1 − S

the capillary

pressure is zero (point C). At this point the residual oil saturation, S

, cannot be

reduced, irrespective of the pressure difference applied between the water and oil

−negative).

The capillary pressure curve can also be interpreted in terms of the elevation of a plane

of constant water saturation above the level at which P

= 0. The analogy is usually

drawn between capillary rise in the reservoir and the simple laboratory experiment,

shown in fig. 10.4, performed with oil and water, the latter being the wetting phase. At

the level interface, application of equ. (10.1) for infinite r

and r

indicates that P

= 0

and therefore at this point p

= p

= p. The water will rise in the capillary tube until it

reaches a height H, above the level interface, when capillary-gravity (hydrostatic)

equilibrium is achieved. If p

and p

are the oil and water pressures on opposite sides

of the curved interface then, in absolute units

gH = p

and

Oil

Water

oil

water

Pressure

elevation

p= p

o w c

= p(P = 0)

capillary tube

Fig. 10.4 Capillary tube experiment for an oil-water system

gH = p

which, on subtraction, give