ECMOR IV - 4th European Conference on the Mathematics of Oil Recovery

This paper describes an approach for modeling geological faults with a locally distorted Cartesian grid. The location of the Cartesian grid points is thereby not changed. Only the shape of grid blocks along fault traces is modified. This modeling technique can be applied both for vertical and slanted faults. It allows the construction of the grid around intersecting faults as well. The resulting grid is strictly orthogonal along the fault, but not orthogonal across the fault. For sealing faults this is not important. For non-sealing faults, however, the calculation of the flux across the faults is modified. Pseudo-points are introduced. They are placed such that the connection across one fault segment is again orthogonal. During the simulation the pressure for the pseudo points is derived by linear interpolation from the pressures of neighboring points. The flow term is implicitly evaluated using the pseudo points. Several examples will show the importance of adequate fault-modeling in reservoir simulation.

Stochastic models of reservoir properties are routinely scaled-up to reduce the number of grid blocks in the input of the flow simulation model. In this process, the heterogeneities in the stochastic model, represented by small-scale grid blocks, are homogenized into much larger grid blocks for flow simulation. As a result, the volume in the larger simulation grids are represented by a single value, and the correlation structure and variance of the small-scale grids are lost. We have conducted an extensive study on the effects of all the parameters determining effective permeabilities through detailed higher-order finite-difference solution of the flow equations. Our results demonstrate the dependence of effective permeability on variance, permeability in principal flow directions, permeability anisotropy, spatial correlation length in the principal flow directions, and spatial correlation anisotropy. We show that the simulation technique employed here may be required to incorporate all factors that affect the value of effective permeability into account. After the scaleup process for each large-scale grid block, the variance of the small-scale permeabilities is removed and the correlation length is maximized. On the other hand, the variance of the large-scale permeabilities is less than the original smallscale permeability distribution, and the spatial correlation is also changed. For these reasons, the design of the large-scale grids should be such that it minimizes the variance and maximizes correlation length of the small-scale grids in each large-scale grid block, and best preserves the spatial distribution and variance of the small-scale grids. In this paper, an efficient simulated annealing algorithm is described that generates near-optimal large-scale grids for scale-up and best preserves the heterogeneities for flow simulation.

Reservoir engineering, despite its name, is more than a mere mechanical discipline where decisions smoothly follow known paths from data through series of standard, well-defined procedures. Describing and understanding reservoirs may sometimes appear as a formidable task, not only because of the complex physics involved in hydrocarbons production, but also, and maybe more importantly, as a result of data shortage.

It has been established in previous papers, that the Truncated Gaussian Random Method is a very efficient tool for simulating sedimentary images in a fluvio-deltaic environment. Moreover, this method allows the introduction of geological knowledge different from the lithofacies, by means of threshold variations 3D matrix. However, there are still some cases which cannot be easily solved by the current Truncated Gaussian Random Function approach, especially when the whole lithofacies cannot be rigorously ordered. It is thus possible to generalize the use of the Truncated Gaussian Random Function by introducing several Gaussian Functions. This allows lithofacies simulations of reservoirs where several sequences overlap. In this paper, we show how to apply this method using mainly two Gaussian Functions when at least three lithofacies must be simulated. Firstly, the linear model of coregionalisation is used to generate correlated Gaussian Functions and the effect of this correlation is presented. Secondly, we use uncorrelated functions to show the effect of the simulation parameters, especially focusing on the anisotropy effect. Thirdly, we present a more complex type of truncation. In conclusion, we begin a discussion on the choice of the simulation parameters.

A method for integrating sub-seismic scale faults as part of a more complete reservoir description is presented. The faults are represented by single planes and displacement of adjacent strata. An established stochastic model for the sub-seismic scale faults, with a simulation algorithm, has been developed further. The model allows for the inclusion of seismically visible faults. They thereby influence the overall fault probability distribution since the model includes interaction between the faults. The interaction includes both repulsion and clustering. This produces fault patterns which are considerably more realistic than those published previously. An example of such a fault pattern realization is provided. Two ways of introducing the effect on fluid flow are used. One method modifies the fine-scale permeability field. A new homnogenizaion procedure has been implemented to interface with the fluid flow simulator. The other introduces the faults on the borders in the fluid flow block system. The second and simpler method is adequate when the simulated faults are fairly large compared to the flow simulation blocks, while the first method should be used when large numbers of small faults are simulated.

Data from different measurement techniques with different spatial resolutions (e.g., well logs and seismic data) are needed to generate geostatistical distributions of reservoir properties. A method based on Fourier transforms is presented to merge data taken at different scales. The Fourier transform retains all the information in an image and segregates the information according to frequency (or inverse length scale). Knowing the spatial resolution of each data set, we can merge information from large- and small-scale data sets. The method is demonstrated for one-dimensional and two-dimensional cases and can be extended to three-dimensional cases.

This paper presents a direct method to determine the uncertainty in reservoir pressure, and other functions, using the time-dependent one phase 2- and 3-dimensional reservoir flow equations. The uncertainty in the solution is modelled as a probability distribution function. This is derived from probability distribution function for input parameters such as permeability. The method involves a perturbation expansion about a mean of the parameters. Coupled equations for second order approximations to the mean at each point and field covariance of the solution, are developed and solved numerically. This method involves only one (albeit complicated) solution of the equations, and contrasts with the more usual Monte-Carlo approach, where many such solutions are required. The procedure is a development of earlier steady-state two-dimensional analyses arid a transient mass-balance analysis using uncertain parameters. These methods can be used to find the risked value of a field for a given development scenario.

A general method for the scale up of highly detailed, heterogeneous reservoir cross sections to coarser scales, for the purposes of flow simulation of displacement processes, is developed and applied. The technique involves the nonuniform coarsening of the fine scale description, with fine resolution introduced in regions of potentially high fluid velocities (typically regions of connected, high permeability) and coarse, homogenized descriptions applied to the remainder of the reservoir. The method is designed to capture both average behavior as well as some important behaviors which are due to the extremes of the permeability field, such as the breakthrough time of the displacing fluid. The method is applied to several example problems, including two actual field examples, and is shown to provide coarsened models (~ 25 x 25) which give simulation results for fractional flows and saturations in close agreement with fine scale (~ 100 x 100) results. These examples demonstrate the ability of the method to capture a wide variety of flow behavior without the need for specific knowledge of the global flow field, indicating that the coarsened reservoir description is to a large degree process independent.

The purpose of this paper is to suggest a framework for analysis of multi-scale permeability distributions. We consider the process of de termining effective representation of the permeability at different length scales. The starting point is the discretizatioii of the pressure-velocity equations using a modified mixed finite element formulation. The discrete pressure and velocity are then subjected to a multiresolution analysis based on the well-known Haar system. The relationship to Additive Schwarz Domain decomposition method is pointed out. Numerical ex periments are an integrated part of the investigations.

By nature, petroleum reservoirs are made of heterogeneous rocks. Fluid flow modelling in these porous media give a system of algebraic equations and partial derivative equations. The heterogeneity is taken into account through parameters varying in space. Because of the large size of the reservoir, it is not practically feasible to take into account all the details in the computation of the solution. In reservoir simulators, the discretisation methods require constant value in space associated to each cell. Here, a rigorous formalism is given to average the absolute permeability. The principle of this method is to control the error done on the flow rate. Different hypotheses are made depending on the hypotheses on the flow surrounding the region to average. In particular, this method allows to average permeabilities around the wells. Another interesting feature is that this method gives averaged permeabilities for any shape of averaged regions in 2D and 3D.