We present two easily computable, equally valid, semianalytic, single-phase, constant-rate solutions to the diffusivity equation for an arbitrarily oriented uniform-flux line source in a 3D, anisotropic, bounded system in Cartesian coordinates. With the addition of superposition, these become inflow solutions for wells of arbitrary trajectory. In addition, we produce analytic time derivatives for pressure-transient analyses (PTAs) of complex wells. If we extract solution components for 2D systems from the general solution, we can construct discrete complex-fracture-inflow and PTA capability for vertical, fully penetrating fractures, suitable for use as the basis solution in modeling complex phenomena, such as pressure-constrained production or development of fracture interference. For a 3D slanted well, the full characterization of dimensionless pressure over 10 decades of dimensionless time behavior can be produced in 1.5 seconds. With a fast-computing analytic solution for pressure anywhere in the system, we can also produce dense pressure maps at scalable resolution where any point could represent an observation well for convolution and enhanced interpretation. Likewise, the pressure derivative and the slope of the logarithmic temporal derivative of pressure can be mapped throughout to indicate local flow regime in a complex system. In particular, we compare and contrast the PTA signatures from symmetrical and asymmetrical horizontal, slanted, and diagonal line sources and examine when the behavior of a thin 3D reservoir collapses to the equivalent of a 2D fully penetrating fracture. Once the reservoir-thickness/length ratio reaches 1:100, all wells with the same projection onto the x–y plane are indistinguishable except for very early time, probably masked by wellbore/fracture-storage effects.
Two algorithms are proposed for isothermal multiphase flash. These are referred to as modified RAND and vol-RAND. The former uses the chemical potentials and molar-phase amounts as the iteration variables, while the latter uses chemical potentials and phase volumes to cosolve a pressure-explicit equation of state (EOS) with the equilibrium equations. Compared with the conventional secondorder approach using Gibbs-energy minimization, these methods are more structured, with all components in all phases treated in the same way. Both have been derived to include chemical reactions for any number of phases along with the possible simplifications for only phase equilibria. The simple structured implementation of these methods is demonstrated for modified RAND and vol-RAND. The rate of convergence of the methods presented is shown to be the same as the conventional second-order method for isothermal flash. It is demonstrated that the use of an association term [cubic plus association (CPA)] adds little additional computational cost when using vol-RAND compared with a simple cubic Soave-Redlich-Kwong (SRK) without association. The RAND methods scale better in terms of the O(n3) operations as more phases are introduced, and are computationally less expensive than the conventional Gibbs minimization method for more than three phases.
Ensemble-based history-matching methods have received much attention in reservoir engineering. In real applications, small ensembles are often used in reservoir simulations to reduce the computational costs. A small ensemble size may lead to ensemble collapse, a phenomenon in which the spread of the ensemble of history-matched reservoir models becomes artificially small. Ensemble collapse is not desired for an ensemble-based history-matching method because it not only deteriorates the capacity in uncertainty quantification, but also forces the ensemble-based method to later stop updating reservoir models. In practice, distance-based localization is thus introduced to tackle ensemble collapse. Distance-based localization works well in many problems. However, one prerequisite in using distance-based localization is that the observations have associated physical locations. In certain circumstances with complex observations, this may not be true, and it thus becomes challenging to apply distance-based localization.
In this work, we propose a correlation-based adaptive localization scheme that does not rely on the physical locations of the observations. Instead, we use the spatial distributions of the correlations between model variables and the corresponding simulated observations. In the course of history matching, we update model variables by only using the observations that have relatively high correlations with them, while excluding those that have relatively low correlations. This is equivalent to introducing a data-selection procedure to the history-matching algorithm. As a result, the threshold values for data selection play an essential role in the proposed adaptive localization scheme, and we develop both ideal and practical approaches to the choices of the threshold values.
We demonstrate the efficacy of the proposed localization scheme using seismic history-matching problems—one 2D and one 3D—in which ensemble collapse is severe in the presence of large amounts of observational data, but distance-based localization may not be applicable because of the lack of physical locations of the seismic data in use. In contrast, correlation-based localization works well to prevent ensemble collapse and also renders good history-matching results. We also note some practical conveniences of the proposed localization scheme, including the applicability to nonlocal observations, the relative simplicity in implementation, the transferability of the same codes among different (either 2D or 3D) case studies, and the adaptivity to different types of observations and petrophysical parameters.
Bidhendi, Mehrnoosh M. (University of Wyoming (now with NALCO Champion, an Ecolab company)) | Garcia-Olvera, Griselda (University of Wyoming (now with PEMEX)) | Morin, Brendon (University of Wyoming) | Oakey, John S. (University of Wyoming) | Alvarado, Vladimir (University of Wyoming)
Injection of water with a designed chemistry has been proposed as a novel enhanced-oil-recovery (EOR) method, commonly referred to as low-salinity (LS) or smart waterflooding, among other labels. The multiple names encompass a family of EOR methods that rely on modifying injection-water chemistry to increase oil recovery. Despite successful laboratory experiments and field trials, underlying EOR mechanisms remain controversial and poorly understood. At present, the vast majority of the proposed mechanisms rely on rock/fluid interactions. In this work, we propose an alternative fluid/fluid interaction mechanism (i.e., an increase in crude-oil/water interfacial viscoelasticity upon injection of designed brine as a suppressor of oil trapping by snap-off). A crude oil from Wyoming was selected for its known interfacial responsiveness to water chemistry. Brines were prepared with analytic-grade salts to test the effect of specific anions and cations. The brines’ ionic strengths were modified by dilution with deionized water to the desired salinity. A battery of experiments was performed to show a link between dynamic interfacial viscoelasticity and recovery. Experiments include double-wall ring interfacial rheometry, direct visualization on microfluidic devices, and coreflooding experiments in Berea sandstone cores. Interfacial rheological results show that interfacial viscoelasticity generally increases as brine salinity is decreased, regardless of which cations and anions are present in brine. However, the rate of elasticity buildup and the plateau value depend on specific ions available in solution. Snap-off analysis in a microfluidic device, consisting of a flow-focusing geometry, demonstrates that increased viscoelasticity suppresses interfacial pinch-off, and sustains a more continuous oil phase. This effect was examined in coreflooding experiments with sodium sulfate brines. Corefloods were designed to limit wettability alteration by maintaining a low temperature (25C) and short aging times. Geochemical analysis provided information on in-situ water chemistry. Oil-recovery and pressure responses were shown to directly correlate with interfacial elasticity [i.e., recovery factor (RF) is consistently greater the larger the induced interfacial viscoelasticity for the system examined in this paper]. Our results demonstrate that a largely overlooked interfacial effect of engineered waterflooding can serve as an alternative and more complete explanation of LS or engineered waterflooding recovery. This new mechanism offers a direction to design water chemistry for optimized waterflooding recovery in engineered water-chemistry processes, and opens a new route to design EOR methods.
Zhang, Haoran (China University of Petroleum, Beijing) | Liang, Yongtu (China University of Petroleum, Beijing) | Yan, Xiaohan (China University of Petroleum, Beijing) | Fang, Limin (China University of Petroleum, Beijing) | Wang, Ning (China University of Petroleum, Beijing)
The settling process of water droplets is significant in the processing technology of crude water-in-oil (W/O) emulsion. The process is complex because it is affected by various microforces on water droplets, which leads to difficulty in studying this issue comprehensively. A large amount of related work has been conducted, but with sparse emphasis being placed on the influence of Brownian motion on the settling process and the heterogeneous system of emulsion. This paper presents a method to study the effect of static settling of the heterogeneous W/O emulsion. A model for calculating water-droplet displacement is fundamentally established by constructing a momentum equation depending on the classical Langevin equation and Stokes formula. It considers dual influences of Stokes gravity settling and Brownian motion as well as microdistribution and interaction of water droplets. Statistical analysis is used to solve the problem of randomness. Combined with the water-cut model and the viscosity-prediction model, and according to the properties of oil phase and the heterogeneous size distribution of water droplets, the migration process of each water droplet can be calculated, and the variation of water cut and viscosity of each layer of the W/O emulsion system as well as the dehydration amount at the bottom can be predicted. In addition, an experiment was carried out to verify the accuracy and practicality of the method. The corresponding results displayed the dehydration amount at five different environment temperatures, and coincided well with the simulation results.
He, Jincong (Chevron Energy Technology Company) | Sarma, Pallav (Chevron Energy Technology Company (now with Tachyus Corporation)) | Bhark, Eric (Chevron Energy Technology Company (now with Chevron Asia Pacific E&P Company)) | Tanaka, Shusei (Chevron Energy Technology Company) | Chen, Bailian (Chevron Energy Technology Company (now with Los Alamos National Laboratory)) | Wen, Xian-Huan (Chevron Energy Technology Company) | Kamath, Jairam (Chevron Energy Technology Company)
Data-acquisition programs, such as surveillance and pilots, play an important role in minimizing subsurface risks and improving decision quality for reservoir management. For design optimization and investment justification of these programs, it is crucial to be able to quantify the expected uncertainty reduction and the value of information (VOI) attainable from a given design. This problem is challenging because the data from the acquisition program are uncertain at the time of the analysis. In this paper, a method called ensemble-variance analysis (EVA) is proposed. Derived from a multivariate Gaussian assumption between the observation data and the objective function, the EVA method quantifies the expected uncertainty reduction from covariance information that is estimated from an ensemble of simulations. The result of EVA can then be used with a decision tree to quantify the VOI of a given data-acquisition program.
The proposed method has several novel features compared with existing methods. First, the EVA method directly considers the data/objective-function relationship. Therefore, it can handle nonlinear forward models and an arbitrary number of parameters. Second, for cases when the multivariate Gaussian assumption between the data and objective function does not hold, the EVA method still provides a lower bound on expected uncertainty reduction, which can be useful in providing a conservative estimate of the surveillance/pilot performance. Finally, EVA also provides an estimate of the shift in the mean of the objective-function distribution, which is crucial for VOI calculation. In this paper, the EVA work flow for expected-uncertainty-reduction quantification is described. The result from EVA is benchmarked with recently proposed rigorous sampling methods, and the capacity of the method for VOI quantification is demonstrated for a pilot-analysis problem using a field-scale reservoir model.
Hui, Mun-Hong (Robin) (Chevron Energy Technology Company) | Karimi-Fard, Mohammad (Stanford University) | Mallison, Bradley (Chevron Energy Technology Company) | Durlofsky, Louis J. (Stanford University)
A comprehensive methodology for gridding, discretizing, coarsening, and simulating discrete-fracture-matrix models of naturally fractured reservoirs is described and applied. The model representation considered here can be used to define the grid and transmissibilities, either at the original fine scale or at coarser scales, for any connectivity-list-based finite-volume flow simulator. For our fine-scale mesh, we use a polyhedral-gridding technique to construct a conforming matrix grid with adaptive refinement near fractures, which are represented as faces of grid cells. The algorithm uses a single input parameter to obtain a suitable compromise between fine-grid cell quality and the fidelity of the fracture representation. Discretization using a two-point flux approximation is accomplished with an existing procedure that treats fractures as lower-dimensional entities (i.e., resolution in the transverse direction is not required). The upscaling method is an aggregation-based technique in which coarse control volumes are aggregates of fine-scale cells, and coarse transmissibilities are computed with a general flow-based procedure. Numerical results are presented for waterflood, sour-gas injection, and gas-condensate primary production for fracture models with matrix and fracture heterogeneities. Coarse-model accuracy is shown to generally decrease with increasing levels of coarsening, as would be expected. We demonstrate, however, that with our methodology, two orders of magnitude of speedup can typically be achieved with models that introduce less than approximately 10% error (with error appropriately defined). This suggests that the overall framework may be very useful for the simulation of realistic discrete-fracture-matrix models.
Stress changes associated with reservoir depletion are often observed in the field. Stress evolution within and surrounding drainage areas can greatly affect further reservoir developments, such as completion of infill wells and refracturing. Previous studies mainly focus on biwing planar-fracture geometry, which limits the possibility of investigating stress evolution caused by complex-fracture geometry. In this paper, we have developed a novel and efficient coupled fluid-flow/geomechanics model with an embedding-discrete-fracture model (EDFM) to characterize stress evolution associated with depletion in unconventional reservoirs with complex-fracture geometry. Coupled geomechanics/fluid flow was developed using the well-known fixed-stress-split method, which is unconditionally stable and computationally efficient to simulate how stress changes during reservoir depletion. EDFM was coupled to the model to gain capability of simulating complex-fracture geometries using structured grids. The model was validated against the classical Terzaghi (1925) and Mandel (1953) problems. Local grid refinement was used as a benchmark when comparing results from EDFM for fractures with 0 and 45° angles of inclination. After that, the model was used to analyze stress distribution and reorientation in reservoirs with three different fracture geometries: planar-fracture (90° angle of inclination), 60° inclination, and nonplanar-fracture geometries. As the pressure decreases, reservoir stresses tend to change anisotropically depending on depletion area. The principal stress parallel to the initial fracture reduces faster than the orthogonal one as a function of time. The decrease rate of principal stresses is distinct for different shapes of depleted areas created by different fracture geometries. The rectangular shape produced by the planar-fracture geometry yields the largest stress-reorientation area for a variety of differential-stress (DS) values (difference between two horizontal principal stresses). The squared shape produced by nonplanar-fracture geometry yields stress reorientation only for low DS. The results indicate that created fracture geometry has a significant effect on stress distribution and reorientation induced by depletion. To the best of our knowledge, this is the first time a coupled fluid-flow/geomechanics model incorporated with EDFM has been developed to efficiently calculate stress evolution in reservoirs with complex-fracture geometry. Characterization of stress evolution will provide critical guidelines for optimization of completion designs and further reservoir development.
We provide analytical solutions for the wellbore pressure during an injection/falloff-test problem under radial-flow conditions in homogeneous porous media where the injected fluid is carbonated water. For both the injection and falloff periods, we assume an isothermal process with thermodynamic equilibrium, a linear adsorption isotherm, and viscosities that depend only on the carbon dioxide (CO2) concentration. We also neglect CO2 diffusion, gravity effects, and capillarity effects. For the injection period, we first determine the saturation and concentration distributions with time in the reservoir by applying the method of characteristics to solve the appropriate system of hyperbolic conservation equations, where we assume incompressible fluids. In solving for water saturation and CO2 concentration in water, we neglect the change in water volume caused by the variation of the CO2 concentration in water. After solving for the saturation and concentration profiles, the pressure solution can be obtained by integrating Darcy’s law, from the wellbore radius to infinity, while assuming an infinite-acting reservoir and invoking the Thompson-Reynolds steady-state theory (Thompson and Reynolds 1997b). Because Darcy’s law does not assume incompressible flow, the pressure solution generated does not assume incompressible flow. To obtain an analytical expression for the wellbore pressure, however, we do assume that for injection and falloff, the total flow-rate profile in the reservoir is constant in a region from the wellbore to a radius greater than the radius of the flood front. The region within this radius increases with time and it is referred to as the steady-state region or zone (Thompson and Reynolds 1997b). During the falloff stage, it is assumed that there is no change in saturation in the reservoir, which is reasonable because we neglect capillary pressure, the gravity force, and fluid compressibilities when determining the saturation profile. Using these assumptions, we generate analytical solutions for a carbonated-water-injection (CWI)/falloff test and compare these solutions with those obtained with a commercial reservoir simulator using very fine spatial grids and very small timesteps. This comparison suggests that the analytical solutions presented can be used reliably to analyze pressure data obtained during CWI/falloff tests.
System-algebraic multigrid (AMG) provides a flexible framework for linear systems in simulation applications that involve various types of physical unknowns. Reservoir-simulation applications, with their driving elliptic pressure unknown, are principally well-suited to exploit System-AMG as a robust and efficient solver method.
However, the coarse grid correction must be physically meaningful to speed up the overall convergence. It has been common practice in constrained-pressure-residual (CPR) -type applications to use an approximate pressure/saturation decoupling to fulfill this requirement. Unfortunately, this can have significant effects on the AMG applicability and, thus, is not performed by the dynamic rowsum (DRS) method.
This work shows that the pressure/saturation decoupling is not necessary for ensuring an efficient interplay between the coarse grid correction process and the fine-level problem, demonstrating that a comparable influence of the pressure on the different involved partial-differential equations (PDEs) is much more crucial.
As an extreme case with respect to the outlined requirement, linear systems from compositional simulations under the volume-balance formulation will be discussed. In these systems, the pressure typically is associated with a volume balance rather than a diffusion process. It will be shown how System-AMG can still be used in such cases.