Rapid Learning-Based and Geologically Consistent History Matching

Various types of data become available at different stages of a reservoir’s life. The production data are integrated into the flow simulation models through a process referred to as history matching. The history-matching process is iterative, and it usually involves a large number of simulation runs. Hence, this process requires significant computational time. In most history-matching methods, the initial geological assumptions in the reservoir model are destroyed or significantly altered in the process. Furthermore, they do not account for the information obtained during the previous trials, and lack learning from the previous failures. In this paper, we introduce a new methodology that maintains the geological realism. The candidate realizations are selected through a learning-based history-matching (LHM) algorithm by which all the previously successful patterns are preserved and used to assist the construction of the next realizations. The various pieces of matching regions are assembled together to make a pool of the successful candidates. Such regions are then utilized for making an auxiliary dataset in a multiscale framework by which the next model is generated. To prevent from trapping in local minima, ideas from the genetic algorithm is adapted. The LHM algorithm can be applied to both categorical and continuous distributions. The LHM provides a conditional map by which the new production data are immediately incorporated into the existing reservoir models. We apply the LHM algorithm to various 2D and 3D examples with very complex binary and continuous properties. The algorithm is shown to produce history-matched models with significantly smaller CPU times.     

A New Approach in Modeling Phase Distribution in Fully Developed Bubbly Pipe Flow

Turbulent two-phase flow equations are derived and solved for fully developed pipe flow using a composite eddy-viscosity model and a new void-fraction equation. The void fraction profile is first specified from experiments and the velocity field is calculated to validate the eddy-viscosity model. Consequently, a new equation is presented for calculation of the void fraction. This void-fraction equation incorporates the gradient of turbulent normal stresses in the radial direction, the conventional lift force, and a contribution from the unsteady drag force. The implications of this new equation, for the bubbly flow regime, are investigated by calculating the void-fraction distribution for a given velocity field. Inclusion of the normal turbulent stresses in the radial direction is shown to simulate correctly the experimentally observed trends of the phase distribution, both for upward and downward bubbly flow, without the need for a fictitious term such as the so called ``lubrication force'. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Steady-State Upscaling Approach for Immiscible Two-Phase Flow

The paper presents a model for computing rate-dependent effective capillary pressure and relative permeabilities for two-phase flow, in 2 and 3 space-dimensions. The model is based on solving the equations for immiscible two-phase flow at steady-state, accounting for viscous and capillary forces, at a given external pressure drop. The computational performance of the steady-state model and its accuracy is evaluated through comparison with a commercial simulator ECLIPSE. The properties of the rate-dependent effective relative permeabilities are studied by way of computations using the developed steady-state model. Examples presented show the dependence of the effective relative permeabilities and capillary pressures, which incorporate the effects of fine scale wettability heterogeneity, on the external pressure drop, and thereby on the dimensionless macro-scale capillary number. The effective relative permeabilities converge towards the viscous limit functions as the capillary number tends to infinity. Special cases, when the effective relative permeabilities are rate-invariant, are also studied. The applicability of the steady-state upscaling algorithm in dynamic displacement situations is validated by comparing fine-gridded simulations in heterogeneous reservoirs against their homogenized counterparts. It is concluded that the steady-state upscaling method is able to accurately predict the dynamic behavior of a heterogeneous reservoir, including small scale heterogeneities in both the absolute permeability and the wettability.     

Thermomechanical Multiscale Constitutive Modeling: Accounting for Microstructural Thermal Effects

In this work we present a thermomechanical multiscale constitutive model for materials with microstructure. In these materials thermal effects at microscale have an impact on the effective macroscopic stress. As a result, it turns out that the homogenized stress depends upon the macroscopic temperature and its gradient. In order to allow this interplay to be thermodynamically valid, we resort to a macroscopic extended thermodynamics whose elements are derived from the microscopic behavior using homogenization concepts. Hence, the thermodynamics implications of this new class of multiscale models are discussed. A variational approach based on the Hill–Mandel Principle of Macro-homogeneity, and which makes use of the volume averaging concept over a local representative volume element (RVE), is employed to derive the thermal and mechanical equilibrium problems at the RVE level and the corresponding homogenization expressions for the effective heat flux and stress. The material behavior at the RVE level is described through standard phenomenological constitutive models. To sum up, the novel contribution of the model presented here is that it allows to include the microscopic temperature fluctuation field, obtained from the multiscale thermal analysis, in the micro-mechanical problem at the RVE level while keeping thermodynamic consistency.     

Pore-scale Network Modeling of Three-Phase Flow Based on Thermodynamically Consistent Threshold Capillary Pressures. II. Results

We use the model described in Zolfaghari and Piri (Transp Porous Media, 2016) to predict two- and three-phase relative permeabilities and residual saturations for different saturation histories. The results are rigorously validated against their experimentally measured counterparts available in the literature. We show the relevance of thermodynamically consistent threshold capillary pressures and presence of oil cusps for significantly improving the predictive capabilities of the model at low oil saturations. We study systems with wetting and spreading oil layers and cusps. Three independent experimental data sets representing different rock samples and fluid systems are investigated in this work. Different disordered networks are used to represent the pore spaces in which different sets of experiments were performed, i.e., Berea, Bentheimer, and reservoir sandstones. All three-phase equilibrium interfacial tensions used for the simulation of three-phase experimental data are measured and used in the model’s validation. We use a fixed set of parameters, i.e., the input network (to represent the pore space) and contact angles (to represent the wettability state), for all experiments belonging to a data set. Incorporation of the MSP method for capillary pressure calculations and cusp analysis significantly improves the agreement between the model’s predictions of relative permeabilities and residual oil saturations with experimental data.     

Pore-Scale Network Modeling of Three-Phase Flow Based on Thermodynamically Consistent Threshold Capillary Pressures. I. Cusp Formation and Collapse

We present a pore-scale network model of two- and three-phase flow in disordered porous media. The model reads three-dimensional pore networks representing the pore space in different porous materials. It simulates wide range of two- and three-phase pore-scale displacements in porous media with mixed-wet wettability. The networks are composed of pores and throats with circular and angular cross sections. The model allows the presence of multiple phases in each angular pore. It uses Helmholtz free energy balance and Mayer–Stowe–Princen (MSP) method to compute threshold capillary pressures for two- and three-phase displacements (fluid configuration changes) based on pore wettability, pore geometry, interfacial tension, and initial pore fluid occupancy. In particular, it generates thermodynamically consistent threshold capillary pressures for wetting and spreading fluid layers resulting from different displacement events. Threshold capillary pressure equations are presented for various possible fluid configuration changes. By solving the equations for the most favorable displacements, we show how threshold capillary pressures and final fluid configurations may vary with wettability, shape factor, and the maximum capillary pressure reached during preceding displacement processes. A new cusp pore fluid configuration is introduced to handle the connectivity of the intermediate wetting phase at low saturations and to improve model’s predictive capabilities. Based on energy balance and geometric equations, we show that, for instance, a gas-to-oil piston-like displacement in an angular pore can result in a pore fluid configuration with no oil, with oil layers, or with oil cusps. Oil layers can then collapse to form cusps. Cusps can shrink and disappear leaving no oil behind. Different displacement mechanisms for layer and cusp formation and collapse based on the MSP analysis are implemented in the model. We introduce four different layer collapse rules. A selected collapse rule may generate different corner configuration depending on fluid occupancies of the neighboring elements and capillary pressures. A new methodology based on the MSP method is introduced to handle newly created gas/water interfaces that eliminates inconsistencies in relation between capillary pressures and pore fluid occupancies. Minimization of Helmholtz free energy for each relevant displacement enables the model to accurately determine the most favorable displacement, and hence, improve its predictive capabilities for relative permeabilities, capillary pressures, and residual saturations. The results indicate that absence of oil cusps and the previously used geometric criterion for the collapse of oil layers could yield lower residual oil saturations than the experimentally measured values in two- and three-phase systems.     

A Thermomechanically Consistent Constitutive Model for Polyoxymethylene

In this article tension, compression and torsion tests are presented using thin-walled tubes of polyoxymethylene (POM). These isothermal experiments show non-linear rate dependence, a tension–compression asymmetry and a pronounced relaxation behaviour. On the basis of the experiments carried out, a constitutive model of viscoplasticity with an equilibrium hysteresis in the small-strain regime is developed. Test calculations using finite elements based on the DAE approach show the capabilities of the thermomechanically consistent model. In particular, a very efficient stress algorithm can be derived which has no iteration on the element level. Moreover, it will be shown that time-adaptive finite elements could be of high importance if rate-dependent constitutive models are applied.     

Operator Splitting Multiscale Finite Volume Element Method for Two-Phase Flow with Capillary Pressure

A numerical method used for solving a two-phase flow problem as found in typical oil recovery is investigated in the setting of physics-based two-level operator splitting. The governing equations involve an elliptic differential equation coupled with a parabolic convection-dominated equation which poses a severe restriction for obtaining fully implicit numerical solutions. Furthermore, strong heterogeneity of the porous medium over many length scales adds to the complications for effectively solving the system. One viable approach is to split the system into three sub-systems: the elliptic, the hyperbolic, and the parabolic equation, respectively. In doing so, we allow for the use of appropriate numerical discretization for each type of equation and the careful exchange of information between them. We propose to use the multiscale finite volume element method (MsFVEM) for the elliptic and parabolic equations, and a nonoscillatory difference scheme for the hyperbolic equation. Performance of this procedure is confirmed through several numerical experiments.     

A Taylor-Galerkin Method for Modeling Unsteady Fluid Flow Under its Own Weight

The free fluid-surface of incompressible creeping flows is analyzed using a finite element method. A pseudo-concentration (PC) function is introduced to determine the position of the free surface. The Taylor-Galerkin finite element method (TGFEM) is applied to solve the equation of the PC function. Nine-node quadratic interpolation is used for both PC and velocity. The unsteady flows of fluids moving of their own weight are analyzed using the proposed method.     

Multiscale, Multiphysics Network Modeling of Shale Matrix Gas Flows

We present a pore network model to determine the permeability of shale gas matrix. Contrary to the conventional reservoirs, where permeability is only a function of topology and morphology of the pores, the permeability in shale depends on pressure as well. In addition to traditional viscous flow of Hagen–Poiseuille or Darcy type, we included slip flow and Knudsen diffusion in our network model to simulate gas flow in shale systems that contain pores on both micrometer and nanometer scales. This is the first network model in 3D that combines pores with nanometer and micrometer sizes with different flow physics mechanisms on both scales. Our results showed that estimated apparent permeability is significantly higher when the additional physical phenomena are considered, especially at lower pressures and in networks where nanopores dominate. We performed sensitivity analyses on three different network models with equal porosity; constant cross-section model (CCM), enlarged cross-section model (ECM) and shrunk length model (SLM). For the porous systems with variable pore sizes, the apparent permeability is highly dependent on the fraction of nanopores and the pores’ connectivity. The overall permeability in each model decreased as the fraction of nanopores increased.     

A Phenomenological Approach to Modeling Transport in Porous Media

The objective of this article is to make use of the phenomenological approach to construct models for the transport of extensive quantities, such as mass of a fluid phase, mass of a component of a fluid phase, momentum of a phase and energy, in porous medium domains. Special attention is devoted to express the fluxes of these extensive quantities, especially the non-advective ones, as functions of their relevant driving forces, obeying the principle of minimum entropy production. It is shown that for each extensive quantity, we have a linear diffusive flux term, a non-linear diffusive term, and a dispersive flux term. The latter is shown to be proportional to the velocity squared. In each case, the number of moduli that describe fluid and porous matrix properties is determined. The momentum balance equation for a porous medium domain, which is the “motion equation,” is analyzed and simplified for special cases, leading to Darcy’s law and to Brinkman’s equation.     

Love Dynamics Between Secure Individuals: A Modeling Approach

A mathematical model composed of two differential equations, which qualitatively describe the dynamics of love between secure individuals, is presented in this paper with two goals. The general goal is to show how dynamic phenomena in the field of social psychology can be analyzed following the modelling approach traditionally used in all other fields of sciences. The specific goal is to derive, from very general assumptions on the behavior of secure individuals, a series of rather detailed properties of the dynamics of their feelings. The analysis shows, in particular, why couples can be partitioned into fragile and robust couples, how romantic relationships are influenced by behavioral parameters and in which sense individual appeal creates order in a community.     

Micro-deep Drawing on Aluminium Crystals in Order to Validate Multiscale Modeling

This paper introduces the application of micro-deep drawing of single and multicrystals of aluminium in order to identify parameters of micro-mechanical models. The information obtained experimentally is abundant: load-displacement curves of the punch, the evolution of grain orientation, mapping deformations throughout the test. All this experimental data was used to validate the multiscale models introduced in a finite element code for loading paths ranging from biaxial tensile to simple tensile.     

Upscaling and Automation: Pushing the Boundaries of Multiscale Modeling through Symbolic Computing

Transport in Porous Media - Macroscopic differential equations that accurately account for microscopic phenomena can be systematically generated using rigorous upscaling methods. However, such...     

Principles of Integrated Flow Modeling

Integrated flow modeling is the combination of a traditional flow simulator with a petrophysical model. By combining a petrophysical model with a traditional flow model, it is possible to perform calculations that improve our ability to monitor fluid movement in porous media. This paper outlines the formulation of an integrated flow model IFLO and its multi-variable, Newton–Raphson IMPES solution procedure. The benefits of integrated flow modeling and the underlying principles involved in the integration of a flow model with a petrophysical model are presented. Results from the IFLO model are used to illustrate the principles.     

The Shape of Moving Boundary of Fluid Flow in Sandstone: Video Microscopic Investigation and Stochastic Modeling Approach

Many engineering problems such as exploitation of petroleum and gas, deposition of nuclear waste, and groundwater contamination by organic liquids are closely related to the movement of fluid in rocks. In this paper, a video microscope is employed to investigate the shape of moving front boundary of fluid flow in sandstone. The experimental results show that the fronts of the moving boundary display a fractal behavior. Based on the experimental results, a stochastic differential equation is proposed to describe the moving boundary. By decomposing the velocity of a given point into a drift term and a fluctuation term, the effect of the mesoscope structure of porous media on fluid flow is taken into account. The stochastic approach is in agreement with the experimental results. The analysis shows that the front of the moving boundary of fluid flow in rocks is a comprehensive result caused by the average tendency of fluid flow, which can be described by the classical Darcys Law, and the fluctuation tendency of fluid flow, which is closely related to the mesoscope structure of rocks.     

A Novel Framework for Integration of Random-Walk Particle-Tracking Simulation in Subsurface Multi-phase Flow Modeling

Coarse-scale models are generally preferred in the numerical simulation of multi-phase flow due to computational constraints. However, capturing the effects of fine-scale heterogeneity on flow and isolating the impacts of numerical (artificial) dispersion, which increases with scale, are not trivial. In this paper, a particle-tracking method is devised and integrated in a scale-up workflow to estimate the conditional probability distributions of multi-phase flow functions, which can be considered as inputs in coarse-scale simulations with existing commercial packages. First, a novel particle-tracking method is developed to solve the saturation transport equation. The transport calculation is coupled with a velocity update, following the implicit pressure, explicit saturation framework, to solve the governing equations of two-phase immiscible flow. Each phase particle is advanced in a deterministic convection step according to the phase velocity, as well as in a stochastic dispersion step based on the random Brownian motion. A kernel-based formulation is proposed for computation of fluid saturation in accordance with the phase particle distribution. A novel aspect is that this method employs the kernel approach to construct saturation from phase particle distribution, which is an important improvement to the conventional box method that necessitates a large number of particles per grid cell for consistent saturation interpolation. The model is validated against various analytical solutions. Finally, the validated model is integrated in a statistical scale-up procedure to calibrate effective, or “pseudo,” multi-phase flow functions (e.g., relative permeability functions). The proposed scale-up framework does not impose any length scale requirement regarding the distribution of sub-grid heterogeneities.     

Negative Saturation Approach for Non-Isothermal Compositional Two-Phase Flow Simulations

This article deals with developing a solution approach, called the non-isothermal negative saturation (NegSat) solution approach. The NegSat solution approach solves efficiently any non-isothermal compositional flow problem that involves phase disappearance, phase appearance, and phase transition. The advantage of the solution approach is that it circumvents using different equations for single-phase and two-phase regions and the ensuing unstable procedure. This paper shows that the NegSat solution approach can also be used for non-isothermal systems. The NegSat solution approach can be implemented efficiently in numerical simulators to tackle modeling issues for mixed CO₂–water injection in geothermal reservoirs, thermal recovery processes, and for multicontact miscible and immiscible gas injection in oil reservoirs. We illustrate the approach by way of example to cold mixed CO₂–water injection in a 1D geothermal reservoir. The solution is compared with an analytical solution obtained with the wave-curve method (method of characteristics) and shows excellent agreement. A complete set of simulations is carried out, which identifies six bifurcations. The two main bifurcations are (1) when the most downstream compositional wave is replaced by a compositional shock and (2) when an extra Buckley–Leverett rarefaction appears. The plot of the useful energy (exergy) versus the CO₂ storage capacity shows a Z-shape. The top horizontal part represents a branch of high exergy recovery/relatively lower storage capacity, whereas the bottom part represents a branch of lower exergy recovery/higher storage capacity.