Table Of ContentAdvanced Reservoir Management and Engineering
Advanced Reservoir
Management and Engineering
Second edition
Tarek Ahmed
D. Nathan Meehan
AMSTERDAM(cid:129)BOSTON(cid:129)HEIDELBERG(cid:129)LONDON
NEWYORK(cid:129)OXFORD(cid:129)PARIS(cid:129)SANDIEGO
SANFRANCISCO(cid:129)SINGAPORE(cid:129)SYDNEY(cid:129)TOKYO
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P R E F A C E
The primary focus of this book is to present the pressured reservoirs, coalbedmethane,tightgas,
basic physics of reservoir engineering using the gas hydrates, and shallow gas reservoirs.
simplest and most straightforward of mathe- Chapter4coversthebasicprincipleofoilrecov-
matical techniques. It is only through having a ery mechanisms and the various forms of the
complete understanding of physics of reservoir material balance equation (MBE). Chapter 5
engineering that the engineer can hope to solve focuses on illustrating the practical application
complex reservoir problems in a practical man- oftheMBEinpredictingtheoilreservoirperfor-
ner. The book is arranged so that it can be used mance under different scenarios of driving
as a textbook for senior and graduate students mechanisms. Chapter 6, is an overview of
or asa reference bookfor practicingengineers. enhanced oil recovery mechanisms and their
Chapter1describesthetheoryandpracticeof application.
well testing and pressure analysis techniques, Chapter 7 covers the fundamentals of oilfield
which is probably one of the most important economic analysis including risk analysis, treat-
subjects in reservoir engineering. Chapter 2 dis- ment of various international fiscal regimes and
cusses various water influx models along with reserve reporting issues. Chapter 8 discusses the
detailed descriptions of the computational steps financial reporting and merger and acquisition
involved in applying these models. Chapter 3 topics relevant to reservoir engineers. Chapter 9
presents the mathematical treatment of uncon- covers petroleum engineering professionalism
ventional gas reservoirs that include abnormally and ethics.
Acknowledgment: D. NathanMeehan would liketoexpress his appreciation toBakerHughes,Incorporated for sup-
portingthedevelopmentoftheadditionstothisbook.
ix
C H A P T E R
1
Well Testing Analysis
1.1 PRIMARY RESERVOIR 1.1.1 Types of Fluids
CHARACTERISTICS
The isothermal compressibility coefficient is
Flow in porous media is a complex phenome- essentially the controlling factor in identifying
non and cannot be described as explicitly as the type of the reservoir fluid. In general, reser-
flow through pipes or conduits. It is easy to voir fluids areclassified into three groups:
measure the length and diameter of a pipe and
compute its flow capacity as a function of pres- (1) incompressiblefluids;
sure; however, flow in porous media is different (2) slightlycompressible fluids;
in that there are no clear-cut flow paths which (3) compressible fluids.
lend themselves tomeasurement.
The isothermal compressibility coefficient c
The analysis of fluid flow in porous media
is described mathematically by the following
has evolved throughout the years along two
two equivalent expressions:
fronts: experimental and analytical. Physicists,
engineers and hydrologists have experimentally
Interms of fluidvolume:
examined the behavior of various fluids as they
flow through porous media ranging from sand
21@V
packs to fused Pyrex glass. On the basis of their c5 (1.1)
V @p
analyses they have attempted to formulate laws
and correlations that can then be utilized to
Interms of fluiddensity:
make analytical predictionsfor similar systems.
The objective of this chapter is to present the 1@ρ
c5 (1.2)
mathematical relationships designed to describe ρ@p
flowbehaviorofreservoirfluids.Themathemat-
ical forms of these relationships will vary where
depending upon characteristics of the reservoir.
V5fluidvolume
The primary reservoir characteristics that must
ρ5fluid density
beconsideredinclude:
p5pressure, psi
(cid:129) types of fluids inthereservoir; c5isothermalcompressibilitycoefficient,Ψ21
(cid:129) flow regimes; Incompressible Fluids. Anincompressiblefluid
(cid:129) reservoir geometry; is a fluid whose volume or density does not
(cid:129) number offlowing fluids in thereservoir. changewithpressure.Thatis:
©2012ElsevierInc.Allrightsreserved. 1
2 CHAPTER 1 WellTesting Analysis
@V @ρ where
50 and 50
@p @p V5volumeat pressure p
Incompressible fluids do not exist; however, ρ5densityat pressure p
this behavior may be assumed in some cases to V 5volume atinitial(reference) pressure p
ref ref
simplify the derivation and the final form of ρ 5densityatinitial(reference) pressure p
ref ref
many flow equations.
It should be pointed out that many crude oil
SlightlyCompressibleFluids. These“slightly”
and water systems fit into this category.
compressible fluids exhibit small changes in vol-
Compressible Fluids. Compressible fluids are
ume, or density, with changes in pressure.
defined as fluids that experience large changes
Knowingthe volume V of a slightly compress-
ref in volume as a function of pressure. All gases
ible liquid at a reference (initial) pressure p ,
ref and gas-liquid systems are considered compress-
the changes in the volumetric behavior of such
ible fluids. The truncation of the series expan-
fluids as a function of pressure p can be mathe-
sion as given by Eq. (1.5) is not valid in this
matically described by integrating Eq. (1.1), t o
category and the complete expansion as given
give:
by Eq.(1.4) is used.
Ð Ð dV The isothermal compressibility of any vapor
2c p dp5 V
pref Vref V phase fluid is described by the following
V (1.3) expression:
exp½cðp 2pÞ(cid:2)5 (cid:2) (cid:3)
ref Vref 1 1 @Z
c 5 2 (1.8)
V5V exp½cðp 2pÞ(cid:2) g p Z @p
ref ref T
where Figures 1.1 and 1.2 show schematic illustra-
p5pressure, psia tions of the volume and density changes as a
V5volumeat pressurep, ft3 function of pressureforallthree types offluids.
p 5initial (reference) pressure, psia
ref
V 5fluidvolumeatinitial(reference)pressure, 1.1.2 Flow Regimes
ref
psia
There are basically three types of flow regimes
The exponential ex may be represented by a that must be recognized in order to describe the
series expansion as: fluidflowbehaviorandreservoirpressuredistri-
bution as a function of time. These three flow
x2 x2 xn
ex511x1 1 1?1 (1.4) regimesare:
2! 3! n!
(1) steady-state flow;
Because the exponent x (which represents the
termc(p 2p))isverysmall,theextermcanbe (2) unsteady-state flow;
ref
(3) pseudosteady-state flow.
approximatedbytruncatingEq.(1.4)to:
Steady-state Flow. The flow regime is identi-
ex511x (1.5) fied as a steady-state flow if the pressure at
every location in the reservoir remains constant,
Combining Eq. (1.5) with(1.3) gives:
i.e., does not change with time. Mathematically,
V5V ½11cðp 2pÞ(cid:2) (1.6) this condition isexpressed as:
ref ref
(cid:2) (cid:3)
A similar derivation is applied to Eq. (1.2), @p
50 (1.9)
to give: @t
i
ρ5ρ ½12cðp 2pÞ(cid:2) (1.7) Thisequationstatesthattherateofchangeof
ref ref
pressurepwithrespecttotimetatanylocationi
CHAPTER 1 Well Testing Analysis 3
Incompressible
Slightly Compressible
e
m
u
ol
V
Compressible
Pressure
FIGURE 1.1 Pressure(cid:3)volumerelationship.
Compressible
y
sit
n
e
D
d
ui
Fl Slightly Compressible
Incompressible
0
Pressure
FIGURE 1.2
Fluiddensityvs.pressurefordifferentfluidtypes.
iszero.Inreservoirs,thesteady-stateflowcondi- position in the reservoir is not zero or constant.
tion can only occur when the reservoir is This definition suggests that the pressure deriva-
completely recharged and supported by strong tive with respect to time is essentially a function
aquiferorpressuremaintenanceoperations. ofboth positioni and time t,thus:
Unsteady-state Flow. Unsteady-state flow (cid:2) (cid:3)
(frequently called transient flow) is defined as @p 5fði;tÞ (1.10)
the fluid flowing condition at which the rate of @t
change of pressure with respect to time at any
4 CHAPTER 1 WellTesting Analysis
Location i
Steady-state Flow
Semisteady-state Flow
e
ur
s
s
e
Pr
Unsteady-state Flow
Time
FIGURE1.3
Flowregimes.
Pseudosteady-state Flow. When the pressure irregular boundaries and a rigorous mathemati-
atdifferentlocations inthereservoirisdeclining cal description of their geometry is often possi-
linearly as a function of time, i.e., at a constant ble only with the use of numerical simulators.
declining rate, the flowing condition is charac- However, for many engineering purposes, the
terized as pseudosteady-state flow. Mathemati- actual flow geometry may be represented by
cally, this definition states that the rate of one ofthe followingflow geometries:
change of pressure with respect to time at every
(cid:129) radial flow;
positionisconstant, or:
(cid:129) linear flow;
(cid:2)@p(cid:3) (cid:129) spherical and hemispherical flow.
5constant (1.11)
@t Radial Flow. In the absence of severe reservoir
i
heterogeneities, flow into or away from a well-
It should be pointed out that pseudosteady- bore will follow radial flow lines at a substan-
state flow is commonly referred to as semistea- tial distance from the wellbore. Because fluids
dy-state flow and quasisteady-state flow and is move toward the well from all directions and
possible for slightly compressible fluids. coverage at the wellbore, the term radial flow is
Figure 1.3 shows a schematic comparison of used to characterize the flow of fluid into the
the pressuredeclines asa function oftime of the wellbore. Figure 1.4 shows idealized flow lines
three flow regimes. and isopotential lines fora radialflow system.
Linear Flow. Linear flow occurs when flow
paths are parallel and the fluid flows in a single
1.1.3 Reservoir Geometry
direction. In addition, the cross-sectional area
The shape of a reservoir has a significant effect to flow must be constant. Figure 1.5 shows an
on its flow behavior. Most reservoirs have idealized linear flow system. A common
CHAPTER 1 Well Testing Analysis 5
Plan View
Wellbore
p
Side View wf Flow Lines
FIGURE 1.4
Idealradialflowintoawellbore.
p p
1 2 Well
Fracture
h Isometric View
A
Plan View
FIGURE 1.5 Linearflow. Wellbore
Fracture
FIGURE 1.6
Ideallinearflowintoverticalfracture.
application of linear flow equations is the fluid
flow into vertical hydraulic fractures as illus-
trated inFigure 1.6.
Spherical and Hemispherical Flow. Depend-
inguponthetypeofwellborecompletionconfig- Wellbore
uration, it is possible to have spherical or
hemispherical flow near the wellbore. A well
with a limited perforated interval could result in Side View p Flow Lines
wf
spherical flow in the vicinity of the perforations
as illustrated in Figure 1.7. A well which only FIGURE1.7 Sphericalflowduetolimitedentry.
partially penetrates the pay zone, as shown in
6 CHAPTER 1 WellTesting Analysis
Wellbore 1.2.1 Darcy’s Law
The fundamental law of fluid motion in porous
media is Darcy’s law. The mathematical expres-
sion developed by Darcy in 1856 states that the
velocity of a homogeneous fluid in a porous
Side View Flow Lines
medium is proportional to the pressure gradient,
FIGURE 1.8 and inversely proportional to the fluid viscosity.
Hemispherical flow in a partially penetrat-
Forahorizontallinearsystem,thisrelationshipis:
ingwell.
q kdp
υ5 52 (1.12a)
A μdx
where
Figure 1.8, could result in hemispherical flow.
The condition could arise where coning of υ5apparent velocity,cm/s
bottomwaterisimportant. q5volumetric flowrate,cm3/s
A5total cross-sectional area of therock, cm2
1.1.4 Number of Fluids Flowing in In other words, A includes the area of the
the Reservoir rock material as well as the area of the pore
channels. The fluid viscosity μ is expressed in
The mathematical expressions that are used to
centipoise units, and the pressure gradient dp/
predict the volumetric performance and pres-
dx is in atmospheres per centimeter, taken in
sure behavior of a reservoir vary in form and the same direction as υ and q. The proportion-
complexity depending upon the number of
ality constant k is the permeability of the rock
mobile fluids in the reservoir. There are gener-
expressedin Darcy units.
ally three cases of flowingsystem:
The negative sign in Eq. (1.12a) is added
(1) single-phaseflow (oil, water,or gas); because the pressure gradient dp/dx is negative
(2) two-phase flow (oil(cid:3)water, oil(cid:3)gas, or inthe direction of flow asshowninFigure 1.9.
gas(cid:3)water); For a horizontal-radial system, the pressure
(3) three-phase flow (oil, water, and gas). gradient is positive (see Figure 1.10) and
Darcy’s equation can be expressed in the fol-
The description of fluid flow and subsequent
lowing generalized radialform:
analysis of pressure data becomes more difficult (cid:2) (cid:3)
q k @p
as thenumberof mobile fluidsincreases. υ5 r 5 (1.12b)
A μ @r
r r
1.2 FLUID FLOW EQUATIONS
Direction of Flow
The fluid flow equations are used to describe p
1
flow behavior in a reservoir take many forms p2
depending upon the combination of variables
presented previously (i.e., types of flow, types ure
s
of fluids, etc.). By combining the conservation es
of mass equation with the transport equation Pr x
(Darcy’s equation) and various equations of
state, the necessary flow equations can be devel-
oped. Since all flow equations to be considered
depend on Darcy’s law, it is important to con-
Distance
siderthis transport relationship first. FIGURE 1.9 Pressurevs.distanceinalinearflow.
