TY - JOUR
AU - Schiozer, Denis, J
AB - 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. 4D seismic, saturation and pressure estimation, petro-elastic inversion, petro-elastic model Nomenclature 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 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 reservoir simulation model. These data are critical in the reservoir 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 al2012). 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 considered different types of uncertainties distributions and compared the results considering the correlation of the parameters. Grana and Rossa (2010) proposed a methodology to integrate statistical 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 uncertainties 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 inversion. 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 properties allowing a more complete analysis of previous studies with various dynamic scenarios and levels of uncertainties in the reservoir. 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 inversion 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 reservoir 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 minimum and maximum 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 influential static variables are looked at more closely in the next steps. After choosing the two main static variables, another sensitivity 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 different dynamic scenarios mimicking possible reservoir locations (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. Figure 1. Open in new tabDownload slide Probabilistic 4D inversion algorithm, modified from Davolio et al (2012). Figure 1. Open in new tabDownload slide Probabilistic 4D inversion algorithm, modified from Davolio et al (2012). 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 represents the true earth model and is further described in the following 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 calculation 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 initial guess of the inversion estimates. At each loop, the optimization process searches for a direction with the minimum gradient, updating the estimates. Since only reservoir saturation 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: OF=||ΔIP(obs)ΔIS(obs)−ΔIP(sim)ΔIS(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 optimization process that come from a base simulation 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 initial 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): σref=∑i=1n(xi−xref)2M, 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 sandstone 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: IP=VPρB,3 IS=VSρB.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: VP=κRsat+43μRsatρB,5 VS=μRsatρB,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 sandstone 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+Sgρg,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 frequency 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: Ksat=K*+(1−K*Kq)2φKfl+1−φKq−K*Kq2,10 μsat=μ*,11 Where K*, Kq and Kfl are the bulk modulus of the porous rock frame, 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: K*=(ϕϕcKHM+43μHM+1−ϕϕcKq+43μHM)−1−43μHM,12 μ*=(ϕϕcμHM+z+1−ϕϕcμq+z)−1−z,13 where: z=μHM6(9KHM+8μHMKHM+2μHM),14 KHM and μHM are the bulk and shear moduli at critical porosity øc according to the Hertz–Mindlin theory, given by: KHM=(n2(1−ϕc)2μmin218π2(1−v)2peff)13,15 μHM=5−4v5(2−v)(3n2(1−ϕc)2μmin22π2(1−v)2peff)13,16 v is the Poisson ratio and Peff is the effective pressure. Wang (2000) defines: peff=Pover−ηP,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 equation (10) is estimated by Woods´s Law, given as: Kfl=(SwKw+SoKo+SgKg)−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 KR=1(NTGKsat+(1−NTG)Kc),20 μV=μsatNTG+μc(1−NTG),21 μR=1(NTGμsat+(1−NTG)μc),22 where the subscript ‘V’ refers to the Voigt upper bound and the subscript ‘R’ to the Reuss lower bound. Kc is the clay bulk modulus. In conclusion, the values of KRsat and μRsat ⁠, required in equations (5) and (6), are given by: KRsat=KV+KR2,23 μRsat=μV+μR2,24 Table 1 presents all PEM primary variables as well as its ‘minimum’, ‘maximum’ and ‘reference’ values used in the sensitivity analyses. 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 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 Open in new tab 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 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 Open in new tab 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. 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 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 Open in new tab 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 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 Open in new tab Model description The application of cases 2 and 3 are done in a reservoir model named BETA. It has a reference simulation model that represents 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. Figure 2. Open in new tabDownload slide Reference model, layer five. Figure 2. Open in new tabDownload slide Reference model, 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 monitor 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 simulation 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 generated 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. Figure 3. Open in new tabDownload slide Delta impedances, considering differences between monitor and base surveys. Figure 3. Open in new tabDownload slide Delta impedances, considering differences between monitor and base surveys. All BETA model results are presented in layer 5. Layer 5 was chosen because every well used in the model had one simulation block completed in this layer with prominent floods and depletions occurring here. Regarding geology, the BETA model consists of two lithofacies (figure 2(c)). One lithofacie has the characteristics of clean sandstone rock, lithofacies 1 with initial water saturation of 0.15, and the other lithofacies possess characteristics of shale sandstone rock, lithofacies 2 with initial water saturation 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 estimations 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 variables represented in three geostatistical images taken as examples. Figure 4. Open in new tabDownload slide Uncertain base variables represented in three geostatistical images taken as examples. Figure 4. Open in new tabDownload slide Uncertain base variables represented in three geostatistical images taken as examples. To get a general idea of how well the base models represents the behavior of the reference model, figure 5 was created. 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. Figure 5. Open in new tabDownload slide Flux simulator, base models versus reference model. Figure 5. Open in new tabDownload slide Flux simulator, base models versus reference model. Results Sensitivity analyses The results of the sensitivity analysis allow us to see the most influential PEM static variables; these are presented in the tornado 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 variables in the IP and IS calculations. Figure 6. Open in new tabDownload slide Tornado graph, influence of the static variables on the impedances calculation. Figure 6. Open in new tabDownload slide Tornado graph, influence of the static variables on the impedances calculation. 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 identify if a specific parameter decreases or increases the impedances 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. Figure 7. Open in new tabDownload slide Dynamic sensitivity analysis. Figure 7. Open in new tabDownload slide Dynamic sensitivity analysis. Considering the mentioned production strategy, positive IP anomalies in water-invaded zones are expected according to figure 7, considering monitor minus base (for more 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 conditions to capture 4D anomalies, since its impedance variations 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. Figure 8. Open in new tabDownload slide 4D sensitivity analysis. Figure 8. Open in new tabDownload slide 4D sensitivity analysis. 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 variation using different values of static variables is associated with a dynamic scenario. Bigger dynamic changes mean more differences in the impedance variations, because the different values of statics variables perturb time zero impedance calculations 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 inversion gives the exact answer (Davolio et al2012). Another feature observed is that the same magnitude of error in the uncertain base variables impacts the inversion estimates differently, 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). Figure 9. Open in new tabDownload slide Case 1 results. 4D petro-elastic inversion considering errors in uncertain base variables under different dynamic scenarios. Figure 9. Open in new tabDownload slide Case 1 results. 4D petro-elastic inversion considering errors in uncertain base variables under different dynamic scenarios. 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). 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 saturation changes and small pressure changes (S1 and S2) are, again, more affected. Note that errors in initial water saturation 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 compares 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 saturation and pressure trends. Only the magnitude of the dynamic changes is different in some areas. Figure 10. Open in new tabDownload slide Reference maps of ΔSw (a) and ΔP (c) values. Mean of all 125 ΔSw (b) and ΔP (d) estimates (case 2). Figure 10. Open in new tabDownload slide Reference maps of ΔSw (a) and ΔP (c) values. Mean of all 125 ΔSw (b) and ΔP (d) estimates (case 2). 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 associated 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 variables are closer to the reference value, so, there is not as much error associated. Figure 11. Open in new tabDownload slide Standard deviation around reference maps of uncertain base variables. Figure 11. Open in new tabDownload slide Standard deviation around reference maps of uncertain base variables. 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 reference 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. Figure 12. Open in new tabDownload slide Case 2 results. σref maps of estimated ΔSw (a) and ΔP (b) values. Figure 12. Open in new tabDownload slide Case 2 results. σref maps of estimated ΔSw (a) and ΔP (b) values. The σref values of the ΔSw estimates in area 1 (figure 12(a)) are lower, even with big errors associated in this area (figures 11(a) and (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 porosities are lower than the reference value, two critical errors to ΔSw estimation according to figure 9. Figure 13. Open in new tabDownload slide Uncertain base variables in block A. Figure 13. Open in new tabDownload slide Uncertain base variables in block A. Figure 14 shows the distribution of the estimated ΔSw and ΔP for block A. For the estimated ΔSw, the values are separated 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. Figure 14. Open in new tabDownload slide Estimated ΔSw and ΔP values in Block A. Figure 14. Open in new tabDownload slide Estimated ΔSw and ΔP values in Block A. 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 (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 variables, the trends of saturation changes stay the same as in the reference maps. Figure 15. Open in new tabDownload slide Mean of all 125 ΔSw and ΔP estimates achieved inverting the noisy seismic impedances. Figure 15. Open in new tabDownload slide Mean of all 125 ΔSw and ΔP estimates achieved inverting the noisy seismic impedances. 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 properties, 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. Figure 16. Open in new tabDownload slide σref maps of estimated ΔSw (a) and ΔP (b) values of case 3, which considers seismic noise. Figure 16. Open in new tabDownload slide σref maps of estimated ΔSw (a) and ΔP (b) values of case 3, which considers seismic noise. Another reservoir model block is chosen to clarify how seismic noise hampered the inversion responses, indicated as point B in figure 16(a). 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 reference values; this is important because low porosity values can render the inversion process unstable. Figure 17. Open in new tabDownload slide Histograms of base variables used in the block B. Figure 17. Open in new tabDownload slide Histograms of base variables used in the block B. Even with a favorable characterization the inversion estimates 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 pressure trend is kept for this block. Figure 18. Open in new tabDownload slide Estimated ΔSw and ΔP values in the block B. Figure 18. Open in new tabDownload slide Estimated ΔSw and ΔP values in the block B. 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. Figure 19. Open in new tabDownload slide Estimated versus reference values, CASE 2 and 3. Figure 19. Open in new tabDownload slide Estimated versus reference values, CASE 2 and 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 inversion estimates are in each case. The cloud in case 2 is more 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 proposed by Davolio et al (2012), combined with a probabilistic approach, is used to show how errors in some reservoir properties 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 probabilistic 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 saturation 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, estimations 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 associated with dynamic changes. The two areas selected showed the relationship between dynamic changes and reservoir property 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 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 estimation 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 inversion 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 estimates than reservoir property errors. It happened that, in some areas, seismic noise led the ΔSw estimates to predict saturation 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 estimates was not completely associated with the dynamic changes. Thus, inversion mismatches can occur where the reservoir characterization 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 , Alvarado V . , 2006 Sensitivity analysis of Gassmann’s fluid equations: some implications in feasibility studies of time-lapse seismic reservoir monitoring , J. Appl. Geophys. , vol. 59 (pg. 47 - 62 ) 10.1016/j.jappgeo.2005.07.006 Google Scholar Crossref Search ADS WorldCat Crossref Batlze M , Wang Z . , 1992 Seismic properties of pore fluids , Geophysics , vol. 57 (pg. 1396 - 1408 ) 10.1190/1.1443207 Google Scholar Crossref Search ADS WorldCat Crossref Davolio A , Maschio C , Schiozer D . , 2012 Pressure and saturations estimation from P and S impedances: a theoretical study , J. Geophys. Eng. , vol. 9 (pg. 447 - 460 ) 10.1088/1742-2132/9/5/447 Google Scholar Crossref Search ADS WorldCat Crossref Gassmann F . , 1951 Uber die Elastizitat poroser Medien , Mitteiluagen aus Inst. fur Geophysik (Zurich) , vol. 17 (pg. 1 - 23 ) 2007 On elasticity of porous media Classics of Elastic Wave Theory Pelissier M A . Tulsa, OK SEG (Engl. transl.) Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Grana D . , 2014 Probabilistic appoach to rock physics modeling , Geophysics , vol. 79 (pg. D123 - D143 ) 10.1190/geo2013-0333.1 Google Scholar Crossref Search ADS WorldCat Crossref Grana D , Rossa E . , 2010 Probabilistic petrophysical-properties estimation integrating statistical rock physics with seismic inversion , Geophysics , vol. 75 (pg. O21 - 37 ) 10.1190/1.3386676 Google Scholar Crossref Search ADS WorldCat Crossref Landro M . , 2001 Discrimination between pressure and fluid saturation changes from time-lapse seismic data , Geophysics , vol. 66 (pg. 836 - 844 ) 10.1190/1.1444973 Google Scholar Crossref Search ADS WorldCat Crossref Mavko G , Mukerji T , Dvorkin J . , 2009 , The Rock Physics Handbook: Tools for Seismic Analysis in Porous Media 2nd edn Cambridge Cambridge University Press Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Smith T , Sondergeld C , Rai C . , 2001 Gassmann fluid substitutions: a tutorial , Geophysics , vol. 68 (pg. 430 - 440 ) 10.1190/1.1567211 Google Scholar Crossref Search ADS WorldCat Crossref Souza M , Machado F , Muneratto P , 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 , Hatchell P . , 2014 Standards in 4D feasibility and interpretation , Leading Edge , vol. 33 (pg. 134 - 140 ) 10.1190/tle33020134.1 Google Scholar Crossref Search ADS WorldCat Crossref Wang Z . , 2000 The Gassmann equation revisited: comparing laboratory data with Gassmann’s predictions , Seismic and Acoustic Velocities in Reservoir Rocks, 3: Recent Developments wang Z , Nur A . Tulsa, OK SEG (pg. 8 - 23 ) Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Wang Z . , 2001 Fundamentals of seismic rock physics , Geophysics , vol. 66 (pg. 398 - 412 ) 10.1190/1.1444931 Google Scholar Crossref Search ADS WorldCat Crossref © 2015 Sinopec Geophysical Research Institute
TI - The impact of rock and fluid uncertainties in the estimation of saturation and pressure from a 4D petro elastic inversion
JF - Journal of Geophysics and Engineering
DO - 10.1088/1742-2132/12/4/686
DA - 2015-08-01
UR - https://www.deepdyve.com/lp/oxford-university-press/the-impact-of-rock-and-fluid-uncertainties-in-the-estimation-of-jKm4zfvCyN
SP - 686
VL - 12
IS - 4
DP - DeepDyve
ER -