Dake L.P. Fundamentals of reservoir engineering

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DARCY'S LAW AND APPLICATIONS 121

Although the above description of the concept of relative permeability has been

restricted to a two phase oil-water system, the same general principle applies to any

two phase system such as gas-oil or gas-water.

4.9 THE MECHANICS OF SUPPLEMENTARY RECOVERY

Supplementary recovery results from increasing the natural energy of the reservoir,

usually by displacing the hydrocarbons towards the producing wells with some injected

fluid. By far the most common fluid injected is water because of its availability, low cost

and high specific gravity which facilitates injection.

The basic mechanics of oil displacement by water can be understood by considering

the mobilities of the separate fluids. The mobility of any fluid is defined as

(4.35)

which, considering Darcy's law, can be seen to be directly proportional to the velocity of

flow. Also included in this expression is the term k

/µ, which is referred to as the

relative mobility.

The manner in which water displaces oil is illustrated in fig. 4.10 for both an ideal and

non-ideal linear horizontal waterflood.

IDEAL

NON-IDEAL

(a)

(b)

1− S

Fig. 4.10 Water saturation distribution as a function of distance between injection and

production wells for (a) ideal or piston-like displacement and (b) non-ideal

displacement

In the ideal case there is a sharp interface between the oil and water. Ahead of this, oil

is flowing in the presence of connate water (relative mobility = k

)/µ

′

/µ

while behind the interface water alone is flowing in the presence of residual oil (relative

mobility = k

=1 - S

)/µ

′

/µ

). This favourable type of displacement will only

occur if the ratio

DARCY'S LAW AND APPLICATIONS 122

rw w

ro o

′

=≤

′

where M is known as the end point mobility ratio and, since both and

′

are

′

the end

point relative permeabilities, is a constant. If M ≤ 1 it means that, under an imposed

pressure differential, the oil is capable of travelling with a velocity equal to, or greater

than, that of the water. Since it is the water which is pushing the oil, there is therefore,

no tendency for the oil to be by-passed which results in the sharp interface between

the fluids.

The displacement shown in fig. 4.10(a) is, for obvious reasons, called "piston-like

displacement". Its most attractive feature is that the total amount of oil that can be

recovered from a linear reservoir block will be obtained by the injection of the same

volume of water. This is called the movable oil volume where,

1 (MOV) = PV(1 – S

− S

)

The non-ideal displacement depicted in fig. 4.10(b), which unfortunately is more

common in nature, occurs when M > 1. In this case, the water is capable of travelling

faster than the oil and, as the water pushes the oil through the reservoir, the latter will

be by-passed. Water tongues develop leading to the unfavourable water saturation

profile.

Ahead of the water front oil is again flowing in the presence of connate water. This is

followed, in many cases, by a waterflood front, or shock front, in which there is a

discontinuity in the water saturation. There is then a gradual transition between the

shock front saturation and the maximum saturation S

= 1−S

. The dashed line in

fig. 4.10(b) depicts the saturation distribution at the time when the shock front breaks

through into the producing well (breakthrough). In contrast to the piston-like

displacement, not all of the movable oil will have been recovered at this time. As more

water is injected, the plane of maximum water saturation (S

= 1−S

) will move slowly

through the reservoir until it reaches the producing well at which time the movable oil

volume has been recovered. Unfortunately, in typical cases it may take five or six

MOV's of injected water to displace the one MOV of oil (as will be demonstrated in

exercises 10.2 and 10.3 of Chapter 10). At a constant rate of water injection, the fact

that much more water must be injected, in the unfavourable case, protracts the time

scale attached to the oil recovery and this is economically unfavourable. In addition,

pockets of by-passed oil are created which may never be recovered.

Mobility control

If the end point mobility ratio for water displacing oil is unfavourable, the injection

project can be engineered to overcome this difficulty. The manner in which this is done

can be appreciated by considering the general expression

rd d

ro o

Mobility of the displacing fluid

Mobility of the displaced fluid

′

(4.36)

DARCY'S LAW AND APPLICATIONS 123

where the subscript "d" refers to the displacing fluid, which need not necessarily be

water. To improve the displacement efficiency, M should be reduced to a value of unity

or less which will have the effect of converting the displacement from the type shown in

fig. 4.10(b), to the ideal type shown in fig. 4.10(a); this is referred to as "mobility

control". The methods by which M can be reduced are.

Polymer flooding (increase, µ

)

Polymers, such as polysaccharide, are dissolved in the injection water, this raises its

viscosity, thus reducing the mobility of the water. Polymer flooding will not only

accelerate the oil recovery but can also increase it, in comparison to a normal water

drive, because the by-passing of oil is greatly reduced.

Thermal methods (decrease, µ

/µ

)

For very viscous crudes the ratio of µ

o /

can be of the order of thousands (which

means that M has the same order of magnitude) and therefore, water drive cannot be

considered as a feasible project (refer Chapter 10, exercise 10.1). In such cases the

viscosity ratio can be drastically reduced by increasing the temperature, as shown in

fig. 4.6(a). This is achieved by one of the following methods:

- hot water injection

- steam injection

- in-situ combustion.

Although mobility control is the primary aim in applying thermal methods, there are

other factors involved than merely the reduction of, µ

o /

(where in this case, µ

is the

viscosity of the hot water or steam and differs from µ

at normal reservoir temperature).

In many cases distillation of the crude occurs, the lighter fractions of the oil being

vapourised and providing a miscible flood in advance of the thermal front. Expansion of

the oil on heating will also add to the recovery. Thermal methods can therefore be

considered as basically secondary recovery processes with some tertiary side effects,

such as the crude distillation, which tends to reduce the residual oil saturation.

Tertiary flooding

Tertiary flooding aims at recovering the oil remaining in the reservoir after a

conventional secondary recovery project, such as a water drive. Oil and water are

immiscible (do not mix) and as a result there is a finite surface tension at the interface

between the fluids. This, in turn, leads to the trapping of oil droplets within each

separate pore which is the normal state after a waterflood.

From a strictly mechanical point of view, the methods commonly employed in tertiary

flooding can be appreciated by considering fig. 4.11, which shows an enlargement of

an oil relative permeability curve (solid line) for water-oil displacement, in the vicinity of

the residual oil saturation point. After a water drive k

is zero when S

= S

, point A,

and the oil will not flow.

Two possibilities for improving the situation are indicated which amount to altering the

oil relative permeability characteristics. The first of these is to displace the oil with a

DARCY'S LAW AND APPLICATIONS 124

fluid which is soluble in it, thus increasing the oil saturation above S

. This is

equivalent to moving from point A to B on the normal relative permeability curve. As a

result k

is finite and the oil becomes mobile.

Alternatively, flooding can be carried out with a fluid which is miscible, or partially

miscible, with the oil thus eliminating surface tension, or in some way modifying the

interfacial properties, between the displacing fluid and the oil. This reduces the residual

oil saturation to a very low value,

′

in fig. 4.11, and alters the oil relative permeability

curve, as shown by the dashed line. In this case, when the displacing fluid contacts the

residual oil left after the waterflood, the effect is that the oil relative permeability

increases from zero to point C and again the residual oil becomes mobile.

S’

Fig. 4.11 Illustrating two methods of mobilising the residual oil remaining after a

conventional waterflood

Obviously the second method appears the more favourable since it creates the

possibility of recovering practically all of the residual oil. In the first case, only part of

each swollen oil droplet is recovered. Tertiary floods generally aim at either total

miscibility or else a combination of the methods described above. The ways in which

such floods can be engineered are many and varied, some of the more popular being,

Miscible (LPG) flooding

The oil is displaced by one of the LPG (Liquid Petroleum Gas) products, ethane,

propane or butane. If the reservoir conditions are such that the LPG is in the liquid

phase then it is miscible with the oil and theoretically all the residual oil can be

recovered.

Carbon Dioxide flooding

Carbon dioxide has a critical temperature of 88°F and is therefore normally injected

into the reservoir as a gas. It is highly soluble in oil and this has two favourable effects.

In the first place the saturation of the oil droplets, containing dissolved CO

, increases

above the residual saturation, S

, the oil permeability becomes finite and oil starts to

flow. Secondly, the viscosity of the oil is reduced resulting in better mobility control. In

addition the carbon dioxide, by extracting light hydrocarbons from the oil, displays

miscible properties.

DARCY'S LAW AND APPLICATIONS 125

Surfactant flooding (Micellar Solution flooding)

Surfactants, or surface acting agents, when dissolved in minute quantities in water

have a significant influence on the interfacial properties between the water and oil

which it is displacing. The surfactant dissolves in the residual oil droplets thus raising

the saturation above the residual level and, in addition, the surface tension between

these enlarged oil droplets and the displacing water is very significantly reduced. Both

these effects are active in reducing the residual oil saturation and, in laboratory tests,

ninety percent residual oil recovery has been observed. The surfactants most

commonly used by the industry are petroleum sulphonates.

The above description of tertiary recovery mechanisms hardly "scratches the surface"

of the subject. For an excellent, simplified description the reader is referred to the set of

papers by Herbeck, Heintz and Hastings

, which cover all aspects of the subject

including the vitally important economic considerations.

The above methods are described as tertiary in that they are capable of recovering

some, if not all, of the residual oil remaining after a waterflood. This does not mean,

however, that they must be preceded by a waterflood. Instead, the two can be

conducted simultaneously. In all tertiary recovery schemes, continuous injection of the

expensive agents is unnecessary. The fluids are injected in batches and frequently the

batches are followed by mobility buffers. For instance, to ensure stable displacement in

a surfactant flood, the chemical slug can be displaced by water thickened with a

polymer, the concentration of which is gradually decreased as the flood proceeds.

REFERENCES

1) King Hubbert, M., 1956. Darcy's Law and the Field Equations of the Flow of

Underground Fluids. Trans. AIME, 207: 222-239.

2) Darcy, H., 1856. Les Fontaines Publiques de la Ville de Dijon. Victor Dalmont,

Paris.

3) Klinkenberg, L.J., 1941. The Permeability of Porous Media to Liquids and Gases.

API Drill. and Prod. Prac.: 200.

4) Cole, F.W., 1961. Reservoir Engineering Manual. Gulf Publishing Co., Houston,

Texas: 19-20.

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

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

6) Campbell, J.M., 1976. Report on Tentative SPE Metrication Standards. Paper

presented to the 51st Annual Fall Conference of the AIME, New Orleans,

October.

7) van Everdingen, A.F., 1953. The Skin Effect and Its Impediment to Fluid Flow

into a Wellbore. Trans. AIME, 198: 171-176.

8) Craig, F.F., Jr., 1971. The Reservoir Engineering Aspects of Waterflooding. SPE

Monograph:

DARCY'S LAW AND APPLICATIONS 126

9) Amyx, J.W., Bass, D.M. and Whiting, R. L., 1960. Petroleum Reservoir

Engineering - Physical Properties. McGraw-Hill: 86-96.

10) Herbeck, E.F., Heintz, R.C. and Hastings, J.R., 1976. Fundamentals of Tertiary

Oil Recovery. Series of articles appearing in the "Petroleum Engineer",

January-September.

CHAPTER 5

THE BASIC DIFFERENTIAL EQUATION FOR RADIAL FLOW IN A POROUS

MEDIUM

5.1 INTRODUCTION

In this chapter the basic equation for the radial flow of a fluid in a homogeneous porous

medium is derived as

1kp p

rr r t

φρ

æö

∂∂ ∂

ç÷

∂∂ ∂

èø

(5.1)

This equation is non-linear since the coefficients on both sides are themselves

functions of the dependent variable, the pressure. In order to obtain analytical

solutions, it is first necessary to linearize the equation by expressing it in a form in

which the coefficients have a negligible dependence upon the pressure and can be

considered as constants. An approximate form of linearization applicable to liquid flow

is presented at the end of the chapter in which equ. (5.1) is reduced to the form of the

radial diffusivity equation. Solutions of this equation and their applications for the flow

of oil are presented in detail in Chapters 6 and 7. For the flow of a real gas, however, a

more complex linearization by integral transformation is required which will be

presented separately in Chapter 8.

5.2 DERIVATION OF THE BASIC RADIAL DIFFERENTIAL EQUATION

The basic differential equation will be derived in radial form thus simulating the flow of

fluids in the vicinity of a well. Analytical solutions of the equation can then be obtained

under various boundary and initial conditions for use in the description of well testing

and well inflow, which have considerable practical application in reservoir engineering.

This is considered of greater importance than deriving the basic equation in cartesian

coordinates since analytical solutions of the latter are seldom used in practice by field

engineers. In numerical reservoir simulation, however, cartesian geometry is more

commonly used but even in this case the flow into or out of a well is controlled by

equations expressed in radial form such as those presented in the next four chapters.

The radial cell geometry is shown in fig. 5.1 and initially the following simplifying

assumptions will be made.

a) The reservoir is considered homogeneous in all rock properties and isotropic with

respect to permeability.

b) The producing well is completed across the entire formation thickness thus

ensuring fully radial flow.

c) The formation is completely saturated with a single fluid.

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 128

r + dr

Fig. 5.1 Radial flow of a single phase fluid in the vicinity of a producing well.

Consider the flow through a volume element of thickness dr situated at a distance r

from the centre of the radial cell. Then applying the principle of mass conservation

Mass flow rate

− Mass flow rate

OUT

= Rate of change of mass in

the volume element

rdr

−

2rhdr

πφ

∂

where 2πrh

dr is the volume of the small element of thickness dr. The left hand side of

this equation can be expanded as

(q )

qdrq2rhdr

ρρ

ρρπφ

∂∂

æö

+−=

ç÷

∂∂

èø

which simplifies to

(q )

2rh

ρρ

πφ

∂∂

(5.2)

By applying Darcy's Law for radial, horizontal flow it is possible to substitute for the flow

rate q in equ. (5.2) since

2 khr p

∂

giving

2khr p

2rh

rrt

ρπφ

æö

∂∂∂

ç÷

∂∂∂

èø

1k p

rr r t

ρρ

ρφ

æö

∂∂∂

ç÷

∂∂∂

èø

(5.3)

The time derivative of the density appearing on the right hand side of equ. (5.3) can be

expressed in terms of a time derivative of the pressure by using the basic

thermodynamic definition of isothermal compressibility

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 129

∂

=−

∂

and since

then the compressibility can be alternatively expressed as

ρρ

æö

∂

ç÷

∂

èø

=− =

∂∂

(5.4)

and differentiating with respect to time gives

∂∂

(5.5)

Finally, substituting equ. (5.5) in equ. (5.3) reduces the latter to

1k p p

rr r t

φρ

æö

∂∂ ∂

ç÷

∂∂ ∂

èø

(5.1)

This is the basic, partial differential equation for the radial flow of any single phase fluid

in a porous medium. The equation is referred to as non-linear because of the implicit

pressure dependence of the density, compressibility and viscosity appearing in the

coefficients k

and

. Because of this, it is not possible to find simple analytical

solutions of the equation without first linearizing it so that the coefficients somehow

lose their pressure dependence. A simple form of linearization applicable to the flow of

liquids of small and constant compressibility (undersaturated oil) will be considered in

sec. 5.4, while a more rigorous method, using the Kirchhoff integral transformation, will

be presented in Chapter 8 for the more complex case of linearization for the flow of a

real gas.

5.3 CONDITIONS OF SOLUTION

In principle, an infinite number of solutions of equ. (5.1 ) can be obtained depending on

the initial and boundary conditions imposed. The most common and useful of these is

called the constant terminal rate solution for which the initial condition is that at some

fixed time, at which the reservoir is at equilibrium pressure p

, the well is produced at a

constant rate q at the wellbore, r = r

. This type of solution will be examined in detail in

Chapters 7 and 8 but it is appropriate, at this stage, to describe the three most

common, although not exclusive, conditions for which the constant terminal rate

solution is sought. These conditions are called transient, semi-steady state and steady

state and are each applicable at different times after the start of production and for

different, assumed boundary conditions.

a) Transient condition

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 130

This condition is only applicable for a relatively short period after some pressure

disturbance has been created in the reservoir. In terms of the radial flow model this

disturbance would be typically caused by altering the well's production rate at r = r

In the time for which the transient condition is applicable it is assumed that the

pressure response in the reservoir is not affected by the presence of the outer

boundary, thus the reservoir appears infinite in extent. The condition is mainly applied

to the analysis of well tests in which the well's production rate is deliberately changed

and the resulting pressure response in the wellbore is measured and analysed during a

brief period of a few hours after the rate change has occurred. Then, unless the

reservoir is extremely small, the boundary effects will not be felt and the reservoir is,

mathematically, infinite.

This gives rise to a complex solution of equ. (5.1) in which both the pressure and

pressure derivative, with respect to time, are themselves functions of both position and

time, thus

and

pg(r,t)

f(r,t)

∂

Transient analysis techniques and their application to oil and gas well testing will be

described in Chapters 7 and 8, respectively.

b) Semi-Steady State condition

q = constant

Pressure

= constant

= 0, at r = r

∂

Fig. 5.2 Radial flow under semi-steady state conditions

This condition is applicable to a reservoir which has been producing for a sufficient

period of time so that the effect of the outer boundary has been felt. In terms of the

radial flow model, the situation is depicted in fig. 5.2. It is considered that the well is

surrounded, at its outer boundary, by a solid "brick wall" which prevents the flow of

fluids into the radial cell. Thus at the outer boundary, in accordance with Darcy's law

0at r r

∂

(5.6)

Furthermore, if the well is producing at a constant flow rate then the cell pressure will

decline in such a way that

∂

≈

∂

constant, for all r and t. (5.7)