An inverse analysis of fluid flow through granular media using differentiable lattice Boltzmann method

This study presents a method for the inverse analysis of fluid flow problems.The focus is put on accurately determining boundary conditions and characterizing the physical properties of granular media,such as permeability,and fluid components,like viscosity.The primary aim is to deduce either constant pressure head or pressure profiles,given the known velocity field at a steady-state flow through a conduit containing obstacles,including walls,spheres,and grains.The lattice Boltzmann method(LBM)combined with automatic differentiation(AD)(AD-LBM)is employed,with the help of the GPU-capable Taichi programming language.A lightweight tape is used to generate gradients for the entire LBM simulation,enabling end-to-end backpropagation.Our AD-LBM approach accurately estimates the boundary conditions for complex flow paths in porous media,leading to observed steady-state velocity fields and deriving macro-scale permeability and fluid viscosity.The method demonstrates significant advantages in terms of prediction accuracy and computational efficiency,making it a powerful tool for solving inverse fluid flow problems in various applications.

Simultaneous Waveform Inverse Modelling for Litho-Fluid Prediction in an Old Marginal, “Agbbo”Field, Onshore Niger Delta, Nigeria

Simultaneous waveform inversion was used to predict lithofacies and fluid type across the field. Very often, characterizing reservoirs in terms of lithology and fluid type using conventional methods is replete with uncertainties, especially in marginal fields. An approach is employed in this study that integrated rock physics and waveform inverse modelling for lithology and fluid-type characterization to appropriately identify potential hydrocarbon saturated zones and their corresponding lithology. Seismic and well-log data were analyzed using Hampson Russel software. The method adopted includes lithofacies and fluid content analysis using rock physics parameters and seismic simultaneous inverse modelling. Rock physics analysis identified 2 broad reservoirs namely: HDZ1 and HDZ2 reservoirs. Results from the inverse modelling showed that low values of acoustic impedance from 19,743 to 20,487 (ft/s)(g/cc) reflect hydrocarbon-bearing reservoirs while medium to high values shows brine and shale respectively, with brine zone ranging from 20,487 to 22,531 (ft/s)(g/cc) and shale above 22,531 (ft/s)(g/cc). Two lithofacies were identified from inversion analysis of Vp/Vs and Mu-Rho, namely: sand and shale with VpVs 1.95 values respectively. Mu-Rho > 12.29 (GPa)(g/cc) and <12.29 (GPa) (g/cc) represent sand and shale respectively. From 3D volume, it was observed that a high accumulation of hydrocarbon was observed to be saturated at the north to the eastern part of the field forming a meandering channel. Sands were mainly distributed around the northeastern to the southwestern part of the field, that tends to be away from Well 029. This was also validated by the volume of rigidity modulus (Mu-Rho) showing high values indicating sands fall within the northeastern part of the field.

Zoeppritz equation-based prestack inversion and its application in fluid identification

Prestack seismic inversion methods adopt approximations of the Zoeppritz equations to describe the relation between reflection coefficients and P-wave velocity, S-wave velocity, and density. However, the error in these approximations increases with increasing angle of incidence and variation of the elastic parameters, which increases the number of inversion solutions and minimizes the inversion accuracy. In this study, we explore a method for solving the reflection coefficients by using the Zoeppritz equations. To increase the accuracy of prestack inversion, the simultaneous inversion of P-wave velocity, S-wave velocity, and density by using prestack large-angle seismic data is proposed based on generalized linear inversion theory. Moreover, we reduce the ill posedness and increase the convergence of prestack inversion by using the regularization constraint damping factor and the conjugate gradient algorithm. The proposed prestack inversion method uses prestack large-angle seismic data to obtain accurate seismic elastic parameters that conform to prestack seismic data and are consistent with logging data from wells.

Fluid identification and effective fracture prediction based on frequency-dependent AVOAz inversion for fractured reservoirs

Fluid and effective fracture identification in reservoirs is a crucial part of reservoir prediction.The frequency-dependent AVO inversion algorithms have proven to be effective for identifying fluid through its dispersion property.However,the conventional frequency-dependent AVO inversion algorithms based on Smith&Gidlow and Aki&Richards approximations do not consider the acquisition azimuth of seismic data and neglect the effect of seismic anisotropic dispersion in the actual medium.The aligned fractures in the subsurface medium induce anisotropy.The seismic anisotropy should be considered while accounting for the seismic dispersion properties through fluid-saturated fractured reservoirs.Anisotropy in such reservoirs is frequency-related due to wave-induced fluid-flow(WIFF)between interconnected fractures and pores.It can be used to identify fluid and effective fractures(fluid-saturated)by using azimuthal seismic data via anisotropic dispersion properties.In this paper,based on Rüger’s equation,we derived an analytical expression in the frequency domain for the frequencydependent AVOAz inversion in terms of fracture orientation,dispersion gradient of isotropic background rock,anisotropic dispersion gradient,and the dispersion at a normal incident angle.The frequency-dependent AVOAz equation utilizes azimuthal seismic data and considers the effect of both isotropic and anisotropic dispersion.Reassigned Gabor Transform(RGT)is used to achieve highresolution frequency division data.We then propose the frequency-dependent AVOAz inversion method to identify fluid and characterize effective fractures in fractured porous reservoirs.Through application to high-qualified seismic data of dolomite and carbonate reservoirs,the results show that the method is useful for identifying fluid and effective fractures in fluid-saturated fractured rocks.

Fluid discrimination incorporating viscoelasticity and frequency-dependent amplitude variation with offsets inversion

Frequency-dependent amplitude versus offset(FAVO)inversion is a popular method to estimate the frequency-dependent elastic parameters by using amplitude and frequency information of pre-stack seismic data to guide fluid identification.Current frequency-dependent AVO inversion methods are mainly based on elastic theory without the consideration of the viscoelasticity of oil/gas.A fluid discrimination approach is proposed in this study by incorporating the viscoelasticity and relevant FAVO inversion.Based on viscoelastic and rock physics theories,a frequency-dependent viscoelastic solid-liquid decoupling fluid factor is initially constructed,and its sensitivity in fluid discrimination is compared with other conventional fluid factors.Furthermore,a novel reflectivity equation is derived in terms of the viscoelastic solid-liquid decoupling fluid factor.Due to the introduction of viscoelastic theory,the proposed reflectivity is related to frequency,which is more widely applicable than the traditional elastic reflectivity equation directly derived from the elastic reflectivity equation on frequency.Finally,a pragmatic frequency-dependent inversion method is introduced to verify the feasibility of the equation for frequency-dependent viscoelastic solid-liquid decoupling fluid factor prediction.Synthetic and field data examples demonstrate the feasibility and stability of the proposed approach in fluid discrimination.

Fluid discrimination incorporating amplitude variation with angle inversion and squirt flow of the fluid

Pre-stack seismic inversion is an important method for fluid identification and reservoir characterization in exploration geophysics. In this study, an effective fluid factor is initially established based on Biot poroelastic theory, and a pre-stack seismic inversion method based on Bayesian framework is used to implement the fluid identification. Compared with conventional elastic parameters, fluid factors are more sensitive to oil and gas. However, the coupling effect between rock porosity and fluid content is not considered in conventional fluid factors, which may lead to fuzzy fluid identification results. In addition,existing fluid factors do not adequately consider the physical mechanisms of fluid content, such as squirt flow between cracks and pores. Therefore, we propose a squirt fluid factor(SFF) that minimizes the fluid and pore mixing effects and takes into account the squirt flow. On this basis, a novel P-wave reflection coefficient equation is derived, and the squirt fluid factor is estimated by amplitude variation with offset(AVO) inversion method. The new reflection coefficient equation has sufficient accuracy and can be utilized to estimate the parameters. The effectiveness and superiority of the proposed method in fluid identification are verified by the synthetic and field examples.

An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid

In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.

Fluids discrimination by ray-path elastic impedance inversion: A successful case from Sulige tight gas field

Existing seismic prediction methods struggle to effectively discriminate between fluids in tight gas reservoirs,such as those in the Sulige gas field in the Ordos Basin,where porosity and permeability are extremely low and the relationship between gas and water is complicated.In this paper,we have proposed a comprehensive seismic fluid identification method that combines ray-path elastic impedance(REI)inversion with fluid substitution for tight reservoirs.This approach is grounded in geophysical theory,forward modeling,and real data applications.We used geophysics experiments in tight gas reservoirs to determine that Brie's model is better suited to calculate the elastic parameters of mixed fluids than the conventional Wood’s model.This yielded a more reasonable and accurate fluid substitution model for tight gas reservoirs.We developed a forward model and carried out inversion of REI.which reduced the non-uniqueness problem that has plagued elastic impedance inversion in the angle domain.Our well logging forward model in the ray-path domain with different fluid saturations based on a fluid substitution model proved that REI identifies fluids more accurately when the ray parameters are large.The distribution of gas saturation can be distinguished from the crossplot of REI(p=0.10)and porosity.The inverted ray-path elastic impedance profile was further used to predict the porosity and gas saturation profile.Our new method achieved good results in the application of 2D seismic data in the western Sulige gas field.

Quantification of Reservoir Fluid Prediction Uncertainty Using AVO Fluid Inversion (AFI)

The AVO fluid inversion (AFI) technique was used to assess for fluids at the target levels of OPL-X in the deepwater Niger Delta, Nigeria. In this study, attempt is made to get a quantitative probability estimate of the possible reservoir fluids in both the shallow and deeper target levels. This was achieved through the development of a stochastic AVO model and an inversion to probability of different fluids using the Bayesian approach. AVO Fluid Inversion (AFI) technique provides a robust and inexpensive method for identifying potential hydrocarbon-filled reservoirs and provides a quantitative estimates of the uncertainties inherent in the prediction.

Direct hydrocarbon identification in shale oil reservoirs using fluid dispersion attribute based on an extended frequency-dependent seismic inversion scheme

The identification of hydrocarbons using seismic methods is critical in the prediction of shale oil res-ervoirs.However,delineating shales of high oil saturation is challenging owing to the similarity in the elastic properties of oil-and water-bearing shales.The complexity of the organic matter properties associated with kerogen and hydrocarbon further complicates the characterization of shale oil reservoirs using seismic methods.Nevertheless,the inelastic shale properties associated with oil saturation can enable the utilization of velocity dispersion for hydrocarbon identification in shales.In this study,a seismic inversion scheme based on the fluid dispersion attribute was proposed for the estimation of hydrocarbon enrichment.In the proposed approach,the conventional frequency-dependent inversion scheme was extended by incorporating the PP-wave reflection coefficient presented in terms of the effective fluid bulk modulus.A rock physics model for shale oil reservoirs was constructed to describe the relationship between hydrocarbon saturation and shale inelasticity.According to the modeling results,the hydrocarbon sensitivity of the frequency-dependent effective fluid bulk modulus is superior to the traditional compressional wave velocity dispersion of shales.Quantitative analysis of the inversion re-sults based on synthetics also reveals that the proposed approach identifies the oil saturation and related hydrocarbon enrichment better than the above-mentioned conventional approach.Meanwhile,in real data applications,actual drilling results validate the superiority of the proposed fluid dispersion attribute as a useful hydrocarbon indicator in shale oil reservoirs.

Predictors of short-term outcome in patients with acute middle cerebral artery occlusion: unsuitability of fluid-attenuated inversion recovery vascular hyperintensity scores

Fluid-attenuated inversion recovery （FLAIR） vascular hyperintensity （FVH） is used to assess leptomeningeal collateral circulation, but clinical outcomes of patients with FVH can be very different. The aim of the present study was to assess a FVH score and explore its relationship with clinical outcomes. Patients with acute ischemic stroke due to middle cerebral artery M1 occlusion underwent magnetic resonance imaging and were followed up at 10 days （National Institutes of Health Stroke Scale） and 90 days （modified Rankin Scale） to determine short-term clinical outcomes. Effective collateral circulation indirectly improved recovery of neurological function and short-term clinical outcome by extending the size of the pial penumbra and reducing infarct lesions. FVH score showed no correlation with 90-day functional clinical outcome and was not sufficient as an independent predictor of short-term clinical outcome.

A high resolution inversion method for fluid factor with dynamic dryrock V_(P)/V_(S) ratio squared

As an important indicator parameter of fluid identification,fluid factor has always been a concern for scholars.However,when predicting Russell fluid factor or effective pore-fluid bulk modulus,it is necessary to introduce a new rock skeleton parameter which is the dry-rock VP/VS ratio squared(DVRS).In the process of fluid factor calculation or inversion,the existing methods take this parameter as a static constant,which has been estimated in advance,and then apply it to the fluid factor calculation and inversion.The fluid identification analysis based on a portion of the Marmousi 2 model and numerical forward modeling test show that,taking the DVRS as a static constant will limit the identification ability of fluid factor and reduce the inversion accuracy.To solve the above problems,we proposed a new method to regard the DVRS as a dynamic variable varying with depth and lithology for the first time,then apply it to fluid factor calculation and inversion.Firstly,the exact Zoeppritz equations are rewritten into a new form containing the fluid factor and DVRS of upper and lower layers.Next,the new equations are applied to the four parameters simultaneous inversion based on the generalized nonlinear inversion(GNI)method.The testing results on a portion of the Marmousi 2 model and field data show that dynamic DVRS can significantly improve the fluid factor identification ability,effectively suppress illusion.Both synthetic and filed data tests also demonstrate that the GNI method based on Bayesian deterministic inversion(BDI)theory can successfully solve the above four parameter simultaneous inversion problem,and taking the dynamic DVRS as a target inversion parameter can effectively improve the inversion accuracy of fluid factor.All these results completely verified the feasibility and effectiveness of the proposed method.

Seismic fluid identification using a nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo method

Elastic impedance inversion with high efficiency and high stability has become one of the main directions of seismic pre-stack inversion. The nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo （MCMC） method is proposed in this paper, combining conventional MCMC method based on global optimization with a preconditioned conjugate gradient （PCG） algorithm based on local optimization, so this method does not depend strongly on the initial model. It converges to the global optimum quickly and efficiently on the condition that effi- ciency and stability of inversion are both taken into consid- eration at the same time. The test data verify the feasibility and robustness of the method, and based on this method, we extract the effective pore-fluid bulk modulus, which is applied to reservoir fluid identification and detection, and consequently, a better result has been achieved.

PARAMETERS INVERSION OF FLUID-SATURATED POROUS MEDIA BASED ON NEURAL NETWORKS

The multi- layers feedforward neural network is used for inversion ofmaterial constants of fluid-saturated porous media. The direct analysis of fluid-saturated porousmedia is carried out with the boundary element method. The dynamic displacement responses obtainedfrom direct analysis for prescribed material parameters constitute the sample sets training neuralnetwork. By virtue of the effective L-M training algorithm and the Tikhonov regularization method aswell as the GCV method for an appropriate selection of regu-larization parameter, the inversemapping from dynamic displacement responses to material constants is performed. Numerical examplesdemonstrate the validity of the neural network method.

Diffusion-weighted imaging and fluid-attenuated inversion recovery sequence in sporadic Creutzfeldt-Jakob disease:One-case report

The diagnosis of sporadic Creutzfeldt-Jakob disease （sCJD） is extremely difficult. Diffusion-weighted imaging has been shown to be the most sensitive technique for the detection of signal alterations in sCJD patients. The present study analyzed the diagnostic value of diffusion-weighted imaging and fluid-attenuated inversion recovery sequence in the early stage of sCJD in one female patient and correlated the clinical symptoms during disease course and magnetic resonance manifestations. Thalamic and basal ganglia hyperintensities were observed on magnetic resonance images in a very early stage, i.e., when the clinical typical manifestations of the disease were not present. With the progression of the disease, cortical and basal ganglia hyperintensities were observed on magnetic resonance images, showing an obvious cerebral atrophy. These findings suggest that diffusion-weighted imaging and fluid-attenuated inversion recovery sequence are helpful in diagnosing sCJD.

Numerical simulation of(T_2,T_1) 2D NMR and fluid responses

One-dimensional nuclear magnetic resonance （1D NMR） logging technology is limited for fluid typing, while two-dimensional nuclear magnetic resonance （2D NMR） logging can provide more parameters including longitudinal relaxation time （71） and transverse relaxation time （T2） relative to fluid types in porous media. Based on the 2D NMR relaxation mechanism in a gradient magnetic field, echo train simulation and 2D NMR inversion are discussed in detail. For 2D NMR inversion, a hybrid inversion method is proposed based on the damping least squares method （LSQR） and an improved truncated singular value decomposition （TSVD） algorithm. A series of spin echoes are first simulated with multiple waiting times （Tws） in a gradient magnetic field for given fluid models and these synthesized echo trains are inverted by the hybrid method. The inversion results are consistent with given models. Moreover, the numerical simulation of various fluid models such as the gas-water, light oil-water, and vicious oil-water models were carried out with different echo spacings （TEs） and Tws by this hybrid method. Finally, the influences of different signal-to-noise ratios （SNRs） on inversion results in various fluid models are studied. The numerical simulations show that the hybrid method and optimized observation parameters are applicable to fluid typing of gas-water and oil-water models.

Seismic AVO statistical inversion incorporating poroelasticity

Seismic amplitude variation with offset(AVO) inversion is an important approach for quantitative prediction of rock elasticity,lithology and fluid properties.With Biot-Gassmann's poroelasticity,an improved statistical AVO inversion approach is proposed.To distinguish the influence of rock porosity and pore fluid modulus on AVO reflection coefficients,the AVO equation of reflection coefficients parameterized by porosity,rock-matrix moduli,density and fluid modulus is initially derived from Gassmann equation and critical porosity model.From the analysis of the influences of model parameters on the proposed AVO equation,rock porosity has the greatest influences,followed by rock-matrix moduli and density,and fluid modulus has the least influences among these model parameters.Furthermore,a statistical AVO stepwise inversion method is implemented to the simultaneous estimation of rock porosity,rock-matrix modulus,density and fluid modulus.Besides,the Laplace probability model and differential evolution,Markov chain Monte Carlo algorithm is utilized for the stochastic simulation within Bayesian framework.Models and field data examples demonstrate that the simultaneous optimizations of multiple Markov chains can achieve the efficient simulation of the posterior probability density distribution of model parameters,which is helpful for the uncertainty analysis of the inversion and sets a theoretical fundament for reservoir characterization and fluid discrimination.

Exact solutions for steady flows of second-grade fluids

This paper aims to investigate exact solutions for a second-grade fluid flow with the inverse method. By assuming the relation between the vorticity field and the streamfunction, the exact solutions of the motion of plane second-grade fluids are investigated and obtained. The solutions obtained include simple Couette flows, slit jet flows and uniform flows over a series of distributed obstacles.

Prestack inversion identification of organic reef gas reservoirs of Permian Changxing Formation in Damaoping area, Sichuan Basin, SW China

Organic reef reservoirs in the platform margin of Kaijiang-Liangping trough in Damaoping area, Sichuan Basin are thin in single layer, fast in lateral variation, and have small P-impedance difference from the surrounding rock, it is difficult to identify and predict the reservoirs and fluid properties by conventional post-stack inversion. Through correlation analysis of core test data and logging P-S wave velocity, this work proposed a formula to calculate the shear wave velocity in different porosity ranges, and solved the issue that some wells in the study area have no S-wave data. AVO forward analysis reveals that formation porosity is the main factor affecting the variation of AVO type, the change of water saturation cannot affect the AVO type, but it has an effect on the change range of AVO. Through cross-plotting analysis of elastic parameters, it is found that fluid factor is a parameter sensitive to gas-bearing property of organic reef reservoir in the study area. By comparing results of post-stack impedance inversion, post-stack high frequency attenuation property, pre-stack simultaneous inversion and AVO anomaly analysis of angle gathers, it is found that the gas-bearing prediction of organic reef reservoirs by using fluid factor derived from simultaneous pre-stack inversion had the highest coincidence rate with actual drilling data. At last, according to the characteristics of fluid factor distribution, the favorable gas-bearing area of the organic reef reservoir in Changxing Formation was predicted, and the organic reef trap at the top of Changxing Formation in Block A of Damaoping area was sorted out as the next exploration target.

The applicability and underlying factors of frequency-dependent amplitude-versus-offset(AVO)inversion

Recently,the great potential of seismic dispersion attributes in oil and gas exploration has attracted extensive attention.The frequency-dependent amplitude versus offset(FAVO)technology,with dispersion gradient as a hydrocarbon indicator,has developed rapidly.Based on the classical AVO theory,the technology works on the assumption that elastic parameters are frequency-dependent,and implements FAVO inversion using spectral decomposition methods,so that it can take dispersive effects into account and effectively overcome the limitations of the classical AVO.However,the factors that affect FAVO are complicated.To this end,we construct a unified equation for FAVO inversion by combining several Zoeppritz approximations.We study and compare two strategies respectively with(strategy 1)and without(strategy 2)velocity as inversion input data.Using theoretical models,we investigate the influence of various factors,such as the Zoeppritz approximation used,P-and S-wave velocity dispersion,inversion input data,the strong reflection caused by non-reservoir interfaces,and the noise level of the seismic data.Our results show that FAVO inversion based on different Zoeppritz approximations gives similar results.In addition,the inversion results of strategy 2 are generally equivalent to that of strategy 1,which means that strategy 2 can be used to obtain dispersion attributes even if the velocity is not available.We also found that the existence of non-reservoir strong reflection interface may cause significant false dispersion.Therefore,logging,geological,and other relevant data should be fully used to prevent this undesirable consequence.Both the P-and S-wave related dispersion obtained from FAVO can be used as good indicators of a hydrocarbon reservoir,but the P-wave dispersion is more reliable.In fact,due to the mutual coupling of P-and S-wave dispersion terms,the P-wave dispersion gradient inverted from PP reflection seismic data has a stronger hydrocarbon detection ability than the S-wave dispersion gradient.Moreover,there is little difference in using post-stack data or pre-stack angle gathers as inversion input when only the P-wave dispersion is desired.The real application examples further demonstrate that dispersion attributes can not only indicate the location of a hydrocarbon reservoir,but also,to a certain extent,reveal the physical properties of reservoirs.