PUC-AULA-17-EQUACOES [MODO DE COMPATIBILIDADE]

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EQUAÇÕES DE SATURAÇÃO
PETROFÍSICA BÁSICA
• Fator de Formação (F)
• Índice de Resistividade (I)
• Equação de Archie
• Parâmetros Rw, m, n, - Perfil e Laboratório
• Fatores petrofísicos que influenciam m, n 
- argilosidade, microporosidade,
- molhabilidade, conectividade
Rw
Ro
Ro
Rw
F = Ro/Rw
Petrofísica Básica – Fator de Formação (F)
Ø = 5 % Ø = 10 % Ø = 15 % Ø = 20 %
F1 F2 F3 F4
F = 1/øm
m = expoente de cimentação 
expoente de porosidade
Log ø
Log F
Petrofísica Básica – Fator de Formação (F)
ARCHIE 1
F = Ro/Rw
Ro = Rw/øm
F = 1/øm Rw= Ro . øm
1a Relação de Archie 
Sw = 100 % Sw = 80 % Sw = 60 % Sw = 40 %
F1 F2 F3 F4
Ro Rt1 Rt2 Rt3
I = Rt/RoLog I
Log Sw
I = 1/Swn
n = expoente de saturação
Petrofísica Básica – Índice de resistividade (I)
I = Rt/ Ro
Swn = Ro/Rt
I = 1/Swn 
2a Relação de Archie 
ARCHIE 2
Swn = Rw / (Rt * øm ) (Equação de Archie)
F = Ro/ Rw
Ro = Rw/øm (Archie 1)
F = 1/øm
I=Rt/Ro
Ro = Rt/I = Rt*Swn (Archie 2)
I = 1/Swn
ARCHIE
Expoentes: m=cimentação n=saturação e a=tortuosidade
Sw2 = 0,81 * Rw / (ø2 * Rt) Arenitos 
ARCHIE
Sw2 = Rw / (ø2 * Rt) Carbonatos
A equação de Archie é indicada para 
rochas sem argilosidade
FORMATION RESISTIVITY FACTOR
1000 Hertz
1
10
100
1000
0,010 0,100 1,000
Porosity, fraction
F
o
r
m
a
t
i
o
n
 
R
e
s
i
s
t
i
v
i
t
y
 
F
a
c
t
o
r
Saturant, ppm: 180.000
Confining Stress, psi: As specified
Brine Resistivity, ohm-m @25°C: 0,0454
Porosity Exponent (m) [Composite]: 2,24
Intercept (a): 1,00
F = a 
φφφφ m
m = 2.24 
a = 1.00
Fator de Formação - laboratório
-
FORMATION RESISTIVITY FACTOR
1000 Hertz
1
10
100
1000
0,010 0,100 1,000
Porosity, fraction
F
o
r
m
a
t
i
o
n
 
R
e
s
i
s
t
i
v
i
t
y
 
F
a
c
t
o
r
Saturant, ppm: 180.000
Confining Stress, psi: As specified
Brine Resistivity, ohm-m @25°C: 0,0454
Porosity Exponent (m) [Composite]: 2,24
Intercept (a): 1,00
F = a 
φφφφ m
m = 2.24 
a = 1.00
Fator de Formação - laboratório
log(1)-log(0,05)/
log(800)-log(1)1 0 1,30103 PHI0,05 -1,30103
800 2,90309 2,90309 2,231378F
1 0
RESISTIVITY INDEX
1000 Hertz
1
10
100
0,010 0,100 1,000
Brine Saturation, fraction Vp
F
o
r
m
a
t
i
o
n
 
R
e
s
i
s
t
i
v
i
t
y
 
I
n
d
e
x
 
(
R
I
)
RI = 1.00
Swn
n = 1.86
Sample Number
Depth, metres
Porosity, fraction: 0,162
Klinkenberg Permeability, md: 6,40
Ìndice de resistividade - laboratório
Saturation Determination
• One of the basic objectives of well log analysis is determining the 
saturation percentages of oil, gas, and/or water occupying the pore 
space of reservoir rocks. Although saturations can be determined by 
any number of methods, most of which require similar log 
measurements, specific circumstances affect or limit the accuracy of 
each method, and it is crucial to use the appropriate method. 
• Saturation is the calculated amount of fluid or gas that occupies pore 
space and is a function of numerous physical, chemical, and biological 
factors. 
• There are no magic numbers for saturation values that definitely 
predict water-free production or water production only. 
• Fractional saturations are a function of:
Saturation Determination
• Several measurements and petrophysical parameters are essential in 
deriving accurate saturation values from log data:
• Reliable and accurate resistivity and temperature values for formation 
water and drilling fluids
• Resistivity values recorded by the appropriate instrument for the 
salinities, porosities, bed thicknesses, etc. encountered
• Accurate determinations of Rt , Rxo , or Ri
• Reliable and accurate porosity determination
• Adequate formation factor to porosity relation
• Adequate exponential for saturation calculations
• Adequate shale volume and resistivity determination
• Awareness of and corrections for any conductive minerals in the 
formation
• Other factors, many of which are possibly still unknown
Saturation Determination
• Type of pore space, connected or isolated
• Amount of pore space
• Size of constituent grain structure
• Homogeneity or heterogeneity of the reservoir matrix and pore 
avenues
• Relation of vertical permeability to horizontal permeability
• In-situ pressures and temperatures
• Capillarity functions
• Wettability of the matrix
• Type of reservoir drive
• Shape of reservoir
• Size of reservoir
• Structural/stratigraphic trap mechanism
Saturation Determination
• Most of the petrophysical numbers (a,m,n) traditionally used in log analysis 
are derived empirically, usually from core and log comparisons. Although 
core data remain the bedrock of petrophysics, core analysis, as well as all 
other analyses and techniques discussed previously in this text, also has its 
imperfections. 
• Numerous advantages and disadvantages of log data have been discussed. 
Limitations of different measuring systems have been addressed, and the 
interpretative efforts applied to many measurements have been openly 
discussed not only for applicability but also for their limitations or inability 
to resolve certain formation evaluation problems. The Archie equation, at 
least three resistivity ratio saturation methods, and two shaly-sand 
saturation techniques were discussed earlier. The test of time has shown 
Archie's formula to stand up well against numerous assaults from the 
formation evaluation discipline.
• Several authors have suggested more laboratory modelling and less 
empirical content. Numerous technical papers have addressed the problem 
of shaliness with shaly-sand saturation equations, whereas only five of them 
are commonly addressed throughout the industry. All revert back to 
Archie's relation when shale content is zero. Attempts to characterize free 
and bound fluids to more accurately estimate total effective porosity, and to 
get a quantitative estimate of producible fluids or gases continue. 
Saturation Determination
Saturation Determination
• Several methods of determining saturation from crossplots or quick-
look methods are discussed in this presentation. 
• These methods also have inherent problems and are affected by 
different circumstances. 
• Quick-look methods of one type may work well in one area but fail 
miserably in another.
• in some cases, performance varies from reservoir to reservoir. Again, 
knowledge of local conditions is important.
• Profiles comparing flushed-zone saturation to virgin-zone saturation 
are very effective in most cases, but their accuracy diminishes with 
decreasing fluid salinities. 
• Nevertheless, movable and residual oil volumes calculated from such 
methods have been amazingly accurate in many controlled studies. 
Saturation Values
• Log data has been conventionally calculated as a percent water in the available 
pore space of a volume of rock. Water saturation (Sw) is a convenient log 
calculation because resistivity devices respond primarily to the conductive 
fluids (water) in pore space. 
• Obviously, water saturation can never exceed 100%; however, the nature of 
different log responses used in the calculations causes some statistical 
fluctuation.
• Vertical resolution and the horizontal investigation distance are somewhat 
different for each resistivity device, and the tools are affected differently by 
borehole size, borehole salinity, and several rock characteristics. 
• Nevertheless, valid log interpretations always reflect water saturations within 
acceptable limits of 100% in known water-bearing horizons.
• At the other extreme, water saturations are probably never at 0%; in fact, 
when Sw < 10%, and accurately determined, the oil in place is virtually never 
producible by conventional well extraction methods. 
Saturation Values
• Accepting these premises, effective log analysis should result in Sw of 
about 10% minimum to 100% maximum. Later in this presentation, 
irreducible water saturation (minimal Sw for a specificreservoir) and 
critical water saturation (lowest Sw at which water influx will occur) 
are discussed. 
• Each reservoir has its own unique identity and is affected by many of 
the variables listed earlier. 
• It is true that many reservoirs have identical characteristics, but 
virtually no two reservoirs can be classified as identical twins. 
• For example, one shale-free carbonate reservoir may have porosity 
and permeability values similar to those of another shale-free 
carbonate rock, but the two reservoirs may have different irreducible 
water saturations (Sir) because of differing Rw values or grain sizes. 
• The numerous variables that affect saturation make it virtually 
impossible to have totally identical characteristics in different 
reservoirs.
TAKING A CLOSER LOOK AT SATURATION 
IMPONDERABLES
• Data acquisition and the methods used to compile and eventually interpret the 
data are less than perfect. On the other hand, it is somewhat amazing that such 
measurements can even be made in the subsurface. 
• Recognize the strength of the downhole information acquired, but always be 
aware that it is difficult to provide 100% quantitative interpretation accuracy. 
• The material in this text is purposely organized to build gradually through 
interpretation steps that provide the data needed to calculate saturation. Of 
course, all data must be relatively accurate if accurate saturation values are to 
be determined. 
• Common sense dictates that when a log analysis is performed from acceptable 
data and several depth levels are calculated at » 120% Sw, one or more of the 
terms in the saturation equation must be in error. 
• The analyst must then backtrack and investigate the accuracy of individual 
terms. For example, Rwmay not have been corrected to formation 
temperature, Rt value may be incorrect, or an assumed petrophysical exponent 
may be incorrect, etc.
• Another situation can occur when several depth levels calculate as 70% Sw in 
known water-bearing horizons. Once the sources of error are found and 
corrected, Sw can again be calculated. 
Saturation Behavior to Reservoir Variables
• Most oil reservoirs are considered water wet; the surfaces of the rock grains 
are coated with a film of water although most pore throat volume is occupied 
by oil. Assume the film of water is uniform in thickness throughout the oil-
bearing portion of the reservoir. 
• Saturation may still vary because of grain size changes, which, in turn, changes 
the amount of surface area coated by the film of water. 
• Perhaps the film of water and grain size remain uniform throughout the 
reservoir but some isolated pores occur. Those isolated pores may contain only 
water because oil was never able to migrate into them. Again, Sw will vary. 
• Some pore throats may be more restrictive than others. Restrictions can be 
caused by clays, fines migration, or in-situ pressure differentials that, in turn, 
affect petrophysical parameters. 
• The a, m, and n values vary somewhat from level to level because of the 
heterogeneous nature of the rock.
• All these important imponderables cannot be accounted for with log analysis 
alone. The most sophisticated computer-processing routines do not normally 
attempt to cope with the petrophysical changes that occur within a reservoir, 
but instead use average values to best estimate saturation results. 
• Core analysis does not totally resolve the problem; plugs from a full core do not 
describe all the rock, but only the portion represented by the plug. 
Furthermore, the core is no longer part of the reservoir. 
• Nevertheless, a piece of the rock and controlled laboratory measurements of it 
are the accepted benchmark of petrophysics. 
• Heterogeneity in reservoir rocks is common and occurs laterally and vertically. 
A value for m or n might vary from one depth level to the next and might have 
considerable variation in value, but most analyses are based on an average 
value. 
• The analyst must be continually aware of such variables and how a change in 
any of the terms used in an equation will affect the resultant saturation 
calculation.
Saturation Behavior to Reservoir Variables
SATURATION IN CLEAN FORMATIONS
• Archie's fundamental saturation experiments were discussed previously. 
Development of the formation factor, the link between resistivity and rock 
porosity, was also discussed along with the development of the Humble 
relationship for sandstones. 
• These fundamental relations generally hold true if the reservoir is shale free 
and contains homogeneous intergranular porosity. However, all clean 
formations are not homogeneous, nor do all clean formations have 
intergranular porosity. 
• This disturbs the simplicity of the basic relations; i.e., parameters do not 
accurately fit a set rule for calculating saturations. 
• Nevertheless, most log analysis uses parameters generally prescribed for 
sandstones or carbonates, and saturation results are satisfactory in most cases. 
• It is not quite accurate to say that unusual rock properties occasionally occur, 
because they probably occur more often than not. 
• A computed saturation profile that demonstrates a sudden Sw change within a 
reservoir's productive zone is often shown little regard although it indicates 
changes in reservoir characteristics. 
• The anomalous behavior of the profile is often explained as a change in the 
invasion profile when it may be a change in m or n caused by differences in 
grain size, sorting, lithology mixtures, cementation, wettability, porosity type, 
permeability, etc.
Formation heterogeneity often affects saturation 
profiles
Archie Equation
• Archie's saturation equation is the basis for essentially all saturation 
determination methods: 
• Where:
• Sw = calculated water saturation percentage,
• n = saturation exponent, 
• Rw = formation water resistivity at formation temperature,
• Rt = the true formation resistivity,
• Ro = representative of true resistivity if the formation is 100% water 
bearing,and 
• F = formation resistivity factor.
• F is typically taken from calculated log values or measured core porosity and 
related to resistivity as follows: F = a/φφφφm
• Where
• a = numerator (considered to be related to tortuosity by some whereas others 
believe it to represent a unit volume of rock and its constituent pore space),
• φφφφ=determined porosity value,and
• m = representative of pore shape or geometry.
Archie Equation
• A similar saturation expression can be written for the flushed zone:
• Where:
• Rmf = mud-filtrate resistivity at formation temperature,
• Rxo = flushed-zone resistivity usually determined with a 
microresistivity device, and
• Sxo = water saturation of the flushed zone.
• The accuracy of calculated saturation depends on the legitimacy of Rw
or Rmf , Rt or Rxo or Ro , and F. 
• For best results, input values should be determined from data 
corrected for borehole signal, bed thickness, invasion, etc. 
• It is also important that the correct resistivity tool is used to resolve 
saturation.
Archie Equation´s Inaccuracy
• As a demonstration, if Rt is near 2000 ohm-m, φφφφ < 1%, and F is 
assumed to be equal to 1/ φφφφ2, water saturation can be calculated at 
100% only if Rw = 0.20 ohm-m at formation temperature. 
• If Rw is given as any value < 0.2 ohm-m, saturation will calculate 
at < 100% water. 
• If Rw = 0.25 ohm-m, saturation will calculate at > 100% water (125%). 
Where is the most logical source of error? 
• Was Rt determined from a laterolog or induction device? 
• Was porosity determined from a single device, two devices, three 
devices, or compared to core? 
• Is Rw accurate, and what is the source of information? 
• Are the correct a, m, and n values imposed? 
• Each input value is subject to question. 
Archie Equation´s Inaccuracy
• In this example, the accuracy of any resistivity measurement in the 2000 ohm-
m range must be questioned. Perhaps φφφφ was determinedfrom ∆∆∆∆t only, and in 
rocks of this type, acoustic logs usually lose their ability to recognize pore space 
unless it is primary. If < 1% pore space is available, there is not much water to 
cause an accurate conductive response to deep-induction measurements. 
• In short, measurement accuracy deteriorates rapidly in these conditions, and 
the analyst must recognize that insofar as reserves are concerned, the result is 
possibly somewhat insignificant. 
• More porous intervals are the true quality check on log analysis. The same 
parameters (Rw = 0.2 ohm-m and Rt = 2000 ohm-m) with 10% pore space 
available are more significant to reserve totals. Again, using a = 1 and m = 2, 
Sw = 1% water in the pore space. 
• Not being a likely number, the analyst should again question the input terms. 
The deep laterolog device may have been a better choice for deep-resistivity 
determination, and density or density-neutron porosity values may be more 
accurate for porosity determination. 
• Most reservoir rocks have resistivities less than 500 ohm-m when Sw is low, and 
it therefore makes sense to check log data at the low end of the resistivity 
spectrum (Ro) if a water-wet zone is available. This opportunity is not always 
available, but use it when it presents itself. 
• Verify input values in as many ways as possible, and investigate other data that 
may provide information on the reservoir character. 
Saturation Determination
• Rwa Technique:
• A real-time Rwa curve has been available for more than 25 years. Knowledge of Rw
in certain reservoir rocks permits a quick comparison of that value to the recorded 
Rwa . When logging through a water-wet horizon, the Rwa value should be similar to 
the known Rw . If Rw is not known, the Rwa curve is often used to establish Rw for 
specific horizons if some or all the reservoir is believed to be 100% water bearing. 
Rwa is simply a mathematical rearranging of the Archie equation:
• If F = a/φm and Ro = F × Rw, then Rw = Ro / F. 
• If Rt > Ro, a similar calculation can be made but an apparent Rw will be calculated if 
the zone is not water bearing: 
• Rwa = Rt / F » Rind / F >> Rind / (1/φ2) >> Rwa = Rind * φφφφ2
• where F is determined from porosity-sensitive log data and the proper formation 
factor-to-porosity relationship. In sandstone reservoirs, the F = 0.62/φ2.15 (or 
F = 0.81/φ2) relationship is commonly input. Deep-induction values are generally 
used as the apparent Rt value. Porosity is often determined from acoustic ∆t , density ρb , or density-neutron crossplot data. 
• An Rwa >> Rw indicates a water saturation less than 100%. 
• Saturation can be calculated easily by using:
Saturation Determination
• Obviously, invasion must be sufficiently shallow such that the deep-resistivity 
measurement is not affected; porosity determination and the formation factor 
relationship must be relatively accurate. In addition, the following requirements 
are necessary in order to successfully implement continuously recorded Rwa
techniques:
1. Rw must be relatively constant or vary in a consistent and predictable manner over 
the interpreted depth intervals.
2. Lithology should be consistent, predictable, and known (sand-shale sequences are 
best).
3. Permeable horizons should be essentially shale free, or at worst, have similar 
shaliness characteristics.
4. Quick estimates of saturation can usually be made if the following Rw to Rwa
comparative values are used:
Quick estimates of saturation
• Sw (%)
• Rwa 2 times the value of Rw: 71
• Rwa 3 times the value of Rw : 57
• Rwa 4 times the value of Rw : 50
• Rwa 8 times the value of Rw : 35
• Rwa 16 times the value of Rw : 25
• Rwa 25 times the value of Rw : 20
• Rwa 40 times the value of Rw : 16
F I M