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Compositional simulation is typically used to study subsurface displacement processes that involve complex physics. Such processes are highly nonlinear and they occur at the interplay of phase thermodynamics (phase stability and split) and rock/fluid interaction (relative permeability). To better characterize the convergence failures of compositional nonlinear solvers, we study the analytical and discrete forms of the component flux functions for the transport problem. We focus on isothermal, two-hydrocarbon phase models in the presence of viscous forces. Our analysis shows that the component flux function exhibits both kinks and inflection points, as described in the literature. We study the kinks due to the crossing of phase boundaries and we show their impact on residual terms. We, then, investigate the location, direction, and evolution of the inflection loci inside the two-phase region. Our findings suggest that these inflection loci may exhibit complex features, as a result of which an a priori determination of such loci is computationally prohibitive. We propose a nonlinear solver that detects phase boundaries and guides the Newton update to avoid crossing these boundaries. Our numerical results show that this phase change detection solver has superior performance in comparison with standard Newton solvers for a wide range of cases. Article Highlights: We study the nonlinearities of compositional flow and transport. Our analysis of the component flux function shows that it exhibits both kinks and inflection loci that have significant impact on the nonlinear convergence of the Newton solver. A new nonlinear solver that detects phase boundaries and adjusts the Newton update was developed. This led to considerable improvement of the nonlinear solution process. [ABSTRACT FROM AUTHOR]

Reservoir simulation using high-fidelity fluid models is typically employed to study subsurface displacement processes that involve complex physics. Such processes are highly nonlinear and occur at the interplay of phase thermodynamics (phase stability and split) and rock/fluid interaction (relative-permeability). Due to its nonlinearity, relative-permeability is an important constitutive of the conservation equations, and has a significant impact on the simulation. Dependence of the phase relative-permeability on fluid compositions, pressure, and temperature is well-documented. In compositional reservoir simulation, however, relative-permeability is typically modeled as a function of phase saturation. Such an approach may lead to serious discontinuities in relative-permeability. To alleviate this issue, several authors proposed models based on phase state indicators (density, parachor, Gibbs Free Energy). However, such techniques cannot represent the complete degrees of freedom that are exhibited by compositional displacements. In this work, we present a relative-permeability model based on a parameterization of the compositional space. The model is independent of the hydrocarbon phase labeling as gas or oil. We show that our proposed model (1) applies regardless of the degrees-of-freedom of the compositional displacement problem, and (2) is guaranteed to yield a continuous relative-permeability function across the entire compositional space. We have implemented this model in our research simulator, and we present test cases using traditional relative-permeability models, as well as, numerical results that compare the nonlinear performance. [ABSTRACT FROM AUTHOR]

Abstract In reservoir simulation, solution of the coupled systems of nonlinear algebraic equations that are associated with fully-implicit (backward Euler) discretization is challenging. Having a robust and efficient nonlinear solver is necessary in order for reservoir simulation to serve as the primary tool for managing the recovery processes of large-scale reservoirs. However, there are several outstanding challenges that are intimately connected to the highly nonlinear nature of the problem. Given a set of sources and sinks, the variation in the total velocity can span many orders of magnitude due to extreme contrasts in the permeability field in large-scale subsurface porous formations. Moreover, multiple and complex saturation fronts must be properly resolved throughout the three-dimensional reservoir model of interest. Add to that numerical simulation studies entail making field-scale predictions over many decades, and the challenge of developing robust and efficient nonlinear solvers across a very wide parameter space becomes clear. Here, we develop a continuation method based on the use of a dissipation operator. We focus on nonlinear two-phase flow and transport in heterogeneous formations in the presence of viscous, gravitational, and capillary forces. The homotopy is constructed by adding numerical dissipation to the coupled discrete conservation equations. A continuation parameter is introduced to control the amount of dissipation. Numerical evidence of multi-dimensional models and detailed analysis of single-cell problems are used to explain how the dissipation operator improves the nonlinear convergence of the coupled system of equations. An adaptive strategy to determine the dissipation coefficient is proposed. The dissipation level is computed locally for each cell interface. We demonstrate the efficiency of the dissipation-based continuation (DBC) nonlinear solver using several examples, including 1D scalar transport and 2D heterogeneous problems with fully-coupled flow and transport. The DBC solver has better convergence properties compared with the standard damped-Newton solvers used in reservoir simulation. Highlights • Develop a continuation method based on dissipation operator. • Numerical evidences and detailed analysis are provided. • An adaptive strategy to determine the optimum dissipation coefficient. • The new solver exhibits superior convergence properties. • The new solver works robustly without parameter tuning. [ABSTRACT FROM AUTHOR]

Supercritical carbon dioxide (sc-CO 2) is an alternative to water for stimulation of low permeability systems such as shale gas and geothermal resources. Previously core-scale experimental studies have compared the behavior of CO 2 to water injection for sample breakdown. Due to differences in experimental setup and core sample preparation, inconsistent or even apparently contradictory conclusions have resulted. To reconcile this contradiction, a phase-field numerical model is applied to understand hydraulic fracturing experiments using Green River shale found in the literature. The finite element numerical model incorporates a rate-dependent phase-field fracture model developed separately to describe fracture initiation and growth. We investigate the impact of various material and fluid properties on the resulting fractures. Most importantly, we study the effect of fluid properties and boundary conditions on the breakdown pressure, including the direction of the resulting fracture plane. Model results predict that (1) sc-CO 2 injection in the laboratory may result in greater breakdown pressure than that of water under no-flow boundary conditions because lower viscosity sc-CO 2 may result in pressure build up at the core boundary that opposes fracture initiation and (2) lower viscosity sc-CO 2 also produces fast-propagating fractures that are less influenced by the bedding plane on their resulting fracture topology. Our model offers a straightforward explanation and reconciliation of existing experimental observations, as well as a means to extrapolate to new conditions. Exploration of field-scale conditions suggests less pronounced or no elevation in breakdown pressure when sc-CO 2 is injected because the pressure build up effect at the system boundary is significantly less or absent at field length scales. • Mechanical model reproduces experimental results for fracturing of Green River shale. • Breakdown pressure is sensitive to boundary conditions at the outer core surface. • CO 2 injection results in fast-propagating fractures due to greater fluid compressibility. • The difference in breakdown pressure between water and CO 2 injection is reconciled. [ABSTRACT FROM AUTHOR]

Abstract: Monte Carlo (MC) simulations of transport in random porous networks indicate that for high variances of the lognormal permeability distribution, the transport of a passive tracer is non‐Fickian. Here we model this non‐Fickian dispersion in random porous networks using discrete temporal Markov models. We show that such temporal models capture the spreading behavior accurately. This is true despite the fact that the slow velocities are strongly correlated in time, and some studies have suggested that the persistence of low velocities would render the temporal Markovian model inapplicable. Compared to previously proposed temporal stochastic differential equations with case‐specific drift and diffusion terms, the models presented here require fewer modeling assumptions. Moreover, we show that discrete temporal Markov models can be used to represent dispersion in unstructured networks, which are widely used to model porous media. A new method is proposed to extend the state space of temporal Markov models to improve the model predictions in the presence of extremely low velocities in particle trajectories and extend the applicability of the model to higher temporal resolutions. Finally, it is shown that by combining multiple transitions, temporal models are more efficient for computing particle evolution compared to correlated CTRW with spatial increments that are equal to the lengths of the links in the network. [ABSTRACT FROM AUTHOR]

Miscible gas injection is one of the most effective enhanced oil recovery techniques. There are several challenges in accurately modeling this process, which occurs in the near-miscible region. The adjustment of relative permeability for near-miscible processes is the main focus of this work. The dependence of relative permeability on phase identification can lead to significant complications while simulating near-miscible displacements. We present an analysis of how existing methods incorporate compositional dependence in relative permeability functions. The sensitivity of the different methods to the choice of reference points is presented with guidelines to limit the modification of the relative permeabilities to physically reasonable values. We distinguish between the two objectives of reflecting near-miscible behavior and ensuring smooth transitions across phase changes. We highlight an important link that combines the two objectives in a more general framework. We make use of Gibbs free energy as a compositional indicator in the generalized framework. The new approach was implemented in an automatic differentiation general purpose research simulator and tested on a set of near-miscible gas-injection problems. We show that including compositional dependencies in the relative permeability near the critical point impacts the simulation results with significant improvements in nonlinear convergence. [ABSTRACT FROM AUTHOR]

There is growing interest in using advanced imaging techniques to describe the complex pore-space of natural rocks at resolutions that allow for quantitative assessment of the flow and transport behaviors in these complex media. Here, we focus on representations of the complex pore-space obtained from X-ray microtomography and the subsequent use of such 'pore-scale' representations to characterize the overall porosity and permeability of the rock sample. Specifically, we analyze the impact of sub-resolution porosity on the macroscopic (Darcy scale) flow properties of the rock. The pore structure of a rock sample is obtained using high-resolution X-ray microtomography $$(3.16^3\,{\upmu } \hbox {m}^{3}/\hbox {voxel})$$ . Image analysis of the Berea sandstone sample indicates that about 2 % of the connected porosity lies below the resolution of the instrument. We employ a Darcy-Brinkman approach, in which a Darcy model is used for the sub-resolution porosity, and the Stokes equation is used to describe the flow in the fully resolved pore-space. We compare the Darcy-Brinkman numerical simulations with core flooding experiments, and we show that proper interpretation of the sub-resolution porosity can be essential in characterizing the overall permeability of natural porous media. [ABSTRACT FROM AUTHOR]

An inversion method for the integration of dynamic (pressure) data directly into statistical moment equations (SMEs) is presented. The method is demonstrated for incompressible flow in heterogeneous reservoirs. In addition to information about the mean. variance, and correlation structure of the permeability, few permeability measurements are assumed available. Moreover, few measurements of the dependent variable are available. The first two statistical measurements of the dependent variable (pressure) are conditioned on all available information directly. An iterative inversion scheme is used to integrate the pressure data into the conditional statistical moment equations (CSMEs). That is, the available in formation is used to condition, or improve the estimates of, the first two moments of permeability, pressure, and velocity directly. This is different from Monte Carlo (MC) -based geostatistical inversion techniques, where conditioning on dynamic data is performed for one realization of the permeability field at a time. In the MC approach, estimates of the prediction uncertainty are obtained from statistical post-processing of a large number of inversions. one per realization. Several examples of flow in heterogeneous domains in a quarter-five-spot setting are used to demonstrate the CSME-based method. We found that as the number of pressure measurements increases, the conditional mean pressure becomes more spatially variable, while the conditional pressure variance gets smaller. Iteration of the CSME inversion loop is necessary only when the number of pressure measurements is large. Use of the CSME simulator to assess the value of information in terms of its impact on prediction uncertainty is also presented. [ABSTRACT FROM AUTHOR]

Abstract The physical mechanisms that control the flow dynamics in organic-rich shale are not well understood. The challenges include nanometer-scale pores and multiscale heterogeneity in the spatial distribution of the constituents. Recently, digital rock physics (DRP), which uses high-resolution images of rock samples as input for flow simulations, has been used for shale. One important issue with images of shale rock is sub-resolution porosity (nanometer pores below the instrument resolution), which poses serious challenges for instruments and computational models. Here, we present a micro-continuum model based on the Darcy–Brinkman–Stokes framework. The method couples resolved pores and unresolved nano-porous regions using physics-based parameters that can be measured independently. The Stokes equation is used for resolved pores. The unresolved nano-porous regions are treated as a continuum, and a permeability model that accounts for slip-flow and Knudsen diffusion is employed. Adsorption/desorption and surface diffusion in organic matter are also accounted for. We apply our model to simulate gas flow in a high-resolution 3D segmented image of shale. The results indicate that the overall permeability of the sample (at fixed pressure) depends on the time scale. Early-time permeability is controlled by Stokes flow, while the late-time permeability is controlled by non-Darcy effects and surface-diffusion. [ABSTRACT FROM AUTHOR]