Integration of Rock Physics Models in a Geostatistical Seismic Inversion for Reservoir Rock Properties

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1 Integration of Rock Physics Models in a Geostatistical Seismic Inversion for Reservoir Rock Properties Amaro C. 1 Abstract: The main goal of reservoir modeling and characterization is the inference of the spatial distribution of petrophysical properties of interest, such as facies, porosity, mineral volumes and fluids. Usually this is a two-step approach where the petrophysical properties of interest are derived from inverted elastic models. This sequential approach does not ensure the propagation of the uncertainty related to the seismic inversion problem into the resulting rock property models. This problem can be tackled by inverting the seismic reflection data directly to petrophysical properties (e.g. porosity, volume of shale and water saturation) ensuring the propagation of the uncertainty and measurement errors into the estimated subsurface models. The purpose of this work is to invert seismic reflection data directly to petrophysical properties, to properly propagate the uncertainty related to the seismic inversion problem and measurement errors into the estimated subsurface elastic models. It is presented a novel methodology that combines rock-physics models and a stochastic inversion with global perturbation method, that can quantify the relationship between geologic processes and the corresponding geophysical signatures. This method has been tested on real well-log data and partially stacked seismic data. The application to a real reservoir converges towards the real seismic data and provided realistic petrophysical models and facies volume with the corresponding elastic models retrieved from the rock-physics modeling process. Keywords: Geostatistical Seismic Inversion, Rock-Physics Models, Facies, Reservoir Modeling INTRODUCTION Reservoir s performance is directly related to the natural heterogeneities of the subsurface geology. Within the exploration and production stages, reservoir modeling plays a crucial role in the assessment of the productive zones. The main goal of reservoir characterization is to identify the spatial distribution of the petrophysical properties of interest, such as facies, porosity, mineral volumes and fluid. Frequently, petrophysical properties are derived from inverted elastic models in a two-step approach. When inverting seismic reflection data directly to petrophysical properties, the uncertainty related to the seismic inversion problem and measurement errors, is propagated into the estimated subsurface models. The integration of rock-physics modeling within the inversion loop allows linking the inverted subsurface rock properties with the corresponding elastic response. The inference of petrophysical properties is based on a perturbation technique that performs a stochastic sequential simulation (DSS; Soares 2001) and cosimulation with joint probability distributions (Horta and Soares 2010) of the model parameter space, ensuring the reproduction of the prior probability distributions, honouring the data values at each location, reproducing the original statistics (mean and variance), as well as reproducing the spatial continuity pattern imposed by the variogram model. Contrary to Sequential Gaussian Simulation (SGS; Deutsch and Journel 1998), the use of DSS allows the distribution of the property to be directly simulated without any transforms, as estimated from the experimental data (i.e. well-log data). The selection of the first property to simulate should consider the quality of the available well-log data. As a bestpractice, this property should be the one associated with a larger uncertainty and smoother. Each set of simulated and co-simulated petrophysical properties generate a facies volume with a Bayesian 1 Petroleum Engineering MSc Candidate, Center for Modeling Petroleum Reservoirs, CERENA/DECivil, Instituto Superior Técnico, Universidade Técnica de Lisboa, Lisbon, Portugal.

2 classification. A key input of the inversion for petrophysical properties and facies, are the prior facies proportions. These proportions are normally estimated from the available well-log data before the inversion procedure and propagate the corresponding uncertainty in zones away from the wells. In most cases (e.g. Grana and Della Rossa 2010) there are three main litho-fluid classes: shale, brine sand and oil sand. To constrain even more the reservoir, other sub-categories can be used, such as the stiffness of the mineral material. In simpler cases, it is easy to identify clusters to constrain the prior probabilities, but when dealing with a complex reservoir more properties can helpful. In cases where the training data, constructed from the well-log data is statistically representative of the reservoir conditions, Bayesian classification is a successful application to classify facies. On the other hand, when few wells are available to represent the lithologies and fluid types, a useful method is to increase the training data with Monte Carlo Simulation. The resulting facies models represent the link between the rock properties and the real subsurface geology and are conditioned not only to the available seismic reflection data but also to the existing well-log data. The generated set of models composed by water saturation, porosity, volume of shale and facies volume are used as input in an ensemble of faciesdependent rock-physics models (RPMs) to predict the seismic velocities (upper and lower bounds of seismic velocities) of a rock and/or facies. RPMs are used to link data from different domains, i.e to improve coherency between the subsurface rock properties and elastic properties. They can be represented by a simple regression based on well-log data or a complex physical model, with a number of elastic parameters to be estimated e.g., elastic moduli of matrix and fluid components, critical porosity, aspect ratio, and/or coordination number (Mavko et al. 2009). There are two fundamental tasks for this method to be consistent: a well-log calibration of the rock-physics models and quality control. Then, the main procedure is to ajust a theoretical model to a trend in the data. The solid phase, is the mineral part of a composite made of the mineral frame and the pore fluid (Dvorkin et al. 2014). Granular media models describe the rock as a collection of separate gains that contact between them with a certain stiffness. This model is usually applicable to sandstones (Simm and Bacon 2014) and based on Hertz-Mindlin contact theory (Dvorkin et al. 1994). A very effective approach starts by the definition of the elastic properties the end-members. At zero porosity, the rock must have the properties of the mineral, and at the high porosity, the elastic contact theory determines the elastic properties. The interpolation of this two end-members is based on upper and lower Hashin-Shtrikman bounds. The upper bound is usually associated with the contact cement (stiff-sand model) and the uncemented or low cemented rock (soft-sand model) is represented by the lower bound (Mavko et al. 2009). The effect of pore fluid is accounted by using Gassmanns equation (1951) which is the most common model, within this setting, to predict fluid substitution effects at low seismic frequencies (Mavko et al. 2009). The soft- and stiff-sand models for V P and V S can be computed for a wet rock, by first calculating the models for a room-dry grain pack to posteriorly use them in Gassmanns fluid substitution equation. Densities for the matrix and fluid can be computed using Woods (1955) formula. The facies-dependent rock-physics models allow the calculation of velocities and densities for each facies individually, to posteriorly generate synthetic seismograms following Shueys (1985) 3-terms approximation. The quality of the inversion results are assessed based on the match between the calculated synthetic seismic data and the real seismic reflection data with a correlation coefficient. Within global inversion approaches, the highest correlation coefficient portions of the generated models are selected and used along with the corresponding local correlation coefficients, to produce new sets of rock property models in a co-simulation. At each iteration, it is performed an update of the set of rock properties based on the trace-by-trace match between the real and synthetic seismic data. 2

3 METHODOLOGY Figure 1 - Schematic representation of the general framework of the proposed methodology. The proposed iterative geostatistical seismic inversion methodology (Figure 1) integrates rockphysics modelling contains the following main steps. First, DSS simulates water saturation models using the available well-log as experimental data and a spatial continuity pattern as revealed by a variogram model. Then, co-dss with joint probability distributions is used to co-simulate porosity models given the previously simulated water saturation models and the available well-log data. The simulation of volume of shale models also recurs to co-dss with joint probability distributions, given the water saturation previously inverted model and the available well-log data. Before applying a classification algorithm, a training data is generated from the well-log data. The facies of interest are identified in a petrophysical domain such as water saturation versus porosity and porosity versus volume of shale, resulting in five facies: stiffshale, soft-shale, stiff-brine sand, soft-brine sand and soft-oil sand. Then a Bayesian classification algorithm uses the previously simulated models and the training data to create a facies volume. A facies dependent rock-physics modelling uses the set of three simulated models (water saturation, porosity, volume of shale) and the facies volume to compute V P, V S and density. The resulting P- and S- velocity models along with density are used to compute angledependent reflection coefficients which are then convolved with angle-dependent wavelets to generate synthetic seismic data, following Shueys' linear approximation After generating the partially stacked synthetic seismic reflection data each synthetic 3 seismic trace is individually compared against the corresponding real seismic trace in terms of correlation coefficient. At each iteration, and from the ensemble of rock properties generated during the first step of the proposed algorithm, the portions of these models that ensure the highest correlation coefficient between real and synthetic seismic for all angles are simultaneously selected with the correlation coefficients. The selection procedure is based on cross-over genetic algorithm where the best genes (portions of the petrophysical models from different realizations that ensure the highest correlation coefficient) of each iteration are then used as seed for the generation of a new family of models during the next iterations. Iterate the entire procedure until a given global correlation coefficient between the angle-dependent synthetic and real seismic data is above a certain threshold. REAL DATA EXAMPLE The available dataset comprises four partial angle stacks, with mean reflection angles of 9º, 15º, 21º and 27º and their corresponding dependent wavelets and a set of four well-logs composed by porosity, volume of shale, water saturation and P- and S-Impedance. The inversion grid has 159x419x128 cells in i, j and k directions respectively. The grid cell size was defined to reproduce the original inline and cross-line spacing and the original seismic sampling interval, 2 ms. The joint distributions between water saturation versus porosity and porosity versus volume of shale are used as conditioning data in the direct sequential co-simulation with joint probability distributions (Horta and Soares 2010). In this way, the inverted petro-elastic models mimic the real reservoir conditions by reproducing the relationships between the primary and secondary variables (Figure 3). The spatial continuity pattern of each property individually is imposed by a variogram model, estimated for all properties in study for the vertical direction. These variograms are part of the geostatistical inversion procedure and are used as

4 conditioning data for the stochastic sequential simulation of the petro-elastic properties of interest. As a geostatistical inversion procedure, each model reproduces the well-log data at its locations, the variogram model imposed during the stochastic sequential simulation and the marginal and joint probability distributions as inferred from the available well-log data. The rock property models (water saturation, porosity and volume of shale; Figure 2) agree with the main structures as interpreted from the original seismic data and from previous inversion studies over this reservoir. The interpretation of the mean model of the ensemble of petrophysical models simulated in the last iteration (Figure 2) is one way of interpreting the results of geostatistical seismic inversion. However, it is important to highlight that the individual realizations have higher variability presenting small- and large-scale details. In the inverted models of water saturation, porosity and volume of shale, the well-log data values are reproduced at well locations, as well as the histograms of each property. The proposed methodology also allows the assessment of the uncertainty of each property individually by for example, the variance model at a given iteration. Notice that assessing the uncertainty related with each property individually is of great importance for better reservoir modeling. Usually, these areas are associated with low signal-to-noise ratio, i.e., areas with higher uncertainty, seismic and the inverted models do not match with the seismic. Areas with high variability are related to more uncertainty regarding the model parameters, and usually correspond to areas far from the wells, where fewer data is available. Before the inversion of petrophysical properties (continuous quantities), facies (discrete quantities) were classified in four facies. Each facies must be defined such as it captures the physics of the reservoir s geology. The facies of interest are identified in a petrophysical domain such as water saturation versus porosity and porosity versus volume of shale. First, three main groups are classified based on the well-log data rock properties (Figure 5a and Figure 5c): shale, brine sand and oil sand. This classification is based on assumptions related to the known geology of the reservoir, i.e. a) d) b) c) Figure 3 - Joint distributions from the well-logs (on the left) and from the best-fit models of iteration 6 (on the right) of water saturation versus porosity (a;c) and porosity versus volume of shale (b;d). The joint distributions reproduce the ones estimated from the well-log data. a) Water Saturation b) Porosity c) Volume of Shale Figure 2 - Vertical section mean models and standard deviation computed from the last 32 models retrieved from the last iteration. From top to bottom: water saturation, porosity and volume of shale. 4

5 facies shale has a shale content threshold of 0.3, while the fluid factor was added by water saturation with a threshold of maximum 0.8 for the oil sands. Then, to isolate the sand reservoir, another classification is performed considering rock-physics. This classification is based on the stiffness of the rock and it splits each shale, brine sand and oil sand into two categories, stiff and soft depending on the porosity and shale content. This classification assumes that shale facies belong to the granular media models, such as sands. According to the available well-log data and the geological setting, the reservoir only has one type of oil sand is present, soft oil sand. A Bayesian classification algorithm is then used to classify all four facies for the entire inversion grid considering the previously simulated petrophysical properties. The resulting facies model reproduce the distributions of water saturation, porosity and volume of shale used as training data (Figure 5b and Figure 5d). The resulting facies volume (Figure 4a) is consistent with the petrophysical models (Figure 2), as well as the structures interpreted in the real seismic. The ensemble of facies models generated during the last iteration may be used to derived facies probability for each facies (Figure 4b), showing the abundance of stiff brine sands where the in the same location where the stiff shale has lower probability of occurrence. The reservoir soft oil sand has high probability of occurrence also in areas where the progradation is located but with less continuity when compared to the predominant facies. It is clear the intercalation between sand and shale facies and the predominance of stiff facies with small-scale details Table 1 - Rock-physics model parameters: density ρ, bulk modulus K, and shear modulus μ of the matrix and fluid components for a Φ c of 0.4, n of 4 and Peff of 20 Mpa. ρ (g/cm 3 ) K (GPa) μ (GPa) which is a consequence of constraining the stochastic simulation and co-simulation to the inverted models to posteriorly generate facies volumes. a) b) Figure 4 -Vertical section extracted from the best-fit facies model (top) and the probability of occurence of soft oil sand (bottom). Where facies 1 corresponds to soft shale, facies 2 to stiff shale, facies 3 to soft brine sand, facies 4 to stiff brine sand and facies 5 to soft oil sand. a) b) c) d) Sand Shale Oil n.a. Water n.a. Figure 5 - Training data for facies modeling using the training data from the wells (on the left); joint distributions from the best-fit inverted models (on the right) of water saturation versus porosity and porosity versus volume of shale. Each joint distribution is color coded by facies. 5

6 b) e) c) f) Figure 6 - Vertical sections of the mean elastic models retrieved from the 32 simulations of the last iteratio (on the left) along with the reproduction of the histogram of the inverted model (filled blue) and well-log data (red). From top to bottom, P- (a) and S- wave velocity (b) and density (c). a) d) The elastic models (Figure 6a, Figure 6b, Figure 6c) calculated from the petrophysical models using the facies dependent rock physics models calibrated with the well-log (Table 1) reproduce the well-log data and the histograms are close to the ones computed from the original well-log data(figure 6d, Figure 6e, Figure 6f). The global correlation coefficient (62%, Figure 8) between the synthetic seismic elastic models is higher than the global correlation coefficient achieved by the mean model (Figure 7). The areas associated with The synthetic seismic reflection data computed from the arithmetic mean elastic models (Figure 9) matches successfully the real partially stacked seismic data. a) Nearstack b) Mid-nearstack c) Midstack d) Mid-farstack Figure 7 Correlation coefficients between the real and synthetic seismic data of all partial angle stacks. From left to right: nearstack, mid-nearstack, midstack and mid-farstack. Figure 8 - Global correlation coefficient evolution per iteration. 6

7 It reproduces the location of the main primary reflections and the amplitude variations versus offset. low degree of uncertainty (low standard deviation) are related to areas where the synthetic seismic converged properly towards the real one, associated with high correlation coefficients. On the other hand, areas with low correlation coefficients are possible related with values not considered in the conditioning dataset, as well as seismic data with low signal-tonoise ratio, very common in real datasets. Within this a) b) Figure 9 Real reflection data (a) and synthetic seismic section of the arithmetic mean of 32 simulations of the last iteration, of the partial angle midstack. inverse methodology, the noisy areas are assigned a higher uncertainty throughout the entire procedure. DISCUSSION After 6 iterations, an the inversion procedure reached a global correlation coefficient between the synthetic seismic and real seismic of 62%. The use of partially stacked seismic data allows the inversion of seismic directly for rock properties to better distinguish lithofluid facies, instead of the traditional acoustic models. All models obtained from the entire iterative geostatistical inversion procedure with rock physics integration, reproduce: the values of the conditioning data at its locations (Figure 2); the joint and marginal distributions of water saturation, porosity and volume of shale and the spatial continuity model of each property imposed within the sequential simulation by a variogram. This method is successful to discriminate lithology parameters and detect hydrocarbon probability of occurrence directly from the petrophysical properties. From the petrophysical properties itself, it is possible to detect some geological features such as the shale trend along the presented horizontal slice, with low porosity and high shale content crossing the horizontal time slice ( Figure 10c) which is also present in Figure 10d, classified mainly as stiff shale a) Water Saturation b) Porosity c) Volume of Shale d) Facies Volume Figure 10 Horizontal time-slices (from left to right) of water saturation (a), porosity (b), volume of shale (c) and facies volume (d) (k=120). 7

8 with soft shale in some portions of that area, showing the clear relationship of rock-physics per facies. The variance models (Figure 2) computed between the models generated during the last iteration show lower variability i.e., lower spatial uncertainty in the volume of shale inverted models, following water saturation and porosity. Usually high values of variance are related to areas where the seismic is noisy and the inverted petrophysical models cannot produce synthetic seismic data that fits the real seismic, or the lack of real petrophysical properties. Most of the spatial uncertainty is related with the location of geological boundaries of facies bodies, because of its similarities with the inverted petrophysical models (Figure 2) and the facies volume (Figure 4). Also, it is important to highlight that the interpretation of reservoir models along with their corresponding uncertainty allows better decision making and risk management. CONCLUSIONS The presented methodology was successfully designed, implemented and applied in a real dataset resulting in a good match between the real and the inverted synthetic seismic data. The novel iterative geostatistical seismic inversion methodology that simultaneously integrates seismic reflection data, well-log data and rock physics models can retrieve directly from the seismic data reliable petrophysical models such as water saturation porosity, volume of shale and facies volumes. The results of the presented inversion method are consistent with other seismic inversions applied to this reservoir, but also add value due to the identification of new geological features and proper fluid/lithology characterization. This approach provides a direct connection between the seismic response and the geological (petrophysical) properties, by the application of Rock Physics Models allowing the propagation of the uncertainty related to the seismic inversion in onestep approach. It is an efficient method to guide and improve qualitative interpretation, as well as avoid ambiguities in seismic interpretation related to fluids/lithology, sand/shale and porosity/saturation. This novel method can be applied to all reservoir where the physical link between the elastic and petrophysical properties can be described by a suitable rock physics model, as well as adapt other rock properties and litho-fluid facies. ACKNOWLEDGEMENTS The authors would like to thanks Schlumberger for the donation of the academic licenses of Petrel and CERENA/IST for supporting this work. CA want to thanks Professor Dario Grana from the University of Wyoming for the valuable input to this work and the Department of Geology and Geophysics for the three months stay. APPENDIX A ROCK-PHYSICS MODELS The soft- and stiff-sand models are based on Hertz- Mindlin grain-contact theory and provide estimation of the bulk and shear moduli of a dry rock, assuming a random pack of identical spherical grains under an effective pressure P, with a certain critical porosity (Φ c ) and a coordination number (Mavko et al. 2009). and 3 2 P K HM = n2 (1 Φ c ) 2 μ mat 18π 2 (1 ν) 2 2 P μ HM = 5 4ν 3 (1 Φ c ) 2 μ mat 10 5ν 3n2 2π 2 (1 ν) 2 A - 1 A - 2 where μ mat is the shear modulus of the solid phase and ν is the grain Poissons ratio. The matrix moduli is calculated using Voigt-Reuss- Hill averages for a matrix with sand and clay materials: 8

9 K mat = 1 2 (V 1 ck c + (1 V c )K s + V c + K ) s K c (1 V c ) and A - 3 μ mat = 1 2 (V 1 cμ c + (1 V c )μ s + V c + μ ) A - 4 s μ c (1 V c ) where V c is the volume of clay, K c, μ c, are the bulk and shear moduli, respectively of the clay and and K s, μ s is the bulk and shear moduli of the sand. The bulk (K HM ) and shear (μ HM ) moduli of a room-dry rock is estimated recurring, for example to Hertz-Mindlin grain-contact theory, under the assumption that the sand frame with a random pack of identical spherical grains is under an effective pressure P, with a certain critical porosity ( Φ c ) and a coordination number (Mavko et al. 2009). and 3 2 P K HM = n2 (1 Φ c ) 2 μ mat 18π 2 (1 ν) 2 μ HM = 5 4ν ν 3n 3 (1 Φ c ) 2 μ2 mat P 2π 2 (1 ν) 2 A - 5 A - 6 where μ mat is the shear modulus of the solid phase and ν is the grain Poissons ratio. For the effective porosity values between zero and the critical porosity, this model that best fits the data (modified lower and/or upper Hashin-Shtrikman bound), interpolates the the two end-members, i.e. the solid-phase elastic moduli (K mat and μ mat ) and the elastic moduli of the dry rock (K HM and μ HM ). At any porosity Φ < Φ c, the main point of the soft connector (modified lower Hashin-Shtrikman bound) is given by the following equations: Φ Φ c K soft = [ K HM μ HM Φ Φ c K mat + 4 ] 4 3 μ 3 μ HM HM Φ Φ c μ soft = [ μ HM ξ μ HM 1 6 ξ HM μ HM, ξ HM = 9K HM + 8μ HM K HM + 2μ HM Φ Φ c μ mat + 4 ] 3 ξ μ HM A - 7 A - 8 For for any porosity Φ > Φ c the modified upper Hashin-Shtrikman bound, or the stiff connector is given by: Φ Φ c K stiff = [ K HM μ mat Φ Φ c K stiff = [ K HM μ mat ξ = 9K mat + 8μ mat K mat + 2μ mat Φ Φ c K mat + 4 ] 4 3 μ 3 μ mat mat A Φ Φ c K mat + 4 ] 4 3 μ 3 μ mat mat A - 10 While density (ρ) is simply the arithmetic average of the various solid and fluid components of the rock (weighted according to their volume fractions) velocity is sensitive (Simm and Bacon 2014). Gassmanns equation (1951) is used to model fluid substitution effects at low seismic frequencies (Mavko et al. 2009). P- and S-wave velocities are estimated using matrix and fluid properties: and K sat = K dry + (1 K 2 dry ) K mat Φ + 1 Φ K fl K mat K dry 2 K mat A - 11 μ sat = μ dry A - 12 From the saturated-rock elastic moduli, velocities can be obtained by; 9

10 V P = K sat μ sat, ρ A - 13 and V S = μ sat ρ. A - 14 REFERENCES Dvorkin, J., Nur, A. and Yin H., Effective properties of cemented granular material: Mechanics of Materials, 18, Dvorkin, J., Gutierrez, M.A., Grana, D., Seismic Reflections of Rock Properties. Cambridge University Press. Gassmann, F., Elasticity of porous media: Uber die elastizitat poroser medien, Vierteljahrsschrift der Naturforschenden Gesselschaft, 96, Grana, D. & Della Rossa, E., Probabilistic petrophysical-properties estimation integrating statistical rock physics with seismic inversion. Geophysics, 75(3), p.o21. Horta, A., & Soares, A., Direct sequential Cosimulation with joint probability distributions. Mathematical Geosciences, 42(3), Mavko, G., Mukerji, T., & Dvorkin, J., The Rock Physics Handbook - Tools for Seismic Analysis of Porous Media (2nd edition). Simm, R., Bacon, M., Seismic Amplitude: An Interpreter s Handbook. Cambridge University Press. Soares, A., Direct Sequential Simulation and Cosimulation. Mathematical Geology, 33(8), Shuey, R. T., 1985, A Simplification of the Zoeppritz equations; Geophysics, Vol. 50, No. 4,

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