Streamline-based Method Research Articles

Parallel implementations of streamline simulators

We discuss various strategies for parallelizing streamline simulators and present a single-phase shared memory implementation. The choice of a shared memory programming model is motivated by its suitability for streamline simulation, as well as the rapid advance of multicore processors, which are readily available at low-cost. We show that streamline-based methods are easily parallelizable on shared memory architectures through their decomposition of the multidimensional transport equations into a large set of independent 1D transport solves. We tested both a specialized explicit load balancing algorithm that optimizes the streamline load distribution across threads to minimize the time that any of the threads are idle, and the dynamic load balancing algorithms provided by OpenMP on the shared memory machines. Our results clearly indicate that built-in schedulers are competitive with specialized load balancing strategies as long as the number of streamlines per thread is sufficiently high, which is the case in field applications. The average workload per thread is nominally insensitive to workload variations between individual streamlines, and any load balancing advantage offered by explicit strategies is not sufficient to overcome associated computational and parallel overhead. In terms of the allocation of streamlines or streamline segments to threads, we investigated both the distributed approach, in which threads are assigned streamline segments, and the owner approach, in which threads own complete streamlines. We found that the owner approach is most suitable. The slight advantage that the distributed approach has in terms of load balancing is not enough to compensate for the additional overheads. Moreover, the owner approach allows straightforward re-use of existing sequential codes, which is not the case for the distributed approach in case of implicit or adaptive implicit solution strategies. The tracing and mapping stages in streamline simulation have low parallel efficiency. However, in real-field models, the computational burden of the streamline solves is significantly heavier than that of the tracing and mapping stages, and therefore, the impact of these stages is limited. We tested the parallelization on three shared memory systems: a 24 dual-core processor Sun SPARC server; an eight-way Sun Opteron server, representative of the state-of-the-art shared memory systems in use in the industry; and the very recently released Sun Niagara II multicore machine that has eight floating point compute units on the chip. We test a single-phase flow problem on three heterogeneous reservoirs with varying well placements (this system gives the worst case scenario as the tracing and mapping costs are not negligible compared to the transport costs). For the SPARC and Opteron system, we find parallel efficiencies ranging between 60 and 75 for the tracer flow problems. The sublinear speedup is mostly due to communication overheads in the tracing and mapping stages. In applications with more complex physics, the relative contributions of these stages will decrease significantly, and we predict the parallel performance to be nearly linear. On the Niagara II, we obtain almost perfect linear scalability even for the single-phase flow problem thanks to the lowered communication costs on these architectures that have a shared cache. This result is all the more satisfactory considering that future server designs will be akin to this system.

3D aquifer characterization using stochastic streamline calibration

In this study, a new inverse approach, stochastic streamline calibration is proposed. Using both a streamline concept and a stochastic technique, stochastic streamline calibration optimizes an identified field to fit in given observation data in a exceptionally fast and stable fashion. In the stochastic streamline calibration, streamlines are adopted as basic elements not only for describing fluid flow but also for identifying the permeability distribution. Based on the streamline-based inversion by Agarwal et al. [Agarwal B, Blunt MJ. Streamline-based method with full-physics forward simulation for history matching performance data of a North sea field. SPE J 2003;8(2):171–80], Wang and Kovscek [Wang Y, Kovscek AR. Streamline approach for history matching production data. SPE J 2000;5(4):353–62], permeability is modified rather along streamlines than at the individual gridblocks. Permeabilities in the gridblocks which a streamline passes are adjusted by being multiplied by some factor such that we can match flow and transport properties of the streamline. This enables the inverse process to achieve fast convergence. In addition, equipped with a stochastic module, the proposed technique supportively calibrates the identified field in a stochastic manner, while incorporating spatial information into the field. This prevents the inverse process from being stuck in local minima and helps search for a globally optimized solution. Simulation results indicate that stochastic streamline calibration identifies an unknown permeability exceptionally quickly. More notably, the identified permeability distribution reflected realistic geological features, which had not been achieved in the original work by Agarwal et al. with the limitations of the large modifications along streamlines for matching production data only. The constructed model by stochastic streamline calibration forecasted transport of plume which was similar to that of a reference model. By this, we can expect the proposed approach to be applied to the construction of an aquifer model and forecasting of the aquifer performances of interest.

Structured flow-based gridding and upscaling for modeling subsurface flow

The combined use of flow-based grid generation and equivalent property calculation offers an effective means for generating subsurface flow simulation models from highly detailed geological descriptions. In this paper, several structured grid generation procedures, specifically a streamline-based method, a Jacobian-based elliptic grid generation approach, and an optimization technique, are investigated for this purpose. The ability of the more advanced methods to resolve both flow information and geological features is illustrated. In addition, both (local) permeability and (global) transmissibility upscaling methods are implemented in conjunction with the gridding approaches. The benefits of structured flow-based grids over uniform Cartesian grids are demonstrated for several three-dimensional cases. It is shown that these grids are most effective when a limited number of features dominate the flow. In addition, consistent with earlier work, global transmissibility upscaling is found to be preferable to permeability upscaling for highly heterogeneous systems.

Reservoir Simulation Technology by Streamline-based Methods

Detailed geological models can now be developed by interpreting different data in a consistent and integrated manner with the advances in computer technology and stochastic methods. Reservoir simulation is important for reducing uncertainties in geological models by history matching with dynamic data. Streamline-based simulation has advantages over conventional finite-difference models in computation speed, accuracy, and visualization. Faster computation is attributed to less frequent solutions of the pressure equation, one-dimensional solutions along streamlines replacing three-dimensional flow calculations, and operator splitting. Streamline-based simulation is particularly suited to repeated history matching for multiple equi-probable geological models. Applications are demonstrated with model development for upscaling, tracer flow simulation, automatic history matching, and dual-porosity modeling.

A Streamline-Based Method for Assisted History Matching Applied to an Arabian Gulf Field

Summary A streamline-based history-matching method is presented that overcomes two of the major limitations of previous approaches: (1) it uses a three-phase compressible streamline simulator for the forward simulation, and so its application is not limited to incompressible waterfloods; and (2) geological constraints, such as regions of differing facies, can be honored. Permeability, porosity, and anisotropy can all be modified using the methodology of Assisted History Matching.1–3 In this paper, the technique is extended with suggested modifications to the reservoir description computed assuming tracer-like flow. This allows the engineer to direct the parameter changes through interactive code. The method is applied to history matching the full-field model for the Southwest Fateh Mishrif field located in an offshore concession in the Emirate of Dubai. The Southwest Fateh Mishrif is a carbonate reservoir containing a highly undersaturated oil. There are currently 42 active producers and 27 active water injectors. First production occurred in 1972, and waterflooding commenced in 1975 prior to the field average reservoir pressure falling below the bubblepoint, with present injection rates of 320,000 BWPD.

An inverse system for incorporation of conditioning to pressure and streamline-based calibration

A streamline-based history matching technique is employed to perform fast and efficient permeability identification and to integrate tracer data into an inverse model. To incorporate tracer data into the inverse model, a given tracer breakthrough curve is interpreted as cumulative breakthrough along independent streamlines. Permeabilities are modified along each streamline to match the tracer breakthrough curve. In this way, there is no explicit computation of sensitivity coefficients, nor any matrix inversion. However, this approach is incomplete by itself. Since the modifications occur along the streamlines, the identified permeability distribution is often incompatible with the actual permeability distribution. Thus, streamlines should be positioned correctly before the streamline-based method is applied. To accomplish this, geostatistical methods such as kriging and sequential Gaussian simulation (SGS) are implemented to provide an appropriate disposition of streamlines at the beginning of the inverse process. Then, permeabilities are iteratively calibrated in a conventional grid system to satisfy pressure and permeability observation data, and simultaneously modified along streamlines to match tracer data. The two independent optimization processes assist mutually and lead to stable convergence to a minimum. By applying the proposed inverse system to synthetic reference fields, it is observed that identified fields satisfactorily reproduce the permeability distribution of the reference fields. In addition, the pressure distributions of the identified and the reference fields are fairly alike, and the identified tracer breakthrough curves are well fitted to those of the reference fields. With regard to spatial patterns of transport behaviors, the streamlines of the identified fields show similar trajectories to those of the reference fields, and the time of flight distributions of the inversed fields are also analogous to those of the reference fields. The proposed inverse system is capable of estimating the future performance of a two-dimensional aquifer from a constrained number of permeability and pressure observation data accompanied by tracer data.

Inversion of dynamic production data for permeability: fast streamline-based computation of sensitivity coefficients of fractional flow rate

Generation of permeability field in a reservoir model that matchs historical dynamic production data requires an inverse calculation. A gradient method is typically used to solve the inverse minimization problem and requires sensitivity coefficients of reservoir responses, e.g. fractional flow rate or pressure, with respect to the change in the permeability. This paper presents a novel semi-analytical streamline-based method for computing such sensitivity coefficients under the framework of two-phase (oil–water) flow conditions. This method is shown to be significantly faster and generate permeability fields with lower objective function than the traditional perturbation method. The method decomposes the multiple-dimensional full flow problem into multiple 1D problems along streamlines. The sensitivity of fractional flow rate at the production well is directly related to the sensitivity of time-of-flight (TOF) along each individual streamline and the sensitivity of pressure at grid cells along the streamline. The sensitivity of TOF of a streamline can be obtained analytically. The sensitivity of pressure is obtained as part of a fast single phase flow simulation. The proposed method is implemented in a geostatistically based inverse technique, called the sequential self-calibration (SSC) method. Results for fractional flow rate sensitivities are presented and compared with the traditional perturbation method. This new method can be easily extended to compute sensitivity coefficients of saturation (concentration) data.

Streamline-Based Method With Full-Physics Forward Simulation for History-Matching Performance Data of a North Sea Field

Summary We present a method for history-matching production data using a streamline simulation that captures all the pertinent physics, including compressible three-phase flow with gravity. We use an approach based on the assumption of 1D flow along streamlines to find the sensitivity of water flow rate at production wells to changes in permeability. Although the computation of the sensitivities is approximate, we show, using data from a North Sea field, that the method can provide a reasonable history match and good predictions of future performance for problems with significant effects caused by compressibility and gravity.

A Geostatistical Approach to Streamline-Based History Matching

Summary History matching large reservoir models under a realistic geological continuity constraint remains an outstanding problem. We propose a new combined geostatistical and streamline-based method potentially capable of history matching large reservoir models, while accounting consistently for geological continuity as provided by a permeability histogram and variogram and by production data. While most existing history-matching techniques somehow rely on the calculation of single-gridblock sensitivity coefficients, the proposed method avoids such calculation entirely by perturbing jointly effective permeabilities along a set of streamlines. Such perturbation results in large changes of the permeability field that reduce significantly error in each iteration of the history-matching procedure. The problem of mapping streamline effective permeability perturbations to single gridblocks is performed under a Gauss-Markov random function constraint. This novel stochastic mapping procedure accounts for the target histogram and variogram while honoring the streamline effective permeability perturbations. Forward flow simulation is achieved by a streamline simulator. The methodology is presented on synthetic cases; it appears to be computationally efficient and robust.

Construction of geostatistical aquifer models integrating dynamic flow and tracer data using inverse technique

Natural aquifers are heterogeneous, and geostatistical methods are widely used to simulate the heterogeneity of aquifer properties. Due to limited available data, it is essential to integrate as much information as possible to reduce the uncertainty in aquifer models and flow predictions. Traditional geostatistical techniques efficiently consider static hard and soft information, such as core data and seismic data. However, dynamic flow and transport data, such as flow rates, pressure, and tracer breakthrough, are important information that are not easily considered with the traditional techniques. Integrating such dynamic data into a model requires the solution of a difficult inverse problem, since dynamic data and aquifer properties are related to each other through the non-linear flow and transport equations. A recently developed geostatistically based inverse technique, the sequential self-calibration (SSC) method, is introduced to integrate those dynamic data. The SSC method is an iterative inverse technique that is coupled with an optimization procedure. It provides for fast generation of multiple realizations of aquifer property models that jointly match pressure and tracer breakthrough data, yet display the same geostatistical characteristics. This method is flexible, computationally efficient, and robust. The main features of SSC include (1) a master point concept that reduces the number of parameters, (2) a perturbation mechanism based on kriging that accounts for the spatial correlation of the aquifer properties, (3) a fast streamline-based tracer flow simulator for integration of tracer data, and (4) a new semi-analytical streamline-based method for computing sensitivity coefficients of tracer breakthrough. Applications of the SSC method are demonstrated with a synthetic data set. Results show that tracer breakthrough data carry important information on the spatial variation of aquifer permeability in the inter-well areas. As a contrast, pressure data provide information at near well-bore areas only. Integrating pressure and breakthrough data jointly leads to significant improvement in the aquifer heterogeneity representation and a reduction in the uncertainty of aquifer model. The accuracy of flow and transport predictions can be dramatically improved by integrating dynamic data.

A streamline-based method for analysing steady state metal forming processes

A new method based on streamlines is proposed for the detailed analysis of steady state metal forming processes (e.g. extrusion and rolling). The field equations are formulated in an Eulerian reference frame and approximated by finite differences. In contrast to most Eulerian formulations, the proposed method automatically tracks the paths of material particles; this is done by letting the kinematic unknowns be the locations of the discrete streamlines. The method of analysis is readily adapted to materials with complicated, history-dependent, constitutive laws and a variety of metal/die interface conditions. The method, which is presented here as it applies to two-dimensional planar flows of an incompressible material, is found to give results that compare well with solutions based on other methods. Generalizations to other flows are discussed.