Evaluation of Gaussian approximations for data assimilation in reservoir models

The Bayesian framework is the standard approach for data assimilation in reservoir modeling. This framework involves characterizing the posterior distribution of geological parameters in terms of a given prior distribution and data from the reservoir dynamics, together with a forward model connecting the space of geological parameters to the data space. Since the posterior distribution quantifies the uncertainty in the geologic parameters of the reservoir, the characterization of the posterior is fundamental for the optimal management of reservoirs. Unfortunately, due to the large-scale highly nonlinear properties of standard reservoir models, characterizing the posterior is computationally prohibitive. Instead, more affordable ad hoc techniques, based on Gaussian approximations, are often used for characterizing the posterior distribution. Evaluating the performance of those Gaussian approximations is typically conducted by assessing their ability at reproducing the truth within the confidence interval provided by the ad hoc technique under consideration. This has the disadvantage of mixing up the approximation properties of the history matching algorithm employed with the information content of the particular observations used, making it hard to evaluate the effect of the ad hoc approximations alone. In this paper, we avoid this disadvantage by comparing the ad hoc techniques with a fully resolved state-of-the-art probing of the Bayesian posterior distribution. The ad hoc techniques whose performance we assess are based on (1) linearization around the maximum a posteriori estimate, (2) randomized maximum likelihood, and (3) ensemble Kalman filter-type methods. In order to fully resolve the posterior distribution, we implement a state-of-the art Markov chain Monte Carlo (MCMC) method that scales well with respect to the dimension of the parameter space, enabling us to study realistic forward models, in two space dimensions, at a high level of grid refinement. Our implementation of the MCMC method provides the gold standard against which the aforementioned Gaussian approximations are assessed. We present numerical synthetic experiments where we quantify the capability of each of the ad hoc Gaussian approximation in reproducing the mean and the variance of the posterior distribution (characterized via MCMC) associated to a data assimilation problem. Both single-phase and two-phase (oil–water) reservoir models are considered so that fundamental differences in the resulting forward operators are highlighted. The main objective of our controlled experiments was to exhibit the substantial discrepancies of the approximation properties of standard ad hoc Gaussian approximations. Numerical investigations of the type we present here will lead to the greater understanding of the cost-efficient, but ad hoc, Bayesian techniques used for data assimilation in petroleum reservoirs and hence ultimately to improved techniques with more accurate uncertainty quantification.     

Nutrient Loading and Transformations in the Columbia River Estuary Determined by High-Resolution In Situ Sensors

The Columbia River estuary is characterized by relatively large tidal currents and water residence times of a few days or less. These and other environmental conditions tend to suppress water column productivity and favor the export of riverborne nutrients to the coastal ocean. However, hotspots of biological activity may allow for significant nutrient transformation and removal within the estuary, but these processes have previously been difficult to quantify due to the challenges of obtaining measurements at appropriate frequency and duration. In this study, nutrient biogeochemical dynamics within the salt-influenced region of the estuary were quantified using high-resolution in situ observations of nutrients and physical water properties. During 2010, three autonomous nutrient sensors (Satlantic SUNA, SubChem Systems Inc. APNA, WET Labs Cycle-PO4) that together measured nitrate?+?nitrite, orthophosphate, ammonium, silicic acid, and nitrite were deployed on fixed observatory platforms. Hourly measurements captured tidal fluctuations and permitted an analysis of river and ocean end-member mixing. The results suggested that during summer, the lower estuary released high concentrations of ammonium and phosphate despite low concentrations in the river and coastal ocean. This was likely a result of organic matter accumulation and remineralization in the estuarine turbidity maximum and the lateral bays adjacent to the main channel.     

Chemical dispersant effectiveness testing: influence of droplet coalescence

Thermodynamic and kinetic investigations were performed to determine the influence of coalescence of chemically dispersed crude oil droplets in saline waters. For the range of pH (4-10) and salinity (10 per thousand, 30 per thousand, 50 per thousand ) values studied, zeta-potential values ranged from -3 to -10 mV. As the interaction potential values calculated using Derjaguin-Landau-Verway-Overbeek (DLVO) theory were negative, the electrostatic barrier did not produce significant resistance to droplet coalescence. Coalescence kinetics of premixed crude oil and chemical dispersant were determined within a range of mean shear rates (Gm = 5, 10, 15, 20 s(-1)) and salinity (10 per thousand, 30 per thousand ) values. Coalescence reaction rates were modeled using Smoluchowski reaction kinetics. Measured collision efficiency values (alpha = 0.25) suggest insignificant resistance to coalescence in shear systems. Experimentally determined dispersant efficiencies (alpha = 0.35) were 10-50% lower than that predicted using a non-interacting droplet model (alpha = 0.0). Unlike other protocols in which the crude oil and dispersant are not premixed, salinity effects were not significant in this protocol. This approach allowed the effects of dispersant-oil contact efficiency eta(contact) to be separated from those of water column transport efficiency (eta(transport)) and coalescence efficiency (eta(coalescence)).     

Measurements of Faraday Rotation Through the Solar Corona During the 2009 Solar Minimum with the MESSENGER Spacecraft

Measurements of Faraday rotation through the solar corona were collected using the radio beacon aboard the MESSENGER spacecraft during the longest solar minimum in a century. As MESSENGER entered superior conjunction, the plane of polarization of its radio signal was observed to rotate as it traversed the circularly birefringent plasma of the Sun’s atmosphere. On time scales of less than three hours, these uncalibrated plane of polarization observations of Faraday rotation can be used to investigate the dynamic processes in the solar plasma, such as magnetohydrodynamic (MHD) waves and coronal mass ejections (CMEs). Here we describe the MESSENGER Faraday rotation experiment, the data processing conducted to obtain the plane of polarization, and the estimation of error.     

Measurement requirements for a Near-Earth Asteroid impact mitigation demonstration mission

A concept for an Impact Mitigation Preparation Mission, called Don Quijote, is to send two spacecrafts to a Near-Earth Asteroid (NEA): an Orbiter and an Impactor. The Impactor collides with the asteroid while the Orbiter measures the resulting change in the asteroid's orbit, by means of a Radio Science Experiment (RSE) carried out before and after the impact. Three parallel Phase A studies on Don Quijote were carried out for the European Space Agency: the research presented here reflects the outcomes of the study by QinetiQ. We discuss the mission objectives with regard to the prioritisation of payload instruments, with emphasis on the interpretation of the impact. The Radio Science Experiment is described and it is examined how solar radiation pressure may increase the uncertainty in measuring the orbit of the target asteroid. It is determined that to measure the change in orbit accurately a thermal IR spectrometer is mandatory, to measure the Yarkovsky effect. The advantages of having a laser altimeter are discussed. The advantages of a dedicated wide-angle impact camera are discussed and the field-of-view is initially sized through a simple model of the impact.     

Ensemble properties of comets in the Sloan Digital Sky Survey

We present the ensemble properties of 31 comets (27 resolved and 4 unresolved) observed by the Sloan Digital Sky Survey (SDSS). This sample of comets represents about 1 comet per 10 million SDSS photometric objects. Five-band (u, g, r, i, z) photometry is used to determine the comets’ colors, sizes, surface brightness profiles, and rates of dust production in terms of the Afρ formalism. We find that the cumulative luminosity function for the Jupiter Family Comets in our sample is well fit by a power law of the form N(<H) ∝ 10^{(0.49±0.05)H} for H < 18, with evidence of a much shallower fit N(<H) ∝ 10^{(0.19±0.03)H} for the faint (14.5 < H < 18) comets. The resolved comets show an extremely narrow distribution of colors (0.57 ± 0.05 in g ? r for example), which are statistically indistinguishable from that of the Jupiter Trojans. Further, there is no evidence of correlation between color and physical, dynamical, or observational parameters for the observed comets.     

Advected fields in maps: II. Dynamo action in the stretch–fold–shear map

The growth of magnetic field is considered in the stretch–fold–shear map in the limit of weak diffusion. Numerical results are given for insulating, perfectly conducting and periodic boundary conditions. The resulting eigenvalue branches and magnetic fields are related to eigenvalue branches for perfect dynamo action, obtained for zero diffusion using a complex variable formulation.

The effect of diffusion on these perfect dynamo modes depends on their structure, growth rate and the diffusive boundary conditions employed. In some cases, the effect of diffusion is a small perturbation, giving a correction going to zero in the limit of weak diffusion, with a scaling exponent given analytically. In other cases weak diffusion can entirely destroy a perfect dynamo branch. Diffusive boundary layers can also generate entirely new branches.

These different cases are elucidated, and within the framework of the asymptotic approximations used (which do not constitute a rigorous proof), it is seen that for all three boundary conditions employed, the stretch–fold–shear map is a fast dynamo.     

Magnetic Helicity in fast dynamos

The behaviour of magnetic helicity in kinematic dynamos at large magnetic Reynolds number is considered. Hughes, et al . [ Phys. Lett. A 223 , 167-172 (1996)] observe that the relative helicity tends to zero in the limit of large magnetic Reynolds number. This paper gives upper bounds on the helicity, by relating the helicity spectrum to the energy spectrum. These bounds are confirmed by numerical simulation and the distribution of helicity over scales is considered. Although it is found that the total helicity becomes small in the limit of high conductivity, there can remain significant, but cancelling, helicity at large and small scales of the field. This is illustrated by considering the evolution of helicity in the stretch-twist-fold dynamo picture.