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Modern Applied Statistics With S

The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

Recent developments in the direct-current geoelectrical imaging method

SUMMARY
The Jacobian of the non-linear mapping from model parameters to observations is a key component in all gradient-based inversion methods, including variants on Gauss–Newton and non-linear conjugate gradients. Here, we develop a general mathematical framework for Jacobian computations arising in electromagnetic (EM) geophysical inverse problems. Our analysis, which is based on the discrete formulation of the forward problem, divides computations into components (data functionals, forward and adjoint solvers, model parameter mappings), and clarifies dependencies among these elements within realistic numerical inversion codes. To be concrete, we focus much of the specific discussion on 2-D and 3-D magnetotelluric (MT) inverse problems, but our analysis is applicable to a wide range of active and passive source EM methods. The general theory developed here provides the basis for development of a modular system of computer codes for inversion of EM geophysical data, which we summarize at the end of the paper.

Computational recipes for electromagnetic inverse problems

ModEM: A modular system for inversion of electromagnetic geophysical data

Induction in electrically conductive seawater attenuates the magnetotelluric (MT) fields and, coupled with a minimum around 1 Hz in the natural magnetic field spectrum, leads to a dramatic loss of electric and magnetic field power on the sea floor at periods shorter than 1000 s. For this reason the marine MT method traditionally has been used only at periods of 103 to 105 s to probe deep mantle structure; rarely does a sea‐floor MT response extend to a 100-s period. To be useful for mapping continental shelf structure at depths relevant to petroleum exploration, however, MT measurements need to be made at periods between 1 and 1000 s. This can be accomplished using ac-coupled sensors, induction coils for the magnetic field, and an electric field amplifier developed for marine controlled‐source applications. The electrically quiet sea floor allows the attenuated electric field to be amplified greatly before recording; in deep (1-km) water, motional noise in magnetic field sensors appears not to be a proble...

/pdf/marine-magnetotellurics-for-petroleum-exploration-part-i-a-3i673tutk7.pdf

Marine magnetotellurics for petroleum exploration Part I: A sea-floor equipment system

A new joint inversion algorithm to directly estimate reservoir parameters is described. This algorithm combines seismic amplitude versus angle (AVA) and marine controlled source electromagnetic (CSEM) data. The rock-properties model needed to link the geophysical parameters to the reservoir parameters is described. Errors in the rock-properties model parameters, measured in percent, introduce errors of comparable size in the joint inversion reservoir parameter estimates. Tests of the concept on synthetic one-dimensional models demonstrate improved fluid saturation and porosity estimates for joint AVA-CSEM data inversion (compared to AVA or CSEM inversion alone). Comparing inversions of AVA, CSEM, and joint AVA-CSEM data over the North Sea Troll field, at a location with well control, shows that the joint inversion produces estimated gas saturation, oil saturation and porosity that is closest (as measured by the RMS difference, L1 norm of the difference, and net over the interval) to the logged values whereas CSEM inversion provides the closest estimates of water saturation.

/pdf/direct-reservoir-parameter-estimation-using-joint-inversion-19adxvto9v.pdf

Direct reservoir parameter estimation using joint inversion of marine seismic AVA and CSEM data

Three-dimensional magnetotelluric characterization of the Coso geothermal field

We develop a Bayesian model to jointly invert marine seismic amplitude versus angle (AVA) and controlled-source electromagnetic (CSEM) data for a layered reservoir model. We consider the porosity and fluid saturation of each layer in the reservoir, the bulk and shear moduli and density of each layer not in the reservoir, and the electrical conductivity of the overburden and bedrock as random variables. We also consider prestack seismic AVA data in a selected time window as well as real and quadrature components of the recorded electrical field as data. Using Markov chain Monte Carlo (MCMC) sampling methods, wedraw a large number of samples from the joint posterior distribution function. With these samples, we obtain not only the estimates of each unknown variable, but also various types of uncertainty information associated with the estimation. This method is applied to both synthetic and field data to investigate the combined use of seismic AVA and CSEM data for gas saturation estimation. Results show th...

A Bayesian model for gas saturation estimation using marine seismic AVA and CSEM data

WedevelopaBayesianmodeltojointlyinvertmarineseismic amplitude versus angle AVA and controlled-source electromagneticCSEMdataforalayeredreservoirmodel.Weconsider the porosity and fluid saturation of each layer in the reservoir, thebulkandshearmodulianddensityofeachlayernotinthereservoir, and the electrical conductivity of the overburden and bedrock as random variables. We also consider prestack seismic AVA data in a selected time window as well as real and quadrature components of the recorded electrical field as data. Using Markov chain Monte Carlo MCMC sampling methods, we drawalargenumberofsamplesfromthejointposteriordistribution function. With these samples, we obtain not only the estimatesofeachunknownvariable,butalsovarioustypesofuncertaintyinformationassociatedwiththeestimation.Thismethodis applied to both synthetic and field data to investigate the combineduseofseismicAVAandCSEMdataforgassaturationestimation. Results show that the method is effective for joint inversion;theincorporationofCSEMdatareducesuncertaintyinfluid saturation estimation compared to inversion of seismic AVA data alone. The improvement in gas saturation estimation obtained from joint inversion for field data is less significant than for synthetic data because of the large number of unknown noise sourcesinherentinthefielddata.

Special Section — Marine Controlled-Source Electromagnetic Methods A Bayesian model for gas saturation estimation using marine seismic AVA and CSEM data

G. Michael Hoversten

Papers

Marine magnetotellurics for petroleum exploration Part I: A sea-floor equipment system

Direct reservoir parameter estimation using joint inversion of marine seismic AVA and CSEM data

Three-dimensional magnetotelluric characterization of the Coso geothermal field

A Bayesian model for gas saturation estimation using marine seismic AVA and CSEM data

Special Section — Marine Controlled-Source Electromagnetic Methods A Bayesian model for gas saturation estimation using marine seismic AVA and CSEM data