Two new Non-Intrusive Reduced Order Modelling approaches to estimate time varying, spatial distributions of variables from arbitrary unseen inputs are introduced. One is a generalization of an existing'dynamic' approach which requires multiple surrogate evaluations to model the solutions at different time instances, the other is a'steady-state' approach that evaluates all time instances simultaneously, reducing the local approximation error. The ability of these approaches to estimate the water saturation distributions expected during a gas flood through a 2D, dipping reservoir is investigating for a range of unseen input parameters. The range of these parameters has been chosen so that a range of flow regimes will occur, from a gravity tongue to a viscous dominated Buckley-Leverett displacement. A number of practically relevant model error measures were employed as opposed to the standard L2 (Euclidean) norm. The influence of the number and the structure of training simulations for the model was also investigated, by employing two simple experimental design methods. The results show that POD based NIROM approaches are prone to significant deviations from the true model. The main sources of error are due to the non-smooth variation of system responses in hyperspace and the transient nature of the flows as well as the underlying dimensionality reduction. Since the first two sources are properties of the physical system modelled it may be expected that similar problems are likely to arise independently of the interpolation method and the reduction process used.
Conventional miscible or near-miscible gasflooding simulation often overestimates oil recovery, mostly because it does not capture a series of physical effects tending to limit interphase compositional exchanges. Those can include microscopic bypassing of oil situated in dead-end pores or blocked by water films, as well as macroscopic bypassing caused by subgrid-size heterogeneities or fingering. We here present a new engineering solution to this problem in the near-miscible case, relying on our in-house research reservoir simulator. The principle is, while using a black-oil or an equation-of-state description, to dynamically decrease the K-value of heavy components and possibly increase the K-value of light components as the oil saturation reaches the desired residual limit; this enables changing the phase boundaries when needed while preserving the original fluid behavior during the initial production stages. The benefits of the proposed solution are demonstrated on a reservoir-conditions tertiary-gas-injection experiment, performed in our laboratories, for which residual saturations as well as oil phase and individual-component production rates have easily and successfully been history matched. Results are then compared with matches obtained by use of saturation exclusion and x-factors methods. As a proof of concept, the suitability of the new method to simulate incomplete revaporization of condensate during gas cycling is also illustrated, on the third SPE comparative solution-project case.
Conventional miscible or near-miscible gas flooding simulation often overestimates oil recovery, mostly because it does not capture a series of physical effects tending to limit interphase compositional exchanges. Those can be for instance microscopic bypassing of oil situated in dead-end pores or blocked by water films, as well as macroscopic bypassing due to sub-grid size heterogeneities or fingering.
We here present a new engineering solution to this problem in the near-miscible case, relying on our in-house research reservoir simulator (IHRRS). The principle is, while using a black-oil or an equation of state description, to dynamically decrease the K-value of heavy components and possibly increase the K-value of light components as the oil saturation reaches the desired residual limit; this enables changing the phase boundaries when needed while preserving the original fluid behavior during the initial production stages.
The benefits of the proposed method are demonstrated on a reservoir conditions tertiary gas injection experiment, performed in our laboratories, for which residual saturations as well as oil phase and individual components production rate have easily and successfully been history matched. Results are then compared to matches obtained using saturation exclusion and α-factors methods. As a proof of concept, suitability of the method to simulate incomplete revaporization of condensate during gas cycling is also illustrated, on the third SPE comparative solution project case.
Conventional reservoir simulation, where complete phase equilibrium is enforced in each cell at each time-step, often overestimates oil production when applied to gas flooding. In miscible conditions, a simulator will predict the presence of a single hydrocarbon phase, either in the entire computational domain or at least at miscibility front, depending on whether the miscibility is first-contact or multi-contact; after a limited amount of pore-volumes injected, complete recovery of the original oil is therefore predicted. In near-miscible conditions, although the oil phase becomes non-mobile at low saturation, its components including heavy ends will eventually vaporize; complete recovery is therefore predicted as well, albeit after a longer injection period. Full recovery is however never observed, be it at the field scale or in laboratory corefloods, mainly because of microscopic (sub-Darcean) as well as macroscopic (Darcean) phenomena tending to limit interphase mass exchange. This comes in addition to the fact that lumping heavy ends leads by itself to an anticipation of ultimate recovery, since the tail of the hydrocarbon distribution is less vaporizable than the heavier pseudo-component it belongs to.
The Microemulsion phase behavior model based on oleic/aqueous/surfactant pseudophase equilibrium, commonly used in chemical flooding simulators, is coupled to Gas/Oil/Water phase equilibrium in our new four-fluid-phase, fully implicit in-house research reservoir simulator (IHRRS) (Moncorge et al. 2012). The method consistsof splitting the equilibrium into two stages, in which all the components other than surfactant are equilibrated first - by use of a black-oil, K-value, or equation of state (EOS) model - and the resulting Gas, Oil, and Water phases are then lumped into pseudophases to be equilibrated by use of the Microemulsion model. This subdivision in stages is conceptual, and at each converged timestep the four phases (Gas, Oil, Water, and Microemulsion, when simultaneously present) will be in equilibrium with each other. The fluid properties (such as densities, viscosities, and interfacial tensions) and rock/fluid properties (such as relative permeabilities) required in the transport equations are evaluated with models from well-known industrial or academic simulators. Surfactant flooding being usually implemented as a tertiary recovery mechanism, on fields for which complete models that we do not wish to modify already exist, particular care is devoted to ensuring continuity of the physics at the onset of surfactant injection. Our code is first validated against a reference academic chemical- flooding simulator, on a 1D, three-fluid-phase (Oil/Water/ Microemulsion) coreflood. Second, as application examples where it is necessary to account for four phases in equilibrium, we consider a scenario where the chemical flood is preceded by a vaporizing Gas drive, as well as a scenario where dissolved gas is released by the Oil during the flooding process. Some aspects of our implementation, such as numerical dispersion vs. timestep length and nonlinear convergence, are also discussed; in particular, we show that numerical performance is not degraded by the four-phase equilibrium.
The dynamic effect of pressure and Oil composition on Microemulsion phase behavior, complementing the key effect of variable salinity, has been implemented in our four-fluid-phase, fully implicit in-house research reservoir simulator. This has been achieved through self-consistent coupling of a traditional Gas/Oil/Water phase equilibrium model, either compositional or generalized black-oil,—providing phase fractions, oleic composition, and aqueous salinity—with a Microemulsion model based on oleic/aqueous/chemical pseudophase equilibrium.
As an application example and validation test case, we consider a hypothetical surfactant/polymer (SP) coreflood of a saturated Oil, interrupted by a progressive depressurization, during which dissolved gas is released, which shifts the Microemulsion phase state from Winsor Type III to Type II-. This proves the good functioning of our new option, and shows, yet on a simple case, that it does not degrade numerical performance, despite the introduction of additional nonlinear dependencies.
Surfactant/polymer (SP) flooding is an enhanced oil recovery process aimed at mobilizing residual Oil to Water1 in mature conventional reservoirs through interfacial tension (IFT) reduction, and sweeping it through mobility ratio improvement (Lake 1989). While most commercial reservoir simulators have long had the capacity of treating polymer, modeled as an additional aqueous component, comprehensive treatment of surfactant has until recently been limited to academic codes. Indeed, above a critical micelle concentration (CMC), Water and Oil become partially soluble, leading to the formation of an additional Microemulsion phase, whose implementation is a rather complicated task. Appropriate modeling of Microemulsion is yet key, as the IFT’s reduction with the excess Oil and/or Water phases is function of the solubilization ratios, defined as the amount of dissolved oil and/or water per amount of surfactant (Huh 1979). In general, those depend on the chemical formulation, Water and Oil phase compositions, pressure, and temperature.
In petroleum reservoirs, it is common to assume that the superficial velocity of a fluid phase equals a mobility coefficient (permeability over viscosity) multiplying the opposite of the pressure gradient. Phase mobility is typically a function of phase saturations, compositions, temperature, and to a lesser extent pressure, but in specific circumstances it may also depend on the velocity itself. For example, in polymer flooding the aqueous phase viscosity is velocity-dependent due to non-Newtonian effects.
The numerical treatment of velocity-dependent mobilities in reservoir simulators based on the finite-volumes method is still an unsettled topic. For a fully implicit discretization, the main difficulty lies in the fact that velocities are not necessarily aligned with the grid; therefore the velocity-dependent mobility governing the flux across a cell face is not merely a function of the normal pressure drop. In addition, the implicit relationship between velocity and pressure gradient needs to be inverted for each flux, at each nonlinear iteration. To get around the above difficulties, several commercial or academic simulators implement a semi-implicit scheme where the pressure gradient driving the flow is evaluated implicitly, while the velocity used in mobility calculations is evaluated explicitly based on the previously converged time step. However, a semi-implicit formulation may be subject to stability restrictions.
In this work, we first review the derivation of a linear stability criterion for the two-level semi-implicit discretization of simplified monophasic, non-Newtonian or non-Darcy flow equations. Based on this criterion, we propose an adaptive-implicit strategy where for each individual flux the velocity argument in the mobility function is evaluated either explicitly or implicitly. We discuss the numerical accuracy of this scheme and its benefits in terms of computational cost. Finally, using a specifically designed MATLAB code, we validate our adaptive-implicit strategy on representative 1D and 2D nonlinear test cases.
Copyright 2013, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Reservoir Simulation Symposium held in The Woodlands, Texas USA, 18-20 February 2013. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright. Abstract Von Neumann analysis is the most commonly used method for determining stability of an explicit-in-time finite difference method. Watts and Ramé (1999) describe a different approach based on the error growth matrix for the finite difference method. In this approach, called herein the matrix method, stability is determined by computing the eigenvalues of the error growth matrix. An advantage of the matrix method is it is it deals rigorously with heterogeneities. However, in a practical application, the growth matrix is large, making the cost of computing the eigenvalues prohibitive.
The Microemulsion phase behavior model based on oleic-aqueous-surfactant pseudo-phase equilibrium, commonly used in chemical flooding simulators, is coupled to Gas-Oil-Water phase equilibrium in our new four-fluid-phase, fully implicit In-House Research Reservoir Simulator (IHRRS). The method consists in splitting the equilibrium in two stages, where all the components other than surfactant are equilibrated first (e.g. using a black-oil, K-value or equation of state model), and the resulting Gas, Oil and Water phases are then lumped into pseudo-phases to be equilibrated using the Microemulsion model. This subdivision in stages is conceptual, and at each converged time-step the four phases (Gas, Oil, Water and Microemulsion, when simultaneously present) will be in equilibrium with each other.
The fluid properties (such as densities, viscosities and interfacial tensions) and rock-fluid properties (such as relative permeabilities), required in the transport equations, are evaluated with models from well-known industrial or academic simulators. Surfactant flooding being usually implemented as a tertiary recovery mechanism, on fields for which complete models that we do not wish to modify already exist, particular care is devoted to ensuring continuity of the physics at the onset of surfactant injection.
Our code is validated against a reference academic chemical flooding simulator, on 1D corefloods where the original hydrocarbons in place form a dead-Oil phase, possibly with free dry-Gas. Some numerical aspects of our implementation such as numerical dispersion versus time-step size and nonlinear convergence performance are also discussed. As an application example of our code where it is necessary to account for four phases in equilibrium, we consider a scenario where the chemical flood is preceded by a vaporizing Gas drive.
Surfactant flooding, whose key principle is to improve pore-level sweep efficiency by reducing the interfacial tension between injected Water and reservoir Oil (Lake, 1989), recently received renewed attention due to the perspective of long-lasting high oil prices. Meaningful simulation of surfactant flooding is challenging, in particular because when the surfactant concentration in the Water phase goes above a Critical Micelle Concentration (CMC), Water and Oil become mutually soluble in a proportion mainly determined by water salinity CS. Three different phase environments should be considered (Nelson & Pope, 1978; Lake, 1989). Near the so-called optimal salinity CSOP the surfactant is similarly hydrophilic and lipophilic, and a middle phase called Microemulsion forms, containing the surfactant in excess of the CMC as well as dispersed oil and water (Winsor III environment). This is the best performing regime because interfacial tensions between Water and Microemulsion and between Microemulsion and Oil are then extremely low. At low salinity the surfactant is typically hydrophilic, hence only Oil and a Microemulsion phase containing dispersed oil coexist (Winsor II- environment); on the contrary at high salinity the surfactant is typically lipophilic, hence only Water and a Microemulsion phase containing dispersed water coexist (Winsor II+ environment).
Numerical modeling of advanced recovery mechanisms at the reservoir scale (e.g. miscible or immiscible Gas flood, chemical flood, steam injection…), typically implemented in a tertiary phase, is essential to reasonably estimate their potential benefit and to rank the various field development options. In this perspective, using a unique advanced-physics simulator for the entire life of an asset is a desirable objective, because in addition to saving engineering time spent in data conversion, it ensures continuity of the models at all times. In our view such a tool, expected to be flexible and allow reactivity for testing new models, should come as a complement to optimized and robust industrial-grade simulators used for prediction and history matching during primary and secondary recovery.
In this paper we present the prototyping framework implemented in our In-House Research Reservoir Simulator (IHRRS), enabling easy integration of new physics for improved recovery processes with the well-known Black-Oil and K-value or Equation of State compositional models. As a demonstration example, we choose Surfactant-Polymer (SP) flooding, possibly requiring an additional Microemulsion phase. The framework is based on a natural variables formulation, solving a coupled system of conservation equations for the hydrocarbon and aqueous components (and optionally for the energy), simultaneously with a set of local thermodynamic constraints. These constraints enforce the equilibrium of hydrocarbon and aqueous components across the different fluid phases, including the equilibrium of Water and Oil with Microemulsion.
To ensure compatibility between the different recovery mechanisms handled by our system, as well as to facilitate their development and benchmarking, special attention has been paid to developing physical options as plug-in functionalities. For this purpose, instead of relying on complex software engineering tools we prefer the approach of using low-level interfaces to communicate between the core and the modules (such as the fluid, the petrophysical, or the surface facility modules). Most of our modules are entirely independent from each other, and can be compiled as stand-alone programs to be called by MATLAB® or Python scripts for instance; symmetrically, they could be replaced by external software in order to test third party functionalities.
As a first benchmark of our IHRRS, we consider a surfactant-polymer flood scenario in a 2D anisotropic quarter five-spot setting, and compare our solutions against those of a reference academic chemical flooding simulator (UTCHEM). The potentialities of our framework will then be demonstrated on a simplified model of a real Middle-Eastern field.
Moncorge, Arthur (Total E&P)
A formulation for the fully coupled simulation of reservoir, well and network for advanced processes is described in this paper.
The governing equations for the reservoir, well and network are generalized into the same finite volume framework. The
approach is close to Shiralkar and Watts (SPE 93073) but it uses natural variables and is generalized to: (1) energy and
aqueous components (polymers, surfactants, salts, …) conservation equations, (2) mass and energy accumulation terms in the
wells and the network to simulate transient effects, (3) acceleration terms in the wellbore. The motivation of this work is to
have a consistent formulation for the reservoir, the well and the network, allowing (1) no limitation in the well/network
architecture, (2) ease of implementation and maintenance, (3) use of advanced linear solvers.
General finite volume formulation
A finite volume formulation for control volume I is described in this section. This is the usual formulation used for reservoir
simulators with "natural variables?? using first-order implicit (backward Euler) time discretization, first-order spatial
discretization with phase-based single-point upstream weighting, and second-order spatial discretization with centered
differences. K is the set of control volumes connected to I.