Philip H. Winterfeld

Percolation and conductivity of random two- dimensional composites

Tessellating an area into convex polygons and labelling a given number of randomly chosen polygons as conducting, the rest not, creates a useful model of a disordered composite. The percolation and conduction properties of composites generated from the random Voronoi tessellation are compared with those from regular tessellations, namely the square and the hexagonal. Percolation properties of interest are the conducting cluster distribution, the percolation threshold, and the percolation probability. Conduction properties are obtained by a lattice network reduction (a finite difference type approximation to conduction) and by finite element analysis.

Coupled Geomechanical and Reactive Geochemical Model for Fluid and Heat Flow: Application for Enhanced Geothermal Reservoir

A major concern in development of fractured reservoirs in Enhanced Geothermal Systems (EGS) is to achieve and maintain adequate injectivity, while avoiding short-circuiting flow paths. The injection performance and flow paths are dominated by the permeability distribution of fracture network for EGS reservoirs. The evolution of reservoir permeability can be affected by rock deformation, induced by the change in temperature or pressure around the injector, and chemical reactions between injection fluid and reservoir rock minerals. Thus the change in permeability due to geomechanical deformation and mineral precipitation/dissolution could have a major impact on reservoir long-term performance. A coupled thermal-hydrologicalmechanical-chemical (THMC) model is in general necessary to examine the reservoir behavior in EGS. This paper presents a numerical model, TOUGH2-EGS, for simulating coupled THMC processes in enhanced geothermal reservoirs. This simulator is built by coupling mean stress calculation and reactive geochemistry into the existing framework of TOUGH2 (Pruess et al., 1999) and TOUGHREACT (Xu et al., 2004a), the well-established numerical simulators for geothermal reservoir simulation. The geomechanical model is fully-coupled as mean stress equations are solved simultaneously with fluid and heat flow equations. After solution of the fluid, heat, and stress equations, the flow velocity and phase saturations are used for reactive geochemical transport simulation in order to sequentially couple reactive geochemistry at each time step. We perform coupled THMC simulations to examine a prototypical EGS reservoir for permeability evolution at the vicinity of the injection well. The simulation results demonstrate the strong influence of rock deformation effects in the short and intermediate term, and long-term influence of chemical effects. It is observed that the permeability enhancement by thermalmechanical effect can be counteracted by the chemical precipitation of minerals, initially dissolved into the low temperature injected water. We analyze the sensitivity of temperature of injected water on the coupled geomechanical and geochemical effects, and conclude that the temperature of injected water could be modified to maintain or even enhance the reservoir permeability and the injection performance. Introduction The successful development of enhanced geothermal systems (EGS) highly depends on the reservoir fracture network of hot dry rock (HDR) and its hydraulic properties, especially the reservoir permeability. The geomechanical processes under subsurface reservoir condition are prevalent in the EGS applications. For example, Tsang (1999) investigated and claimed that hydraulic properties of fracture rocks are subjected to change under reservoir mechanical effects. Rutqvist et al. (2002) presented the correlations between reservoir in-situ stress and the porosity, permeability and capillary pressure. It is also well known that the EGS production processes, such as the cold water injection and steam or hot fluid extraction, have strong thermo-poro-elastic effects on EGS reservoirs. On the other hand, the strong impacts of geochemical reaction on the EGS reservoirs have been observed in the commercial EGS fields and studied for carbon dioxide (CO2) based geothermal system in the past few years. Kiryukhin et al. (2004) modeled the reactive chemical process based on the field data from tens of geothermal fields in Kamchatka (Russia) and Japan. In addition, Xu et al. (2004b) presented the reactive transport model of injection well scaling and acidizing at Tiwi field in Philippines. Montalvo et al. (2005) studied the calcite and silica scaling problems with exploratory model for Ahuachapan

A Fully Coupled Model of Nonisothermal Multiphase Flow, Solute Transport and Reactive Chemistry in Porous Media

Over the past decades, geochemical reaction has been identified through experiments in different processes, e.g. the CO2 EOR process, the CO2 sequestration, the enhanced geothermal system. Research has gradually led to the recognition that chemical reactions between injected fluid and mineral rock have significant impacts on fluid dynamics and rock properties in these processes. However, for the majority of the reactive transport simulators, the sequential calculation processes of fluid flow, solute transport, and reactive geochemistry result in numerical instability and computation efficiency problems. In this paper, we present a fully coupled computational framework to simulate reactive solute transport in porous media for mixtures having an arbitrary number of phases. The framework is designed to keep a unified computational structure for different physical processes. This fully coupled simulator focuses on: (1) the fluid flow, solute transport, and chemical reactions within a threephase mixture, (2) physically and chemically heterogeneous porous and fractured rocks, (3) the non-isothermal effect on fluid properties and reaction processes, and (4) the kinetics of fluid-rock and gas-rock interactions. In addition, a system of partial differential equations is formed to represent the physical and chemical processes of reactive solute transport. A flexible approach of integral finite difference is employed to to obtain the residuals of the equation system. Jacobin matrix for NewtonRaphson iteration is generated by numerical calculation, which helps the future parallelization of the fully coupled simulator. Finally, the fully coupled model is validated using the TOUGHREACT simulator. Examples with practical interests will be discussed, including CO2 flooding in a reservoir, supercritical CO2 injection into a saline aquifer, and cold water injection into a natural geothermal reservoir. This type of simulation is very important for modeling of physical processes, especially for CO2 EOR and storage, and geothermal resources development. Inroduction Reactive fluid flow and geochemical species transport that occur in subsurface reservoirs have been of increasing interest to researchers in the subjects of CO2 geological sequestration, CO2 EOR process, enhanced geothermal system, or even waterflooding and other EOR processes. The chemical reaction path has been observed in these processes when subjected to fluid injection in the subsurface reservoir. The nonisothermal reactive solute transport phenomena involed in these processes are thermal-hydrological-chemical (THC) processes. However, the reaction paths may be slightly different due to the different fluid flow mechanisms related to these processes. CO2 geological sequestration and CO2 EOR are two effective solutions to store CO2 from burning fossil fuels in geological formations and petroleum reservoirs. Saline aquifers and petroleum reservoirs have the largest capacity among the many options for long-term geological sequestration. They are large underground formations saturated with brine water or hydrcarbons, and are often rich in dissolved minerals. CO2 is injected into these formations as a supercritical fluid with a liquid-like density and a gas-like viscosity. It is believed that geochemical reaction between CO2 and rock minerals in the aqueous–based system dominates the long-term fate of CO2 sequestrated in geological formations. Two types of geochemical reactions between CO2 and rock minerals have been identified by experiments, i.e. reactions between dissolved CO2 and rock minerals, and reactions between supercritical CO2 and rock minerals. The chemical mechanism between dissolved CO2 and rock mineral has been well understood. The acid H2CO3 is formed by the dissolution of CO2 in an aqueous solution, and it dissociates in the brine to release H + . The carbonate minerals are dissolved into the aqueous phase under this weak acid

Effects of Shear Planes and Interfacial Slippage on Fracture Growth and Treating Pressures

The equations used in current hydraulic fracture simulators are based on plane-strain solutions, or use a complete surface integral solution for tlacture width. Assumptions inherent in these solutions control the smess field surrounding the fkacture tip and the stress intensity developed at the tip which, in turn, controls the rate of tlacture growth and containment and the pre&cted net pressnre. The overriding assumption made in these solutions is that the entire rock mass is elastically coupled so that all stresses and deformations interact. Many reservoirs that are hydraulically fractured are susceptible to complex tincturing which can invalidate the assumption of elastic coupling. Microseismic monitoring of tiacture growth indicates that energy is lost to shear failures around the fracture. During hydraulic fracturing high fluid pressures, often exceeding both the minimum and maximum horizontal stress (fissure opening pressure), result in the reduction of the normal stress acting across natural fissures. This allows free shear or slippage along naturat fracture planes m reservoirs or cleats in coal. When shear or slippage occurs elastic coupling in the rock mass is lost and each shear block &forms as a separate unit. This shear decoupling resuIts in tremendous reduction in created fkacture width and leads to high frictional pressures (tow transmissibility) and difficulty in placing proppant, especially large proppant. The purpose of this work is to suggest that current fracture models are missing what could be a dominant containment mechanism in the tlacturing of fissured reservoirs, coals, and soft rocks and that further work is required to filly understand the implications of slippage and shear failure on treatment designs.

Geomechanics Coupling Simulation of Fracture Closure and Its Influence on Gas Production in Shale Gas Reservoirs

A sharp initial decline in the production rate is being experienced in many shale gas plays. One reason is the closure of natural micron and micro fractures. The natural fractures, which widely exist in the over-pressurized source rock, react sensitively to the change of subsurface stress. The change in stress can be caused by the decrease of gas pressure during production. These fracture closure or re-open phenomena have significant effects on reservoir permeability and gas production. In this paper, a fully coupled geomechanics and multiphase fluid flow model is presented to accurately simulate the fields of stress and fluid flow in shale gas reservoirs. Several relationships between fracture closure and applied stress are incorporated in this model, based on the experimental data from literatures. Therefore, the stress dependency of shale natural fractures is quantified and modeled in its full complexity. The natural fractures in this model are characterized as stiff, self-propped, and prone to closure. It represents an extension of our earlier “hybrid-fracture model” (DFN for hydraulic fractures, doubleporosity for natural fractured domain inside the SRV, and single porosity or dual-continuum outside the SRV).

Parallel Simulation of CO2 Sequestration with Rock Deformation in Saline Aquifers

We developed a massively parallel reservoir simulator for modeling CO2 flow and transport in saline aquifers coupled with geomechanical processes that would occur during CO2 sequestration. These geomechanical processes, namely rock deformation, affect rock permeability, porosity, and bulk volume. The theories of hydrostatic and linear poroelasticity, and correlations from the literature, are used to obtain dependencies of rock permeability, porosity, and bulk volume on pore pressure and average stress. These dependencies are incorporated into the simulator mass and energy conservation equations, which determine pressure, temperature, and fluid composition. The simulator is based on the TOUGH2-MP code (Zhang et al. 2008) with the ECO2N module (Pruess, 2005), which calculates properties of H2O-NaCl-CO2 mixtures. These mixtures consist of one aqueous and one CO2-rich phase, salt can precipitate and dissolve in the aqueous phase, and the mixture temperature may change. The simulator is fully implicit, three-dimensional, the grid is unstructured, and the linear equations associated with the conservation equations are solved in porous and fractured media. The simulator code is parallelized and uses MPI for processor communication, the METIS software package for domain partitioning, and the Aztec solver package for linear equation solution. The simulator formulation, numerical implementation, and computational efficiency are verified using example problems from the literature, and options for the above dependencies are also illustrated. One example problem (Birkholzer et al., 2008) is a study of the volume influenced during and after CO2 injection into a saline aquifer. Another (Rutqvist and Tsang, 2002) addresses coupled hydromechanical changes during CO2 injection into an aquifer–caprock system. In this example, geomechanical processes were modeled by coupling two simulators, TOUGH2 (Pruess et al., 1999), which simulates fluid flow and heat transport, and FLAC3D (FLAC3D, 1997), which simulates rock and soil mechanics with thermomechanical and hydromechanical interactions. Finally, the last example problem (Kumar et al., 2005) is a simulation of a prototypical CO2 sequestration project in a deep saline aquifer using CMGs GEM simulator (Nghiem, 2002). Introduction Geological sequestration of CO2 in deep saline aquifers is a primary option for reducing anthropogenic CO2 emissions into the atmosphere. Deep saline aquifers are widely distributed, contain high salinity water that is unfit for human consumption, and can accommodate large volumes of CO2. CO2 is injected into these aquifers as a supercritical fluid with a liquid-type density and a gas-type viscosity. The critical point of CO2, 31.1 °C and 7.4 MPa, corresponds to an aquifer depth of about 800 m. Sequestration is achieved by trapping supercritical CO2 in the aquifer pore spaces (either as a separate phase trapped beneath impermeable rock or through capillary forces where CO2 is an immobile phase), dissolution of CO2 in the saline aqueous phase (which is dependent on pressure, temperature, and salinity), and reaction of CO2 with minerals present in the rock. Reactions with minerals require a much longer time scale than the other sequestration mechanisms and are not considered in this paper. Numerical simulation of CO2 sequestration is an active research area. Two simulators that have been used extensively for CO2 sequestration modeling are GEM (Ngheim et al., 2001) and the TOUGH2 family of codes (Pruess et al. 1999; Pruess, 2005; Zhang et al. 2007). These simulators model flow in porous media with convection and dispersion, and equilibrium between phases and components. Geomechanical effects are also important to consider in CO2 sequestration simulation because injection results in pressure increases and changes in reservoir stresses and strains that effect rock transport properties and could cause rock to fracture, allowing CO2 to leak into the surroundings. Ngheim et al. (2004), Settari and

An Efficient Adaptive Nonlinearity Elimination Preconditioned Inexact Newton Method for Parallel Simulation of Thermal-Hydraulic-Mechanical Processes in Fractured Reservoirs

In this paper, we introduce a physics-based nonlinear preconditioner, based on the Inexact Newton method, to accelerate the highly nonlinear thermal-hydraulic-mechanical (THM) simulation of fractured reservoirs. Inexact Newton method has become a popular iterative solver for solution of partial differential equations (PDE). Instead of solving the PDEs exactly with the expensive Newton method, the Inexact Newton method finds a direction for the iteration and solves the equations inexactly. The Inexact Newton method is very efficient when the initial guess is close to the objective solution. However, when the equations are not smooth enough, especially when local discontinuities exits, the Inexact Newton method may be slow or even stagnant. Local discontinuities are commonly encountered during oil and gas flow in reservoirs. One problem that involves lots of local discontinuities is the simulation of multiphase flow in fractured reservoirs. Fractures in petroleum reservoirs are typically sensitive to the change of pressure and stress. The thermal-hydraulic-mechanical (THM) effects of injection and production can dramatically change the properties of fractures, resulting in a huge variation in the permeability, which adds nonlinearity to the governing partial differential equations. Considering these characteristics of fractured reservoirs, applying the Inexact Newton method directly to them faces severe difficulties. In this work, we have proposed and studied a nonlinear preconditioner to resolve the above problem. In this nonlinear preconditioner, a restricted additive Schwarz approach is used to coarsen the problem and the Inexact Newton method is used as a global iterative solver. We have developed a physics-based strategy to adaptively identify the highly nonlinear zones by computing and comparing the gradient of the stress/temperature-permeability correlations of the fractured zones. These nonlinear zones are treated as a subspace problem, which is solved locally. The results of the subspace problem are used to modify the global residual. By conducting the above operations, the local nonlinearity is eliminated and the global iteration is accelerated. An adaptive strategy is adopted to dynamically choose between the Inexact Newton method and the Newton method. The above algorithm has the advantage of remarkable scalability and is easy to implement in massively parallel reservoir simulators. We have programed the algorithm and implemented it into our fully coupled, fully implicit THM reservoir simulator to study the effects of cold water injection on fractured reservoirs. The numerical and parallel framework of the simulator has been described by Wang et al. (2014). Previously, the cold water injection problem suffered from slow convergence at the injection zone where fracture permeability changes rapidly. The results of this work show that after the implementation of this nonlinear preconditioner, the iterative solver has become significantly more robust and efficient.

Simulation of Coupled Thermal-Hydrological-Mechanical Phenomena in Porous and Fractured Media

For processes such as production from low permeability reservoirs and storage in subsurface formations, reservoir flow and the reservoir stress field are coupled and affect one another. This paper presents a thermal-hydrological-mechanical (THM) reservoir simulator that is applicable to modeling such processes. The fluid and heat flow portion of our simulator is for general multiphase, multicomponent, multi-porosity systems. The geomechanical portion consists of an equation for mean stress, derived from linear elastic theory for a thermo-poro-elastic system, and equations for stress tensor components that depend on mean stress and other variables. The integral finite-difference method is used to solve these equations. The mean stress and reservoir flow variables are solved implicitly and the remaining stress tensor components are solved for explicitly. Our simulator is verified using analytical solutions for stress and strain tensor components and is compared to published results.

A Multi-Porosity, Multi-Physics Model to Simulate Fluid Flow in Unconventional Reservoirs

Gas flow in shales is complicated by the highly heterogeneous and hierarchical rock structures (i.e., ranging from organic nanopores, inorganic nanopores, less permeable micro-fractures, more permeable macro-fractures, to hydraulic fractures). The dominant fluid flow mechanism varies in these different flow regimes, and properties of these rock structures are sensitive to stress changes with different levels. Although traditional single-porosity and double-porosity models can simulate certain time range of reservoir performance with acceptable accuracy, they are not generally applicable for the prediction of long-term performance and have limitations to improve our understandings of enhanced hydrocarbon recovery. In this paper, we present a multi-domain, multi-physics model, aiming to accurately simulate the fluid flow in shale gas reservoirs with more physics-based formulations. An idealized model has been developed for the purpose of studying the characteristic behavior of a fractured nanopore medium, which contains five regions: organic nanopores, inorganic nanopores, local micro-fractures, global natural fractures, and hydraulic fractures. Fluid flow governing equations in this model vary according to the different dominant fluid flow mechanisms in different regions. For example, the apparent permeability, which is the intrinsic permeability multiplied by a correction factor, is used to account for the gas slippage through nanopores of shale matrix; while the organic and inorganic nanopores in this matrix have different capacities for gas adsorption. On the other hand, for fluids flow in natural fractures and hydraulic fractures with high velocity, the non-Darcy flow model is used to capture the strong inertia when is comparable to viscous force. Numerical studies with practical interests are discussed. Several synthetic, but realistic test cases are simulated. Input parameters in these cases are evaluated using either the laboratory or theoretical work. Our results demonstrate that this model is able to capture the typical production behavior of unconventional reservoirs: a great initial peak, the sharp decline in the first few months, followed by a long flat production tail. A series of sensitivity analyses, which address the organic matter content, organic matter connectivity, natural fracture density, and hydraulic fracture spacing, will also be conducted.

A Compositional Model Fully Coupled with Geomechanics for Liquid-Rich Shale and Tight Oil Reservoir Simulation

The pore sizes of unconventional reservoir rocks, such as shale and tight rocks, are on the order of nanometers. The thermodynamic phase behavior of in-situ hydrocarbon mixtures in such small pores is significantly different from that of bulk fluids in the PVT cells, primarily due to effect of large capillary pressure. For example, it has been recognized that the phase envelop shifts and bubble point pressure is suppressed under subsurface condition in tight oil reservoirs. On the other hand, it has been observed that the pore sizes, especially the sizes of pore-throats, are subject to change due to rock deformation induced by the fluid depletion from over-pressurized unconventional reservoirs. As the fluids are being produced from the pore space, the effective stress on reservoir rock increases, resulting in reduction of the pore and pore-throat sizes. This reduction on pore spaces again affects the fluid flow through impacts on the thermodynamic phase behavior, as well as stress induced changes in porosity and permeability. Thus a coupled flow-geomechanics model capturing in-situ reservoir phase behavior is in general necessary to model tight and shale reservoir performance. In this paper, we propose a multiphase, multidimensional compositional reservoir model, fully coupling fluid flow with geomechanics for tight and shale reservoirs. The fluid flow model is a compositional model, based on general mass conservation law for each component, incorporating both Darcy flow and molecular diffusions. The geomechanical model is derived from the thermo-poro-elasticity theory extended to multiple porous and fractured media systems; mean normal stress as the stress variable is solved simultaneously with mass conservation equations. The vapor-liquid equilibrium (VLE) calculation is performed with Peng-Robinson Equation of State (EOS) including the effects of capillary pressure on phase behaviors. The finite-volume based numerical method, integrated finite difference method, is used for space discretization for both mass conservation and stress equations. The formulations are solved fully implicitly to assure the stability. This compositional model integrates key subsurface behaviors of unconventional shale reservoirs, such as rock compaction effect, stress-induced changes of rock properties, and stress-dependent capillary effects on VLE. We take the Eagle Ford tight oil as an example to illustrate the effects of stress-dependent capillary pressure on VLE and in-situ fluid properties. This model can be generally applied to both dew-point (gas condensate) and bubble-point (tight oil) systems of tight and shale reservoirs. Eventually it could improve the forecast accuracy for long-term production rate and recovery factors of unconventional petroleum reservoirs.