Model Calibration and Efficient Reservoir Imaging (MCERI)


I.             Past Work

3-D streamline simulation is widely considered as one of the major developments in petroleum reservoir simulation and performance forecasting in the last decade. The technology has been rapidly assimilated by the industry for highly detailed flow simulation, reservoir management, history matching and uncertainty assessment. Several commercial reservoir simulators have been developed based on the streamline ‘time of flight’ concept introduced by Dr. Datta-Gupta and King (1995). With the advancement in high resolution data acquisition and seismic technolgies, geologic models now routinely consist of multimillion cells. This resulted in a widening gap between geologic modeling, flow simulation and uncertainty assessement. Streamline simulation has effectively bridged this gap. Our work developing streamline flow simulation and history matching of high resolution geologic models has been pivotal to these developments.

Links to Previous Research Works

Π.  Ongoing work

 ·        Fast Marching Methods: Application to Unconventional Reservoirs

The concept of depth of investigation is fundamental to well test analysis and is routinely used to design well tests and to understand the reservoir volume investigated. The depth of investigation can also be useful in identifying new well locations and planning, designing and optimizing hydraulic fractures in unconventional reservoirs. One way to calculate the depth of investigation is through numerical simulation. The numerical approach is very general and can be applied to arbitrary reservoir and well conditions. The computation time and expenses, however, make the numerical simulation approach unfeasible for routine applications.

As an alternative to numerical simulation, we can directly solve for the propagation equation for the pressure ‘front’ defined as the maximum of the pressure response for an impulse source.  Such propagation equation can be derived using asymptotic ray theory which has been used extensively in electromagnetic and seismic wave propagation (Virieux et al., 1994). The asymptotic method draws upon an analogy between the propagating pressure ‘front’ and a propagating wave front, and many of the concepts such as rays and propagating fronts have their counterparts in Petroleum Engineering (Datta-Gupta and King, 2007). Specifically, a high frequency asymptotic solution of the diffusivity equation leads to the following equation for a propagating pressure ‘front’ for an impulse source or sink (Vasco et al., 2000; Datta-Gupta and King, 2007):

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In the above equation, is the diffusivity (consisting of  = permeability,  = porosity, µ = fluid viscosity, ct = total compressibility) which can be function of position and  is the propagation time of the pressure ‘front’ (also called a ‘diffusive time of flight’). It simply tells us that the pressure ‘front’ propagates in the reservoir with a velocity given by the square root of diffusivity. This propagation depends on reservoir and fluid properties and is independent of flow rate. Also,  has unit of square root of time which is consistent with the scaling behavior of pressure diffusion. In fact,  is related to physical time through a simple expression of the form  where the pre-factor c depends on the specific flow geometry (Kim et al., 2009). For example, for radial flow, c = 1/4, for linear flow, c = 1/2 and for spherical flow, c = 1/6.

The pressure ‘front’ equation is a form of the Eikonal equation which appears in many contexts such as elastic and electromagnetic wave propagation; its properties are well developed in the literature (Sethian, 1999). Most importantly, the Eikonal equation can be solved very efficiently (in seconds opposed to hours of simulation time for comparable problems) by a class of solution called Fast Marching Methods (FMM) (Sethian, 1999). Following are three examples of depth of investigation: 1) generalization to heterogeneous medium; 2) generalization to anisotropic permeability and unstructured grid; and 3) a horizontal well with multistage hydraulic fractures.


·         Rapid Compositional Flow Simulation in Shale Oil Reservoirs using Fast Marching Method

Reliable performance assessment of unconventional reservoirs requires accurate modeling of inter-porosity flow characteristics in hydraulic fractures, microfractures and reservoir rock. In addition, phase behavior in nano-porous rocks plays an important role in reservoir performance. High resolution flow simulation incorporating the complex underlying physics and detailed reservoir heterogeneity is computationally expensive. Our proposed simulation approach is based on a high frequency asymptotic solution of the diffusivity equation in heterogeneous and fractured reservoirs. The high frequency solution leads to the Eikonal equation which is solved for a ‘diffusive time of flight’ (DTOF) that governs the propagation of the ‘pressure front’ in the reservoir. Our approach consists of two decoupled steps: calculation of the DTOF using the Fast Marching Method (FMM) and fully-implicit compositional simulation using DTOF as a spatial coordinate. The computational efficiency is achieved by reducing the 3-D compositional flow equation into 1-D equation using the DTOF as spatial coordinate, leading to orders of magnitude faster computation over full 3-D compositional simulation. The savings in computation time increases significantly with grid refinement and for high resolution models. 


·         Data-Driven Fracture Flow Diagnostics and Characterization

Multistage hydraulically fractured horizontal wells provide an effective means to exploit unconventional reservoirs. The current industry practice in the interpretation of field response often utilizes empirical decline curve analysis or pressure/rate transient analysis (PTA/RTA) for the characterization of these reservoirs and fractures. These analytical tools are based on many simplifying assumptions and cannot provide a detailed description of the evolving reservoir drainage volume from a well, the understanding of which is essential for unconventional reservoir and fracture assessment and optimization. We propose a novel “data-driven” methodology for the production analysis of shale gas and shale oil reservoirs. The approach is based on the high frequency asymptotic solution of the diffusivity equation in heterogeneous reservoirs. It allows us to determine the drainage volume from a well, and the instantaneous recovery ratio (IRR), which is defined as the ratio of the produced volume to the drainage volume, directly from the production data. In addition, in a manner analogous to the diagnostic plot in PTA, a new w(τ) plot has been proposed to provide better insight into the depletion mechanisms and the fracture geometry. The major advantages of this current approach are the model free analysis without presumptions of flow regimes, and a simple and intuitive understanding of the drainage volume and fracture conductivity. The results of the analysis are useful for well and hydraulic fracturing operation design optimization and matrix and fracture parameter estimation.


·         Flow Simulation of Complex Fracture Systems with Unstructured Grids using Fast Marching Method

With the current industry practice of reduced cluster spacing and increased fracturing proppant/fluid volume, the hydraulic fracture treatment tends to generate more complex fracture systems, where an unstructured computational grid, instead of a Cartesian or corner point grid, is preferred to accurately model the fracture geometry. With unstructured grids, the reservoir performance is generally simulated with finite volume simulation, for which one major issue is the potentially heavy computational cost. A novel approach has recently been introduced to provide a rapid simulation of unconventional reservoirs, which first captures the drainage volume during the transient pressure propagation process using the Fast Marching Method (FMM) and then rapidly solves fluid flow equation in an equivalent 1D domain. However, this application is currently limited to calculating the reservoir response with Cartesian or corner-point grids.

·         Multiscale Parameterization and Model Calibration with Grid Connectivity Based Transforms (GCT)

The motivation of this topic is to expand upon the current state of the art in parameterization by linear transformation for history matching. In addition to theoretical developments, an emphasis is placed on the development of parameterization methods that are applicable to large reservoir models, on par with current industry standards and computational modeling capabilities, and also varying levels of prior geologic complexity.


Methods of linear transform dependent on prior model assumptions have commonly utilized the parameter covariance matrix. The Karhunen-Loeve transform (KLT) or principle component analysis (PCA) has been widely used [Gavalas et al., 1976; Karhunen, 1947; Loéve, 1978; Reynolds et al., 1996; Li and Cirpka, 2006; Ma et al., 2008; Jafarpour and McLaughlin, 2009]. In this approach the estimable property defined at any grid cell is represented as the linear expansion of the weighted eigenvectors of the property covariance matrix. The eigenvectors form the transform basis and are typically ranked by their corresponding eigenvalues, from largest to smallest, that are related to the variance contribution of each eigenvector to the total parameter variance. The expansion is optimal in the mean-squared-error sense among the linear class of transformations; therefore, the KLT coefficients present the fewest number of parameters that capture the maximum amount of variation for any low-rank approximation. Accordingly, the sorted eigenvectors convey larger to smaller scales of spatial parameter variation. A limitation of this approach is that in realistic problems with high-resolution models, the covariance can be unknown (or uncertain), resulting in misleading basis functions when prior model assumptions are incorrect [Jafarpour and McLaughlin, 2009]. Eigendecomposition of the covariance matrix, required for each parameter update, is also prohibitively expensive for high-resolution models relative to current computational capability. Last, the preservation of moments beyond second-order (i.e., covariance), as well as complex continuous geologic structures (e.g., channel systems), are not guaranteed in the KL expansion during the parameter updates.

A new model-independent basis constructed from grid-connectivity information is developed here. The parameterization is applicable to any grid structure and domain geometry. The development of the grid-connectivity-based transform (GCT) basis begins from first principles, merging discrete Fourier analysis and spectral graph theory. The basis functions represent the modal shapes or harmonics of the grid, are defined by a modal frequency, and converge to the discrete cosine transforms (DCT) for certain grid geometries and boundary assumptions; therefore, reservoir model calibration is performed in the spectral domain. Using an adaptive multiscale workflow, the GCT parameterization is successfully applied for history matching of several synthetic reservoir models of varying geometry and geologic complexity and also of a field case.


·        A Hierarchical Multi-scale Approach to History Matching

We develop a hierarchical multiscale history matching approach with two levels of hierarchy for inversion in parameter space and geometric space. The approach follows a sequence of calibrations from global to local parameters in coarsened and fine scales. At first, we identify the ‘heavy hitters’ in the large scale static and dynamic parameters and then calibrate them using an evolutionary algorithm. This global parameter calibration, matching and balancing field level energy, is followed by streamline-assisted multiscale inversion to match well by well production history with local parameter updates. Starting with a relatively coarse grid, we match the production data at wells by dynamically refining the reservoir grid. This multiscale data integration offers not only benefits of enhancement in computation efficiency and but also effective iterative minimization. The approach has been successfully applied to a variety of field cases in collaboration with operating companies.


·        Robust Streamline Tracing in Embedded Discrete Fracture Models

The streamline based technology has proven to be effective for various subsurface flow and transport modeling problems including reservoir simulation, model calibration and optimization. For naturally fractured systems, current streamline models are well suited for dual porosity single permeability systems because streamlines exist need to be traced only for the fracture system. However, complications arise for dual porosity dual permeability (DPDP) systems because streamlines need to be traced for both fracture and matrix systems. Also, the streamlines in the two systems may interact. We present a robust streamline tracing framework for use in the DPDP models via an embedded discrete fracture model (EDFM) framework.

·        Flood Rate Optimization Using Streamlines

It is well recognized that field-scale rate optimization problems often involves highly complex reservoir model, production and facility related constraints and geological uncertainty. Deployment of smart well completions with inflow control valves (ICV) to control production/injection rates for various segments along the wellbore further compounds the complexity of the problem. All these make optimal reservoir management via rate control difficult without efficient optimization algorithms. Alhuthali et al. (2007, 2008, 2010) proposed a streamline-based approach to maximize waterflood sweep efficiency. The approach focuses on equalizing arrival times of the waterfront at all producers within selected sub-regions of a waterflood project. This results in delayed water breakthrough and reduces field water cut after water breakthrough. The optimization was performed under production and facility constraints using a sequential quadratic programming approach. A major advantage of the streamline-based approach is the analytical computation of the sensitivities of the waterfront arrival times at the producers with respect to well production/injection rates and also, analytic calculations of the gradient and Hessian of the objective function. This makes it computationally efficient and suitable for large field cases. It also accounts for geological uncertainty via a stochastic optimization framework based on a combination of the expected value and variance of a performance measure from multiple realizations. We have also studied the tradeoff between maximizing sweep efficiency and accelerating production (maximizing NPV) using an augmented objective function. In addition to waterflooding, we have applied this method to EOR processes like polymer and CO2 flooding.



·        Diffuse Source Upscaling of Geologic Models

The motivation for Diffuse Source (DS) upscaling is to address the assumption of pressure equilibrium while coarsening. The assumption that weakly connected and disconnected pay would have the same equilibrium pressure as the well-connected pay inside a coarse block is error prone. Existing flow based upscaling methods are based on Steady State (SS) concepts of flow and cannot distinguish between well-connected and weakly connected pay in the upscaling region. The current work looks at the use of diffusive time of flight and drainage volume concepts to build basis functions that effectively capture the subgrid heterogeneity that is well-connected especially in high contrast media. The application of so called Diffuse Source basis functions to set up local flow problems leads to non-uniform pressure gradients even for homogeneous systems. This forced us to revisit the construction of transmissibility. A new construction of transmissibility based on Pseudo Steady State (PSS) flow concept is utilized which does not require the explicit specification of cell centers, instead the total PSS pressure drop across the volume is used as the reference pressure drop for the transmissibility construction. The DS formulation is an extension of PSS concepts to high contrast media through the use of diffusive time of flight. 

Figure 1 shows the pressure map comparison of the fine scale and a vertically coarsened case for the 7 million cell Amellago carbonate geologic model [Amellago carbonate model provided by Dr. Sebastian Geiger and the International Centre for Carbonate Reservoirs, Institute of Petroleum Engineering, Heriot-Watt University]. The coarsening design is generated from the statistical layering algorithm in SWIFT giving a factor of 6 coarsening in this case and the upscaling of properties is done by the Diffuse Source algorithm. Figure 2 shows the water saturation maps for the same and in Figure 3, we have the quantitative comparison of cumulative oil volumes for Steady State and Diffuse Source upscaling algorithms. The field water cut comparison is given in Figure 4.

Figure 1: Pressure comparison between fine scale (left) and 1x1xN coarse models (right).

Figure 2: Water saturation comparison between fine scale (left) and 1x1xN coarse models (right).


                                                  Figure 3: Field water cut vs. Pore Volume Injected                              Figure 4: Field oil preoduction vs. Pore Volume Injected