24 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

109 εμφανίσεις





Tatiana Lipovetsky



Petroleum reservoirs with high permeability lenses consist of concentrated coarse grains deposits, forming
vertically and horizontally limited “lenses”, localized in a low permeability matrix. In carbonantes, for example,
such lenses may be formed by eithe
r depositional or diagenetic processes. A way of verifying a reservoir
performance is by measuring its flow and pressure at the well, for a given condition. During a certain test
period, after opening the well, pressure in formation is monitored through ti
me. Then, the well is shut in and
pressure continues being measured until the steady state is reached. This pressure analysis is commonly done by
Well Testing Analysis (WTA) and

with diagnosis of the correct model can provide information on size and
of formation heterogeneities and their influence on fluid production efficiency.

The objective of this work is to describe, using computational numerical analysis, the behaviour of pressure in
reservoirs with high permeability lenses intersecting the well
bore, for multiphase flow composed of oil and gas.
Therefore, Wolfram Mathematica was chosen to be the computational programming platform used for the
proposed model.

The Finite Volume Method (FVM) will be applied to a bi
dimensional model. The FVM was ch
osen due to the
fact that most of commercial softwares use the Finite Difference Method (FDM), at which the properties and
equations are applied to discrete points, and not to discrete volumes as done with FVM. Hence, the FVM is
applied to the IMPES Method

(Implicit Pressure, Explicit Saturation Method), which provides means to forecast
pressure and flow conditions at well bottom and pressure and saturation distributions throughout the reservoir.
Thereunto, some problems must be “contoured”, such as the siz
e of the finite volumes containing the well which
is, usually much greater than the well diameter, not correctly representing its bottom pressure. This way, an
accurate method of analysing and representing a reservoir and the well in it will be discussed i
n this paper.


Differences in grain size, permeability, porosity, grain shape, sorting, packing, orientation, diagenesis, among
other grain characteristics, can cause to reservoirs great heterogeneities. The so called reservoir lenses happen
when there is a channel concentration of coarse grains, causing high permeability (k), situated in a low
permeability matrix. These lenses are vertically and laterally limited. This work intends to report, through Well
Test Analysis, the pressure behaviour

when the wellbore is intersected by lenses as show in Figure 1.

Figure 1

Example of Reservoir Lenses, extracted from Sagawa et al., 1999.

To do such study, a radial
geometry reservoir (as shown in Figure 2) consisted of oil, gas and residual water
considered. The well was placed in its centre and the reservoir boundaries were considered to be of no
type. The mathematical model used for the well was the numerical model and the developed reservoir
can be classified as Black Oil, whi
ch is used when recovery processes are intensive to compositional changes in
the reservoir fluids.

Figure 2

Scheme of Radial Reservoir

Well Test Analysis

Well Test Analysis (WTA) consists of temporary changes in production rates, causing transient pressure
responses, usually measured at the well bottom while the flow rate is measured at surface (standard conditions).
Before opening the well, the initial pre
ssure is the static reservoir pressure, which is constant and uniform.
When the well is opened to flow, the drawdown pressure is measured, and when the well is shut, one can
measure the build
up pressure. The well response is generally monitored during a r
elative short period of time
compared to the life of the reservoir.

WTA provides information on reservoir and well, which, associated with geological and geophysical data, are
used to build a reservoir model for prediction of the field behaviour and flui
d recovery to different scenarios.
For well evaluation, tests don’t need to be run for more than a couple of days, while for reservoir evaluation, the
tests might last up to several months. (BOURDET, 2002)



nate is a type of chemical/biochemical sedimentary rock, originated by precipitation of minerals from
water through various chemical or biochemical processes. Chemical/biochemical sedimentary rocks can be
distinguished from siliclastic sedimentary rocks by

its chemistry, mineralogy and texture. Carbonaceous
sedimentary rocks, such as coals and oil shales, make up a further special group of rocks that contain abundant
nonskeletal organic matter in addition to various am amounts of siliclastic or chemical (e.
g. carbonate)
constituents. (BOGGS, 2006)

The carbonate rocks are the most abundant kind of chemical/biochemical sedimentary rock, and can be divided
on the basis of mineralogy into limestones and dolomites. Limestones are mainly composed of calcite miner
while dolomites are mainly composed of dolomite minerals, and both act as reservoir rocks for more than one
third of the world’s petroleum reserves.

Numerical Carbonate Reservoir Model

The main objective of this study is to characterize a given geo
logical model, through a numerical simulator,
looking at well
test responses to different geological phenomena. According to the geology of the reservoir,
trends in the well
test responses can be set as a family of pressure diagnostic plots, or the so call
ed “Geotype
curves”, as explained by CORBETT et al., 2011.

The reason why a carbonate reservoir has been chosen in this work is that it can be a great challenge to be
modelled, concerning numerical and gridding issues, due to its pore systems, which are c
omplex through
depositional and diagenetic processes, and fractures are often present.


When well tests analyses are being made, analytical models are recommended to do, for example, pressure
transient analysis. The
analytical models represent exact solutions to simplified problems, meaning that the
physics of the problem is preserved. This way the analytical methods are often used to determine how the
reservoir parameters affect its performance.

The equations derive
d during the formulation process, if solved analytically, would give the pressure, saturation,
and production rates as continuous functions of time and location. Because of the non
linear nature of these
equations, analytical techniques cannot be used and
solutions must be obtained with numerical methods.

This means that, reservoir problems which are more complicated than an analytical model could stand, demand
the problem to be simplified, which will not properly represent the reservoir and the perforated

well. Numerical
models are more acceptable to this kind of problem, since they yield approximate solutions to exact problems.

Usually, commercial softwares use the Finite Differece Method (FDM), but as an alternative, in this work, the
chosen numerical m
odel is the Finite Volume Method (FVM). The main reason for this change is that while the
FDM discretizes the reservoir medium in spaced points, the FVM uses volume discretization, “embracing”, or
comprehending, the whole reservoir volume (no spaces betwee
n volumes, unlike points), which, theoretically,
brings the problem representation closer to reality.

The Finite Volume Method applied to the reservoir.

The finite volume method is a method largely used in computational fluid mechanics to represent and e
partial differential equations in the form of algebraic equations. To start the calculation, a meshed geometry is
created and values are calculated for each volume of this mesh, then initial and boundary conditions are applied
to the problem.

In t
his case, a Neumann Problem is being used to formulate the basic equations for the multi
phase flow in the
reservoir. The Neumann Problem consists of specifying a pressure gradient normal to the boundary, so the flux
(or velocity) normal to the boundary ca
n be prescribed. This means that a constant flow rate specification at the
wellbore is equivalent to specifying the pressure gradient at the wellbore sandface.

A schematic figure of the problem being treated is show in Figure 3.

Figure 3

Meshed geomet
ry of the reservoir perforated by the well

Well Representation

According to Ertekin et al. (2001), the ultimate goal of reservoir
simulation study is to forecast well flow rates
and/or flowing bottomhole pressures accurately and to estimate pressure and
saturation distributions, not
forgetting to mention reservoir description. Well treatment in reservoirs simulators presents difficulties that
require special considerations:

The block hosting the well is usually large compared to the size of the well, so
the pressure of the block
computed by the reservoir simulator is a poor estimate of the flowing well pressure;

Coupling of the complex interaction between the reservoir and the wellbore is often problematic,
particularly in the case of multi
layered wells;

Allocating phase production rates in multiphase flow when single
phase or total rate from the well is

Also important to notice, when it comes to single well in reservoir with radial coordinates analysis, the well
specifications have to be impl
emented as boundary conditions, not requiring an additional source term.

IMPES Method

The basic idea of the IMPES Method consists of solving a partial differential coupled system for two
phase flow
in a porous medium, by separating the computation of pre
ssure from that of saturation. Basically, the IMPES
method allows to implicitly calculate the pressure in the porous medium, and then, through using the obtained
pressure values, to explicitly calculate the saturations of the two fluids in the porous mediu
m. Therefore, The
IMPES method applied to the FVM is a quick and accurate alternative to the analytical method and to the FDM.


The problem here discussed consists of a radial reservoir, intersected by the wellbore in its centre, and the

flowing fluids are oil and gas. Water is considered to be residual.

The equations that govern the steady flow composed of oil and gas in a porous medium are:






is the medium porosity,
is the oil saturation,
is the Oil Formation Volume Factor,
is the permeability
is the oil
relative permeability,
is the oil viscosity,

is the oil pressure,

is the oil density, g is gravity,
z is distance to datum,
is oil flow per volume unit in standard condition
is gas saturation,
is gas formation
volume factor,
is gas solubility in oil ratio,
is gas
relative permeability,
is gas viscosity,
is gas pressure,
is gas density,
is free gas flow per volume unit in standard conditions,

is associated gas flow

per unit
in standard conditions,
is oil
gas capillary pressure.

These equations are, then, written in the bi
dimensional radial coordinate system (r,z) and the hydrostatic term is
not considered:



is horizontal permeability,
vertical permeability.

The equations (1) to (6) are, then, discretized into the FVM, and applied to IMPES as a Neumann Problem, so
pressure and saturation fields in the porous medium can be obtained. After that, oil, free gas and associated gas
flows can
be calculated. The representing mathematical problem was implemented in Wolfram Mathematica®,
following the shown diagram
Annexe 1.


For starters, the mathematical model developed here did not include high permeability len
ses. In sequence, a
high permeability l
ens intersecting the wellboreis
s introduced in the reservoir matrix, which has lower
permeability, intercepting the well
its mid length, and reaching until ½ of the reservoir radius
. The

permeability is higher

than the matrix permeability.

The tests consist of measuring pressure
transient, and observing its behaviour throughout the reservoir,
characterizing a well test.

It is expected, according to the literature, that for a uniform reservoir, the pressure pr
ofile into the reservoir is
described by a simple pressure gradient (Figure 4). Departures from this profile lead to negative or positive skin.
The negative skin results in positive contribution to oil and gas production. The negative skin occurs when the
reservoir has fractures, but it can also happen when the reservoir has high permeability lenses. Another example
of behaviour is that the radial flow regime is characterized by a horizontal line of zero slope on a log
log plot of
pressure derivative versus

time. Linear flow and depletion are seen to give diagnostic half and unit slopes as can
be observed, for example, for a well in a regular linear channel sandstone. The geological signature of the
reservoir, therefore, comes (in the pressure response) from

the particular sequence of flow regimes. The radial
flow regime is used to determine the transmissivity which will ideally match the transmissivity from the core
data over the perforated interval. (CORBETT, 2012)

Figure 4

Definition sketch of the
pressure profile away from the wellbore showing skin as an increase or decrease in
pressure in the immediate wellbore region. As the reservoir pressure drops from P1 to P2 in the expected, radial uniform
case, the pressure profile drops as shown. If the pr
essure measured at the wellbore is higher than expected from this profile
then this defines a negative pressure drop and an increase in production. If the pressure measured in the wellbore is much
lower than this gives a positive skin and results in reduce
d production. (CORBETT, 2012)


Analysing the influence of matrix heterogeneity on pressure transient required the following steps:


Build a mathematical model for multiphase (oil and gas) flow;


Represent the mathematical model numerically;


Build the numerical computational code;


Simulate draw


Implicitly calculate pressure field, explicitly calculate saturation field;


Calculate oil rate, free gas rate and dissolved gas rate at the well head, in standard conditions;


Analyse the resulting

numerical pressure transients;


Correlate the pressure transients to the known geological features;



Model 1

Homogeneous Reservoir

Many tests with different griddings were done in this kind of reservoir, showing that a poor grid (too f
ew finit
volumes) doesn´t deliver
accurate result, or doesn´t deliver

any results at all. Therefore,

the more one

up a reservoir, the better it will be represented, due to the calculations that can be more accurately
executed. When the software accuse
s a poorly conditioned matrix of reservoir and well coefficients, for
instance, significant numerical errors may occur, disturbing the problem´s answers. Discretizing and
incorporating the well to the reservoir is an issue that, if not carefully done, may
bring significant errors to the
solution of the problem.

The wellbore pressure distribution can be obtained, iteration by iteration, so it can be plotted against time,
characterizing a WTA.

Some results obtained are:

Figure 5

Pressure distributi
on in the first numerical iteration

Figure 6

Gas Pressure distribution in the first numerical iteration

Figure 7

Saturation distribution in the first numerical iteration

Figure 8


saturation in the first numerical iteration



Oil saturation in the first numerical iteration

Model 2

Reservoir Intersected by lens at the wellbore

Differently from Model 1,
Model 2 has a high permeability (k) lens i
ntersecting the wellbore, at its

mid length.

The same wellbore pressure d
istribution analysis is done,

and a similar plot is obtained, therefore not being
plotted here.


far, due to numerical issues,
accurate result

of pressure and saturation distribution
could not have been

for more than one iterati
. This is probably happening because of the lack of precise information on
reservoir properties.

Though, it is possible to notice that the pressure fields stabilize far

from the wellbore, and
drop near the wellbore, as expected.



BOGGS Jr., S. Principles of Sedimentology and Stratigraphy, Fourth edition, 2006

BOURDET, D. Well Testing and Interpretation, 2002.

CORBETT, P. W. M. Well Testing and Interpretation Tutoring Classes, 2012. COPPE/UFRJ

, L., BORGES L., GARAYEV M., VALDEZ C.The third porosity system: understanding the role
of hidden pore systems in well
test interpretation in carbonates, 2012.

COUTO, P. Notas de aula: Simulação de Reservatórios, Professor Paulo Couto, 2012.

KASSEM, J. H.; KING, G.R. Basic Applied Reservoir Simulation. SPE, 2001.

HAMDI, H. Illumination of Channelised Fluvial Reservoir Using Geological Well Testing and Seismic Modelling, Heriot
University, 2012.

HORNE, R. N.. Modern Well Tes
t Analysis: A Computer Aided Approach, 1995.

MALISKA, C. R. Transferência de Calor e Mecânica dos Fluidos Computacional. Ed. LTC, 2004.

ROSA, A. J.; CARVALHO, R. S.; XAVIER, J.A.D. Engenharia de Reservatórios de Petróleo.
Ed. Interciência, 2006.

.; CORBETT, P. W. M.; DAVIES, D. R. Pressure transient analysis of reservoirs with a high permeability lens
intersected by the Wellbore, 1999.

SCHLUMBERGER, Well Test Interpretation, 2008.