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1.
Carbon capture and storage is a viable greenhouse gas mitigation technology and the Sleipner CO2 sequestration site in the North Sea is an excellent example. Storage of CO2 at the Sleipner site requires monitoring over large areas, which can successfully be accomplished with time lapse seismic imaging. One of the main goals of CO2 storage monitoring is to be able to estimate the volume of the stored CO2 in the reservoir. This requires a parametrization of the subsurface as exact as possible. Here we use elastic 2D time‐domain full waveform inversion in a time lapse manner to obtain a P‐wave velocity constrain directly in the depth domain for a base line survey in 1994 and two post‐injection surveys in 1999 and 2006. By relating velocity change to free CO2 saturation, using a rock physics model, we find that at the considered location the aquifer may have been fully saturated in some places in 1999 and 2006.  相似文献   

2.
CO2 injection and storage in deep saline aquifers involves many coupled processes, including multiphase flow, heat and mass transport, rock deformation and mineral precipitation and dissolution. Coupling is especially critical in carbonate aquifers, where minerals will tend to dissolve in response to the dissolution of CO2 into the brine. The resulting neutralization will drive further dissolution of both CO2 and calcite. This suggests that large cavities may be formed and that proper simulation may require full coupling of reactive transport and multiphase flow. We show that solving the latter may suffice whenever two requirements are met: (1) all reactions can be assumed to occur in equilibrium and (2) the chemical system can be calculated as a function of the state variables of the multiphase flow model (i.e., liquid and gas pressure, and temperature). We redefine the components of multiphase flow codes (traditionally, water and CO2), so that they are conservative for all reactions of the chemical system. This requires modifying the traditional constitutive relationships of the multiphase flow codes, but yields the concentrations of all species and all reaction rates by simply performing speciation and mass balance calculations at the end of each time step. We applied this method to the H2O–CO2–Na–Cl–CaCO3 system, so as to model CO2 injection into a carbonate aquifer containing brine. Results were very similar to those obtained with traditional formulations, which implies that full coupling of reactive transport and multi-phase flow is not really needed for this kind of systems, but the resulting simplifications may make it advisable even for cases where the above requirements are not met. Regarding the behavior of carbonate rocks, we find that porosity development near the injection well is small because of the low solubility of calcite. Moreover, dissolution concentrates at the front of the advancing CO2 plume because the brine below the plume tends to reach high CO2 concentrations quite rapidly. We conclude that carbonate dissolution needs not to be feared.  相似文献   

3.
This paper presents the analytic element modeling approach implemented in the software AnAqSim for simulating steady groundwater flow with a sharp fresh‐salt interface in multilayer (three‐dimensional) aquifer systems. Compared with numerical methods for variable‐density interface modeling, this approach allows quick model construction and can yield useful guidance about the three‐dimensional configuration of an interface even at a large scale. The approach employs subdomains and multiple layers as outlined by Fitts (2010) with the addition of discharge potentials for shallow interface flow (Strack 1989). The following simplifying assumptions are made: steady flow, a sharp interface between fresh‐ and salt water, static salt water, and no resistance to vertical flow and hydrostatic heads within each fresh water layer. A key component of this approach is a transition to a thin fixed minimum fresh water thickness mode when the fresh water thickness approaches zero. This allows the solution to converge and determine the steady interface position without a long transient simulation. The approach is checked against the widely used numerical codes SEAWAT and SWI/MODFLOW and a hypothetical application of the method to a coastal wellfield is presented.  相似文献   

4.
Geologic carbon sequestration (GCS) is being considered as a climate change mitigation option in many future energy scenarios. Mathematical modeling is routinely used to predict subsurface CO2 and resident brine migration for the design of injection operations, to demonstrate the permanence of CO2 storage, and to show that other subsurface resources will not be degraded. Many processes impact the migration of CO2 and brine, including multiphase flow dynamics, geochemistry, and geomechanics, along with the spatial distribution of parameters such as porosity and permeability. In this article, we review a set of multiphase modeling approaches with different levels of conceptual complexity that have been used to model GCS. Model complexity ranges from coupled multiprocess models to simplified vertical equilibrium (VE) models and macroscopic invasion percolation models. The goal of this article is to give a framework of conceptual model complexity, and to show the types of modeling approaches that have been used to address specific GCS questions. Application of the modeling approaches is shown using five ongoing or proposed CO2 injection sites. For the selected sites, the majority of GCS models follow a simplified multiphase approach, especially for questions related to injection and local‐scale heterogeneity. Coupled multiprocess models are only applied in one case where geomechanics have a strong impact on the flow. Owing to their computational efficiency, VE models tend to be applied at large scales. A macroscopic invasion percolation approach was used to predict the CO2 migration at one site to examine details of CO2 migration under the caprock.  相似文献   

5.
Fluid depletion within a compacting reservoir can lead to significant stress and strain changes and potentially severe geomechanical issues, both inside and outside the reservoir. We extend previous research of time‐lapse seismic interpretation by incorporating synthetic near‐offset and full‐offset common‐midpoint reflection data using anisotropic ray tracing to investigate uncertainties in time‐lapse seismic observations. The time‐lapse seismic simulations use dynamic elasticity models built from hydro‐geomechanical simulation output and a stress‐dependent rock physics model. The reservoir model is a conceptual two‐fault graben reservoir, where we allow the fault fluid‐flow transmissibility to vary from high to low to simulate non‐compartmentalized and compartmentalized reservoirs, respectively. The results indicate time‐lapse seismic amplitude changes and travel‐time shifts can be used to qualitatively identify reservoir compartmentalization. Due to the high repeatability and good quality of the time‐lapse synthetic dataset, the estimated travel‐time shifts and amplitude changes for near‐offset data match the true model subsurface changes with minimal errors. A 1D velocity–strain relation was used to estimate the vertical velocity change for the reservoir bottom interface by applying zero‐offset time shifts from both the near‐offset and full‐offset measurements. For near‐offset data, the estimated P‐wave velocity changes were within 10% of the true value. However, for full‐offset data, time‐lapse attributes are quantitatively reliable using standard time‐lapse seismic methods when an updated velocity model is used rather than the baseline model.  相似文献   

6.
In the Norwegian North Sea, the Sleipner field produces gas with a high CO2 content. For environmental reasons, since 1996, more than 11 Mt of this carbon dioxide (CO2) have been injected in the Utsira Sand saline aquifer located above the hydrocarbon reservoir. A series of seven 3D seismic surveys were recorded to monitor the CO2 plume evolution. With this case study, time‐lapse seismics have been shown to be successful in mapping the spread of CO2 over the past decade and to ensure the integrity of the overburden. Stratigraphic inversion of seismic data is currently used in the petroleum industry for quantitative reservoir characterization and enhanced oil recovery. Now it may also be used to evaluate the expansion of a CO2 plume in an underground reservoir. The aim of this study is to estimate the P‐wave impedances via a Bayesian model‐based stratigraphic inversion. We have focused our study on the 1994 vintage before CO2 injection and the 2006 vintage carried out after a CO2 injection of 8.4 Mt. In spite of some difficulties due to the lack of time‐lapse well log data on the interest area, the full application of our inversion workflow allowed us to obtain, for the first time to our knowledge, 3D impedance cubes including the Utsira Sand. These results can be used to better characterize the spreading of CO2 in a reservoir. With the post‐stack inversion workflow applied to CO2 storage, we point out the importance of the a priori model and the issue to obtain coherent results between sequential inversions of different seismic vintages. The stacking velocity workflow that yields the migration model and the a priori model, specific to each vintage, can induce a slight inconsistency in the results.  相似文献   

7.
In the context of geological carbon sequestration (GCS), carbon dioxide (CO2) is often injected into deep formations saturated with a brine that may contain dissolved light hydrocarbons, such as methane (CH4). In this multicomponent multiphase displacement process, CO2 competes with CH4 in terms of dissolution, and CH4 tends to exsolve from the aqueous into a gaseous phase. Because CH4 has a lower viscosity than injected CO2, CH4 is swept up into a ‘bank’ of CH4‐rich gas ahead of the CO2 displacement front. On the one hand, this may provide a useful tracer signal of an approaching CO2 front. On the other hand, the emergence of gaseous CH4 is undesirable because it poses a leakage risk of a far more potent greenhouse gas than CO2 if the cap rock is compromised. Open fractures or faults and wells could result in CH4 contamination of overlying groundwater aquifers as well as surface emissions. We investigate this process through detailed numerical simulations for a large‐scale GCS pilot project (near Cranfield, Mississippi) for which a rich set of field data is available. An accurate cubic‐plus‐association equation‐of‐state is used to describe the non‐linear phase behavior of multiphase brine‐CH4‐CO2 mixtures, and breakthrough curves in two observation wells are used to constrain transport processes. Both field data and simulations indeed show the development of an extensive plume of CH4‐rich (up to 90 mol%) gas as a consequence of CO2 injection, with important implications for the risk assessment of future GCS projects.  相似文献   

8.
CO2 saturations are estimated at Sleipner using a two-step imaging workflow. The workflow combines seismic tomography (full-waveform inversion) and rock physics inversion and is applied to a two-dimensional seismic line located near the injection point at Sleipner. We use baseline data (1994 vintage, before CO2 injection) and monitor data that was acquired after 12 years of CO2 injection (2008 vintage). P-wave velocity models are generated using the Full waveform inversion technology and then, we invert selected rock physics parameters using an rock physics inversion methodology. Full waveform inversion provides high-resolution P-wave velocity models both for baseline and monitor data. The physical relations between rock physics properties and acoustic wave velocities in the Utsira unconsolidated sandstone (reservoir formation) are defined using a dynamic rock physics model based on well-known Biot–Gassmann theories. For data prior to injection, rock frame properties (porosity, bulk and shear dry moduli) are estimated using rock physics inversion that allows deriving physically consistent properties with related uncertainty. We show that the uncertainty related to limited input data (only P-wave velocity) is not an issue because the mean values of parameters are correct. These rock frame properties are then used as a priori constraint in the monitor case. For monitor data, the Full waveform inversion results show nicely resolved thin layers of CO2–brine saturated sandstones under intra-reservoir shale layers. The CO2 saturation estimation is carried out by plugging an effective fluid phase in the rock physics model. Calculating the effective fluid bulk modulus of the brine–CO2 mixture (using Brie equation in our study) is shown to be the key factor to link P-wave velocity to CO2 saturation. The inversion tests are done with several values of Brie/patchiness exponent and show that the CO2 saturation estimates are varying between 0.30 and 0.90 depending on the rock physics model and the location in the reservoir. The uncertainty in CO2 saturation estimation is usually lower than 0.20. When the patchiness exponent is considered as unknown, the inversion is less constrained and we end up with values of exponent varying between 5 and 20 and up to 33 in specific reservoir areas. These estimations tend to show that the CO2–brine mixing is between uniform and patchy mixing and variable throughout the reservoir.  相似文献   

9.
We present a numerical study for 3D time‐lapse electromagnetic monitoring of a fictitious CO2 sequestration using the geometry of a real geological site and a suite of suitable electromagnetic methods with different source/receiver configurations and different sensitivity patterns. All available geological information is processed and directly implemented into the computational domain, which is discretized by unstructured tetrahedral grids. We thus demonstrate the performance capability of our numerical simulation techniques. The scenario considers a CO2 injection in approximately 1100 m depth. The expected changes in conductivity were inferred from preceding laboratory measurements. A resistive anomaly is caused within the conductive brines of the undisturbed reservoir horizon. The resistive nature of the anomaly is enhanced by the CO2 dissolution regime, which prevails in the high‐salinity environment. Due to the physicochemical properties of CO2, the affected portion of the subsurface is laterally widespread but very thin. We combine controlled‐source electromagnetics, borehole transient electromagnetics, and the direct‐current resistivity method to perform a virtual experiment with the aim of scrutinizing a set of source/receiver configurations with respect to coverage, resolution, and detectability of the anomalous CO2 plume prior to the field survey. Our simulation studies are carried out using the 3D codes developed in our working group. They are all based on linear and higher order Lagrange and Nédélec finite‐element formulations on unstructured grids, providing the necessary flexibility with respect to the complex real‐world geometry. We provide different strategies for addressing the accuracy of numerical simulations in the case of arbitrary structures. The presented computations demonstrate the expected great advantage of positioning transmitters or receivers close to the target. For direct‐current geoelectrics, 50% change in electric potential may be detected even at the Earth's surface. Monitoring with inductive methods is also promising. For a well‐positioned surface transmitter, more than 10% difference in the vertical electric field is predicted for a receiver located 200 m above the target. Our borehole transient electromagnetics results demonstrate that traditional transient electromagnetics with a vertical magnetic dipole source is not well suited for monitoring a thin horizontal resistive target. This is due to the mainly horizontal current system, which is induced by a vertical magnetic dipole.  相似文献   

10.
Brine migration and saltwater intrusion into freshwater aquifers are among the hazards which may result from injecting CO2 into deep saline formations. Comprehensive risk assessment should include estimates of the salinization of freshwater aquifers, preferably based on numerical simulation results. A crucial task is to choose an appropriate conceptual model and relevant scenarios. Overly conservative assumptions may lead to estimation of unacceptably high risks, and thus prevent the implementation of a CO2 storage project unnecessarily. On the other hand, risk assessment should not lead to an underestimation of hazards. This study compares two conceptual model approaches for the numerical simulation of brine-migration scenarios through a vertical fault and salt intrusion into a fresh water aquifer. The first approach calculates salt discharge into freshwater using an immiscible two-phase model with constant salinity in the brine phase. The second approach takes compositional effects into account and considers salinity as a variable parameter in the water phase. A spatial model coupling is introduced to adapt the increased model complexity to the required complexity of the physics. The immiscible two-phase model is applied in the CO2 storage reservoir and spatially coupled to a single-phase (water) two-component (water, salt) model, where salt mass fraction is a variable. A Dirichlet–Neumann technique is used for the coupling conditions at the interface of the two models. The results show that the predicted salt discharges can vary by orders of magnitude depending on the choice of the model. The implications of the results for risk assessment are discussed.  相似文献   

11.
The CO2 and H2S concentration in the Solfatara atmosphere has been measured. The concentrations of both gases are higher neraby the more active areas and decrease away from them. A sharp horizontal and vertical gradient of the CO2 content has been recognized.Such gradient is assumed to result from a diffusion of gas from the ground to the atmosphere.The total output of CO2 has been computed based on a turbulent diffusion model. The obtained value is in good agreement with previously abserved values (Italiano et al., 1984).The feasibility of monitoring the atmosphere of Solfatara for either gas hazard and surveillance of volcanic activity has also been evaluated.  相似文献   

12.
This paper presents an erosion model, ARMOUR, which simulates time‐varying runoff, erosion, deposition and surface armour evolution down a hillslope either as a result of a single erosion event or as the cumulative impact of many events over periods up to decades. ARMOUR simulates sediment transport for both cohesive and non‐cohesive soil and dynamically differentiates between ‘transport‐limited’ and ‘source‐limited’ processes. A variety of feasible processes for entrainment of different size classes can be modelled and evaluated against data. The generalized likelihood of uncertainty estimation (GLUE) technique was used to calibrate and validate ARMOUR using data collected during rainfall simulator experiments at two contrasting sites: (1) non‐cohesive stony sediments at Ranger Uranium Mine, Northern Territory, Australia; and (2) cohesive silty sediments at Northparkes Gold Mine, NSW, Australia. The spatial and temporal variations of model predictions within the individual runoff events showed that some entrainment processes could not model the spikes in concentration and subsequent depletion, while the hiding model of Andrews and Parker best simulated the concentration trends for both calibrated and independent runoff events. ARMOUR also successfully captured the coarsening of the surface material, though small, over the duration of the rainfall simulator trials. This was driven by the depletion of the finest size class of the soil. For a constant discharge, ARMOUR simulated higher sediment flux at the start of the storm with the sediment flux and concentration diminishing with time. For natural rainfall a power law relationship between sediment flux and discharge was observed. The calibration exercise showed that sediment concentration and discharge alone are insufficient to calibrate all aspects of the physics, in particular the armour depth. This appears to be because the armouring during the short duration events is driven by depletion of the finest classes of the sediments (diameters less then 62·5 mm), which are not normally measured. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

13.
Time‐lapse seismic analysis is utilized in CO2 geosequestration to verify the CO2 containment within a reservoir. A major risk associated with geosequestration is a possible leakage of CO2 from the storage formation into overlaying formations. To mitigate this risk, the deployment of carbon capture and storage projects requires fast and reliable detection of relatively small volumes of CO2 outside the storage formation. To do this, it is necessary to predict typical seepage scenarios and improve subsurface seepage detection methods. In this work we present a technique for CO2 monitoring based on the detection of diffracted waves in time‐lapse seismic data. In the case of CO2 seepage, the migrating plume might form small secondary accumulations that would produce diffracted, rather than reflected waves. From time‐lapse data analysis, we are able to separate the diffracted waves from the predominant reflections in order to image the small CO2 plumes. To explore possibilities to detect relatively small amounts of CO2, we performed synthetic time‐lapse seismic modelling based on the Cooperative Research Centre for Greenhouse Gas Technologies (CO2CRC) Otway project data. The detection method is based on defining the CO2 location by measuring the coherency of the signal along diffraction offset‐traveltime curves. The technique is applied to a time‐lapse stacked section using a stacking velocity to construct offset‐traveltime curves. Given the amount of noise found in the surface seismic data, the predicted minimum detectable amount of CO2 is 1000–2000 tonnes. This method was also applied to real data obtained from a time‐lapse seismic physical model. The use of diffractions rather than reflections for monitoring small amounts of CO2 can enhance the capability of subsurface monitoring in CO2 geosequestration projects.  相似文献   

14.
Common shot ray tracing and finite difference seismic modelling experiments were undertaken to evaluate variations in the seismic response of the Devonian Redwater reef in the Alberta Basin, Canada after replacement of native pore waters in the upper rim of the reef with CO2. This part of the reef is being evaluated for a CO2 storage project. The input geological model was based on well data and the interpretation of depth‐converted, reprocessed 2D seismic data in the area. Pre‐stack depth migration of the ray traced and finite difference synthetic data demonstrate similar seismic attributes for the Mannville, Nisku, Ireton, Cooking Lake, and Beaverhill Lake formations and clear terminations of the Upper Leduc and Middle Leduc events at the reef margin. Higher amplitudes at the base of Upper‐Leduc member are evident near the reef margin due to the higher porosity of the foreslope facies in the reef rim compared to the tidal flat lagoonal facies within the central region of the reef. Time‐lapse seismic analysis exhibits an amplitude difference of about 14% for Leduc reflections before and after CO2 saturation and a travel‐time delay through the reservoir of 1.6 ms. Both the ray tracing and finite difference approaches yielded similar results but, for this particular model, the latter provided more precise imaging of the reef margin. From the numerical study we conclude that time‐lapse surface seismic surveys should be effective in monitoring the location of the CO2 plume in the Upper Leduc Formation of the Redwater reef, although the differences in the results between the two modelling approaches are of similar order to the effects of the CO2 fluid replacement itself.  相似文献   

15.
The injection of CO2 at the Ketzin pilot CO2 storage site started in June 2008 and ended in August 2013. During the 62 months of injection, a total amount of about 67 kt of CO2 was injected into a saline aquifer. A third repeat three‐dimensional seismic survey, serving as the first post‐injection survey, was acquired in 2015, aiming to investigate the recent movement of the injected CO2. Consistent with the previous two time‐lapse surveys, a predominantly west–northwest migration of the gaseous CO2 plume in the up‐dip direction within the reservoir is inferred in this first post‐injection survey. No systematic anomalies are detected through the reservoir overburden. The extent of the CO2 plume west of the injection site is almost identical to that found in the 2012 second repeat survey (after injection of 61 kt); however, there is a significant decrease in its size east of the injection site. Assessment of the CO2 plume distribution suggests that the decrease in the size of the anomaly may be due to multiple factors, such as limited vertical resolution, CO2 dissolution, and CO2 migration into thin layers, in addition to the effects of ambient noise. Four‐dimensional seismic modelling based on dynamic flow simulations indicates that a dynamic balance between the newly injected CO2 after the second repeat survey and the CO2 migrating into thin layers and being dissolved was reached by the time of the first post‐injection survey. In view of the significant uncertainties in CO2 mass estimation, both patchy and non‐patchy saturation models for the Ketzin site were taken into consideration.  相似文献   

16.
An integrated multiscale seismic imaging flow is applied to dense onshore wide‐aperture seismic data recorded in a complex geological setting (thrust belt). An initial P‐wave velocity macromodel is first developed by first‐arrival traveltime tomography. This model is used as an initial guess for subsequent full‐waveform tomography, which leads to greatly improved spatial resolution of the P‐wave velocity model. However, the application of full‐waveform tomography to the high‐frequency part of the source bandwidth is difficult, due to the non‐linearity of this kind of method. Moreover, it is computationally expensive at high frequencies since a finite‐difference method is used to model the wave propagation. Hence, full‐waveform tomography was complemented by asymptotic prestack depth migration to process the full‐source bandwidth and develop a sharp image of the short wavelengths. The final traveltime tomography model and two smoothed versions of the final full‐waveform tomography model were used as a macromodel for the prestack depth migration. In this study, wide‐aperture multifold seismic data are used. After specific preprocessing of the data, 16 frequency components ranging from 5.4 Hz to 20 Hz were inverted in cascade by the full‐waveform tomography algorithm. The full‐waveform tomography successfully imaged SW‐dipping structures previously identified as high‐resistivity bodies. The relevance of the full‐waveform tomography models is demonstrated locally by comparison with a coincident vertical seismic profiling (VSP) log available on the profile. The prestack depth‐migrated images, inferred from the traveltime, and the smoothed full‐waveform tomography macromodels are shown to be, on the whole, consistent with the final full‐waveform tomography model. A more detailed analysis, based on common‐image gather computations, and local comparison with the VSP log revealed that the most accurate migrated sections are those obtained from the full‐waveform tomography macromodels. A resolution analysis suggests that the asymptotic prestack depth migration successfully migrated the wide‐aperture components of the data, allowing medium wavelengths in addition to the short wavelengths of the structure to be imaged. The processing flow that we applied to dense wide‐aperture seismic data is shown to provide a promising approach, complementary to more classical seismic reflection data processing, to quantitative imaging of complex geological structures.  相似文献   

17.
Tile‐drain response to rainfall events is determined by unsaturated vertical flow to the water table, followed by horizontal saturated water movement. In this study, unsaturated vertical movement from the redistribution of water is modelled using a sharp‐front approximation, and the saturated horizontal flow is modelled by an approximate solution to the Boussinesq equation. The unsaturated flow component models the fast response that is associated with the presence of preferential flow paths. By convoluting the responses of the two components, a transfer function is developed that predicts tile‐drain response to unit amounts of infiltrated water. It is observed that the unsaturated flow component can be cast in a form that is linear in a power function of the infiltrated depth. Since the approach is process based, model parameter definitions are easily identified with soil properties at the field scale. Furthermore, it is demonstrated that the transfer function model parameters can be estimated from moment analysis. Using superposition, the transient tile‐drain response to arbitrary amounts of infiltrated water can be constructed. Comparison with data measured from the Water Quality Field Station show that this approach provides a promising method for generating tile‐drain response to rainfall events. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

18.
Steady interface flow in heterogeneous aquifer systems is simulated with single‐density groundwater codes by using transformed values for the hydraulic conductivity and thickness of the aquifers and aquitards. For example, unconfined interface flow may be simulated with a transformed model by setting the base of the aquifer to sea level and by multiplying the hydraulic conductivity with 41 (for sea water density of 1025 kg/m3). Similar transformations are derived for unconfined interface flow with a finite aquifer base and for confined multi‐aquifer interface flow. The head and flow distribution are identical in the transformed and original model domains. The location of the interface is obtained through application of the Ghyben‐Herzberg formula. The transformed problem may be solved with a single‐density code that is able to simulate unconfined flow where the saturated thickness is a linear function of the head and, depending on the boundary conditions, the code needs to be able to simulate dry cells where the saturated thickness is zero. For multi‐aquifer interface flow, an additional requirement is that the code must be able to handle vertical leakage in situations where flow in an aquifer is unconfined while there is also flow in the aquifer directly above it. Specific examples and limitations are discussed for the application of the approach with MODFLOW. Comparisons between exact interface flow solutions and MODFLOW solutions of the transformed model domain show good agreement. The presented approach is an efficient alternative to running transient sea water intrusion models until steady state is reached.  相似文献   

19.
We present an upscaled model for the vertical migration of a CO2 plume through a vertical column filled with a periodic layered porous medium. This model may describe the vertical migration of a CO2 plume in a perfectly layered horizontal aquifer. Capillarity and buoyancy are taken into account and semi-explicit upscaled flux functions are proposed in the two following cases: (i) capillarity is the main driving force and (ii) buoyancy is the only driving force. In both cases, we show that the upscaled buoyant flux is a bell-shaped function of the saturation, as in the case of a homogeneous porous medium. In the capillary-dominant case, we show that the upscaled buoyant flux is the harmonic mean of the buoyant fluxes in each layer. The upscaled saturation is governed by the continuity of the capillary pressure at the interface between layers. In the capillary-free case, the upscaled buoyant flux and upscaled saturation are determined by the flux continuity condition at the interface. As the flux is not continuous over the entire range of saturation, the upscaled saturation is only defined where continuity is verified, i.e. in two saturation domains. As a consequence, the upscaled buoyant flux is described by a piecewise continuous function. Two analytical approximations of this flux are proposed and this capillary-free upscaled model is validated for two cases of heterogeneity. Upscaled and cell averaged saturations are in good agreement. Furthermore, the proposed analytical upscaled fluxes provide satisfactory approximations as long as the saturation set at the inlet of the column is in a range where analytical and numerical upscaled fluxes are close.  相似文献   

20.
Common‐midpoint moveout of converted waves is generally asymmetric with respect to zero offset and cannot be described by the traveltime series t2(x2) conventionally used for pure modes. Here, we present concise parametric expressions for both common‐midpoint (CMP) and common‐conversion‐point (CCP) gathers of PS‐waves for arbitrary anisotropic, horizontally layered media above a plane dipping reflector. This analytic representation can be used to model 3D (multi‐azimuth) CMP gathers without time‐consuming two‐point ray tracing and to compute attributes of PS moveout such as the slope of the traveltime surface at zero offset and the coordinates of the moveout minimum. In addition to providing an efficient tool for forward modelling, our formalism helps to carry out joint inversion of P and PS data for transverse isotropy with a vertical symmetry axis (VTI media). If the medium above the reflector is laterally homogeneous, P‐wave reflection moveout cannot constrain the depth scale of the model needed for depth migration. Extending our previous results for a single VTI layer, we show that the interval vertical velocities of the P‐ and S‐waves (VP0 and VS0) and the Thomsen parameters ε and δ can be found from surface data alone by combining P‐wave moveout with the traveltimes of the converted PS(PSV)‐wave. If the data are acquired only on the dip line (i.e. in 2D), stable parameter estimation requires including the moveout of P‐ and PS‐waves from both a horizontal and a dipping interface. At the first stage of the velocity‐analysis procedure, we build an initial anisotropic model by applying a layer‐stripping algorithm to CMP moveout of P‐ and PS‐waves. To overcome the distorting influence of conversion‐point dispersal on CMP gathers, the interval VTI parameters are refined by collecting the PS data into CCP gathers and repeating the inversion. For 3D surveys with a sufficiently wide range of source–receiver azimuths, it is possible to estimate all four relevant parameters (VP0, VS0, ε and δ) using reflections from a single mildly dipping interface. In this case, the P‐wave NMO ellipse determined by 3D (azimuthal) velocity analysis is combined with azimuthally dependent traveltimes of the PS‐wave. On the whole, the joint inversion of P and PS data yields a VTI model suitable for depth migration of P‐waves, as well as processing (e.g. transformation to zero offset) of converted waves.  相似文献   

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