Parameterization of element balance formulation in reactive compositional flow and transport

We present a novel nonlinear formulation for modeling reactive-compositional transport in the presence of complex phase behavior related to dissolution and precipitation in multi-phase systems. This formulation is based on the consistent element balance reduction of the molar (overall composition) formulation. To predict a complex phase behavior in such systems, we include the chemical equilibrium constraints to the multiphase multicomponent negative flash calculations and solve the thermodynamic and chemical phase equilibrium simultaneously. In this solution, the phase equilibrium is represented by the partition coefficients whereas the chemical equilibrium reaction is represented by the activity coefficients model. This provides a generic treatment of chemical and thermodynamic equilibrium within an EOS SSI loop by modification of the multiphase flash to accommodate chemical equilibrium. Using the Equilibrium Rate Annihilation matrix allows us to reduce the governing unknowns to the primary set only while the coupling between chemical and thermodynamic equilibrium is captured by a simultaneous solution of modified multiphase flash equations. An input in this thermodynamic computation is an element composition of the mixture when an output contains fractions of components in each phase, including solids. This element balance molar formulation along with the modified formulation for multiphase flash has been tested in a simple transport model with dissolution and precipitation reactions. The same approach will be later used to model a system involving kinetic reactions. The simulation of more general practical models is performed using the recently developed Operator-Based Linearization (OBL) technique. In the modified version of the OBL, the nonlinear element based governing equations are formulated in terms of space and state-dependent parameters constrained by the solution of the extended multiphase flash based on molar element compositions. This approach helps us to add equilibrium reaction capabilities to the computationally efficient OBL technique. </p

Robust and accurate formulation for modeling of acid stimulation

Accurate representation of processes associated with energy extraction from subsurface formations often requires models which account for chemical interactions between different species in the presence of multiphase flow. In this study, we focus on modeling of acid stimulation in the near-well region. For the chemical processes which include a dissolution of rock material, an issue arises with the predictive representation of flow. Taking into account the spatial scale of discretization, some of simulation control volumes can have values of porosity close to 1, which makes an application of Darcy's law inconsistent and requires employing a true momentum equation such as the Darcy-Brinkman-Stokes (DBS) equation. The DBS equation automatically switches the description between Darcy equation in control volumes with low porosity and Stokes equation in grid blocks with high porosity. For chemical reactions, we propose a local nonlinear solution technique that allows solving the balance of solid species separately yet retaining the full coupling with rest of the equations. Finally, we study the impact of multiphase flow. The DBS approach is not well established for multiphase flow description. Therefore we employ a hybrid approach, where we assume that the single-phase DBS flow and the multiphase Darcy flow occur in separate regions. We test the accuracy and performance of both approaches on realistic models of practical interest.</p

Multi-level discrete fracture model for carbonate reservoirs

The main challenge for predictive simulation of carbonate reservoirs is associated with large uncertainties in the geological characterization with multiple features including fractures and cavities. This type of reservoirs requires robust and efficient forward-simulation capabilities to apply data assimilation or optimization technique under uncertainties. The interaction between reservoir matrix and various features introduces a complex multi-scale flow response driven by global boundary conditions. The Discrete Fracture Models (DFM), which represent fractures explicitly, is capable to accurately depict all important features of flow behavior. However, these models are constrained by many degrees of freedom when the fracture network becomes complicated. The Embedded DFM, which represents the interaction between matrix and fractures analytically, is an efficient approximation. However, it cannot accurately reproduce the effect of local flow conditions, especially when the secondary fractures are present. In this study, we applied a numerical upscaling of DFM a triple continuum model where large features are represented explicitly using the numerical EDFM and small features are upscaled as a third continuum. In this approach, we discretize the original geo-model with unstructured grid based on DFM and associate the mesh geometry with large features in the model. Using the global solution, we generate local boundary conditions for the model capturing the response of primary features to the flow. Applying local boundary conditions, we resolve all secondary features using a fine scale solution and update the local boundary conditions. This procedure is applied iteratively using the local-global-upscaling formalism. To demonstrate the accuracy of the Multi-Level Discrete Fracture Model, several realistic cases have been tested. By comparing with fine scale DFM solution and the traditional EDFM technique, we demonstrate that the proposed model is accurate enough to capture the flow behavior in complex fractured systems with advanced computational efficiency.</p

Modelling of reactive flow and transport in the presence of a complex phase transition phenomena

We present a novel simulation approach for modeling of reactive flow and transport in multiphase multicomponent mixtures that include light gases, hydrocarbon components, and different ions present in an aqueous electrolyte phase. The phase behavior in these systems involves both thermodynamically-driven phase transitions (e.g. between supercritical vapor and liquid phases) and chemically driven precipitation and dissolution of solid (mineral) phases. All phases are modeled using the multi-scale Gibbs-Helmholtz Constrained Equation of State (GHC EoS), which up-scales molecular length scale information from a priori Monte Carlo simulations to help build accurate estimates of the energy parameter. Our proposed approach is implemented in the combined software system included the Automatic Differentiation General Purpose Research Simulator (ADGPRS) developed at Stanford University and the GFLASH library developed at University of Rhode Island. The extended variable substitution schema for a natural fully implicit formulation is designed to support the potential coexistence of an arbitrary set of phases in the flow. The classical reduction in the number of conservation equations based on element balances is combined with specific local constraints describing simultaneous thermodynamic and chemical equilibrium. Rigorous flash solutions for detecting phase changes in each grid block are computed using phase splitting and phase/chemical equilibrium to ensure equality of component chemical potentials and by monitoring the Gibbs free energy of the system to guarantee a global minimum is found. We present examples that cover a wide range of physical processes related to CO2 sequestration in saline aquifers

Optimization Of CO2 injection using multi-scale reconstruction of compositional transport

The current situation with green gas emission requires the development of low carbon energy solutions. However, a significant part of the modern energy industry still relies on fossil fuels. To combine these two contradictory targets, we investigate a strategy based on a combination of CO2 sequestration with Enhanced Oil Recovery (EOR) in the hydrocarbon reservoirs. In such technology, the development of miscibility is the most attractive strategy from both technological and economic aspects. Modeling of this process involves solving complex nonlinear problem describing compositional flow and transport in highly heterogeneous porous media. An accurate capture of the miscibility development usually requires an extensive number of components to be present in the compositional problem which makes simulation run-time prohibitive for optimization. Here, we apply a multi-scale reconstructing of compositional transport to the optimization of CO2 injection. In this approach, a prolongation operator, based on the parametrization of injection and production tie-lines, is constructed following the fractional flow theory. This operator is tabulated as a function of pressure and pseudocomposition which then is used in the Operator-Based Linearization (OBL) framework for simulation. As a result, a pseudo two-component solution of the multidimensional problem will match the position of trailing and leading shocks of the original problem which helps to accurately predict phase distribution. The reconstructed multicomponent solution can be used then as an effective proxy-model mimicking the behavior of the original multicomponent system. Next, we use this proxy-model in the optimization procedure which helps to improve the performance of the process in several folds. An additional benefit of the proposed methodology is based on the fact that important technological features of CO2 injection process can be captured with lower degrees of freedom which makes the optimization solution more feasible.</p

Multiscale reconstruction of compositional transport

A compositional formulation is a reliable option for understanding the complex subsurface processes and the associated physical changes. However, this type of model has a great computational cost, since the number of equations that needs to be solved in each grid block increases proportionally with the number of components employed, thereby making them computationally demanding. In an effort to enhance the solution strategy of the hyperbolic problem, we herewith propose a multiscale reconstruction of compositional transport problem. Until recently, multiscale techniques have been seldom implemented on transport equations. Here, the ideology consists of two stages, wherein two different sets of restriction and prolongation operators are defined based on the dynamics of compositional transport. In the first stage, an operator restricting the arbitrary number of components to single transport equation is implemented with the objective of reconstructing the leading and trailing shock positions in space. The prediction of front propagation is the most critical aspect of the approach, as they involve a lot of uncertainty. Once their positions are identified, the full solution lying in the regions outside the shocks can be conservatively reconstructed based on the prolongation interpolation operator. Subsequently, the solution for the multicomponent problem (full system) in the two-phase region is reconstructed by solving just two transport equations with the aid of restriction operator defined based on an invariant thermodynamic path (based on Compositional Space Parameterization technique). We demonstrate applicability of the approach for the idealistic 1D test cases involving various gas drives with different number of components. Further, the first stage reconstruction was tested successfully on more realistic problems based on implementation in recently developed Operator-Based Linearization (OBL) platform.</p

Tie-simplex parametrization for operator-based linearization for non-isothermal multiphase compositional flow in porous

As oil production continues worldwide, more oil fields require complex EOR methods to achieve outlined recovery factors. Reservoir engineers are dealing more often with problems involving thermal multiphase multi-component flow models tightly coupled with complex phase behavior. Such modeling implies the solution of governing laws describing mass and energy transfer in the subsurface, which in turn requires the linearization of strongly nonlinear systems of equations. The recently proposed Operator-Based Linearization (OBL) framework suggests an unconventional strategy using the discrete representation of physics. The terms of governing PDEs, discretized in time and space, which depend only on state variables, are approximated by piece-wise multilinear operators. Since the current physical state fully defines operators for a given problem, each operator can be parametrized over the multidimensional space of nonlinear unknowns for a given distribution of supporting points. Onwards, the values of operators, along with their derivatives with respect to nonlinear unknowns, are obtained from the parametrization using multilinear interpolation and are used for Jacobian assembly in the course of a simulation. Previously, the distribution of supporting points was always uniform, requiring higher parametrization resolution to provide accurate and consistent interpolation of an operator around its most nonlinear regions in parameter space. In addition, when the resolution is low, the system can lose hyperbolicity causing convergence issues. In this work, we apply the prior knowledge of underlying physics to distribute the supporting points according to the tie-simplex behavior of the multiphase mixture in parameter space. The approach allows to decrease the parametrization resolution keeping the same accuracy. In addition, the OBL framework is extended to describe multisegment wells working under different controls. We test the accuracy of the developed framework for truly multi-component systems of practical interest.</p