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Summary Mineral compositional analysis of rocks is important for developing shale resources because the relationships between mineral compositions and petrophysical properties are critical for resource evaluation and completion optimization. Elementary properties are now routinely analyzed at wells in evaluating shale reservoirs. However, these properties have not been modeled in the three-dimensional (3D) reservoir. This is because an elemental composition has a physical constraint that is relatively easily adhered to in data analysis for wells but not in 3D modeling of reservoirs. A critical condition of elemental composition is that the sum of its components is equal to 100% to honor the mass-preservation principle. Traditional modeling methods do not satisfy this physical condition, sometimes producing nonphysical values, such as negative porosity values and fluid-saturation values greater than 100%. To date, only the compositional-modeling methods using a log-ratio transform can consistently satisfy this physical constraint. This paper presents modeling methods using additive log-ratio transform for modeling mineral compositions.

By using a database of 4,634 cold patches (high density and low electron temperature) and 4,700 hot patches (high density and high electron temperature) from Defense Meteorological Satellite Program F16 in 2005–2018 winter months (October–March), we present a statistical survey of the distributions of polar cap patches for different interplanetary magnetic field (IMF) orientations and ionospheric convection geometries. We investigate the dependence of cold and hot patches on local plasma transport and soft-electron precipitation. Our results indicate that: in winter, (a) more cold and hot patches occur in the stronger anti-sunward flow organized by different IMF orientations. (b) cold patches are frequent near the central polar cap, while hot patches are closer to the auroral oval. (c) enhanced anti-sunward flow (E × B drift) mainly contributes to cold patch occurrence under Bz < 0, and soft-electron precipitation contributes to hot patch occurrence both under southward and northward IMF. publishedVersion

Close attention was paid to contact effect between dam body and dam foundation of high earth rockfill dams. A soil–bedrock interface model is developed based on hyperbolic hardening rule of coarse-...

The weakly bound nuclei near the drip-line have many interesting properties which have been the hot spot for current nuclear physics research. Carbon isotopes can reach a high neutron to proton ratio and could be a good example to investigate the weakly bound and unbound nuclei. The drip-line nucleus 22C is the heaviest Borromean nucleus that people have observed and corresponds to a sub-shell at neutron number N =16 which is also observed in nitrogen and oxygen isotopes. For nuclei close to drip-line, the coupling of bound-, resonant- and continuum states becomes particularly important. But the traditional shell models with harmonic-oscillator basis are difficult to describe the resonant- and continuum states because the harmonic-oscillator potential always has a localized wave function and bound-eigenstates. One of the ways to include the continuum effects is employing the Berggren basis which is generated by finite potential like Woods-Saxon potential. With Berggren basis, the Schrodinger equation is expanded to complex-momentum space and the bound-, resonant- and continuum states could be taken into account on an equal footing. In addition, the necessity of chiral three-body forces has been shown in many ab-initio calculations like the reproducing of oxygen drip-line and low-lying spectrum of beryllium and boron isotopes. So it is meaningful to have a theoretical research of the carbon isotopes with considering both continuum effects and three-body forces. In this paper, we calculate the neutron-rich carbon isotopes with realistic Gamow shell model which starts from chiral N3LO two-body and chiral N2LO three-body forces. The three-body forces are included by normal-ordering approach. We use Woods-Saxon potential with 14C as a core to generate the Berggren basis. The parameters of Woods-Saxon potential are chosen to reproduce the single-particle energies close to experimental data. In Berggren basis, the continuum states are lying on the integral contour within the fourth quarter of complex-momentum space and discretized by Gauss-Legendre quadrature method for the further calculations. The effective interactions are built by Q ^ -box folded-diagram method within the model space of {1s1/2-bound, 0d5/2-bound, 0d3/2-resonance, d3/2-continuum, s1/2-continuum}. In order to deal with the large dimensions caused by the introducing of continuum states, we take an approximation that only two or less particles are allowed to occupy the continuum states. The ground states energies and the first 2+ excitation energies are calculated for even-mass carbon isotopes from A =16 to A =24 and achieve a good agreement with the experimental data. With our calculations, the 22C is shown to be the last bound even- A carbon isotope. In addition, by investigating the evolution of the first 2+ states, our results agree with the experiment that the sub-shell at N =14 in oxygen isotopes is observed to disappear in carbon isotopes, meanwhile the sub-shell at N =16 is reproduced by our calculations. Furthermore, we compare the ground states energies calculated with and without three-body forces and find out that the three-body forces will make the results more weakly bound and more consistent with experimental values especially for 22C and 24C. Meanwhile, the resonance widths which are calculated self-consistently by realistic Gamow shell model become broader. So our calculations give a support that the three-body forces will introduce a repulsive effect in neutron rich carbon isotopes and play a key role when describing the nuclear drip-line.

We employ the shell model with the chiral two- and three-nucleon forces. The effective Hamiltonian relevant to the valence-model space is computed microscopically. This framework is applied to the study of the neutron-drip line of the calcium isotopes. Our simulation shows that the calculated two-neutron separation energies are consistent with those of currently available experiment and $$^{70}\mathrm {Ca}$$ is possibly bound, as other theoretical attempts predict.

Background: The half-life of the famous $^{14}$C $\beta$ decay is anomalously long, with different mechanisms: the tensor force, cross-shell mixing, and three-body forces, proposed to explain the cancellations that lead to a small transition matrix element. Purpose: We revisit and analyze the role of the tensor force for the $\beta$ decay of $^{14}$C as well as of neighboring isotopes. Methods: We add a tensor force to the Gogny interaction, and derive an effective Hamiltonian for shell-model calculations. The calculations were carried out in a $p$-$sd$ model space to investigate cross-shell effects. Furthermore, we decompose the wave functions according to the total orbital angular momentum $L$ in order to analyze the effects of the tensor force and cross-shell mixing. Results: The inclusion of the tensor force significantly improves the shell-model calculations of the $\beta$-decay properties of carbon isotopes. In particular, the anomalously slow $\beta$ decay of $^{14}$C can be explained by the isospin $T=0$ part of the tensor force, which changes the components of $^{14}$N with the orbital angular momentum $L=0,1$, and results in a dramatic suppression of the Gamow-Teller transition strength. At the same time, the description of other nearby $\beta$ decays are improved. Conclusions: Decomposition of wave function into $L$ components illuminates how the tensor force modifies nuclear wave functions, in particular suppression of $\beta$-decay matrix elements. Cross-shell mixing also has a visible impact on the $\beta$-decay strength. Inclusion of the tensor force does not seem to significantly change, however, binding energies of the nuclei within the phenomenological interaction., Comment: 8 Pages,7 Figures

The even-even Ti isotopic chain, from A = 42 to 70, has been studied within the nuclear shell-model framework by employing an effective Hamiltonian which is derived by way of many-body perturbation theory from a chiral potential with two- and three-body forces, and includes three-body contributions which account for Pauli principle violations in nuclei with more than two valence particles. We consider 40Ca as a closed core and a model space spanned by the neutron and proton 0f1p orbitals with the addition of the 0g9/2 orbital for neutrons. Calculated two-neutron separation energies and excitation energies of the yrast 2+ states are reported and compared with the experimental data, which are available up to 62Ti. The present study intends to investigate the effects of the adopted effective interactions on the evolution of the shell structure., Comment: 9 pages, 6 figures

We report in this paper a study in terms of the nuclear shell model about the location of the calcium isotopes drip line. The starting point is considering the realistic two-body potential derived by Entem and Machleidt within chiral perturbation theory at next-to-next-to-next-to-leading order (N3LO), as well as a chiral three-body force at next-to-next-to-leading order (N2LO) whose structure and low-energy constants are consistent with the two-body potential. Then we construct the effective single-particle energies and residual interaction needed to diagonalize the shell-model Hamiltonian. The calculated two-neutron separation energies agree nicely with experiment until 56Ca, which is the heaviest isotope whose mass has been measured, and do not show any sign of two-neutron emission until 70Ca. We discuss the role of the choice of the model space in determining the neutron drip line, and also the dependence of the results on the parameters of the shell-model Hamiltonian., 9 pages, 9 figures, 3 tables, to be published in Physical Review C

Three-nucleon force and continuum play important roles in reproducing the properties of atomic nuclei around driplines. Therefore it is valuable to build up a theoretical framework where both effects can be taken into account to solve the nuclear Schrödinger equation. To this end, in this letter, we have expressed the chiral three-nucleon force within the continuum Berggren representation, so that bound, resonant and continuum states can be treated on an equal footing in the complex-momentum space. To reduce the model dimension and computational cost, the three-nucleon force is truncated at the normal-ordered two-body level and limited in the sd-shell model space, with the residual three-body term being neglected. We choose neutron-rich oxygen isotopes as the test ground because they have been well studied experimentally, with the neutron dripline determined. The calculations have been carried out within the Gamow shell model. The quality of our results in reproducing the properties of oxygen isotopes around the neutron dripline shows the relevance of the interplay between three-nucleon force and the coupling to continuum states. We also analyze the role played by the chiral three-nucleon force, by dissecting the contributions of the 2π exchange, 1π exchange and contact terms. Keywords: Three-nucleon forces, Continuum, Gamow shell model, Unstable nuclei, Binding energy, Spectra

Gamow shell model (GSM) is usually performed within the Woods-Saxon (WS) basis in which the WS parameters need to be determined by fitting experimental single-particle energies including their resonance widths. In the multi-shell case, such a fit is difficult due to the lack of experimental data of cross-shell single-particle energies and widths. In this paper, we develop an {\it ab-initio} GSM by introducing the Gamow Hartree-Fock (GHF) basis that is obtained using the same interaction as the one used in the construction of the shell-model Hamiltonian. GSM makes use of the complex-momentum Berggren representation, then including resonance and continuum components. Hence, GSM gives a good description of weakly bound and unbound nuclei. Starting from chiral effective field theory and employing many-body perturbation theory (MBPT) (called nondegenerate $\hat Q$-box folded-diagram renormalization) in the GHF basis, a multi-shell Hamiltonian ({\it sd-pf} shells in this work) can be constructed. The single-particle energies and their resonance widths can also been obtained using MBPT. We investigated $^{23-28}$O and $^{23-31}$F isotopes, for which multi-shell calculations are necessary. Calculations show that continuum effects and the inclusion of the {\it pf} shell are important elements to understand the structure of nuclei close to and beyond driplines., Accept by Physics Letters B

Summary One of the most-important bases for field-development planning is the estimate of hydrocarbon initially in place (HIIP), which has been traditionally estimated either deterministically or by Monte Carlo simulation. The classical volumetric calculation is the most-common deterministic method, and it requires the use of the averages of the reservoir variables and thus does not model the correlations of the input variables. It is well-known that ignoring the correlations among the reservoir variables can lead to incorrect estimations of hydrocarbon volumetrics. The Monte Carlo method uses the input means in the volumetric equation for random simulation of hydrocarbon volumes, yet allows modeling of the correlation between the input parameters. However, using the means and modeling the correlation of the properties are theoretically conflicting. This paper presents new parametric equations for volumetric calculation using mathematical correlation. Unlike the classical volumetric calculation, these equations are the exact expressions of the rigorous hydrocarbon volumetric equation. We discuss how these new equations enable the quantification of inaccuracy in hydrocarbon volumetric estimation by the classical method. Our examples will further show that the magnitude of inaccuracy of the traditional volumetrics depends on the reservoir characteristics; the inaccuracy is generally more significant for heterogeneous, low-quality, and tight reservoirs than for relatively homogeneous high-quality reservoirs.

HI intensity mapping is a new observational technique to survey the large-scale structure of matter using the 21 cm emission line of atomic hydrogen (HI). In this work, we simulate BINGO (BAO from Integrated Neutral Gas Observations) and SKA (Square Kilometre Array) phase-1 dish array operating in auto-correlation mode. For the optimal case of BINGO with no foregrounds, the combination of the HI angular power spectra with Planck results allows $w$ to be measured with a precision of $4\%$, while the combination of the BAO acoustic scale with Planck gives a precision of $7\%$. We consider a number of potentially complicating effects, including foregrounds and redshift dependent bias, which increase the uncertainty on $w$ but not dramatically; in all cases the final uncertainty is found to be $\Delta w < 8\%$ for BINGO. For the combination of SKA-MID in auto-correlation mode with Planck, we find that, in ideal conditions, $w$ can be measured with a precision of $4\%$ for the redshift range $0.35 < z < 3$ (i.e., for the bandwidth of $\Delta \nu = [350, 1050]$ MHz) and $2\%$ for $0 < z < 0.49$ (i.e., $\Delta \nu = [950, 1421]$ MHz). Extending the model to include the sum of neutrino masses yields a $95\%$ upper limit of $\sum m_\nu < 0.24$ eV for BINGO and $\sum m_\nu < 0.08$ eV for SKA phase 1, competitive with the current best constraints in the case of BINGO and significantly better than them in the case of SKA., Comment: 18 pages, 7 figures, 6 tables. Updated to match version accepted by MNRAS

We probe nuclear multipole resonances in the framework of the random-phase approximation by using the interaction obtained from the chiral effective field theory. The three-nucleon force is included in a form of the in-medium two-nucleon interaction which was derived from the chiral three-nucleon force. The isoscalar monopole, isoscalar dipole, isovector dipole and isoscalar quadrupole resonances of the closed-shell $^{56,68,78}$Ni have been investigated. The calculations reasonably reproduce the experimental multipole resonances of $^{56,68}$Ni, and well describe the pygmy dipole resonance and dipole polarizability measured in $^{68}$Ni. The multipole resonances of $^{78}$Ni, including pygmy dipole resonance and dipole polarizability, are predicted. The detailed effects of the tensor force and three-body force are analyzed by dissecting the chiral interaction. We find that in general the tensor force effect on electric giant resonances is not as significant as the effect from the three-body force, although the tensor force provides more than half of the binding energy. The effect from three-body force is strong in light nuclei. Particularly, three-body force is crucial for the formation of the pygmy resonance in calculations., Accepted by PHYSICAL REVIEW C

Starting from chiral two-nucleon (2NF) and chiral three-nucleon (3NF) potentials, we present a detailed study of 17Ne, a Borromean system, with the Gamow shell model which can capture continuum effects. More precisely, we take advantage of the normal-ordering approach to include the 3NF and the Berggren representation to treat bound, resonant and continuum states on equal footing in a complex-momentum plane. In our framework, 3NF is essential to reproduce the Borromean structure of 17Ne, while the continuum is more crucial for the halo property of the nucleus. The two-proton halo structure is demonstrated by calculating the valence proton density and correlation density. The astrophysically interesting 3 / 2 − excited state has its energy above the threshold of the proton emission, and therefore the two-proton decay should be expected from the state.

Cu substituted TiNi alloys have been investigated as hydrogen storage material for Ni-MH batteries by experiments and first principle calculations. The amount of Cu (x in TiNi1−xCux) is varied from 0.1 to 0.3. All of samples were prepared by mechanical alloying using a planetary high-energy ball mill and subsequent heat treatment at 750 °C for 0.5 h. The structural transformation was characterized by X-ray diffraction method (XRD) and scanning electron microscopy (SEM). It indicated that mechanical alloyed TiNi1−xCux alloys possessed broad diffraction peaks related to BCC structure. After heat treatment, studied Ti Ni Cu materials consisted of Ti(Ni,Cu) B2 phase as main phase. The cell parameter of B2 phase increased linearly with increasing Cu content. The influence of Cu substitution for Ni in TiNi cubic phases was investigated by first principle calculation. It is found that TiNi0.7Cu0.3 possessed the most negative enthalpy of formation. The stability of TiNi1−xCux phase increased with Cu content. The width of valence bands was enlarged by substituting Cu atom. The discharge capacities of annealed samples were tested by electrochemical measurements at galvanostatic conditions. The results showed that the substitution of Cu for Ni seemed to deteriorate the activation properties of TiNi based alloys. Among all of studied materials, unmodified TiNi showed the highest discharge capacity which is equal to 154 mAh/g. Accompanying the substitution of Cu for Ni, the discharge capacity and cycling abilities of Cu substituted TiNi alloys declined and increased, respectively. All of the substituted samples exhibited at least 97% retaining rate after 20 cycles.

The relative spatial position of each chain link and the phase angle of engaged tooth are not concerned in traditional sprocket designs. As such, the chain shifting might fail regularly. In this study, an optimized path of one chain for up-shifting is derived. Such path is formed by several bended chain links sustain yaw and roll. Empirically, appropriate bending, yaw and torsion angles for tooth chamfer, could be shorten the up-shifting distance and phase angle required, eventually facilitate not only more ascending points compactly arranged on sprocket but larger chain laterodeviation which curtails sprockets clearance and integrates the sprocket unit. Then again, the optimized shifting path promotes efficiency and decreases required movement yielded by derailleur tappet pressure.

Apatite-type lanthanum silicates draw researchers' attention due to their good performance as electrolyte materials for IT-SOFC (intermediate temperature solid oxide fuel cells). In this paper we present the synthesis of Sr-doped La10Si6O27 obtained by optimizing a water-based sol–gel process. The relevant synthesis parameters have been investigated to get pure, highly crystalline powders. The mechanisms occurring in the sol–gel reactions are discussed to improve the process of the sol formation. Using this optimized sol–gel process, pure apatite powders have been obtained by calcination from a temperature as low as 800 °C for 2 h and characterized using scanning electron microscopy and X-ray diffraction. The sintering treatment was performed at the temperature of 1500 °C leading to well-crystallized electrolytes likely to be used in fuel cell applications. Ionic conductivities have been measured in order to investigate the effect of the Sr-doping. The results show that the ionic conductivity is thermally activated and its value lies between 3 × 10−2 and 1 × 10−6 S cm−1 at 873 K as function of the composition and powder preparation conditions.

This chapter presents methods for constructing reservoir-model frameworks. A reservoir model framework is a representation of reservoir architecture, and it incorporates geological variables that segregate large heterogeneities in the reservoir. A framework without using faults is termed unfaulted framework and its main inputs are stratigraphic elements. A framework constructed with faults is termed faulted framework and it incorporates both stratigraphic elements and faults. A reservoir-model framework is also termed geocellular model because the 3D model is composed of discretized cells that are subsequently filled with reservoir properties. Heterogeneities of petrophysical properties of a reservoir cannot be accurately described without a 3D geocellular model framework.

Regression is one of the most commonly used multivariate statistical methods. Multivariate linear regression can integrate many explanatory variables to predict the target variable. However, collinearity due to intercorrelations in the explanatory variables leads to many surprises in multivariate regression. This chapter presents both basic and advanced regression methods, including standard least square linear regression, ridge regression and principal component regression. Pitfalls in using these methods for geoscience applications are also discussed.

This chapter first gives an overview of the main characteristics of seismic data and basic analytics using seismic data. It then presents identifications of facies and mapping of continuous reservoir properties using seismic data through mathematical correlation-based methods. The presentation emphasizes analytics in reservoir characterization using seismic attributes. In the last two to three decades, the generation of many seismic attributes from various methods, such as amplitude versus offset (AVO), inversion, and signal analysis, has become common, making seismic attributes part of geoscience big data. There is also an increasing trend in data-analytical methods for treating attribute data. The traditional use of seismic data for structural and stratigraphic interpretations of reservoirs is presented for construction of reservoir-model frameworks in Chap. 15.

Before the arrival of big data, statistical methods used in science and engineering were dominantly model-based with an emphasis on estimation unbiasedness. Although many traditional statistical methods work well with small datasets and a proper experimental design, they are less effective in handling some of the problems that have arisen out of big data. Artificial intelligence (AI) has led the way to data mining for discovering patterns and regularities from big data and for making predictions for scientific and technical applications. Although the movement was initially led by computer scientists, statisticians, scientists and engineers are now all involved, thus strengthening the trend.

This chapter presents geostatistical estimation methods for modeling continuous variables. These include several kriging techniques, namely, simple kriging, ordinary kriging, kriging with varying mean, collocated cokriging, and factorial kriging. Geoscientists who want only a basic understanding of kriging estimation can skip the advanced kriging methods, but simple kriging is necessary for understanding stochastic simulation. Similarly, collocated cokriging will be useful for understanding collocated cosimulation. Stochastic (co)simulation is presented in Chap. 17.

This chapter presents statistical methods and their applications to geoscience data analysis. These include descriptive statistics and change of scale problem in characterizing rock and petrophysical properties, and mitigations of sampling bias in exploration and production.

This chapter presents methods for classifying lithofacies from well logs. Lithofacies are a discrete variable that describes categories of the rock quality, defined as having two or more states. Lithofacies represent small- to intermediate-scale heterogeneities in geological analysis of subsurface formations. Different lithofacies often have different petrophysical properties and can impact subsurface fluid flow. Cores are generally limited, and lithofacies data are often derived from well logs in reservoir characterization.

One of the most important bases for field development planning is the estimate of hydrocarbon initially in place. The volumetric estimation impacts reservoir management and investment decision. When the estimate is too optimistic, it can lead to excessive investments; when the estimate is too pessimistic, it can lead to under investments or inopportune asset disposition. This chapter presents two approaches for volumetric estimations: parametric methods and model-based methods.

This chapter presents an overview of petrophysical analysis, mainly from the viewpoint of data analytics. Petrophysical analysis is critical in a reservoir study because it provides a primary source of input data for integrated reservoir characterization and resource evaluation. Wireline logging provides various recordings of subsurface formation properties and well logs are the main sources for petrophysical analysis. Logging records are first used for single-well evaluations and then extended to fieldwide resource evaluation and reservoir modeling.

Geological processes and reservoir properties are not random; why should one use probabilistic analytics in geosciences? Probability is a useful theory not just for dealing with randomness but also for dealing with non randomness and uncertainty. Many geoscience problems are indeterministic, meaning that it is impossible to perfectly describe them by a deterministic function. This is due to the complexity of physical processes that took place in geological time and limited data, which leads to uncertainties in their analysis and prediction.

A reservoir model is a computer-based digital representation of the subsurface formation and its rock and petrophysical properties. Building a reservoir model includes the construction of a structural and stratigraphic model and determining the spatial distributions of facies and various petrophysical properties in the model. Constructing a good reservoir model requires multidisciplinary analyses and integration of geological, geophysical, petrophysical, and reservoir engineering data using scientific and statistical inferences. This chapter presents an introduction to reservoir modeling, including the aims, principles, and general workflows. The subsequent chapters present various modeling methods and their applications.

This chapter presents geostatistical characterizations of geospatial data, focusing on the description of the spatial continuity/discontinuity of reservoir properties using variogram, covariance or correlation function. Other geostatistical methods used for modeling reservoir properties are presented in Chaps. 16, 17 and 18.

Facies analysis typically focuses on geological descriptions and physical characteristics of deposits. It has typically been emphasized in exploration and appraisal, including prospect generation and reservoir delineation. Facies modeling can be useful in both exploration and field development. In practice, because of limited core and well-log-derived facies data, there have been disconnects between facies analysis and modeling, which can lead to a geologically unrealistic model. It is important to integrate facies analysis into facies modeling for field development.

Because facies are nominal variables, their modeling methods are different from the modeling methods for continuous variables. Kriging and stochastic simulation methods presented in Chaps. 16 and 17 cannot be directly used for construction of a facies model; they can be modified for facies modeling, or totally different methods are used. Although facies are often modeled before modeling petrophysical variables, modeling methods for continuous variables were presented in the earlier chapters because it is easier to understand facies modeling methods after understanding kriging and stochastic simulation for continuous variables. This chapter presents several facies modeling methods, including indicator kriging, sequential indicator simulation and its variations, object-based modeling, truncated Gaussian and plurigaussian simulations, and simulation using multipoint statistics.

This chapter presents model upscaling, validation and history match. Upscaling is necessary when a reservoir model is made at a very fine scale, and its cell count is excessively large for dynamic simulators. Many geocellular models range from tens of millions to hundreds of millions of cells and cannot be simulated numerically in a reasonable time with the current mathematical algorithms and computing technology. The main principle of upscaling is the accurate representation of the fine-scaled model by the upscaled model, including the preservation of volumetrics and the equivalencies in flow and production profile between the fine and coarse models.

This chapter presents methods for modeling water saturations of subsurface formations. It first gives an overview of basic physics of fluid distributions in porous media. It then presents several methods for 3D modeling of water saturation, including various capillarity-related saturation-height functions and a geostatistical method.

We present a study of the role played by realistic three-body forces in providing a reliable monopole component of the effective shell-model Hamiltonian. To this end, starting from a nuclear potential built up within the chiral perturbation theory, we derive effective shell-model Hamiltonians with and without the contribution of the three-body potential and compare the results of shell-model calculations with a set of observables that evidence shell-evolution properties. The testing ground of our investigation are nuclei belonging to fp shell, since the shell evolution towards shell closures in 48Ca and 56Ni provides a paradigm for shell-model Hamiltonians. Our analysis shows that only by including contributions of the three-body force the monopole component of the effective shell-model Hamiltonian is then able to reproduce the experimental shell evolution towards and beyond the closure at N=28., 13 pages, 22 figures, to be published in Phys. Rev. C

This chapter presents geostatistical methods for stochastically simulating continuous geospatial properties. For facilitating the presentations, it uses many temporal data in stochastic simulations benefiting from the 1D simplification. The commonly used simulation methods for spatial data include sequential Gaussian simulation and spectral simulation. Unlike estimation methods (e.g., regression and kriging), one main goal of stochastic simulation is to model the heterogeneities of physical properties.

Heterogeneity is one of the most complex problems in subsurface formations, and it is ubiquitous in many geoscience disciplines. Fluid storage and flow in porous media are governed by a variety of geological and petrophysical variables, including structure, stratigraphy, facies, lithology, porosity, and permeability. These variables all contribute to subsurface heterogeneities and have different scales, often in a hierarchical scheme.