UNIVERSIDADE ESTADUAL DE CAMPINAS SISTEMA DE BIBLIOTECAS DA UNICAMP
REPOSITÓRIO DA PRODUÇÃO CIENTIFICA E INTELECTUAL DA UNICAMP
Versão do arquivo anexado / Version of attached file:
Versão do Editor / Published Version
Mais informações no site da editora / Further information on publisher's website:
https://academic.oup.com/jge/article/12/4/686/5110768
DOI: 10.1088/1742-2132/12/4/686
Direitos autorais / Publisher's copyright statement:
©2015
by Oxford University Press. All rights reserved.
DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
CEP 13083-970 – Campinas SP Fone: (19) 3521-6493
Journal of Geophysics and Engineering
Nomenclature
Δ Time lapse difference API Degree API
BHP Bottom-hole pressure GG Gas gravity
IP P impedance
IS S impedance
K Bulk modulus
Kdry Effective bulk modulus of dry rock
Ksat Effective bulk modulus of the rock with pore fluid
μ Shear modulus
μdry Effective shear modulus of dry rock
μsat Effective shear modulus of the rock with pore fluid η Coefficient of internal deformation
NTG Net to gross OF Objective function
ρ Bulk density
ρfl Fluid density P Pore pressure
Peff Effective pressure Pover Overburden pressure
The impact of rock and fluid uncertainties
in the estimation of saturation and pressure
from a 4D petro elastic inversion
Bruno Pazetti, Alessandra Davolio, Denis J Schiozer and UNICAMP Department of Petroleum Engineering, FEM, UNICAMP/CEPETRO, São Paulo, Brazil
E-mail: [email protected], [email protected] and [email protected] Received 23 October 2014, revised 18 March 2015
Accepted for publication 9 June 2015 Published 26 June 2015
Abstract
The integration of 4D seismic (4DS) attributes and reservoir simulation is used to reduce risks in the management of petroleum fields. One possible alternative is the saturation and pressure domain. In this case, we use estimations of saturation and pressure changes from 4D seismic data as input in history matching processes to yield more reliable production predictions in simulation models. The estimation of dynamic changes from 4DS depends on the knowledge of reservoir rock and fluid properties that are uncertain in the process of estimation. This paper presents a study of the impact of rock and fluid uncertainties on the estimation of saturation and pressure changes achieved through a 4D petro-elastic inversion. The term impact means that the saturation and pressure estimation can be perturbed by the rock and fluid uncertainties. The motivation for this study comes from the necessity to estimate uncertainties in saturation and pressure variation to incorporate them in the history matching procedures, avoiding the use of deterministic values from 4DS, which may not be reliable. The study is performed using a synthetic case with known response from where it is possible to show that the errors of estimated saturation and pressure depend on the magnitude of rock and fluid uncertainties jointly with the reservoir dynamic changes. The main contribution of this paper is to show how uncertain reservoir properties can affect the reliability of pressure and saturation estimation from 4DS and how it depends on reservoir changes induced by production. This information can be used in future projects which use quantitative inversion to integrate reservoir simulation and 4D seismic data.
Keywords: 4D seismic, saturation and pressure estimation, elastic inversion, petro-elastic model
(Some figures may appear in colour only in the online journal)
B Pazetti et al
Printed in the UK 686
jge
© 2015 Sinopec geophysical Research Institute 2015 12 j. geophys. eng. jge 1742-2132 10.1088/1742-2132/12/4/686
Papers
4journal of geophysics and engineering IOP
Journal of Geophysics and Engineering
686
701
doi:10.1088/1742-2132/12/4/686 J. Geophys. Eng. 12 (2015) 686–701 doi:10.1088/1742-2132/12/4/686 J. Geophys. Eng. 12 (2015) 686–701
Rs Solution gas-oil ratio Sw Water saturation Swi Initial water saturation So Oil saturation S Salinity T Temperature Ø Porosity Øc Critical porosity VP P-wave velocity VS S-wave velocity Subscript
obs Attribute derived from the reference model sim Attribute derived from petro-elastic inversion
Introduction
During the lifetime of a hydrocarbon reservoir, the fluid flow simulator forecasts production data. Bottom-hole pressure, oil, water and gas production rates are estimated using a reser-voir simulation model. These data are critical in the reserreser-voir management process. Thus, the estimated financial return is closely linked to the reliability of the simulation model.
In general, observed well production data is used to update the simulation model through history matching processes. However, other complementary information can be used, such as 4D seismic data, to interpret saturation and pressure changes to improve the simulation model. There are different ways to estimate reservoir saturation and pressure changes from 4D seismic data in the literature, as authors have studied this subject from different approaches (Landro 2001, Souza
et al2010, Davolio et al 2012).
In some methods of reservoir property estimation a petro-elastic model is necessary. Thus, the model uncertainties are expected to affect the estimations. Artola and Alvarado (2006) discussed the sensitivity of seismic responses to uncertainties in the physical parameters of the reservoir rock. They consid-ered different types of uncertainties distributions and compared the results considering the correlation of the parameters. Grana and Rossa (2010) proposed a methodology to integrate statis-tical rock physics and Bayesian elastic inversion to compute the probability distributions of petrophysical properties. Grana (2014) presented a statistical method to analytically compute posterior probability functions. They showed applications in Gassmann’s equation as well as in other rock physics models; the methodology can be used to evaluate rock and fluid uncer-tainties associated with reservoir characterization.
This paper investigates the relationship between rock and fluid uncertainties with the estimations of reservoir saturation and pressure changes obtained through a 4D petro-elastic inver-sion. Firstly, we study the petro-elastic model (PEM) used to perform the inversion. Through sensitivity analyses, we show which PEM variables are the most influential and how they affect the impedances calculation under different dynamic changes. Secondly, we analyze how uncertainties in the PEM’s most influential variables affect the estimations of reservoir
saturation and pressure changes, assuming specific reservoir locations. Next, we apply the petro-elastic inversion throughout a reservoir model assuming uncertainties in the rock proper-ties allowing a more complete analysis of previous studies with various dynamic scenarios and levels of uncertainties in the res-ervoir. Finally, we add seismic noise to be inverted throughout the reservoir model to compare the impact of seismic noise and reservoir uncertainties with the inversion estimates.
This paper assumes the following:
• No presence of gas, the reservoir pressure is above the bubble point.
• Seismic impedances are synthetic, computed through a forward modeling; there is no elastic or acoustic inver-sion to compute seismic impedances.
• The generation of synthetic impedances is performed in the simulation scale to avoid upscaling and downscaling allowing a better quantification of the influence of rock and fluid uncertainties.
These assumptions provide a controlled case, with known response. We can then compare results with expected values to better understand the process for further studies of real cases.
Methodology Sensitivity analysis
The first step in studying the PEM is to identify which variables most affect its results. The PEM is a set of equations that link reservoir rock and fluid properties to seismic attributes, such as impedances. Therefore, there is a relationship between reser-voir rock and fluid properties, and the calculated impedances. However, it is difficult to study this relationship analytically.
This paper identifies all PEM input variables as primary variables such as; porosity, mineral bulk and shear modulus, temperature and fluid saturations. The primary variables that are considered unchanged in the reservoir lifetime are named static variables (e.g. porosity, mineral properties). The primary variables that do change in the reservoir lifetime are named dynamic variables (fluid saturation and pressure). To perform the sensitivity analysis, three different feasible values are defined for each primary variable. Two of those are the min-imum and maxmin-imum limits and the third, reference, defines an intermediate value between the minimum and maximum.
The relationship between static variables and calculated impedances is analyzed as following; firstly, a reference value of P and S impedances (IP and IS) is calculated using the PEM primary variables with their reference values. After that, each static variable is individually perturbed alternately using its minimum and maximum values. The variation in IP and IS due to these perturbations is measured by comparing these new values with the reference impedances. The two most influen-tial static variables are looked at more closely in the next steps. After choosing the two main static variables, another sensi-tivity analysis is performed to exhibit the trend of each dynamic variable to increase or decrease the impedances according to the variation. The static variables are kept constant with the reference values; while the pressure and saturation (the only
two dynamic variables) are varied one at a time. The twenty-one values within their minimum and maximum limits are used to calculate how each perturbation impacts the impedance calculation, this step is called dynamic sensitivity analysis.
The last step regarding the relationship between primary variables and calculated impedances is to develop one more sensitivity analysis-4D sensitivity analysis. The aim is to verify the impact of the two most influential static variables when reservoir saturation and pressure vary together. The 4D seismic data considered here encompass two seismic surveys: time zero (base survey) and time one (monitor survey).
To perform the 4D sensitivity analysis we create four dif-ferent dynamic scenarios mimicking possible reservoir loca-tions (far from/near to injectors/producers). In each dynamic scenario we compare how the impedances in time one varied from time zero in percentage because of the dynamic change. In this analysis, we also consider variations in the two most influential static variables.
Petro-elastic inversion
The 4D petro-elastic inversion proposed by Davolio et al (2012), figure 1, is used to estimate reservoir saturation and pressure at time one from seismic impedances. The estimated
values are then subtracted from the dynamic variables at time zero to calculate the inverted dynamic deltas ΔSw and ΔP. The
values of the dynamic variables at time zero are estimated by the flux simulator. The values of ΔSw and ΔP represent the
variation of water saturation and pore pressure between base and monitor seismic surveys.
This study uses synthetic seismic data that are generated through forward modeling using a petro-elastic model with the properties of a reference simulation model, which repre-sents the true earth model and is further described in the fol-lowing section. Note that any elastic inversion was used. The seismic impedances were calculated using the petro-elastic model with the properties of the reference simulation model as input. The green box to the left of figure 1 refers to the cal-culation of these synthetic impedances. The right side shows the optimization process. The inversion is carried out using a gradient-type optimization algorithm that starts with an ini-tial guess of the inversion estimates. At each loop, the opti-mization process searches for a direction with the minimum gradient, updating the estimates. Since only reservoir satura-tion and pressure at time one are estimated, all other variables are held constant during the inversion process. The inversion algorithm advances iteratively searching for the minimum of the objective function, which is given as:
Figure 1. Probabilistic 4D inversion algorithm, modified from Davolio et al (2012).
= ΔΔ ( )−ΔΔ ( ) ( ) ( ) IP IS IP IS OF obs , obs sim sim (1) where ΔIP and ΔIS are the time lapse difference of P and S impedances, respectively. The subscript ‘obs’ refers to the observed seismic data and the subscript ‘sim’ refers to the seismic data calculated inside the inversion procedure using the simulation data as input.
In the inversion algorithm of figure 1, the PEM is used to calculate the synthetic seismic impedances that represent the observed data and also to perform the optimization process, so no uncertainties in the modeling are considered. Concerning the PEM primary variables, the term reference is designated to the PEM primary variables used in the calculation of the synthetic impedances to be inverted, whereas the term base is designated to the PEM primary variables used in the optimiza-tion process that come from a base simulaoptimiza-tion model.
A base variable is considered uncertain when it is used with a different value than the reference value. Thus, the term error is used to mention how far the uncertain base variable is from the reference. The base variables, which are considered uncertain, are the two most influential static variables and ini-tial water saturation.
In this paper, the inversion is employed for three cases. • Case 1 (1 grid cell): in the first case, four synthetic ΔIP
and ΔIS pairs are calculated using four different dynamic scenarios, the same used in the 4D sensitivity analysis. A set of inversions is performed for every ΔIP and ΔIS pair. Each inversion using the uncertain base variables has different magnitudes of error. For every saturation and pressure estimate, we take the difference between the estimate and the reference values, which means the true answer. This study allows us to relate the inversion estimates and errors in the uncertain base variables within different possible dynamic changes faced in a single reservoir location. Note that this is possible because, as proposed by Davolio et al (2012), the optimization is performed for each reservoir block independently. • Case 2 (probabilistic full reservoir): in the second case, the
inversion is run for the whole reservoir model to provide a more general analysis of the several combinations of static and dynamic variables. As we want to consider errors in the uncertain base variables, for the second case, the inversion is performed several times. Each time considers a different base simulation model with the uncertain base variables generated through geostatistical techniques. This procedure is shown in red in the figure 1. After applying this procedure, every block in the simulation model has
M saturation and pressure estimated values. An indicator is used to measure the errors of the several estimations and the uncertain variables. This is defined as the sample standard deviation around the reference value (σref):
∑
σ = ( − ) = x x M , i n i ref 1 ref 2 (2)where xi is a sample of the desired base variable or inversion
estimate, xref is the reference value and M is the number of the
inversions.
• Case 3 (probabilistic full reservoir/noisy seismic data): in this case the inversion is run M times for the whole reservoir model, the same procedure used in case 2. This time, a random noise is added to the seismic impedances to be inverted. The noise added to the seismic impedances mimics real situations where the impedances values are not totally correct due to problems related to seismic signals such as seismic noise, modeling and errors caused by processing procedures.
Application
Petro-elastic model (PEM)
The PEM represents a reservoir rock comprised of clean sand-stone rock interspersed with shale. The PEM presented is a simplification of this composition, for a more accurate PEM see Mavko (2009).
The P (IP) and S (IS) impedances can be calculated as follows: ρ = IP VP B, (3) ρ = IS VS B. (4) In equations (3) and (4) ρB is the bulk density, i.e. the
density of the medium where the waves are propagating. VP
and VS are P and S wave velocities, respectively, and can be
defined as: κ μ ρ = + VP R , 4 3 B R sat sat (5) μ ρ = VS R , B sat (6) where the subscript ‘Rsat’ refers to the saturated reservoir rock
already mixed with the shale content. ρB is the bulk density
and can be written as:
ρB= ( −1 NTG)ρshale+ρsandNTG,
(7) where ρshale is the clay density, ρsand is the filled reservoir rock
density and NTG is the net-to-gross ratio. Since only the sand-stone rock contributes to production in our model ρsand can be
calculated as:
ρsand= ( − ) +1 ϕ ρq ϕρfl,
(8) where ø is the rock porosity, ρq is the quartz density and ρfl is
the density of the fluid mixture calculated by: ρfl=Soρo+Swρw+Sg ,ρg
(9)
So, Sw and Sg are the saturation of the oil, water and gas,
respectively. ρo, ρw and ρg are the pore fluid densities. These
densities can be achieved theoretically using the Batlze and Wang (1992) equations.
To compute VP and VS presented in equations (5) and (6),
it is also necessary to calculate the saturated reservoir rock bulk and shear modulus (i.e. KRsat and μRsat) mixed with the
shale content. It can be done in two steps, firstly ‘flooding’ the sandstone rock with a desired fluid mixture using the low fre-quency Gassmann’s equation (Gassmann 1951), then, mixing it with clay content using the Voigt–Reuss–Hill average.
The Gassmann’s equation allows the calculus of different pore fluid scenarios for the same reservoir rock, solving the fluid substitution problem. As stated in Smith (2001) Gassman’s equation can be written as:
( )
= + − + − φ φ * * − * K K 1 , K K K K K K sat 2 1 q fl q q2 (10) μsat= *,μ (11) Where K*, Kq and Kfl are the bulk modulus of the porous rockframe, the quartz and the pore fluid mixture. And μ* is the
shear modulus of the porous rock frame. To calculate K* and
μ* the Hertz–Mindlin contact theory and a heuristic modified
Hashin–Strikman lower bound were used, according to the equations: ⎛ ⎝ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟⎟ μ μ μ = + + − + − ϕ ϕ ϕϕ * − K K K 1 4 3 , HM 43 HM q 43 HM 1 HM c c (12) ⎛ ⎝ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟⎟ μ μ μ = + + −+ − ϕ ϕ ϕ ϕ − * z z z 1 , HM q 1 c c (13) where: ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ μ μ μ = ++ z K K 6 9 8 2 , HM HM HM HM HM (14)
KHM and μHM are the bulk and shear moduli at critical porosity øc according to the Hertz–Mindlin theory, given by:
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ϕ μ π = ( − ) ( − ) K n v p 1 18 1 e , HM 2 c2 min2 2 2 ff 1 3 (15) ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ μ ϕ μ π = 5( − )−4vv n ( − )( − )v p 5 2 3 1 2 1 , HM 2 c2 min2 2 2 eff 1 3 (16)
v is the Poisson ratio and Peff is the effective pressure. Wang
(2000) defines:
η
= −
peff Pover P,
(17) where Pover is the overburden pressure and P is the pore
pressure. The scalar η is called ‘effective-stress coefficient’,
Mavko (2009). Its value is less than one, showing that some energy can be lost when pore pressure opposes the overburden pressure (Wang 2001).
The bulk modulus of pore fluid mixture used in equa-tion (10) is estimated by Woods´s Law, given as:
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = + + − K S K S K S K , fl w w o o g g 1 (18)
Kw, Ko and Kg are the bulk modulus of the fluids water, oil and
gas, respectively. These values as well as the densities of the fluids can be calculated using the equations given by Batlze and Wang (1992).
Therefore, to calculate KRsat and μRsat, the Voigt–Reuss–Hill
average is used, mixing the saturated sandstone rock and the clay content weighted by the NTG value. To use the Voigt– Reuss–Hill average, first it is necessary to calculate their upper and lower bounds. Applied to the bulk and shear modulus it can be written as:
= + ( − ) KV KsatNTG Kc 1 NTG , (19)
(
)
= +( − ) K 1 , K K R NTG 1 NTG sat c (20) μV=μsatNTG+ ( −μc1 NTG ,) (21)(
)
μ = + μ ( −μ ) 1 , R NTG 1 NTG sat c (22) where the subscript ‘V’ refers to the Voigt upper bound and the subscript ‘R’ to the Reuss lower bound. Kc is the clay bulkmodulus. In conclusion, the values of KRsat and μRsat, required
in equations (5) and (6), are given by:
= + K K K 2 , Rsat V R (23) μRsat = μV+2 μR, (24) Table 1 presents all PEM primary variables as well as its ‘minimum’, ‘maximum’ and ‘reference’ values used in the sensitivity analyses.
The dynamic scenarios considered in the 4D sensitivity analysis represent specific reservoir locations between an injector and producer well from the simulation model described in the following sections. Scenario S1 is closer to an injector well whereas scenario S4 is closer to a producer well. The other scenarios are placed within S1 and S4. They are presented in table 2.
Model description
The application of cases 2 and 3 are done in a reservoir model named BETA. It has a reference simulation model that repre-sents its perfect characterization. Beyond validating the results, this model generates the synthetic impedances to be inverted. The reference and base models are created using a Cartesian grid with dimensions 90 × 110 × 9 blocks, of which 41085 are active. Each block measures 60 × 60 × 6.67 m. Figure 2
shows the reference maps of NTG, porosity and initial water saturation in layer five.
Two sets of synthetic impedances are calculated through forward modeling using a petro-elastic model, the reference model properties are input, representing the base and mon-itor survey. The time difference between the two surveys is five years, in which the reservoir was drained by 11 producer and 8 water injector wells. Figure 3 shows the calculated ΔIP and ΔIS, comparing the impedance maps with and without seismic noise. The noise is generated through a Gaussian sim-ulation process to create random but horizontally correlated values. The goal is to distort the 4D anomalies so that we lose some detail but keep the main features. The noise is gener-ated through a normal distribution centered at 0, with standard deviation of 25% of the average of the anomalies observed in ΔIP and 30% of the average of the anomalies observed in ΔIS.
All BETA model results are presented in layer 5. Layer 5 was chosen because every well used in the model had one sim-ulation block completed in this layer with prominent floods and depletions occurring here.
Regarding geology, the BETA model consists of two litho-facies (figure 2(c)). One lithofacie has the characteristics of clean sandstone rock, lithofacies 1 with initial water satura-tion of 0.15, and the other lithofacies possess characteristics of shale sandstone rock, lithofacies 2 with initial water satura-tion equals 0.35.
The inversion boundaries defined for case 2 are based on lithofacies 1. The lower saturation limit is defined by its initial water saturation, of 0.15. The upper saturation defined limit
is based on the residual oil saturation of lithofacies 1. Since the strategy of production in the BETA model does not allow that gas might be liberated from oil, only water and oil phases are considered. Thus, the maximum possible water saturation is 1 − Sor. Based on this, the upper water saturation limit in
lithofacies 1 became equal to 0.9. For the pore pressure esti-mations the lower and upper pressure limits are defined based on the minimum and maximum pore pressure values present in the simulation results of the reference model. These values are 23.9450 MPa and 33.1954 MPa, for the minimum and maximum, respectively.
All base variables are necessary to start the inversion algorithm. In this paper the uncertain base variables come from different simulation models that represent the different possible representations of the reference model. Thus, 125 base models are generated with different uncertain base variable distributions provided by geostatistical images. All base variables that have values equal to the reference values are considered certain. The uncertainties are described in the next section. Figure 4 shows the uncertain base vari-ables represented in three geostatistical images taken as examples.
To get a general idea of how well the base models repre-sents the behavior of the reference model, figure 5 was cre-ated. This figure presents the oil and water production curves as well as pressure and water injection curves. The red lines represent the base models responses, whereas the green line represents the reference model response. It is possible to note that many qualities of representation are achieved. However, the reference response is within the representations.
Results
Sensitivity analyses
The results of the sensitivity analysis allow us to see the most influential PEM static variables; these are presented in the tor-nado graph (figure 6). In this figure the importance of each static variable follows the order top to bottom. It is apparent that NTG and porosity are the two most influential static vari-ables in the IP and IS calculations.
Still, in figure 6 every bar length shows how much each static variable impacts impedance calculation, and, the bar color indicates if the variable is being used in its minimum or maximum value. The minimum value is indicated in blue and the maximum value is indicated in green. We can iden-tify if a specific parameter decreases or increases the imped-ances values. For instance, when the NTG value increases the impedance values decrease.
Figure 7 shows the results of the sensitivity analysis of the PEM dynamic variables. It provides the base for 4D seismic data interpretation, giving an idea of what to expect from 4D seismic impedances for the sand reservoir considered, in which water is injected to maintain the pore pressure above the bubble point.
Considering the mentioned production strategy, positive
IP anomalies in water-invaded zones are expected according to figure 7, considering monitor minus base (for more
Table 2. Dynamic scenarios considered.
Scenario
Time 0 Time 1 Δ
Sw P [MPa] Sw P [MPa] Sw P [MPa]
S1 0.15 32.27 0.85 30.27 0.7 − 2 S2 0.15 32.27 0.65 28.27 0.5 − 4 S3 0.15 32.27 0.45 26.27 0.3 − 6 S4 0.15 32.27 0.25 24.27 0.1 − 8
Table 1. Primary variables used in the PEM. Variable Min Reference Max Unit Type
Sw 0.15 0.5 0.85 Dynamic P 22 32 42 MPa ρ (quartz) 2.62 2.65 2.67 g cm−3 Static μ (quartz) 44 45 45.6 GPa к (quartz) 36.5 37 37.9 GPa ρ (clay) 2.55 2.6 2.65 g cm−3 μ (clay) 7 8 9 GPa к (clay) 21 23 25 GPa Ø 0.2 0.25 0.3 v/v Øc 0.36 0.4 0.4 v/v NTG 0 0.5 1 η 0.9 0.95 1 T 70 75 80 °C API 24 28 32 Rs 50 100 150 GG 0.62 0.64 0.68 L/L S 0.044 0.056 0.067 Fraction of one p over. 44 55 66 MPa
information about standards see Stammeijer and Hatchell
2014). This behavior is expected because in the water-invaded zones the water saturation is supposed to increase, increasing the values of IP in the monitor survey. If the pore pressure decreases in this area it would reinforce the effect of water saturation on increased IP. If pore pressure increases in this area the effect of decreasing the IP values would be lower than the effect of water saturation of increasing IP.
Regarding the S impedance, the anomalies sign would depend on whether the pore pressure in water-invaded zones goes up or down because the effect of pore pressure is much more prominent on IS than IP. If the pore pressure decreases it would generate positive IS anomalies and if the pore pressure increases it would generate negative IS anomalies. So, the IS anomalies can be a good indicator of what is happening with the reservoir pressure.
Figure 8 shows the 4D sensitivity analysis in each dynamic scenario (table 2). As expected, each scenario has different impedance variations. For example, S1 shows the best con-ditions to capture 4D anomalies, since its impedance varia-tions are higher. If a 4D feasibility study was prepared for scenario S1 and the criterion of acceptance for a new seismic acquisition had more than 6% of IP variation the criterion of acceptance would only be met when the NTG value was 0.9. However, considering a reservoir location with a lower NTG, such as 0.5, the new seismic acquisition would not have the expected response.
The impedance variations among the different values of static variables within each dynamic scenario in figure 8 are different. This is because the effect on the impedance vari-ation using different values of static variables is associated with a dynamic scenario. Bigger dynamic changes mean more
Figure 2. Reference model, layer five.
Figure 3. Delta impedances, considering differences between monitor and base surveys.
Figure 4. Uncertain base variables represented in three geostatistical images taken as examples.
Figure 5. Flux simulator, base models versus reference model.
differences in the impedance variations, because the different values of statics variables perturb time zero impedance calcu-lations differently than those in time one. So, the perturbation of different static values is not completely compensated when considering the time difference.
Petro-elastic inversion
In the next topics the two most influential static variables in the IP and IS calculation, NTG and porosity, are considered uncertain. The initial reservoir saturation is also an important static property considered as uncertain, as will be clarified in the case 1 results below. Thus, the inversions performed (case 1 to case 3) considered these three properties as uncertain in the process.
Case 1. Figure 9 shows the results for case 1. When no errors are assumed (error = 0 in the x-axis), the inversion yields the
correct value, showing that for an ideal case (no errors in rock/ fluid properties and in the petro elastic modeling) the inver-sion gives the exact answer (Davolio et al 2012). Another feature observed is that the same magnitude of error in the uncertain base variables impacts the inversion estimates dif-ferently, depending on the dynamic scenario. For example, the same error magnitude in NTG of 0.1 impacts the ΔSw
estimates more in scenarios with big saturation changes and small pressure changes (S1 and S2). However, in these same scenarios the ΔP estimates are less affected. The ΔP estimates are more affected in scenarios with small saturation changes and big pressure changes (S3 and S4).
Errors in porosity show the same behavior observed for NTG, for the ΔSw estimates and positive porosity error values.
However, this behavior does not occur when the porosity error is negative, where the scenario with the lowest saturation change (S4) has more errors in ΔSw estimations than that with
the biggest saturation change (S1).
Figure 6. Tornado graph, influence of the static variables on the impedances calculation.
Figure 7. Dynamic sensitivity analysis.
Errors in initial water saturation impact the ΔSw estimates
much more than the ΔP estimates; compare figures 9(e) and (f). In this situation it is clear that the scenarios with big satu-ration changes and small pressure changes (S1 and S2) are, again, more affected. Note that errors in initial water satura-tion cause the biggest impact on ΔSw estimates compared with
errors in NTG and porosity.
Case 2. The first result of the inversion applied throughout the BETA model (case 2) is presented in the figure 10. It com-pares the reference maps of ΔSw and ΔP with the mean of all
125 ΔSw and ΔP estimates. Figure 10 shows that the mean of
all 125 ΔSw and ΔP estimates captures the reservoir
satura-tion and pressure trends. Only the magnitude of the dynamic changes is different in some areas.
Note how the dynamic changes occurred in the BETA model from base to monitor survey because, as shown in figure 9, reservoir locations with high dynamic changes tend to be more sensitive to errors in static properties. The left side of figure 10 shows that in the case of the saturation changes, the areas with big variations are around the injector wells in the water invaded zones; in the rest of the model there are no saturation changes. There is a confined pressure area; colored in red in the ΔP maps, on the bottom of the model. There, the pressure changes do not reach less than −2 MPa, outside this area the pressure changes are lower (the magnitude of the change is bigger).
The only cause of inversion mismatches for this case is the use of uncertain base variables. Each simulation block has 125 values of each base variable considered uncertain (NTG,
porosity and initial water saturation). Figure 11 shows the variability around the reference values, σref, of the uncertain
base variables throughout layer 5 of the BETA model. The higher the σref value, the higher the magnitude of errors
asso-ciated with each simulation block, because more uncertain base variables are farther from the reference value. Two areas are selected to better understand the influence of errors on the inversion estimates. These areas are marked by two squares in figure 11. Area 1 presents high errors in the three uncertain base variables. Alternatively, area 2 presents low σref values.
Low σref values indicate that most of the uncertain base
vari-ables are closer to the reference value, so, there is not as much error associated.
Figure 12 shows the σref maps of the estimated ΔSw and
ΔP values for case 2. In these maps, high σref values indicate
that most of the inversion estimates are far from the refer-ence value suggesting errors in the estimations. It is clear that the areas with big errors of estimated ΔSw are placed
around the injector wells in the flooded zones, as expected from the sensitivity analysis previously presented. The areas with big errors of estimated ΔP values occur where there is high reservoir depletion. The areas where pressure is confined have lower estimated ΔP errors. Again, this occurs because of the magnitude of the pressure changes in each area. Where the pressure changes are lower the impact of errors in uncertain base variables is lower and where the pressure changes are higher the impact of wrong variables is higher.
The σref values of the ΔSw estimates in area 1 (figure 12(a))
are lower, even with big errors associated in this area (figures
Figure 8. 4D sensitivity analysis.
11(a)–(c)), confirming the results presented in figure 9. This is because in area 1 there is a small saturation change, so the ΔSw estimates are much less affected by errors in uncertain
base variables. In this same area the ΔP estimates are affected by the errors, as indicated by the big values of σref. This is due
to big pressure changes (see figure 10(c)).
Regarding area 2, the ΔSw estimates are affected even by
small errors in the uncertain base variables, because in this area a big saturation change occurred amplifying the impact of errors. In the case of the ΔP estimates the errors in the uncertain base variables are less prominent, because there were only small pressure changes. Another feature observed is that the map in figure 12(a) does not correlate with the error maps of figure 9. The opposite is observed for pressure as the data in figure 12(b) closely resembles the NTG errors (figure
9(a)); see, for instance, the high error zone between wells P4 and P8.
One block of the BETA model clarifies how the uncertain base variables affect the estimated values. The block chosen has a big σref value in the ΔSw estimates map and it is marked
in figure 12(a) as block A.
The distributions of each uncertain base variable at block A are plotted in figure 13. The red mark in the histograms shows the reference values. We see that most of the used base initial
water saturation values are wrong and most of the base porosi-ties are lower than the reference value, two critical errors to ΔSw estimation according to figure 9.
Figure 14 shows the distribution of the estimated ΔSw and
ΔP for block A. For the estimated ΔSw, the values are
sepa-rated into two groups. The number of responses in the group on the left side is the same as the number of the inversions made with the wrong initial water saturation. This occurred because the impact of the error in initial water saturation on the estimated ΔSw values is big (see figure 9(e)), so, the
responses of this group of inversions are far from those that do not have errors in initial water saturation. In the group at the right the distribution reflects only that of the wrong NTG and porosity around the reference value.
The ΔP inverted values (figure 14(b)) follow a smooth distribution. This reflects the errors in NTG and porosity since errors in initial water saturation do not impact the ΔP estimates (see figure 9(f)). Even though the majority of ΔP inverted values achieved the reference value, there are many ΔP inverted values far from the correct answer, being the maximum difference of 5 MPa in modulus.
Case 3. Figure 15 shows the mean of all 125 ΔSw and ΔP
estimates achieved by inverting the noisy seismic impedances
Figure 9. Case 1 results. 4D petro-elastic inversion considering errors in uncertain base variables under different dynamic scenarios.
Figure 10. Reference maps of ΔSw (a) and ΔP (c) values. Mean of all 125 ΔSw (b) and ΔP (d) estimates (case 2).
Figure 11. Standard deviation around reference maps of uncertain base variables.
Figure 12. Case 2 results. σref maps of estimated ΔSw (a) and ΔP (b) values.
(case 3). Here, the inversion estimates can capture the main dynamic changes. However, in the saturation mean map some areas present saturation changes where they do not actually occur (compare figure 10(a) and (c) with figures 15(a) and (b)). The same is seen in the pressure mean map (figure
15(d)). The mean pressure map indicates, in some areas in the region with confined pressure, positive changes where in fact there is negative change. This does not happen in case 2 estimates, where despite errors in the uncertain base vari-ables, the trends of saturation changes stay the same as in the reference maps.
As with case 2 results, the σref maps of the estimated
ΔSw and ΔP are plotted to show the error of the inversion
responses around the reference values (figure 16). This time,
where seismic noise was considered, the error of the inverted responses increased throughout the model when compared withcase 2 results. Also, big values of σref occur in areas with
small dynamic changes, which does not happen in case 2. The map of errors in figure 16 presents strong marks that are related to the noise considered (compare with figure 3). The correlation between pressure estimation and the static proper-ties, especially NTG, observed for case 2 are not seen now in case 3. This indicates that the errors in the observed data are more problematic when estimating pressure and saturation than errors in the static properties.
Another reservoir model block is chosen to clarify how seismic noise hampered the inversion responses, indicated as point B in figure 16(a).
Figure 13. Uncertain base variables in block A.
Figure 14. Estimated ΔSw and ΔP values in Block A.
Figure 15. Mean of all 125 ΔSw and ΔP estimates achieved inverting the noisy seismic impedances.
The distribution of the uncertain base variables in block B are presented in figure 17. Note that this block presents a good characterization of the initial water saturation. Furthermore, NTG base values are near to the reference value and the majority of the base porosity values are higher than the refer-ence values; this is important because low porosity values can render the inversion process unstable.
Even with a favorable characterization the inversion esti-mates achieved in block B cannot forecast the reference values, figure 18. For the ΔSw estimates all inversions indicate
an increase in the amount of the water saturation, diverging from the correct value. In the case of the ΔP estimates the values are lower than the reference value. However, the pres-sure trend is kept for this block.
All results so far show the results achieved in layer five of the BETA model. To show how the inversion behaved in all blocks of BETA model, figure 19 was plotted. In all plots of figure 19 the x-axis represents the reference values of ΔSw and
ΔP; the y-axis represents the inverted values. Figures 19(a) and (b) show the results of case 2 and figures 19(c) and (d) the results of case 3.
The ideal relationship between inverted and reference values would be represented by a 45° curve, plotted as a green line in figure 19. The linear adjust (dashed black line) made from case 2 results, where no seismic noise is considered, is closer to the 45° curve than case 3 results. Furthermore, the points’ dispersion gives us an idea of how wrong the inver-sion estimates are in each case. The cloud in case 2 is more
Figure 17. Histograms of base variables used in the block B.
Figure 18. Estimated ΔSw and ΔP values in the block B.
Figure 16. σref maps of estimated ΔSw (a) and ΔP (b) values of case 3, which considers seismic noise.
concentrated than in case 3, as expected. This is because seismic noise leads the inversion to wrong predictions in many blocks in case 3, leading to incorrect interpretations of the reservoir dynamic trend in some blocks.
Conclusions
In this paper, the impact of rock and fluid uncertainties in the estimation of saturation and pressure variation from 4DS is analyzed. The deterministic petro-elastic inversion pro-posed by Davolio et al (2012), combined with a probabilistic approach, is used to show how errors in some reservoir prop-erties affect the inversion estimation for ΔSw and ΔP from 4D
seismic data for each block within a model. This procedure allows us to relate errors of reservoir properties to the proba-bilistic estimation for saturation and pressure.
NTG and porosity are the static variables that most affect PEM computations. The dynamic sensitivity analysis showed how IP and IS values are affected by changes in reservoir sat-uration and pressure separately. The 4D sensitivity analysis showed what happens to IP and IS variation due to changes in reservoir saturation and pressure assuming different static values for NTG and porosity.
Case 1 results consider different error magnitudes in NTG, porosity and initial water saturation in the inversion process under different dynamic scenarios. It showed that the
reservoir properties errors can impact differently on the ΔSw
and ΔP estimates; it also depends on dynamic changes. In our case, the estimation of ΔSw was more affected by errors
in dynamic scenarios with big saturation changes, and, the estimation of ΔP was more affected by errors in dynamic scenarios with big pressure changes. Considering case 1 as a possible configuration of locations for a single reservoir, we saw that the reservoir uncertainties did not prevent the inversion from capturing the main dynamic trends. However, depending on the magnitude of the uncertainties associated with the real dynamic change, the magnitude of changes was prevented from being precisely recovered. So, in an ideal case without seismic noise the qualitative interpretation of a 4D map would provide perfect information about where the dynamic changes occurred.
Regarding the inversion applied throughout the BETA model (case 2 results), as expected from case 1 results, esti-mations of ΔSw were more affected by errors in areas with big
saturation changes, and, estimations of ΔP were more affected by errors in areas with big pressure changes. Because of that, the errors in the inversion estimates in this case are associ-ated with dynamic changes. The two areas selected showed the relationship between dynamic changes and reservoir prop-erty errors. In area 1 (figure 11), where uncertainties occurs in NTG, porosity and initial water saturation the ΔSw estimates
are largely unaffected, because of small saturation changes in this area, and the ΔP estimates are affected because of big
Figure 19. Estimated versus reference values, CASE 2 and 3.
pressure changes in this area. In area 2, where NTG, porosity and initial water saturation are well characterized, the ΔSw
estimates are affected, because of big saturation changes, and the ΔP estimates are largely unaffected, because of small pressure changes.
There are two lithofacies with different initial values of water saturation in the BETA model. Errors in this parameter are critical to the estimations of ΔSw. So the ΔSw
estima-tion appeared divided between two groups in the histograms of inverted values. One group encompassed the estimations made with the right value of initial water saturation value and the other group, the estimations made with wrong values of initial water saturation. It is clear that one group of the ΔSw
estimates were made with the wrong characterization of the lithofacies. Since the two lithofacies are very different, a 3D reservoir characterization could be done to better understand the lithofacies occurrence, and so, better distribute the inver-sion estimates.
Case 3 results are important to bring a more realistic problem to the previous studies (cases 1 and 2) because an absolutely correct value of impedance is unfeasible in real cases. It showed that seismic noise can be more critical to the inversion esti-mates than reservoir property errors. It happened that, in some areas, seismic noise led the ΔSw estimates to predict
satura-tion changes where there were none. The same happened with the ΔP estimates to a lesser extent. Furthermore, in the case 3 results the variability around the reference to the inversion esti-mates was not completely associated with the dynamic changes. Thus, inversion mismatches can occur where the reservoir char-acterization is good and where small dynamic changes occur. Because of that, we can expect seismic noise to adversely affect the information about where the dynamic changes occur in real cases, leading to incorrect interpretations.
References
Artola V and Alvarado V 2006 Sensitivity analysis of Gassmann’s fluid equations: some implications in feasibility studies of time-lapse seismic reservoir monitoring J. Appl. Geophys. 59 47–62 Batlze M and Wang Z 1992 Seismic properties of pore fluids
Geophysics57 1396–408
Davolio A, Maschio C and Schiozer D 2012 Pressure and saturations estimation from P and S impedances: a theoretical study J. Geophys. Eng. 9 447–60
Gassmann F 1951 Uber die Elastizitat poroser Medien Mitteiluagen aus Inst. fur Geophysik (Zurich) 17 1–23; 2007 On elasticity of porous media Classics of Elastic Wave Theory ed M A Pelissier (Tulsa, OK: SEG) (Engl. transl.)
Grana D 2014 Probabilistic appoach to rock physics modeling Geophysics79 D123–43
Grana D and Rossa E 2010 Probabilistic petrophysical-properties estimation integrating statistical rock physics with seismic inversion Geophysics 75 O21–37
Landro M 2001 Discrimination between pressure and fluid saturation changes from time-lapse seismic data Geophysics 66 836–44 Mavko G, Mukerji T and Dvorkin J 2009 The Rock Physics
Handbook: Tools for Seismic Analysis in Porous Media 2nd edn (Cambridge: Cambridge University Press)
Smith T, Sondergeld C and Rai C 2001 Gassmann fluid substitutions: a tutorial Geophysics 68 430–40
Souza M, Machado F, Muneratto P and Schiozer D 2010 Iterative history matching technique for estimation reservoir parameters from seismic data Ann. Conf. and Exhibition (Barcelona, Spain, 14–17 June 2010) SPE131617 EUROPEC/EAGE Stammeijer J and Hatchell P 2014 Standards in 4D feasibility and
interpretation Leading Edge 33 134–40
Wang Z 2000 The Gassmann equation revisited: comparing laboratory data with Gassmann’s predictions Seismic and Acoustic Velocities in Reservoir Rocks, 3: Recent Developments ed Z wang and A Nur (Tulsa, OK: SEG) pp 8–23
Wang Z 2001 Fundamentals of seismic rock physics Geophysics 66 398–412
