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2 2Basic Concepts in Reservoir Engineering

1. INTRODUCTION

2. MATERIAL BALANCE

2.1 Introduction to Material Balance (MB)

2.2 Derivation of Simplied Material

Balance Equations

2.3 Conditions for the Validity of Material

Balance

3. SINGLE PHASE DARCY LAW

3.1 The Basic Darcy Experiment

3.2 Mathematical Note: on the Operators

“gradient” ∇ and “divergence” ∇

.

3.3 Darcy’s Law in 3D - Using Vector and

Tensor Notation

3.4 Simple Darcy Law with Gravity

3.5 The Radial Darcy Law

4. TWO-PHASE FLOW

4.1 The Two-Phase Darcy Law

5. CLOSING REMARKS

6. SOME FURTHER READING ONRESERVOIR ENGINEERING

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LEARNING OBJECTIVES:

Having worked through this chapter the student should:

• be familiar with the meaning and use of all the usual terms which appear in

reservoir engineering such as, Sw, S

o, B

o, B

w, B

g, R

so, R

sw, c

w, c

o, c

f , k

ro, k

rw, P

cetc.

• be able to explain the differences between material balance and reservoir

simulation.

• be aware of and be able to describe where it is more appropriate to use material

balance and where it is more appropriate to use reservoir simulation.

• be able to use a simple given material balance equation for an undersaturated oil

reservoir (with no inux or production of water) in order to nd the STOOIP.

• know the conditions under which the material balance equations are valid.

• be able to write down the single and two-phase Darcy Law in one dimension (1D)

and be able to explain all the terms which occur (no units conversion factors need

to be remembered).

• be aware of the gradient (∇) and divergence (∇.) operators as they apply to

the generalised (2D and 3D) Darcy Law (but these should not be committed to

memory).

• know that pressure is a scalar and that the pressure distribution, P(x, y, z) is a

scalar eld; but that ∇P is a vector.

• know that permeability is really a tensor quantity with some notion of what this

means physically (more in Chapter 7).

• be able to write out the 2D and 3D Darcy Law with permeability as a full tensor

and know how this gives the more familiar Darcy Law in x, y and z directions when

the tensor is diagonal (but where we may have kx ≠ k

y ≠ k

z).

• be able to write down and explain the radial Darcy Law and know that the pressure

prole near the well, ΔP(r), varies logarithmically.

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Institute of Petroleum Engineering, Heriot-Watt University 3


REVIEW OF BASIC CONCEPTS IN RESERVOIR ENGINEERING

Brief Description of Chapter 2This module reviews some basic concepts of reservoir engineering that must be

familiar to the simulation engineer and which s/he should have covered already. We

start with Material Balance and the denition of the quantities which are necessary

to carry out such calculations: φ, co, c

f , B

o, S

wietc. This is illustrated by a simple

calculator exercise which is to be carried out by the student. The same exercise is then

repeated on the reservoir simulator. Alternative approaches to material balance are

discussed briey. The respective roles of Material Balance and Reservoir Simulation

are compared.

The unit then goes on to consider basic reservoir engineering associated with uid ow: the single phase Darcy law (k), tensor permeabilities, k , two phase Darcy Law

- relative permeabilitites (kro

, krw

) and capillary pressures (Pc).

Note that many of the terms and concepts reviewed in this section are summarised in

the Glossary at the front of this chapter.

1. INTRODUCTIONIt is likely that you will have completed the introductory Reservoir Engineering part

of this Course. You should therefore be fairly familiar with the concepts reviewed

in this section. The purpose of doing any review of basic reservoir engineering isas follows:

(i) Between them, the review in this section and the Glossary make this course more

self-contained, with all the main concepts we need close at hand;

(ii) This allows us to emphasise the complementary nature of “conventional” reservoir

engineering and reservoir simulation;

(iii) We would like to review some of the ow concepts (Darcy’s law etc.), in a

manner of particular use for the derivation of the ow equations later in this course

(in Chapter 5).

An example of point (ii) above concerns the complementary nature of Material Balance

(MB) and numerical reservoir simulation. At times, these have been presented as

almost opposing approaches to reservoir engineering. Nothing could be further from

the truth and this will be discussed in detail below. Indeed, a MB calculation will be

done by the student and the same calculation will be performed using the reservoir

simulator.

In addition to an introductory review of simple material balance calculations, we will

also go over some of the basic concepts of ow through porous media. These ow

concepts will be of direct use in deriving the reservoir simulation ow equations in

Chapter 5. Again, most of the concepts are summarised in the Glossary.

Exercises are provided at the end of this module which the student must carry out.

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The following concepts are dened in the Glossary and should be familiar to you:viscosity (μ

o, μ

w, μ

g), density (ρ

o, ρ

w, ρ

g), phase saturations (S

o, S

wand S

g), initial or

connate water saturation (Swi

or Swc

), residual oil saturation (Sor). In addition, you

should also be familiar with the basic reservoir engineering quantities in Table 1

below:

Bg Bo

Bw

Pb

FVF

P

Pb

Rso

Rso

Rso

Bo

P

STC = Stock Tank Conditions (60°F; 14.7 psi).Likewise for water (usually const.) and gas; Pb= bubble point pressure below.

Symbol Name Field Units Meaning / Formulae

Bo, Bw, BgFormation volume bbl/STBfactors (FVF) for oil, or RB/STBwater and gas

=

=

= = -

Rso, Rsw Gas solubility factors SCF/STBor solution gas oilratios

co, cw, cg Isothermal fluid psi-1

compressibilities ofoil water and gas

Vol. oil + dissolved gas in reservoir

Vol. oil at STC

Vol. dissolved gas in reservoirVol. gas at STC

ρk and Vk - density and volume of phase k;

k = o, w, g

ck1ρk

∂ρk∂P

∂Vk∂P

1Vk

2. MATERIAL BALANCE

2.1 Introduction to Material Balance (MB)The concept of Material Balance (MB) has a central position in the early history of

reservoir engineering. MB equations were originally derived by Schilthuis in 1936.

There are several excellent accounts of the MB equations and their application to

different reservoir situations in various textbooks (Amyx, Bass and Whiting, 1960;

Craft, Hawkins and Terry, 1991; Dake, 1978, 1994). For this reason, and because

this subject is covered in detail in the Reservoir Engineering course in this series,

we only present a very simple case of the material balance equation in a saturated

reservoir case. The full MB equation is presented in the Glossary for completeness.

Our objectives in this context are as follows:

• To introduce the central idea of MB and apply it to a simple case which we

will then set up as an exercise for simulation;

Table 1: Basic reservoir

engineering quantities to

revise

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• To demonstrate the complementary nature of MB and reservoir simulation

calculations.

Material balance has been used in the industry for the following main purposes:

1. Determining the initial hydrocarbon in place (e.g. STOIIP) by analysing mean

reservoir pressure vs. production data;

2. Calculating water inux i.e. the degree to which a natural aquifer is supporting

the production (and hence slowing down the pressure decline);

3. Predicting mean reservoir pressure in the future, if a good match of the early

pressure decline is achieved and the correct reservoir recovery mechanism has beenidentied.

Thus, MB is principally a tool which, if it can be applied successfully, denes the

input for a reservoir simulation model (from 1 and 2 above). Subsequently, the mean

eld pressure decline as calculated in 3 above can be compared with the predictions

of the numerical reservoir simulation model.

Before deriving the restricted example of the MB equations, we quote the introduction

of Dake’s (1994) chapter on material balance.

Material Balance Applied to Oilelds(from Chapter 3; L. P. Dake, The Practice of Reservoir Engineering, Developments

in Petroleum Science 36, Elsevier, 1994.) Dake says:

It seems no longer fashionable to apply the concept of material balance to oilelds,

the belief being that it has now been superseded by the application of the more

modern technique of numerical reservoir simulation modelling. Acceptance of this

idea has been a tragedy and has robbed engineers of their most powerful tool for

investigating reservoirs and understanding their performance rather than imposing

their wills upon them, as is often the case when applying numerical simulation directly

in history matching.

As demonstrated in this chapter, by dening an average pressure decline trend for a

reservoir, which is always possible, irrespective of any lack of pressure equilibrium,

then material balance can be applied using simply the production and pressure

histories together with the uid PVT properties. No geometrical considerations

(geological models) are involved, hence the material balance can be used to calculate

the hydrocarbons in place and dene the drive mechanisms. In this respect, it is the

safest technique in the business since it is the minimum assumption route through

reservoir engineering. Conversely, the mere act of construction of a simulation

model, using the geological maps and petrophysically determined formation properties

implies that the STOIIP is “known”. Therefore, history matching by simulation can

hardly be regarded as an investigative technique but one that merely reects the input

assumptions of the engineer performing the study.

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There should be no competition between material balance and simulation, instead they

must be supportive of one another: the former dening the system which is then used as input to the model. Material balance is excellent at history matching production

performance but has considerable disadvantages when it comes to prediction, which

is the domain of numerical simulation modelling.

Because engineers have drifted away from oileld material balance in recent years,

the unfamiliarity breeds a lack of condence in its meaningfulness and, indeed, how

to use it properly. To counter this, the chapter provides a comprehensive description of

various methods of application of the technique and included six fully worked exercises

illustrating the history matching of oilelds. It is perhaps worth commenting that in

none of these elds had the operators attempted to apply material balance, which

denied them vital information concerning the basic understanding of the physics of

reservoir performance.

Notes on Dake’s comments

1. The authors of this Reservoir Simulation course would very much like to echo

Dake’s sentiments. Performing large scale reservoir simulation studies does not

replace doing good conventional reservoir engineering analysis - especially MB

calculations. MB should always be carried out since, if you have enough data to

build a reservoir simulation model, you certainly have enough to perform a MB

calculation.

2. Note Dake’s comments on the complementary nature of MB in dening the

input for reservoir simulation, as we discussed above.

3. Take careful note of Dake’s comment on where a reservoir simulation model

is used for history matching. The very act of setting up the model means that you

actually input the STOIIP, whereas, this should be one of the history matching

parameters. The reservoir engineer can get around this to some extent by building

a number of alternative models of the reservoir and this is sometimes, but not

frequently, done.

2.2 Derivation of Simplied Material Balance EquationsMaterial balance (MB) is simply a volume balance on the changes that occur in the

reservoir. The volume of the original reservoir is assumed to be xed. If this is so,then the algebraic sum of all the volume changes in the reservoir of oil, free gas,

water and rock, must be zero. Physically, if oil is produced, then the remaining oil,

the other uids and the rock must expand to ll the void space left by the produced

oil. As a consequence, the reservoir pressure will drop although this can be balances

if there is a water inux into the reservoir. The reservoir is assumed to be a “tank”

- as shown in Figure 5 Chapter 1. The pressure is taken to be constant throughout

this tank model and in all phases. Clearly, the system response depends on the

compressibilities of the various uids (co, c

wand c

g) and on the reservoir rock formation

(crock

). If there is a gas cap or production goes below the bubble point (Pb), then the

highly compressible gas dominates the system response. Typical ranges of uid and

rock compressibilities are given in Table 2:

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Fluid or formation Compressibility (10-6

psi-1

)

Formation rock, crock 3 - 10

Water, cw 2 - 4

Undersaturated oil, co 5 - 100

Gas at 1000psi, cg 900 - 1300

Gas at 5000psi. cg 50 - 200

The simple example which we will take in order to demonstrate the main idea of

material balance is shown in Figure 1 where the system is simply an undersaturated

oil, with possible water inux.

Water influx Water influx We

Oil

N

NBoi = Vf.(1-Swi)

Water, Swi

W = Vf.Swi

Oil

(N - Np)Bo

NBoi = Vf.(1-Swi)

Water, Swi

W + We - Wp

Water, Wp

Oil, Np

Initial conditionspressure = poAfter production (Np)pressure = p

Denitions:

N = initial reservoir volume (STB)

Boi

= initial oil formation volume factor (bbl/STB or RB/STB)

Np

= cumulative produced oil at time t, pressure p (STB)

Bo

= oil formation volume factor at current t and p (bbl/STB)

W = initial reservoir water (bbl)

Wp = cumulative produced water (STB)B

w= water formation volume factor (bbl/STB)

We

= water inux into reservoir (bbl)

cw

= water isothermal compressibility (psi-1)

Δ P = change in reservoir pressure, p - po

Vf

= initial void space (bbl); Vf = N.B

oi/(1- S

wi); W = V

f .S

wi

Swi

= initial water saturation (of whole system)

cf

= void space isothermal compressibility (psi-1); cV

V

p f

f

f =∂∂

1

( NB: (i) bbl = reservoir barrels, sometimes denoted RB; and (ii) in the gures

above, the oil and water are effectively assumed to be uniformly distributed

throughout the system)

Table 2: Typical rock and

uid compressibilities (from

Craft, Hawkins and Terry,

1991)

Figure 1.

Simplied system for

material balance (MB) in

a system with anundersaturated oil above the

bubble point and possible

water inux.

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Denitions of the various quantities we need for our simplied MB equation for the

depletion of an undersaturated oil reservoir above the bubble point (Pb) are givenin Figure 1. (NB a more extensive list of quantities required for a full material

balance equation in any type of oil or gas reservoir is given in the Glossary for

completeness).

In going from initial reservoir conditions shown in Figure 1 at pressure, po, to pressure,

p, volume changes in the oil, water and void space (rock) occur, ΔVo, ΔV

w, ΔV

void

(ΔVvoid

= - ΔVrock

). The pressure drop is denoted, Δ P = p - po. The volume balance

simply says that:

∆ ∆ ∆ ∆ ∆ ∆V V V V V Vo w rock o w void+ + = + − = 0

(1)

Each of these volume changes can be calculated quite straightforwardly as

follows:

Oil volume change, ΔVo

Initial oil volume in reservoir = N.Boi

(bbl = RB)

Oil volume t time t, pressure p = (N - Np). B

o(bbl)

Change in oil volume, ΔVo

= N.Boi

- (N - Np). B

o(bbl) (1)

Water volume change, ΔVw

Initial reservoir water volume = W (bbl)

Cumulative water production at time = Wp

(STB)

Reservoir volume of cumulative water production at time

= Wp.B

w(bbl)

Volume of water inux into reservoir

= We

(bbl)

Water volume change due to compressibility

= W.cw. Δ P (bbl)

Change in water volume, ΔVw

= W - (W - Wp

Bw

+ We

+ W.cw. Δ P )

(bbl)

ΔVw

= Wp

Bw

- We

- W.cw. Δ P (2)

Change in the void space volume, ΔVvoid

Initial void space volume = Vf

(bbl)

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It is convenient to rearrange equation 8 above as follows:

11

−

= −

+−

N

N

B

B

B

B

S c c

S P

p oi

o

oi

o

wi w f

wi

∆

(9)

We then identify 1-(Np/N) as the fraction of the initial oil still in place. We can then

plot this quantity vs. -∆ P shown in Figure 2 (we take -∆ P since it plots along the

positive axis, since ∆ P is negative).

-∆P

1-

1

0

Np

N

"almost" straight linefor w/o systems

As noted in Figure 2, this decline plot is not necessarily a straight line but for oil water

systems, it is very close in practice. Figure 2 suggests a way of applying a simple

material balance equation to the case of an undersaturated oil above the bubble point

(with no water inux or production). This is a pure depletion problem driven by the

oil (mainly), water and formation compressibilities. Suppose we know the pressure

behaviour of B0

(i.e. B0(P)) as shown in Figure 3.

1.4

1.3

4000 P (psi) 5500

Oil FVF

Bo

Bo(P) = m.P + c

If we draw the reservoir pressure down by an amount ∆ P (known or measured) and

we know that to do this we had to produce a volume Np

(STB) of oil. This point of

depletion is shown in Figure 4.

-∆P

1-

1

0

Y

XNp

N

Figure 3

B0

as a function of pressure

for a black oil.

Figure 4

Reservoir depletion on a

plot following equation 9.

Figure 2

Plot of remaining oil,

1−

− N

N vs P

p. ∆

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We know Y ( it is ∆ P ), we can calculate X (the RHS of equation 9). X is equal to 1-

(Np/N) and we know Np (the amount of oil we had to produce to get drawdown ∆ P ).Hence, we can nd N the initial oil in place. An exercise to do this is given below.

2.3 Conditions for the Validity of Material BalanceThe basic premise for the material balance assumptions to be correct is that the

reservoir be “tank like” i.e. the whole system is at the same pressure and, as the

pressure falls, then the system equilibrates immediately. For this to be correct, the

pressure communication through the system must at least be very fast in practice

(rather than instantaneous which is strictly impossible). For a pressure disturbance

to travel very quickly through a system, we know that the permeability should

be very high and the uid compressibility should be low (pressure changes a re

communicated instantaneously through and incompressible uid). Indeed, we willshow later (Chapter 5) that pressure equilibrates faster - or “diffuses” through the

system faster - for larger values of the “hydraulic diffusivity”, which is given by

k/(φ µc) (Dake, 1994, p.78).

Dake (1994, p.78), also points out two “necessary” conditions to apply material

balance in practice as follows:

(i) We must have adequate data collection (production/pressures/PVT); and

(ii) we must have the ability to dene an average pressure decline trend i.e. the more

“tank like”, the better and this is equivalent to having a large k/(φμc) as discussedabove.

EXERCISE 1.

Material Balance problem for an undersaturated reservoir using equation 8 above.This describes a case where production is by oil, water and formation expansionabove the bubble point (P

b) with no water inux or production.

Exercise:

Suppose you have a tank - like reservoir with the uid properties given below (and

in Figure 4). Plot a gure of 1−

N

N

pvs. -∆ P over the rst 250 psi of depletion

of this reservoir. Suppose you nd that after 200 psi of depletion, you haveproduced 320 MSTB of oil. What was the original oil in place in this reservoir?

Input data: The initial water saturation, Swi

= 0.1. The rock and watercompressibilities are, as follows:

cf= 5 x 10-6 psi-1; c

w= 4 x 10-6 psi-1.

The initial reservoir pressure is 5500 psi at which Boi

= 1.3 and the bubble point isat P

b= 4000 psi where B

o= 1.4. That is, the oil swells as the pressure drops as

shown in Figure 4.

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3. SINGLE PHASE DARCY LAW

We review the single phase Darcy Law in this section in order to put our own particular

“slant” or viewpoint to the student. This will prove to be very useful when we derive

the ow equations of reservoir simulation in Chapter 5. We also wish to extend

the idea of permeability (k) somewhat further than is covered in basic reservoir

engineering texts. In particular, we wish to introduce the idea of permeability as

a tensor property, denoted by k . Some useful mathematical concepts will also be

introduced in this section associated with vector calculus; in particular, the idea of

gradient ∇ and divergence ∇• will be discussed in the context of the generalised

formulation of the single phase Darcy law. Note that for reference, many of the terms

discussed here are also summarised in the Glossary.

3.1 The Basic Darcy ExperimentDarcy in 1856 conducted a series of ow tests through packs of sands which he took

as approximate experimental models of an aquifer for the ground water supply at

Dijon. A schematic of the essential Darcy experiment is shown in Figure 5 where

we imagine a single phase uid (e.g. water) being pumped through a homogeneous

sand pack or rock core. (Darcy used a gravitational head of water as his driving force

whereas, in modern core laboratories, we would normally use a pump.)

The Darcy law given in Figure 5, is in its “experimental” form where a conversion

factor, β, is indicated that allows us to work in various units as may be convenient

to the problem at hand. In differential form, a more useful way to express the Darcy

Law and introducing the Darcy velocity, u, is as follows:

=

= − = −

∂∂

u

Q

A

k P

L

k P

x . .∆

µ µ (9)

where the minus sign in equation 9 indicates that the direction of uid ow is down

the pressure gradient from high pressure to low pressure i.e. in the opposite direction

to the positive pressure gradient.

∆P

L

Q = β.k.A

µ∆P

L

Q Q

.

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Definitions:

Symbol Dimensions Meaning Consistent Unitsc.g.s lab. field SI -

field

Q L3 /T Volumetric cm

3 /s cm

3 /s bbl/day m

3 /day

flow rate

L L Length of cm cm ft. m

system

A L2

Cross - sectional cm2

cm2

ft.2

m2

area

µ Viscosity cP cP cP Pa.s

∆P M.L.T.2

Pressure drop atm dyne/cm2

psi Pa

(Force/Area)

k L2

Permeability#

darcy darcy mD mD

β dimensionless Conversion 1.00 9.869x10-6

1.127x10-3

8.527

factor x10-3

# permeability - dimensions L2; e.g. units m

2, Darcies (D), milliDarcies (mD); 1 Darcy

= 9.869 x 10-9

cm2

= 0.98696 x 10-12

m2 ≈ 1 µm

2.

Note on Units Conversion for Darcy’s Law: the various units that are commonly used

for Darcy’s Law are listed in Figure 2 above. Sometimes, the conversion between

various systems of units causes confusion for some students. Here, we briey explain

how to do this using the examples in the previous gure; that is, we go from c.g.s.

(centimetre - gram - second) units where β = 1, indeed, the Darcy was dened such

that β = 1. Starting from the Darcy Law in c.g.s. units:

Q cm sk Darcy A cm

cp

P atm

L cm=

µ

( / )( ) . ( )

( ).

( )

( )

∆1.003

2

Suppose we now wish to convert to eld units as follows:

Q bbl dayk Darcy A ft

cp

P psi

L ft=

µ

( / )( ) . ( )

( ).

( )

( .)

∆

??2

How do we nd the correct conversion factor for these new units? Essentially, we

convert it unit by unit starting from the c.g.s. expression where we know that β = 1.

We do need to know a few conversion factors as follows: 1 ft. = 30.48 cm (exact),

14.7 psi = 1 atm., 1 bbl = 5.615 ft3 = 5.615 x 30.483 cm3 = 1.58999 x 105 cm3, 1 day

= 24 x 3600 s = 8.64 x 104 s. Thus, we now convert everything in the eld units to

c.g.s. units as follows (except for cp. which are the same):

Figure 5.

The single phase Darcy Law

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Qbbl

day.

1.58999xx 5

x4

10

108 64

1000 30 48 14 7

30 48

2 2

.

( ) . ( . ). .

( ). .

( .). .

= ( )

k

mD A ft

cp

P

psi

L ftµ

∆

Thus, collecting the numerical factors together we obtain:

x4

x5

10

1000 . 1.58999 10

8 64 30 48

14 7 30 48

2. . .

. . . .

=

Qbbl

day

( )

k Darcy A ft

cp

P psi

L ft

( ) . ( . )

( ).

( .)

2

µ∆

which simplies to

= ( )

1.126722 10x-3Q

bbl

day

k Darcy A ft

cp

P psi

L ft

( ) . ( . )

( ).

( .)

2

µ∆

and hence β = 1.127 x 10-3 for these units (as given in Figure 5).

3.2 Mathematical Note: on the Operators “gradient” ∇ and

“divergence” ∇•

Before generalising the Darcy Law to 3D, we rst make a short mathematical digressionto introduce the concepts of gradient and divergence operators. These will be used

to write the generalised ow equation of single and two phase ow in Chapter 5.

Gradient (or grad) is a vector operation as follows:

∇ =∂∂

+∂

∂+

∂∂

xi

y j

zk

where i, j and k are the unit vectors which point in the x, y and z directions,

respectively. The gradient operation can be carried out on a scalar eld such as

pressure, P, as follows:

∇ =∂∂

+∂∂

+∂∂

PP

xi

P

y j

P

zk

where ∇P is sometimes written as grad P. The quantity ∇P is actually a vector of

the pressure gradients in the three directions, x, y and z as follows:

∇ =

∂∂

∂∂

∂∂

P

P

xi

Py j

P

zk

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This is shown schematically in Figure 6 where the three components of the vector

∇P, i.e. ∂∂

∂∂

Px

i Py

j and, , ∂∂

Pz

k , and are shown by the dashed lines.

Figure 3: The denition of grad P or

Unit vectorsz

y

x

k

j

i

P ∆

Divergence (or div) is the dot product of the gradient operator and acts on a vector

to produce a scalar. The operator is denoted as follows:

∇ = ∂∂

∂∂

∂∂

. . . .

xi

y j

zk

For example, taking the divergence of the Darcy velocity vector, u, gives the

following:

∇ =∂

∂∂

∂∂∂

. . . .

ux

iy

jz

k

u i

u j

u k

x

y

z

where we can expand the RHS of the above equation by multiplying out the rst

(1x3) matrix by the second (3x1) matrix to obtain a “1x1 matrix” which is a scalar

as follows:

∇ =∂

∂∂

∂∂∂

. . . .

ux

iy

jz

k

u i

u j

u k

x

y

z

=∂∂

+∂∂

+∂∂

u

xi i

u

y j j

u

zk kx y z. . .

where we use the relationships, = = =i i j j k k. . .

1

, to obtain:

∇ =∂∂

+∂∂

+∂∂

u

u

x

u

y

u

z

x y z.

Figure 6

The denition of grad P or

∇ P

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which multiplies out as follows:

∇ =

∂∂

∂∂

k P

k k k

k k k

k k k

P

x

P

y

xx xy xz

yx yy yz

zx zy zz

.

∂∂

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

P

z

kP

xk

P

yk

P

z

kP

xk

P

yk

P

z

xx xy xz

yx yy yz

+ +

+ +

∂∂

∂∂

∂∂

kP

xk

P

yk

P

zzx zy zz+ +

giving the nal result:

∇ =

∂∂

∂∂

k P

kP

xk

P

yxx xy

+

.

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+

+ +

+ +

kP

z

kP

xk

P

yk

P

z

kP

xk

P

yk

P

z

xz

yx yy yz

zx zy zz

Using the above concepts from vector calculus (div. and grad), we can extend the

Darcy Law (in the absence of gravity) to 3D as follows by introducing the tensor

permeability, k:

u k P

k k k

k k k

k k k

P

x

P

y

P

z

kP

x

xx xy xz

yx yy yz

zx zy zz

xx

= ∇ =

∂∂

∂∂

∂

∂

=

∂∂

-1

-1

-1

.µ µ µ

++ +

+ +

+ +

kP

yk

P

z

kP

xk

P

yk

P

z

kP

xk

P

yk

xy xz

yx yy yz

zx zy zz

∂∂

∂∂

∂∂

∂∂

∂∂

∂

∂

∂

∂

∂∂

∂

P

z

which we may write as:

=

=

∂∂

∂∂

∂∂

u

u

u

u

kP

xk

P

yk

P

z

k

x

y

z

xx xy xz

-1

+ +

µ yxyx yy yz

zx zy zz

P

xk

P

yk

P

z

k Px

k Py

k Pz

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+ +

+ +

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18

and we can identify the three components of the velocity as follows:

=∂∂

∂∂

∂∂

=∂∂

∂∂

∂∂

=

u kP

xk

P

yk

P

z

u kP

xk

P

yk

P

z

u

x xx xy xz

y yx yy yz

z

-1

+ +

-1

+ +

-1

µ

µ

µµk

P

xk

P

yk

P

zzx zy zz

∂

∂

∂

∂

∂

∂

+ +

If the permeability tensor is diagonal i.e. the cross-terms are zero as follows:

k

k

k

k

xx

yy

zz

=

0 0

0 0

0 0

then the various components of the Darcy law revert to their normal form and :

u kP

x

u kP

y

u kP

z

x xx

y yy

z zz

=∂∂

=∂∂

=∂

∂

-1

-1

-1

µ

µ

µ

3.4 Simple Darcy Law with GravityIn the presence of gravity the 1D Darcy Law becomes:

u kP

xg

z

xx xx=

∂∂

∂∂

-1

µρ-

where, in the case of a simple inclines system at a slope of θ, as shown in Figure 7,

z

x

∂

∂

=

θcos , as shown in the gure above and:

u kP

xgx xx=

∂∂

-1

µρ- . ..cosθ

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x

θ

Note that:

= cos θ∂z

∂x

3.5 The Radial Darcy LawIn the above discussion, in both 1D and 3D we considered the Darcy Law in normal

Cartesian coordinates (x, y and z). In Chapter 6, we will explain how wells are

treated in reservoir simulation. Because a radial (r/z) geometry is appropriate for

the near-well region, it is useful to consider the Darcy Law in radial coordinates. In1D, this simply involves the radial coordinate, r. In fact, the radial form of the Darcy

law can be derived from the linear form as shown in Figure 8.

rh

Qdr

Area, A = 2π.r.h

Radial Darcy Law is:

Q =k.A

µ

dPdr

2πkhrµ

dPdr

=

Notation: Q = volumetric ow rate of uid into well

r = radial distance from well

h = height of formation

dP = incremental pressure drop from r→ (r + dr) i.e. over dr

A = area of surface at r = 2π.r.h

μ = uid viscosity

k = formation permeability

rw = wellbore radiusdr = incremental radius

Starting from the radial form of the Darcy Law, as follows:

πµ

=

Q

khr dP

dr

2

we can rearrange this to obtain:

µπ= dPQ

kh

dr

r 2

Figure 8

Single phase Darcy Law in

an inclines system - effect of

gravity

Figure 7

Radial form of the single-

phase Darcy Law

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20

Taking rw

as the wellbore radius and r some appropriate radial distance, we can easily

integrate the above equation to obtain the radial pressure prole in a radial systemas follows:

dPQ

kh

dr

r r

r

r

r

w w

∫ ∫ =

µπ2

which gives:

P r Q

kh

r

r w∆ =

µπ2

( ) ln

where we have denoted the radial pressure drop (or increase for a producer) from rw

to

r as,ΔP(r). Note that, unlike the linear Darcy Law, the pressure prole is logarithmic

in the radial case. This means that pressure drops are much higher closer to the well.

This is exactly what we expect physically since the area is decreasing with r as we

approach the well and Q is the same; therefore, the pressure drop, dP, over a given

dr is higher. This is shown schematically for an injector and a producer in Figure 9.

The formulae and the ideas developed here will be used later in Chapter 4 on well

modelling in reservoir simulation and we will not discuss this further here.

QQ

Producer

Pwf

Pwf

rw rrw r

∆P(r)∆P(r)

∆P(r) = P(r) - Pwf

∆P(r) = Pwf - P(r)

Injector

4. TWO-PHASE FLOW

4.1 The Two-Phase Darcy LawDarcy’s Law was originally applied to single phase ow only. However, in reservoir

engineering, it has been convenient to extend it to describe the ows of multiple

phases such as oil, water and gas. To do this, the Darcy Law has been modied

empirically to include a term - the relative permeability - which is intended to describe

the impairment of the ow of one phase due to the presence of another. A schematic

representation of a steady-state two phase Darcy type (relative permeability) experiment

is shown in Figure 10, where all of the quantities are dened. Examples of the relative

permeability curves which can be measured in this way are also shown schematically

in Figure 10 and actual experimental examples are given for rock curves of different

wettability states in the Glossary.

Figure 9

Pressure proles, Δ P(r), in

radial single-phase ow; Pwf

is the well owing pressure

(at r w )

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00 1

kro

krw

Qw

Sw

Qo

Qw

QoL

∆Pw

∆Po

1

Schematic of relativepermeabilities, krw and kro

Rel.Perm.

Qw = k.krw.Aµw

Qo =k.kro.Aµo

∆PwL

.

∆PoL

.

The two - phase Darcy Law is as follows:

At steady - state flow conditions, the oil and water flow rates in and out,Qo and Qw, are the same:

Where:

Qw

and Qo

= volumetric ow rates of water and oil;

A = cross-sectional area;L = system length;

µw

and µo

= water and oil viscosities;

k = absolute permeabilities;

ΔPw

and ΔPo

= the pressure drops across the water and oil phases at

steady-state ow conditions

krw

and kro

= the water and oil relative permeabilities

NB the Units for the two-phase Darcy Law are exactly the same as those in Figure

5.

The differential form of the two phase Darcy Law in 1D, again including gravitywhich is taken to act in the z-direction, is as follows:

uk k P

xg

z

x

uk k P

xg

z

x

wrw w

w

oro o

o

=∂∂

∂∂

=∂∂

∂∂

µρ

µρ

.-

.-

-

-

where we note that the ow of the two phases (water and oil, in this case) dependson the pressure gradient in that phase; i.e. on

P

xandw∂

∂

∂

PP

x

o

∂

.

Figure 10

The two-phase Darcy Law

and relative permeability

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22

The phase pressures, Po

and Pw, at a given saturation, S

w(S

o= 1 - S

w), are generally

not equal. However, they are related through the capillary pressure, as follows:

P S P Pc w o w( ) = −

More strictly, the capillary pressure is the difference between the non-wetting

phase pressure and the wetting-phase pressure; P S P Pc w non wett wett( ) = −−

. .. We

can think of the capillary pressure as a constraint on the phase pressures. That is, if

we know the capillary pressure function - from experiment , say - then, if we have

Po

at a given saturation, we can calculate Pw. Examples of capillary pressure curves

are also shown in the Glossary.

Note that, as in the single-phase Darcy Law, we may generalise the two-phase Darcy

expressions to 3D. Dening the combination of absolute permeability in its full

tensor form,k

, with the phase relative permeabilities gives:

k k k

k k k

w rw

o ro

=

=

where k and kw o

are the effective phase permeability tensors of water and oil,

respectively. Using this notation, the Darcy velocity vectors for the water and oil, uw

and uo, may be written in 3D as follows:

u k P g z

u k P g z

w

w

w w w

o

o

o o o

= − ∇ − ∇( )

= − ∇ − ∇( )

1

1

µρ

µρ

.

.

This form of these equations is particularly useful in deriving the two-phase ow

equations in their most general form (this will done in Chapter 5).

5. CLOSING REMARKS

The purpose of Chapter 2 is to review some key concepts in reservoir engineering

which impact directly on the subject matter of reservoir simulation. The topics

reviewed specically involved:

- Material balance and its particular relationship with reservoir simulation;

- The single-phase Darcy law and its extension using vector calculus terminologyto a 3D version of the Darcy Law including tensor permeabilities;

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- The two-phase Darcy Law and the related concepts that arise in two-phase

flow e.g. relative permeabilities (kro and krw), phase pressures (Po and Pw),capillary pressure (P

c(S

w) = P

o- P

w), etc.

Ideas and concepts developed here will be used in other parts of this course.

6. SOME FURTHER READING ON RESERVOIR ENGINEERING

A full alphabetic list of References which are cited in the course is presented in

Appendix A. Many excellent texts have appeared over the years covering the basics

of Reservoir Engineering. Some of these are listed below, although this list is far

from comprehensive.

Amyx, J W, Bass, D M and Whiting, R L: Petroleum Reservoir Engineering, McGraw-

Hill, 1960. This is still an excellent petroleum engineering text although the coverage in

some areas a little old fashioned. It has a very good chapter on material balance.

Archer, J S and Wall, C: Petroleum Engineering: Principles and Practice, Graham

and Trotman Inc., London, 1986. This book offers a good overview of petroleum

engineering and covers many of the basics of reservoir engineering. This book is

also one of the earliest proponents of the importance of integrating the geology within

the reservoir model.

Craft, B C, Hawkins, M F and Terry, R E: Applied Petroleum Reservoir Engineering,

Prentice Hall, NJ, 1991. The original text by Craft and Hawkins was already an

early classic. This was revised and updated by Terry and reissued in 1991. This has

very good clear coverage of material balance and its application in various reservoir

systems.

Craig, F F: The Reservoir Engineering Aspects of Waterooding, SPE monograph,

Dallas, TX, 1979. This text is conned to the underlying principles and reservoir

engineering applications of waterooding. It is an excellent monograph on the

subject and an essential reference text for the reservoir engineer who is interested in

the traditional analytical methods for assessing waterooding.

Dake, L P: The Fundamentals of Reservoir Engineering, Developments in Petroleum

Science 8, Elsevier, 1978. This has become a modern classic on the basics of reservoir

engineering. It is very widely referenced and draws on Dake’s vast experience of

teaching reservoir engineering basics. It has particularly good coverage of material

balance and Buckley-Leverett theory.

Dake, L P: The Practice of Reservoir Engineering, Developments in Petroleum

Science 36, Elsevier, 1994. This book is a modern plea for the continued application

traditional reservoir engineering principles and techniques in performance analysis and

prediction. It gives central place to the interpretation of well testing, the application

of material balance and the use of Buckley Leverett theory. It has many examplesfrom the hundreds of reservoirs that Dake himself worked on. This book also makes

a number of interesting and controversial observations on reservoir simulation (not

all of which the authors agree with!).

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24

Solution To Exercises

EXERCISE 1:

Material Balance problem for an undersaturated reservoir using equation 8 above.

This describes a case where production is by oil, water and formation expansion

above the bubble point (Pb) with no water inux or production.

Exercise: For the input data below, do the following:

(i) Plot the function (1 - N/Np) as calculated by equation 8 vs. -ΔP.

As a reminder equation 8 is 1 11

− = − + +− N N

B B

B B

S cS

P p oi

o

oi

o

wi f

wi

∆

This is shown below

Series 1

(1-Np /N) vs. -DP

0.999

0.997

0.995

0.993

0.991

0.989

0.987

0.985

0 50 100 200 300150 250

( 1

- N p

/ N )

-∆p (psi)

(ii) Note from the graph (or from your numerical calculation when plotting the

graph) that, at - ΔP = 200 psi, then (1 - Np/N) = 0.991. However, we know by eld

observation that this 200 psi drawdown was caused by the production of 320 MSTB.

That is, we know that Np = 320 MSTB. Hence,

(1 - 320/N) = 0.991 => N = 35555.5 MSTB ≈ 35.6 MMSTB

Answer: the STOOIP = 35.6 MMSTB.

Input data:The initial water saturation, Swi

= 0.1. The rock and water compressibilities

are, as follows:

crock

= 5 x 10-6 psi-1; cw

= 4 x 10-6 psi-1.

The initial reservoir pressure is 5500 psi at which Boi

= 1.3 and the bubble point is

at Pb

= 4000 where Bo

= 1.4. That is, the oil swells as the pressure drops as shown

below:

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1.4

1.3

4000 P (psi) 5500

Oil FVF

Bo

Bo(P) = m.P + c

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