Faults and complex wells are two important types of internal boundaries to resolve in reservoir simulation. Faults are physical boundaries which may form local barriers or conduits to fluid flow. In structured-grid simulation, fault surfaces are typically represented as zig-zag cell edges where the depths may be shifted across the fault face. The better representation of fault traces using unstructured gridding has been the subject of research in the petroleum literature for over two decades. The use of long horizontal and multi-branch complex wells for production from tight and heterogeneous reservoirs is also common practice nowadays. These wells can be densely populated which make classical local grid refinement (LGR) methods difficult to apply. It is highly desirable to represent the perforation inflow and the near-wellbore flow more accurately in full-field simulation.
The paper extends the Voronoi gridding method (
Following an introduction of unstructured-grid methods in reservoir simulation, the gridding algorithm is discussed in details. This is followed by simulation examples, which includes a full-field compositional simulation example of a faulted gas-condensate reservoir completed with many deviated and horizontal wells. An in-house parallel reservoir simulator is used to run the models. Simulation results using both the structured corner-pointed-geometry (CPG) grid and the unstructured-grid method are compared. The advantages of unstructured approach in complex field-scale simulation are demonstrated.
The surrogate model of choice in this investigation uses Trajectory Piecewise Linearization as numerical complexity reduction technique and Proper Orthogonal Decomposition for dimension reduction. The stability of the model is assured through the use of Petrov-Galerkin left projector in finding the reduced space solution of the linearized problem. The TPWL/POD approximation is further refined by the addition of a kriging correction model. The high fidelity model is a two-phase, 3D, fully implicit simulator that uses mass fraction-based formulation. The approximate model is used to expedite a waterflooding optimization problem where design variables are BHP controls to maximize the lifecycle Net Present Value. The proposed optimization strategy is based on a trust region framework. It decomposes the original problem into a sequence of local problems performed on the surrogate model bounded by a trust region whose extent is adaptively managed by the strategy during the optimization process depending on surrogate accuracy. Should surrogate accuracy deteriorate as a result of changes in well controls during the optimization process TPWL/POD model must be retrained to incorporate new state snapshots corresponding to those controls. This work proposes TPWL/POD retraining criteria based on the trust region accuracy parameter and an error indicator that represents the average distance between stored snapshots and the corresponding simulation states. The proposed optimization strategy is applied to a 24000 cell reservoir based on SPE-10 problem with two injector and four producer wells considering four control cycles. Differences in fluid densities and fluid compressibilities are take into account increasing problem nonlinearity. Different trends are used for the correction model. A parameter study is conducted to fine tune the proposed correction criteria. Excellent results are obtained with NPV values exceeding those obtained by coupling the SQP optimizer directly to the simulator as well as the TPWL/POD approximation with no correction or retraining. The strategy proved to be very effective because of the reuse of previous computation through stored Jacobians and continuous refinement of the kriging correction model. Unnecessary retraining simulations were avoided by the criteria which can be used by other surrogate based strategies making use of TPWL/POD approximation.
Modeling the flow regime of well completions in unconventional plays presents a particular challenge for obtaining accurate solutions in the vicinity of fractures for both linear and complex geometries. To this end, a hybrid grid-generation method has been developed that automatically generates computational grids with prescribed spatial resolution normal to the fracture faces to capture the extreme gradients in the solution while still conforming to the geometry of the fracture, even when the fracture is arbitrarily aligned and intersecting other fractures.
The method can be applied to both hydraulically induced fractures as well as discrete fracture networks, with full 3-D cells being used to capture the flow physics. While significant portions of the computational grid in the near-fracture region are hexahedra, prismatic cells are used to fill in the remainder of the volume, forming a hybrid grid.
The use of the hybrid grid is demonstrated for complex systems of wellbores, hydraulic fractures, and natural fractures. While much of the grid is K-orthogonal, some portions are not. To correct for this, the grid is solved in a simulator in which multi-point flux-approximation (MPFA) schemes have been implemented. The MPFA can be applied regionally along with normal two-point flux-approximation (TPFA) schemes. This provides us with the confidence that we can mix K-orthogonal and non-K-orthogonal grid areas. The results are compared with results using TPFA everywhere, both in terms of solution and computational speed.
The resulting combination of new gridding algorithms and discretization techniques provides us with the combination of improved accuracy in regions where the accuracy is required, but without the performance penalties often seen for uniformly fine grid solutions and/or oddly shaped grids used to conform to complex wells and geologic structures.
Low Salinity Waterflood (LSW) is an emerging Enhanced Oil Recovery (EOR) method that is simple to implement and has been shown to yield substantial increase in oil recovery over conventional waterflood, especially in oil-wet sandstone reservoirs. The mechanism for increased oil recovery is a wettability change from oil-wet to water-wet, which is induced by ion exchange between the injected fluid and the clay surface.
This paper presents the geochemistry that is required to model LSW and a method for shifting the relative permeabilities from oil-wet conditions to water-wet conditions. Excellent matches of core displacement data were achieved with a compositional simulator that incorporates those processes.
As ion exchange with the clay surface is important for the success of LSW, a method for modeling clay distribution and content based on facies is proposed. A new concept for LSW optimization based on well placement is introduced and demonstrated with reservoir simulation. As there are uncertainties associated with the geological modeling of the clay distribution, this paper shows how robust optimization can be applied to reduce uncertainties in the LSW optimization through well placement.
We propose a numerically stable algorithm for coupled flow and finite-strain multiplicative elastoplastic geomechanics in this study, extending small deformation to large deformation problems. The proposed algorithm first solves flow, being energy dissipative, and then solve mechanics. Energy-dissipation at the flow step can be achieved by fixing the first Piola-Kirchhoff total stress field. In this sense, the proposed algorithm is an extension of the fixed stress sequential method in coupled flow and geomechanics. Although fixing the first Piola-Kirchhoff total stress field provides theoretical unconditional stability, we fix the second Piola-Kirchhoff total stress field in this study, based on the assumption that the difference between the two is small, because the constitutive relations are formulated by the second Piola-Kirchhoff total stress.
In space discretization, we use the finite element method for mechanics with the total Lagrangian approach scheme, while employing the finite volume method for flow. Geometrical nonlinearity from the total Lagrangian approach results in full-tensor permeability even though the initial permeability is isotropic. To deal with full-tensor permeability, we use the multipoint flux approximation in flow. In time discretization, the backward Euler method is used.
We show by the energy method that the proposed algorithm is unconditionally stable, i.e., the proposed operator splitting and sequential algorithm are contractive and B-stable, respectively. Then, we present numerical examples of coupled finite-strain geomechanics and flow.
In this article, two formulations of multiphase compositional Darcy flows taking into account phase transitions are compared. The first formulation is the so called natural variable formulation commonly used in reservoir simulation, the second has been introduced by Lauser et al. and uses the phase pressures, saturations and component fugacities as main unknowns. We will discuss how the Coats and the Lauser approaches can be used to solve a compositional multiphase flow problem with cubic equations of state of Peng and Robinson. Then, we will study results of several synthetic cases that are representative of petroleum reservoir engineering problems and we will compare their numerical behavior.
A multilevel optimization procedure, in which optimization is performed over a sequence of upscaled models, is developed for use in combined well placement and control problems. The multilevel framework, which can be incorporated with any type of optimization algorithm, is implemented here with a derivative-free Particle Swarm Optimization – Mesh Adaptive Direct Search (PSO–MADS) hybrid technique. An accurate global transmissibility upscaling procedure is applied to generate the coarse-model parameters required at each grid level. Distinct upscaled models are constructed using this approach for each candidate solution evaluated by the optimization algorithm. We demonstrate that the coarse models are able to capture the basic ranking of the candidate well location and control scenarios, in terms of objective function, relative to the ranking that would be computed using fine-scale simulations. This enables the optimization algorithm to appropriately select and discard candidate solutions. Two- and three-dimensional example cases are presented, one of which involves optimization over multiple geological realizations. The multilevel procedure is shown to provide optimal solutions that are comparable, and in some cases better, than those from the conventional (single-level) approach, but with computational speedups of about an order of magnitude.
Mostaghimi, Peyman (The University of New South Wales) | Kamali, Fatemeh (The University of New South Wales) | Jackson, Matthew D. (Imperial College London) | Muggeridge, Ann H. (Imperial College London) | Pain, Christopher C. (Imperial College London)
Viscous fingering is a major concern in the waterflooding of heavy oil reservoirs. Traditional reservoir simulators employ low-order finite volume/difference methods on structured grids to resolve this phenomenon. However, their approach suffers from a significant numerical dispersion error due to insufficient mesh resolution which smears out some important features of the flow. We simulate immiscible incompressible two phase displacements and propose use of unstructured control volume finite element (CVFE) methods for capturing viscous fingering in porous media. Our approach uses anisotropic mesh adaptation where the mesh resolution is optimized based on the evolving flow features. The adaptive algorithm uses a metric tensor field based on solution interpolation error estimates to locally control the size and shape of elements in the metric. We resolve the viscous fingering patterns accurately and reduce the numerical dispersion error significantly. The mesh optimization, generates an unstructured coarse mesh in other regions of the computational domain where a high resolution is not required. We analyze the computational cost of mesh adaptivity on unstructured mesh and compare its results with those obtained by a commercial reservoir simulator based on the finite volume methods.
Petvipusit, Kurt R. (Department of Earth Science and Engineering, Imperial College London) | Elsheikh, Ahmed H. (Institute of Petroleum Engineering, Heriot-Watt university, UK) | King, Peter R. (Department of Earth Science and Engineering, Imperial College London) | Blunt, Martin J. (Department of Earth Science and Engineering, Imperial College London)
The successful operation of CO2 sequestration relies on designing optimal injection strategies that maximise economic performance while guaranteeing long-term storage security. Solving this optimisation problem is computationally demanding. Hence, we propose an efficient surrogate-assisted optimisation technique with three novel aspects: (1) it relies on an ANOVA-like decomposition termed High-Dimensional Model Representation; (2) component-wise interactions are approximated with adaptive sparse grid interpolation; and (3) the surrogate is adaptively partitioned closer to the optimal solution within the optimisation iteration.
A High-Dimensional Model Representation (HDMR) represents the model output as a hierarchical sum of component functions with different input variables. This structure enables us to select influential lower-order functions that impact the model output for efficient reduced-order representation of the model. In this work, we build the surrogate based on the HDMR expansion and make use of Sobol indices to adaptively select the significant terms. Then, the selected lower-order terms are approximated by using the Adaptive Sparse Grid Interpolation (ASGI) approach. Once the HDMR is built, a global optimizer is run to decide: 1) the domain shrinking criteria; and 2) the centre point for the next HDMR building. Therefore, this proposed technique is called a walking Cut-AHDMR as it shrinks the search domain while balancing the trade-off between exploration and exploitation of the optimisation algorithm.
The proposed technique is evaluated on a benchmark function and on the PUNQ-S3 reservoir model. Based on our numerical results, the walking Cut-AHDMR is a promising approach: not only does it require substantially fewer forward runs in building the surrogate of high dimension but it also effectively guides the search towards the optimal solution. The proposed method provides an efficient tool to find optimal injection schedules that maximise economic values of CO2 injection in deep saline aquifers.
Rahmani, Amir Reza (Now with Quantum Reservoir Impact (QRI)) | Bryant, Steven L. (Now with University of Calgary) | Huh, Chun (The University of Texas at Austin) | Ahmadian, Mohsen (The University of Texas at Austin) | Zhang, Wenji (Duke University) | Liu, Qing Huo (Duke University)
As surface-coated superparamagnetic nanoparticles are capable of flowing through micron-size pores across long distances in a reservoir, with modest retention in rock, they have novel use potential in subsurface applications. These particles change the magnetic permeability of the flooded region, and thus can be used to enhance images of the subsurface and characterize hydrocarbon reservoirs. We earlier demonstrated the feasibility of using magnetic nanoparticles to track flood-front in waterflood and EOR processes in a homogeneous reservoir. In this paper, we model the propagation of a “ferrofluid” slug in a heterogeneous reservoir and its response to a crosswell magnetic tomography system. Specifically, we highlight the magnetic response at a low frequency (10 Hz) to the magnetic excitations generated by a vertical magnetic dipole source positioned at the injection well. The “ferrofluid” alters only the magnetic permeability of the domain occupied by the fluid and is thus distinct from methods that rely on contrasts in electrical conductivity. The flow behavior of the magnetic nanoparticles is coupled with time-lapse magnetic measurements through applying appropriate mixing laws and effective medium theory. Fluid flow is computed with a reservoir simulator; the electromagnetic response is computed with an electromagnetic (EM) simulator developed at Duke University for the overburden/reservoir/underburden system.
The approach to monitoring fluid movement within a reservoir is built on established electromagnetic conductivity monitoring technology. Here we investigate the detectability of a contrast in magnetic permeabilities between injected and resident fluids. At the low frequency studied here, the induction effect is small, the casing effect is manageable, the crosswell response originates purely from the magnetic contrast in the formation, and changes in fluid conductivities are irrelevant. This approach thus offers a new and independent mechanism for tracking flood fronts.
Numerical simulations indicate that the influence of areal and vertical reservoir permeability heterogeneity on flood fronts can be detected. For areal permeability heterogeneity, we use a five-spot reservoir model (with injector in the center) and incorporate high- and low-permeability ellipsoidal features with two orientations. The most detectable heterogeneity is a low permeability feature perpendicular to the streamlines. For vertical heterogeneity, we devise a two-layer reservoir model with single-well radial injection with a variable thickness for the high-permeability layer and study the evolution of time-lapse magnetic tomography maps. The tomography maps are shown to be capable of detecting the vertical heterogeneity in different stages of the flood. This is particularly helpful for identifying thief zones. In all the cases, the magnetic response is sensitive to the pattern and distribution of streamlines; therefore, permeability heterogeneity could be deduced from time-lapse magnetic measurements.
By adding magnetic nanoparticles into the injection fluids for waterflood and EOR processes and utilizing the established EM crosswell tomography technique, we show the feasibility of inferring the major features of reservoir heterogeneity, as well as of tracking the injectant bank front, from the time-lapse magnetic responses. This can substantially improve the management and optimization of such floods.