03 dake fundofreseng cap5

8/12/2019 03 Dake FundofResEng Cap5

1/9

CHAPTER 5

T H E B A S IC D I F F E R E N T I A L E Q U A T IO N F OR R A D I A L FL OW I N APOROUS M E D I U M

5.1 I N T R O D U C T I O N

In this chapter the basic equation for the radial flow of a f luid in a homogeneousporous medium is derived as

This equation i s non-linear since the coefficients on both sides are themselvesfunctions of the dependent variable, the pressure. In order to obtain analyticalsolutions, it i s first necessary to linearize the equation by expressing it in a form inwhich the coefficients have a negligible dependence upon the pressure and can beconsidered as constants. An approximate form of linearization applicable to liquidflow i s presented a t the end of the chapter in which equ. (5.1) i s reduced to theform of the radial diffusivity equation. Solutions of this equation and their applica-tions for the f low of oi l are presented in detail i n Chapters 6 and 7 . For the flow ofa real gas, however, a more complex linearization by integral transformation i srequired which wi ll be presented separately in Chapter 8.

5.2 D E R I V A T I O N OF T H E B A SIC R A D I A L D I F F E R E N T I A L E Q U A T IO N

The basic differential equation will be derived in radial form thus simulating theflow of fluids in the vicinity of a well. Analytical solutions of the equation can thenbe obtained under various boundary and initial conditions for use in the descriptionof well testing and well inflow, which have considerable practical application inreservoir engineering. This is considered of greater importance than deriving thebasic equation in Cartesian coordinates since analytical solutions of the latter areseldom used in practice by field engineers. In numerical reservoir simulation, how-ever, Cartesian geometry i s more commonly used but even in this case the flow in toor out of a well i s controlled by equations expressed in radial form such as thosepresented in the next four chapters. The radial cell geometry i s shown in fig. 5.1 andinitia lly the following simplifying assumptions wi ll be made.a) The reservoir i s considered homogeneous in a l l rock properties and isotropic

with respect to permeability.b) The producing well is completed across the entire forn.ation thickness thus

ensuring fu ll y radial flow.


2/9

132 RADIAL DIFFERENTIAL EQUATION FOR FLUID F LO W

c) The formation i s completely saturated wi th a single fluid.

Fig. 5.1 Radial f low o a single phase fluid in the vic ini ty of a producing well.

Consider the flow through a volume element of thickness dr situated a t a distancer from the centre of the radial cell. Then applying the principle of mass conserva-tion

Mass flow rate Rate of change of mass inOUT the volume elementass flow rateI NaPa t

2nrhq5dr

where 2nrh$dr is the volume of the small element of thickness dr. The lef t handside of this equation can be expanded as

which simplifies to

(5.2)

By applying Darcys Law for radial, horizontal flow it i s possible to substitute forthe flow rate q in equ. 5.2) since

giving

or


3/9

RADI AL DIFFERENTIAL EQUATION FOR FLUID FLOW 133

The time derivative of the density appearing on the right hand side of equ. 5.3)anbe expressed in terms of a time derivative of the pressure by using the basic thermo-dynamic definition of isothermal compressibility

and since

then the compressibility can be alternatively expressed as

m aP aPand differentiating with respect to time gives

aP aPa t a tcp- =

Finally, substituting equ. 5.5)n equ. 5.3)educes the latter t o

5.4)

5.5)

This i s the basic, partial differential equation for the radial flow of any single phasefluid in a porous medium. The equation i s referred to as non-linear because of theimplicit pressure dependence of the density, compressibility and viscosity appearingin the coefficients p lp and 'cp. Because of this, it i s not possible to find simpleanalytical solutions of the equation without first linearizing it so that the coeffi-cients somehow lose their pressure dependence. A simple form of linearizationapplicable to the flow of liquids of small and constant compressibility (under-saturated oi l) wil l be considered in sec. 5.4,while a more rigorous method, using theKirchhoff integral transformation, will be presented in Chapter 8 for the morecomplex case of linearization for the flow of a real gas.

5 3 CONDITIONS OF SOLUTION

In principle, an infin ite number of solutions of equ. 5.1)an be obtained dependingon the initial and boundary conditions imposed. The most common and useful ofthese i s called the constant terminal rate solution for which the init ial condition i sthat a t some fixed time, a t which the reservoir i s a t equilibrium pressure pi, thewell i s produced a t a constant rate q a t the wellbore, r= rw . This type of solutionwill be examined in detail in Chapters 7 and 8 but it i s appropriate, a t this stage, todescribe the three most common, although not exclusive, conditions for which theconstant terminal rate solution i s sought. These conditions are called transient, semisteady s ta te and steady sta te and are each applicable a t different times after the startof production and for different, assumed boundary conditions.


4/9

34 R A D I A L D I F F E R E N T I A L E Q U A T I ON F OR F L U I D F L O W

a) Transient condit ion

This condition i s only applicable for a relatively short period after some pressuredisturbance has been created in the reservoir. In terms of the radial flow model thisdisturbance would be typically caused by altering the wells production rate a tr=r,. In the time for which the transient condition i s applicable it i s assumed thatthe pressure response in the reservoir i s not affected by the presence of the outerboundary, thus the reservoir appears infinite in extent. The condition i s mainlyapplied to the analysis of well tests in which the wells production rate i s deliberate-ly changed and the resulting pressure response in the wellbore i s measured andanalysed during a brief period of a few hours after the rate change has occurred.Then, unless the reservoir i s extremely small, the boundary ef fec ts will not be feltand the reservoir is, mathematically, infinite.

This gives rise to a complex solution of equ. (5.1) in which both the pressure andpressure derivative, with respect to time, are themselves functions of both positionand time, thus

P = g(r,t)and f(r,t)a tTransient analysis techniques and their application to oi l and gas well testing will bedescribed in Chapters 7 and 8, respectively.

b) Semi Steady State condit ion

= 0 , a t r

Fig. 5 2 Radial flow under semi steady state con dit io ns.

This condition i s applicable to a reservoir which has been producing for a sufficientperiod of time so that the effect of the outer boundary has been felt. In terms of theradial flow model, the situation i s depicted in fig. 5.2. I t s considered that the welli s surrounded, a t i t s outer boundary, by a solid brick wall which prevents the flowof fluids into the radial cell. Thus a t the outer boundary, in accordance with Darcyslaw


5/9

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW

- 0 a t r = r ,Par

135

(5.6)

Furthermore, if the well is producing a t a constant flow rate then the cell pressurewill decline in such a way that

constant, for all r and ta t (5.7)

The constant referred to in equ. (5.7) can be obtained from a simple materialbalance using the compressibility definition, thus

which for the drainage of a radial cell can be expressed as

(5.10

This i s a condition which will be applied in Chapter 6, for oil flow, and in Chapter8, for gas flow, to derive the well in flow equations under semi steady sta te con-ditions, even though in the latter case the gas compressibility i s not constant.

One important feature of this stabilized type of solution, when applied to a deple-tion type reservoir, has been pointed out by Matthews, Brons and Hazebroek' and i sillustrated in fig. 5.3. This i s the fact that, once the reservoir i s producing under thesemi steady s ta te condition, each well will drain from within i t s own no-flow bound-ary quite independently of the other wells.For this condition dp/dt must be approximately constant throughout the entirereservoir otherwise flow would occur across the boundaries causing a re-adjustmentin their positions unti l stabil ity was eventually achieved. In this case a simple tech-nique can be applied to determine the volume averaged reservoTr pressure

(5.11 )

in whichViand pi

Equation (5.9) implies that since dp/dt i s constant for the reservoir then, if thevariation in the compressibility is small

= the pore volume of the ithdrainage volume= the average pressure within the ithdrainage volume.


6/9

136 R A D I A L D I F F E R E N T I A L E Q U A T I O N FO R F L U I D F L OW

Fig. 5 3 Reservoir deplet io n und er semi steady state con dit io ns.

qi Vi 5.12)and hence the volume average in equ. 5.1 1 ) can be replaced by a rate average, asfollows

5.13)

and, whereas the Vi's are difficult to determine in practice, the qi's are measuredon a routine basis throughout the lifetime of the field thus facili tating the calcula-tion of Pres,which i s the pressure a t which the reservoir material balance i s evaluate-ed. The method by which the individual p i s can be determined will be detailed inChapter 7. sec. 7.

c ) Steady State cond i t io n

q onstante c o n s t a n t

l u id InfluxP w f r - e

Fig. 5 4 Radial f l o w und er steady state condit ions .


7/9

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 137

The steady s ta te condition applies, after the transient period, to a well draining a cel lwhich has a completely open outer boundary. I t s assumed that, for a constant rateof production, fluid withdrawal from the ce l l wil l be exactly balanced by f lu id entryacross the open boundary and therefore,

p = pe = constant, a t r = re 5.14)

and ap = 0 forall r and ta t 5.15)This condition i s appropriate when pressure i s being maintained in the reservoir dueto either natural water i'nflux or the injection of some displacing fluid (refer ChapterI O .

I t should be noted that the semi steady s ta te and steady sta te conditions may neverbe fully realised in the reservoir. For instance, semi steady s ta te flow equations arefrequently applied when the rate, and consequently the position of the no-flowboundary surrounding a well, are slowly varying functions of time. Nevertheless, thedefining conditions specified by equs. 5.7)and 5.15)are frequently approximatedin the field since both production and injection facilities are usually designed tooperate a t constant rates and it makes li ttle sense to unnecessarily alter these. I f theproduction rate of an individual well i s changed, fo r instance, due to closure forrepair or increasing the rate to obtain a more even flu id withdrawal pattern through-out the reservoir, there will be a brief period when transient flow conditions prevailfollowed by stabilized flow for the new distribution of individual well rates.

5 4 T H E L I N E A R I Z A T I O N O F EQUATION 5 1 FOR FLUIDS OF S M A L L A N DCONSTANT COMPRESSIBI L l TY

A simple linearization of equ. 5.1) can be obtained by deletion of some of theterms, dependent upon making various assumptions concerning the nature of fluidfor which solutions are being sought. In this section the fluid considered will be aliquid which, in a practical sense, will apply to the flow of undersaturated oil. Exp-anding the left hand side of equ. 5.1),using the chain rule for differentiation gives

I[?( ) a P r a P + k~ r2_P+kPr C P ] = +cpZPr ar pr % ar ar ar ar*and differentiating equ. 5.4)with respect to r gives

which when substituted into equ. 5.16)changes the lat ter to

5.16)

5.17)

-1- -)a k pr- p k cpr ( )r ar ar p ar p ar2 + c p g 5.18)


8/9

138 R A D I A L D I F F E R E N T I A L E Q U A T IO N F O R F L U I D F L OW

For liquid flow, the following assumptions are conventionally madethe viscosity / L i s practically independent o f pressure and may be regarded as aconstantthe pressure gradient ap/ar i s small and therefore, terms of the order (ap/ar)*can be neglected.

These two assumptions eliminate the first two terms in the lef t hand side of equ.(5.181, reducing the latter to

which can be more conveniently expressed as

(5.19)

(5.20)

Making one final assumption, that the compressibility i s constant, means that thecoefficient &c/k i s also constant and therefore, the basic equation has beeneffectively linearized.

For the flow of liquids the above assumptions are quite reasonable and have beenfrequently applied in the past. Dranchuk and Quon, however, have shown that thissimple linearization by deletion must be treated with caution and can only beapplied when the product

cp


9/9

RADI AL DIFFERENTIAL EQUATION F O R FLUID FLOW 139

the compressibility of the gas alone may, to a first approximation, be expressedas the reciprocal of the pressure and the cp product, equ. (5.21), will itself beunity. The linearization of equ. (5.1) under these circumstances will be described inChapter 8, secs. 2-4.

Before leaving the subject of compressibility, it should be noted that the product ofr and c in all the equations, in this and the following chapters, i s conventionallyexpressed as

(5.23)

since it was assumed in deriving equ. (5.1) that the porous medium was completelysaturated with a single fluid thus implying the use of the absolute porosity. Alterna-tively, allowing for the presence of a connate water saturation, the @c product canbe interpreted as

(5.24)in which ( l - S w c ) is the effective, hydrocarbon porosity, and the com-pressibility i s equivalent to that derived in Chapter 3, equ. (3.19), which is used inconjunction with the hydrocarbon pore volume. In either event, the products ex-pressed in equs. (5.23) and (5.24) have the same value, the reader must only becareful not to mix the individual terms appearing in the separate equations.

Equation (5.20) i s the radial dif fusivity equation in which the coefficient k/r$pcis called the diffusivity constant. This is an equation which is frequently applied inphysics, for instance, the temperature distribution due to the conduction of heat inradial symmetry would be described by the equation

in which T is the absolute temperature and K the thermal diffusivity constant.Because of the general nature of equ. (5.20) it i s no t surprising that many reservoirengineering papers, when dealing with complex solutions of the diffusivity equation,make reference to a text book entitled "Conduction of Heat in Solids", by Carslawand Jaeger3, which gives the solutions of the equation for a large variety of bound-ary and initial conditions and i s regarded as a standard text in reservoir engineering.

REFERENCES1) Matthews, C.S., Brons, F. and Hazebroek, P., 1954. A Method fo r Determination of

Average Pressure in a Bounded Reservoir. Trans. AIME. 201: 182-191.Dranchuk, P.M. and Quon, D., 1967. Analysis of the Da'rcy Continuity Equation.Producers Monthly, October: 25-28.Carslaw, H S and Jaeger, J.C., 1959. Conduction of Heat in Solids. Oxford a t theClarendon Press, (2nd edit ion).

2)

3

03 dake fundofreseng cap5

Documents