ECMOR III - 3rd European Conference on the Mathematics of Oil Recovery

Most statistical models which are used when simulating oil reservoir performance employ a correlation function. We have found that fields in which correlation, as a function of direction, is represented only as a positive variable do not give an adequate representation of immiscible displacements in realistic geological formations. A better representation can be achieved by introducing negative correlation in one or more directions. We describe the water/oil displacement efficiency in a selection of deterministic fields, based on typical sediment bedform structures, and then show how this performance may be reproduced in random correlated fields with varying amounts of positive and negative correlation. Negative correlation needs to be considered when the heterogeneity displays significant periodicity; for example, in layered systems. Positive correlation, which represents the tendency for (local) similarity, results in favourable viscous-capillary interactions and better displacement efficiency. Negative correlation results in poorer displacement efficiency. Clastic sedimentary formations, which are characterised by contrasting layers, are better represented, statistically, by anisotropic positively/negatively correlated permeability fields in which displacement efficiency is strongly directiondependent. The implications of negative correlation for numerical flow models are also assessed.

A new approach for conditioning stochastic fields is proposed. In this paper, the Simulated An nealing Method (SAM) is applied to laboratory and field permeability data. A generated field of a desired rock property preserves the spatial distribution and honors the actual measured data from field and/or laboratory. In this approach, in contrast to the traditional conditional simula tion, no interpolation technique is used to smooth data at unknown locations. The least-squares objective function is the difference between the spatial variability of the generated field and that of the actual spatial structure of the known data. Hence, theoretical models for fitting experimental variograms are no longer required in this approach. The simulated annealing algorithm is tested on an exhaustive permeability data measured by minipermeameter on a slabbed carbonate core. Comparison of the generated permeability field with the actual data indicates that this approach is efficient and the conditioning is sensitive to the spatial distribution of the known points. A field application is also presented in this study. The anisotropic permeability field generated by SAM on a North Sea Oil Reservoir indicates the capability of SAM to preserve the spatial variability and to represent the uncertainty that exists more realistically than the smoothed representation used in most geostatistical approaches.

Stochastic simulations of sandstone deposits in hydrocarbon reservoirs should honour the available well data and geological knowledge about their setting. In this paper, the problem of conditioning the simulations on well data is considered. This is done in the case where a Boolean model is used to describe the spatial distribution of sandstone deposits in a reservoir (object-based modelling). It is argued that the conventional simulation algorithms generate sandstone deposits that do not possess the correct statistics for their size at the conditioning wells. The reason for this is closely related to the so-called waiting-time paradox. A statistically correct simulation approach is proposed for the problem of simulating random arcs on a circle, given that certain points on the circle are covered by arcs while other points remain uncovered. Intensity functions for the arc centres and probability distributions of an arc radius are derived for the conditional distribution of arcs on the circle. The results obtained for this problem might be partly used to solve the 2D and 3D conditional simulation problem.

This paper describes an application of optimal control meth ods to a wide range of history matching problems arisen dur ing oil and gas field development. A conjugate gradient met hod is used to minimize an objective function value. To det ermine a step value along the corresponding conjugate direc tion, a new high effective procedure is set up. The procedu re significantly increases the convergence rate. A new, more accurate form of adjoint problem is obtained that allows to improve the gradient accuracy calculation for the problems with high unlinearities. The parameterization of k(x) and m(x) distributions is considered and the results of numeric al testing of the algorithms are presented.

The reliability of flow and transport models depends on the understanding of the characteristics and impacts of spatial structures of subsurface formations. Preferential pathways are controlled by the geometries of these units. In subsurface characterization, it is the current practice to use both interpolation and extrapolation using a series of observations from well logs. Classical techniques, such as weighted interpolation and least-squares polynomial, are not recommended because they assume independence of sample data. Geostatistics methods bring a new horizon for reservoir engineering studies. Spatial dependence or autocorrelation of data exist, given the fact that neighbouring points tend to present similarities. In addition to this fact, reservoir heterogeneities are caused by usually known geologic processes such as deposition, sediment diagenesis and fracturing. Thus geologic data and models should also be helpful in characterizing spatial variability of flow properties. This work applies a Bayesian Kriging technique which includes an expertise guess with a given uncertainty in the estimation procedure. This technique assures that observation data (hard data) prevail in areas close to field measurements, whereas in areas without observations the expert’s experience (soft data) have greater influence. Maps of the estimates and the associated uncertainties are shown to be key tools in reflecting the quantity and quality of the available. Simple Kriging and Universal Kriging become subsets of this procedure.

This paper considers the two-phase multicomponent displacement in porous media. We focus on the ideal systems (mixture volume and entropy are additive). This assumption is extensively used for oil-gas and gas-condensate systems at low pressure. For the actual n-component system the problem of the approximation by ideal mixture is solved - density and entropy of every component is found as a phenomenological parameter from the equation of state. Explicit formulae for the Riemann invariants, shock and rarefaction waves are found. Invariant solutions for one-dimensional displacement problem is obtained. Description of the displacement of oil by gases is given.

An original approach for modeling the dispersion of a chemical tracer through heterogeneous porous media is described in this paper. It is shown that, instead of introducing a time dependent dispersion coefficient, the presence of heterogeneities strongly modifies the structure of the transport equation. The presentation and the solution of this complicated equation are simplified by using the concept of fractional derivatives. Fractional derivatives are generalizations of the standard integration for a noninteger order q. In this study, the one-dimensional equation describing the transport of a tracer in a strongly heterogeneous medium (correlation at all the length’s scales) is demon strated. In such a medium, characterized by a parameter a (which can be related to a fractal dimension), the spreading width of a tracer varies as tα/², instead of t¹/² for standard dispersion, or t for pure convection in a layered medium. A gen eral transport equation for the concentration is derived using Laplace and Fourier transforms, which yield derivatives of a fractional order a in time and a second order in space. This form of transport equation can be reduced to the two limiting cases: (1) dispersive flow, which gives the standard convection-diffusion equation (a = 1), and (2), convective flow in layered media, which gives a wave equation (a = 2). The general equation represents a continuous crossover between dispersive and convective flow with only one tuning parameter (1

Some new results are presented for systems in Vertical Equilibrium (VE) which extend our recent work on this subject (Yortsos, 1991). The linearized stability of a prototypical miscible displacement at VE is considered and shown to correspond to the short wavelength regime of previous studies in unbounded geometries. Concepts from VE are applied to model general displacement processes in various geometries. Under conditions of gravity-segregated flow we investigate and suggest simple methods for the construction of the solution. In particular, we extend our previous work to flows involving dip and show that the latter dominates the solution. Prepared for The 3rd European Conference on the Mathematics of Oil Recovery, 17-19 June 1992, Delft University of Technology, The Netherlands.

We present a new approach to permeability modelling and prediction in sandstones. By simulating a range of geological processes which operate to form a sandstone, we obtain physically representative model porous media. The geometry of the models is completely specified, so that permeability can be calculated directly using a flow path network approach. In contrast to previous efforts to predict permeability, our approach is based on first principles. There are no adjustable parameters in the calculations, and no requirement for additional measurements or correlations (e.g. capillary pressure data or pore system data from thin sections). Model predictions match measurements on Fontainebleau sandstone samples whose permeabilities span nearly five orders of magnitude. We have also been able to correctly predict pore throat size distributions to match mercury injection measurements. Pore-scale geometrical features of the model are found to be spatially correlated, and this departure from randomness significantly affects macroscopic properties. The agreement between predictions and measurements suggests that spatial correlation is inherent in granular porous media, and that uncorrelated (or arbitrarily correlated) models of transport in such media are unlikely to be physically representative.

Current reservoir simulators are generally inefficient due to the use of low order accurate schemes on regular grids. The numerical diffusion inherent in the singlepoint upstream weighting scheme can cause large errors in the predicted solutions, which can only be removed by the use of fine grids with many grid blocks. The aim of this work is to develop a method which can produce results of high order accuracy efficiently. A high order Godunov scheme is combined with grid adaptivity, where grid blocks are inserted in highly active regions of the flow field thereby concentrating the computational effort where it is most needed. The results presented demonstrate the benefits of the method in reservoir simu lation. The superiority of the adaptive high order scheme over that of the first order scheme is demonstrated. The quality of the results obtained by the adap tive scheme is comparable with those of the standard scheme, while great savings in computer time are obtained. The adaptive high order scheme also uses less computer time than the corresponding adaptive low order scheme.