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ABSTRACT Seismic velocities of rocks depend strongly on confining pressure, a behavior often explained by changes in the compliances of cracks within the rock. It is important to have an accurate model relating confining pressure and seismic velocities for applications such as time-lapse reservoir characterization. Previously developed solutions include a variety of models. One approach is to represent the cracks as ellipsoidal voids (penny-shaped cracks) with varying aspect ratios, and effective medium theories based on this approach describe the pressure-dependence of the elastic velocities using a distribution of aspect ratios closing at different pressures. Another approach is to characterize the cracks as rough surfaces, and it is the increase in asperity contact area with higher pressures that causes the increase in velocity instead of closure of penny-shaped cracks. We propose a new model that combines an existing theoretical framework that represents the compliance associated with cracks or grain surfaces in a rock in terms of crack density tensors (Sayers, 2002) with new and revised analytic solutions for the pressure-dependence of the compliance of rough surfaces (asperity-deformation models). Specifically, previous results are available for the normal compliance B, N, of a fracture surface, and we derive a new model for the tangential compliance B, T, . These asperity-deformation models represent the surfaces with a power-law distribution of asperity heights, and using these compliances in the crack density tensors naturally leads to simple expressions for P- and S-wave velocities with seven unknown parameters. We apply joint least squares inversion of P- and S-wave velocity laboratory data to show that the model fits data very well.
- Geology > Geological Subdiscipline > Geomechanics (1.00)
- Geology > Rock Type > Sedimentary Rock (0.70)
Accurate generation of seismograms on fractured reservoirs
Thore, Pierre (Total) | Tarrass, I. (Total) | Gibson, Richard L. (Texas A&M U.) | Lisitsa, Vadim V. | Reshetova, Galina V. | Tcheverda, Vladmir A.
Abstract A major challenge in carbonates environments is to map heterogeneities which have a strong impact on oil and gas production. For example in many carbonate reservoirs, matrix porosity contains the oil in place but the permeability is mainly provided by fracture corridors. In some carbonate reservoirs, such as Kashagan field in Kazakhstan, the oil in place is essentially contained primarily in karstic caves and conduits. Therefore being able to precisely locate these heterogeneities and possibly characterize their properties is essential in these environments. Various techniques have been developed recently to locate and characterize these heterogeneities, by detailed analysis of the elastic signals recorded in seismic experiments. Among them one can think of the scattering index presented by Willis et al (2004) or all the imaging techniques developed recently under the generic name of interferometry. The first step in all processing or inversion procedures is to be able to accurately simulate the forward problem, and it is therefore of paramount importance to be able to accurately model the scattering related to fractures and caves. The numerical and computer constrains even on very large clusters limit the resolution at which a model can be described. The current practice for finite difference modeling is to use 20m grid cells for acoustic modeling and spatial discretization leading to cells 10m in size for elastic modeling. In these cases, heterogeneity associated with fracturing on smaller scales must be upscaled and distributed in an equivalent medium, which will reproduce varaiations in traveltimes and changes in reflection coefficient but will neglect the scattered wavefields that are the subject of the recently developed methods for characterizing fracture distributions. In this paper, we present a full waveform modeling technique based on a new approach of finite differences which allows simultaneous simulation of both macro- and micro-scale heterogeneities. This algorithm is based on a multi grid approach that models the seismic response of a shot point over a large size model containing micro scale heterogeneities in a reasonable time frame (i.e., several hours on a modern cluster). We show various examples of the signals related to fractures and their importance in terms of heterogeneities interpretation, and we also examine results of simulations using 3-D modeling based on the Born approximation to help interpret and explain the synthetic seismograms. These new numerical results will have significant advantages over previous results that rely on approaches that replace the detailed spatial variations of elastic properties with media with altered velocities and densities.
SUMMARY Upscaling of velocity logs is a common problem in correlating surface seismic and log data and in the generation of coarse scale models from log data to use in modeling or data processing. Backus averaging allows computation of the long wavelength effective elastic moduli of a medium comprised of a stack of thin, isotropic and homogeneous layers. However, it is not obvious how to accurately choose the depth interval over which the average should be applied and where to assign interface depths between upscaled layers. Though running Backus average methods have been proposed, they produce smoothed models, even though it is clear that many settings such as unconformities are best modeled with velocity discontinuities. Here we apply a Markov Chain Monte Carlo (MCMC) method to optimize the depths of interfaces in upscaled log data and to quantify the uncertainty in the resulting models. Application to log data from the North Sea shows that the method provides useful, objective guidelines for assigning model properties. Histograms showing the estimated distribution functions of interface depths indicate both where interfaces should be located and how large is the uncertainty in each location. These results also suggest a simple means for choosing the minimum number of interfaces that should be incorporated into the final model to generate the simplest solution. INTRODUCTION Upscaling of velocity log measurements, the prediction of velocities for low frequencies given high frequency data, is an important and challenging problem for several reasons. For example, it is well known that a medium composed of isotropic layers behaves as a transversely isotropic effective medium when the layers are thin compared to a seismic wavelength, and the Backus averaging method predicts the relevant effective elastic moduli (Backus, 1962). When the materials are laterally heterogeneous, the effective medium can also be transversely isotropic (Sayers, 1998). While there can be fundamental changes in the elastic response of earth materials as wavelength ? increases from ?ห 0.1 m to values in surface seismic exploration, ? ห 100 m, there are also important practical issues for interpretation and processing. De et al. (1994) performed a systematic comparison of log and vertical seismic profile (VSP) data, showing that VSP velocity measurements were lower than log values. The Backus averaging method is used frequently for upscaling velocity measurements (e.g.,Stovas and Arntsen, 2006; Partyka et al., 2000; Lindsay and Van Koughnet, 2001; Rio et al., 1996; Rich 2006; Tiwary, 2007). This has the distinct advantage of creating a coarser earth model that facilitates modeling, imaging and inversion applications, and the velocities can be specified on the same length scale as the surface seismic data. However, it is not always obvious how the averaging should be applied to log data, as the choice of depths for the interfaces of the layers in the upscaled, โblockedโ model is often arbitrary. One solution is to apply sequential Backus averaging, where the properties are defined by a running average scheme that produces an upscaled and smoothed velocity model (Lindsay and Koughnet, 2001; Rio et al., 1996).
- North America > United States > Oklahoma (0.29)
- Europe > United Kingdom > North Sea (0.25)
- Europe > Norway > North Sea (0.25)
- (2 more...)
SUMMARY Theoretical models and laboratory measurements both suggest that the interaction of solids and fluids in a reservoir formation can lead to strong attenuation measured by a low quality factor Q. When the reservoir layer is relatively thin, the loss of energy from a propagating seismic wave may be negligible. However, the low Q in such a layer can still have a measurable influence on reflections as it makes the reflection coefficient complex. A numerical modeling method that can simulate such effects rapidly for complex, laterally heterogeneous media will help to quantify the effects on reflected waveforms. Born scattering theory is useful for such problems, though it does not seem to have been applied in previous studies to modeling including finite Q values represented by complex elastic moduli. Here therefore we introduce a new implementation of the Born algorithm that does allow this type of modeling. Validation against full waveform results shows that results are accurate, and the seismograms also show that phase changes indicate the presence of low Q even when the reflecting reservoir layer is thin. The instantaneous phase seismic attribute can help to identify these effects, providing a potentially useful hydrocarbon indicator. INTRODUCTION Many direct hydrocarbon indicators have been proposed to make use of seismic data to better locate economic concentrations of oil or gas in the subsurface. For example, amplitude variation with offset (AVO) methods often provide useful results (Shuey, 1985; Castagna et al., 1993; Ross, 2000; Stovas et al., 2006). This method, and other approaches based on detecting velocity changes associated with hydrocarbons, typically assume purely elastic wave propagation. On the other hand, some recent investigations suggest that attenuation may also be an important process that affects seismic data (Goloshubin and Korneev, 2000; Goloshubin et al., 2002; Castagna et al., 2003). The propagation of waves through an highly attenuating layer will cause preferential loss of high frequencies, motivating the search for changes in frequency content that may provide additional hydrocarbon indicators. This is especially true since careful laboratory measurements suggest that the quality factor Q may approach values on the order of 10 in some cases (Batzle et al., 2005). In addition, theoretical models for the effective properties of seismic velocities also suggest that rocks with pore fluids can lead to attenuation. An interesting example is fractures with fluids (Pointer et al., 2000), as characterizing the distribution, orientation and concentration of fractures is an important task for reservoir characterization. However, there is another interesting problem that may arise is practical settings, since many reservoirs are not very thick, perhaps 10 m or less. Under such conditions, the distance of propagation through the attenuating medium with a low Q value is only a small fraction of the seismic wavelength, which is on the order of 100 m. Therefore the loss of energy associated with the layer will be small and other measures of seismic waveforms must be considered to attempt to detect the low Q region.
SUMMARY Knowledge of the orientation and spatial distribution of fractures in rocks is important for predicting the flow of fluids. Masihi et al. (2007) developed a new method of modeling these distributions beginning with theoretical results from the physics of fracturing. We implemented and extended this modeling technique to generate models that better incorporate field observations. The method starts with an energy function based on the pair-wise spatial correlation of fractures that also serves as an objective function for a simulated annealing algorithm (SA) that generates realizations of correlated fracture networks. We improved this technique by incorporating periodic boundary conditions, including criteria to limit maximum range of the pair-wise calculations, and by suggesting methods to constrain models to match field data. For most subsurface rocks (with Poisson ratio ? = 0.25), this method generates orthogonal sets of fractures, a pattern that is commonly observed during basin formation or subsidence. This new method is compared with conventional discrete fracture network (DFN) modeling by computing the fractal dimension of the networks. We also examine the implications for seismic reservoir characterization by computing effective seismic velocities and the resulting synthetic seismograms. The new approach can be considered better than DFN as DFN generates realizations based on only statistical distributions, without any knowledge of physics of fracturing. INTRODUCTION The quantification of the spatial concentrations and orientations of fractures in low permeability rocks is essential since they control the nature of fluid flow in those rocks. Generally, these spatially distributed fractures form complex networks that can either act as fluid carriers or barriers depending upon fracture connectivity. Therefore, understanding the connectivity pattern, and areas of high and low fracture density zones, is essential to characterize flow inside the earth. To date much research has considered the effect of geometrical properties of fractures such as length (Berkowitz, 1995; Bour and Davy, 1997) and orientation (Balberg et al., 1984; Masihi et al., 2005) on the scaling laws of the connectivity of fractures. However, fewer studies have examined the spatial correlation of quantities such as length, orientation and position of fractures, though some of the studies examined the long-range density correlations using fractal geometry (Berkowitz et al., 2000; Darcel et al., 2003). These spatial correlation parameters are important as they affect the connectivity of fractures. A common approach used to model fractures is the discrete fracture network (DFN) method. Generally DFN modeling specifies the statistical distributions of several parameters such as fracture density, orientation, location, size, etc. to generate several realizations for production estimation and reservoir planning (Al-Harbi et al., 2004). Here, we implement and extend a new model of the spatial distribution of fractures based on the physics of the fracturing process (Masihi et al., 2007; see also Shekhar, 2008) which is not explicitly considered in DFN modeling. The idea for modeling is based on the assumption that the elastic free energy associated with the fracture density follows the Boltzmann distribution.
- Geophysics > Seismic Surveying > Seismic Processing (0.54)
- Geophysics > Seismic Surveying > Seismic Modeling > Velocity Modeling (0.48)
Summary Seismic data provide essential information for guiding reservoir development. Improvements in data quality hold the promise of improving performance even further, provided that the value of these data exceeds their cost. Previous work has demonstrated value-of-information (VOI) methods to quantify the value of seismic data. In these examples, seismic accuracy was obtained by means of expert assessment instead of being based on geophysical quantities. In addition, the modeled seismic information was not representative of any quantity that would be observed in a seismic image. Here we apply a more general VOI model that includes multiple targets, budgetary constraints, and quantitative models relating poststack seismic amplitudes and amplitude-variation-with-offset (AVO) parameters to the quantities of interest for reservoir characterization, such as porosity and reservoir thickness. Also, by including estimated changes in data accuracy based on signal-to-noise ratio, the decision model can provide objective estimates of the reliability of measurements derived from the seismic data. We demonstrate this methodology within the context of a west Texas 3D land survey. This example demonstrates that seismic information can improve reservoir economics and that improvements in seismic technology can create additional value. Introduction Reservoir characterization makes heavy use of seismic data both for selecting a target for drilling and, with time-lapse data, for monitoring the fluid movements in the reservoir to optimize production of hydrocarbons. Reservoir characterization requires good-quality seismic data for optimal results. Improvements in aspects of seismic acquisition, such as signal-to-noise ratio, bandwidth, receiver positioning, or maximum offset, may help improve images or AVO analyses, thereby increasing the level of knowledge about reservoir structure or properties. However, modifications to acquisition procedures to estimate rock properties better or to improve subsalt images, for example, may increase expense of data acquisition and possibly experiment duration. The improved data quality must always be weighed against the additional cost. Previous work has addressed valuing seismic data using the decision-analysis concept of VOI, including Stibolt and Lehman (1993), Waggoner (2000b, 2002), Begg et al. (2002), Pickering and Bickel (2006), and Bickel et al. (2006). Ballin et al. (2005) and Steagall et al. (2005) provide examples of actual seismic projects where VOI analyses shaped management decisions significantly. See Bratvold et al. (2007) for a review of VOI papers in the SPE literature. One challenge of implementing VOI methodologies to value seismic data is the assessment of seismic accuracy. The studies discussed in the preceding paragraph rely on expert assessment and model seismic information at a high level. In many cases, these assessments are not tied directly to observable seismic signals. For example, some studies assess the probability that the seismic survey will report "success," "unswept," or "large reservoir," even though the actual signal from a seismic survey may be an amplitude reading. This gap between what seismic surveys actually report and what is needed in decision making makes the implementation of VOI techniques problematic (Bratvold et al. 2007). To address these concerns, several authors have performed historical look-backs to document the impact of seismic information [e.g. see Aylor (1999) and Waggoner (2000a)]. Another difficulty is appropriately modeling the decision-making environment and the role seismic information plays. Many authors implicitly embed downstream decisions in the seismic-accuracy assessment by assuming the chance of geologic success can only go up after commissioning a seismic survey (Head 1999; Waggoner 2000b, 2002). This mixing of probability assessments and decision making makes it difficult to understand the value of seismic in a specific situation. Houck (2004) addressed some of these concerns by valuing seismic's ability to inform estimates of porosity in the context of a multiwell drilling program and tying the accuracy of seismic data to directly observable seismic signals. This paper also extends previous VOI studies by considering multiple targets and budgetary constraints. We extend Houck's results by investigating the accuracy and value of AVO and peak amplitude. Furthermore, we examine the ability of seismic information to inform estimates of multiple reservoir properties simultaneously (e.g., porosity, thickness, and water saturation). The resulting models allow quantification of the accuracy of information provided by seismic data and quantification of the information's economic value. The contributions of this paper are three-fold. First, we illustrate a VOI method that directly relates observable seismic signals to reservoir properties and reservoir-management decisions. Second, we develop a seismic model that allows us to quantify objectively the accuracy of seismic information across a range of acquisition and processing techniques. Third, we quantify both the absolute value of seismic information and the relative value of improved seismic information within the context of a 3D land example situated in a hypothetical carbonate reservoir modeled after the McElroy field in west Texas.
- North America > United States > Texas > Upton County (0.25)
- North America > United States > Kentucky > Butler County (0.24)
- Geology > Geological Subdiscipline (0.87)
- Geology > Rock Type > Sedimentary Rock (0.67)
- Geophysics > Seismic Surveying > Surface Seismic Acquisition (1.00)
- Geophysics > Seismic Surveying > Seismic Processing > Seismic Migration (1.00)
- Geophysics > Seismic Surveying > Seismic Modeling > Velocity Modeling (1.00)
- Geophysics > Seismic Surveying > Seismic Interpretation > Seismic Reservoir Characterization > Amplitude vs Offset (AVO) (1.00)
SUMMARY Changes in the seismic velocities of fractured reservoirs detected by time-lapse seismic experiments provide important constraints on changes in fluid saturations and pressures. Effectivemodels for variations in elastic properties as a function of pressure are important for properly interpreting such field data. Here we investigate the validity of a model that explains changes in elastic moduli as a result of the contact of rough fracture surfaces with asperities of varying heights (Gangi, 1978, 1981). Increases in effective pressure cause an increase in fracture stiffness as the asperities are brought into contact and deform. Since the asperities are modeled as cylindrical rods to derive the analytic solution, the result is sometimes referred to as the โbed-of-nailsโ model. Inverting measurements of velocity as a function of pressure for a large set of clastic, carbonate and igneous rocks shows that while the model predicts the general trend in velocity, it is less accurate for some rocks, especially for some clastics, and does not predict detailed variations in velocity well. The original, analytic solution assumes a power-law distribution of asperity heights, so to determine whether this is the source of error, we generalized the model to allow for arbitrary distributions. Inversions with this new model reduce residual error to negligible values, suggesting that this is a much better description of the rock properties. The new model therefore has the potential to facilitate modeling and interpretation of applications such as time-lapse seismic investigations of fractured reservoirs. INTRODUCTION Quantifying the influence of pressure changes on elastic properties is important for optimal use of time-lapse seismic surveys of fractured reservoirs, such as applications to to carbondioxide (CO2) sequestration. Specifically, a valid model should relate changes in effective pressure and reservoir pore pressure, to changes in the seismic parameters, and it should be easily applicable to reservoir-scale models for straightforward use in large seismic studies. Finally, simulations of field-scale fluid flow and seismic reflection data require a model that is computationally simple enough for fast and easy implementation in conjunction with reservoir simulation. Models for velocity variation with pressure are common for granular media such as sandstone formations (e.g., (Dvorkin and Nur, 1996;Endres and Knight, 1997)), but most recently developed models for the effective seismic properties of fractured media do not include the influence of pressure (Pointer et al., 2000; Liuet al., 2000; Chapman, 2003). Other solutions do attempt to include pressure effects by modeling the closure of fractures depending on their orientation with respect to stress fields (Zatsepin and Crampin, 1997; Crampin and Zatsepin, 1997; Angerer, 2002). However, all of these approaches tend to require he specification of many parameters that would be difficult, or impossible, to constrain on the scale of a complete reservoir formation. Therefore, we have investigated the utility of a simpler model that allows a relatively straightforward calculation of the influence of pressure on seismic velocities (Gangi, 1978, 1981). Noting that fractures in rocks have complex, irregular surfaces that are in contact at many asperities.
- Geology > Geological Subdiscipline > Geomechanics (1.00)
- Geology > Rock Type > Sedimentary Rock > Clastic Rock > Sandstone (0.58)
SUMMARY A hybrid ray-propagator matrix approach can be an efficient, accurate and robust method to seismic modeling of complex stratified reservoirs. The hybrid approach combines wavefront construction (WFC) ray tracing with composite reflection coefficients to generate synthetic seismograms of such reservoir models. The WFC method will efficiently simulate the wave propagation in the geologic layer over a target reservoir, while the composite reflection coefficient obtained by the propagator matrix will model the amplitude of the signal reflected by the stratified reservoir model. The approach is applied to compute synthetic seismograms for test models of turbidite reservoirs in the Ursa field, Gulf of Mexico, validating the new results against exact calculations using the discrete wavenumber method. Comparison of synthetic seismograms between the hybrid approach and exact solutions demonstrates an excellent agreement. Since the total thickness of the turbidite reservoir is relatively thin to the seismic wavelength, the central frequency of the source at 10 and 60 Hz is conducted to discuss the frequency dependence of waveforms in the synthetic seismograms of a thin and complex stratified reservoir. As the source frequency goes higher, the more complex waveforms in the synthetic seismograms. They suggest the internal fluctuation of layering is detectable and the reflections from the top and bottom interfaces are differentiable. The new approach, however, can also be used to generate synthetic seismograms for the laterally heterogeneous, complex stratified reservoir models. INTRODUCTION Amplitude variation with offset (AVO) is a very important technique to locate hydrocarbons because the reflection amplitude at different angles of incidence changes significantly when fluids in a formation change from brine to gas (Domenico, 1976; Ostrander, 1984; Murphy, 1984; Rutherford andWilliams, 1989; Castagna et al., 1998; Ross, 2000; Smith and Sondergeld, 2001). However, some limitations often lead to false conclusions. Classic AVO analysis based on the approximation of the Zoeppritz equation is valid only for the P-wave reflection from the interface between two solid half-spaces. Though the thin layer tuning effect can be included in the AVO modeling (Almoghrabi and Lange, 1986; Juhlin and Young, 1993; Bakke and Ursin, 1998; Liu and Schmitt, 2003), it is more difficult to understand the composite reflection by the conventional AVO analysis when there are many layers in a turbidite reservoir model. Also, the classic AVO analysis assumes that the reflections of each interface are independent from the others generated by other interfaces. Only reflections are considered, and transmissions, conversions, and multiple waves are all neglected. However, these wavefields may include useful information to detect hydrocarbon in the rock. Wave propagation through multi-layered media including reflection, transmission and conversion complicates seismic modeling. The propagator matrix method has been introduced to solve for reflection and transmission coefficients in media with horizontally stratified isotropic layers (Thomson, 1950; Haskell, 1953; Gilbert and Backus, 1966). Instead of the direct computation of the plane layer responses, the reflectivity method (Kennett and Kerry, 1979; Booth and Crampin, 1983; Mรผller, 1985) is also applied to model wave propagation for such layer stacked models.
- North America > United States > Texas (0.28)
- North America > United States > Gulf of Mexico > Central GOM (0.25)
- North America > United States > Gulf of Mexico > Central GOM > East Gulf Coast Tertiary Basin > Mississippi Canyon > Block 854 > Ursa Field (0.99)
- North America > United States > Gulf of Mexico > Central GOM > East Gulf Coast Tertiary Basin > Mississippi Canyon > Block 853 > Ursa Field (0.99)
- North America > United States > Gulf of Mexico > Central GOM > East Gulf Coast Tertiary Basin > Mississippi Canyon > Block 852 > Ursa Field (0.99)
- (3 more...)
Wavefront construction is an effective tool for the rapid calculation of ray fields in anisotropic media. The method explicitly tracks the propagation of a wavefront through a model, mapping it to a computational mesh that is interpolated when accuracy criteria based on paraxial ray methods are violated. Takeoff angles are used often to define the initial ray directions, but uniform sampling in the two angles leads to oversampling of the ray field in the direction of the axis. Such sampling can lead also to numerical instability associated with vanishing derivatives with respect to the azimuthal angle. We suggest a new wavefront mesh definition using the cubed-sphere mesh, which is a coordinate system used to solve partial differential equations in spherical geometries. When using this mesh, ray directions are assigned by mapping points on a regular discretization of the faces of a cube surrounding the source to corresponding rays. This scheme produces a nearly uniform distribution of rays with minimal effort and using the cubed-sphere coordinates as ray parameters to calculate partial derivatives completely eliminates the singularities that arise when takeoff angles are used as ray parameters. Numerical results for quantities related to seismic amplitudes confirm that this new mesh does provide more stable and reliable results.
Seismic Modeling of Compositional And Geochemical Effects In CO2 Sequestration
Shekhar, Ravi (Texas A&M Univ., College Station, TX) | Gibson, Richard L. (Texas A&M Univ., College Station, TX) | Kumar, Ajitabh (Texas A&M Univ., College Station, TX) | Datta-Gupta, Akhil (Texas A&M Univ., College Station, TX)
ABSTRACT Time-lapse seismic monitoring of CO injection into a hydrocarbon reservoir can be important for either enhanced recovery or CO sequestration tasks. In the latter case, over long time periods, the interaction of CO with in-situ brine and host rock minerals generates a variety of geochemical reactions that significantly affect reservoir rock properties and reservoir fluid properties. Dissolution of CO in brine to attain gas-brine equilibria alters brine density, which changes the bulk properties of reservoir fluids. Furthermore, slow mineral reactions between CO reservoir fluids and host rock minerals change porosity and the salinity of the reservoir fluids. Here we present results of a modeling study that combines direct simulation of geochemical processes with fluid flow and seismic models. The results show that the CO injection leads to P-wave velocity reduction of up to 12% for the first 10 years, while chemical effects largely associated with salinity changes become observable seismically only after longer time periods of hundreds of years, producing small velocity changes of about 2%. In general, the results suggest that it will be difficult to distinguish mineral reactions and intra-aqueous reactions from the reduction in bulk modulus caused by injection of CO into brine, especially with noisy data. Therefore, this type of geochemical reaction may not be too important for monitoring of sequestration efforts.
- Research Report > New Finding (0.55)
- Research Report > Experimental Study (0.55)
- Reservoir Description and Dynamics > Storage Reservoir Engineering > CO2 capture and sequestration (1.00)
- Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)
- Reservoir Description and Dynamics > Fluid Characterization > Fluid modeling, equations of state (1.00)