Quantitative methods taking into account the sedimentological characteristics will answer the needs of reservoir engineers. We propose here a geostatistical method for the conditional modelling of the facies of a sedimentary fluvio-deltaic series. This model was elaborated jointly by I.F.P. and the Paris School of Mines, with the aim of modelling reservoir heterogeneities. From the sedimentological study contained in the paper by C. Ravenne et al., we present several simulations, conditioned by "drill-core" taken from the outcrop. The block permeabilities are then calculated from the values given to the facies. Introduction Reservoir engineers have, for a long time, been asking what type of models could be entered into reservoir simulators. The geological models that are normally used are essentially qualitative and so it is difficult to numerize them, except by correlating the drillholes facies, which is not always self evident. This often leads to models with too many constraints for simulating reservoirs. Their dynamic behaviour worsens considerably going from very continuous layers of sandstone to disseminated lenses. (Fig. 1). This raises the question of how to characterize the geometry of the sandstone given the drillhole data, the geologist's interpretation and also other measurements (e.g. seismic recordings), and how to model the reservoir levels to suit this shape. As well as this, the models must match the lithology along the drillholes. In this article, we present a method for conditionally modelling the lithology which is designed for sedimentary processes. This approach was tested using the geological section of a cliff-face in Yorkshire (England) which shows a fluvio-deltaic environment similar to some of the levels in the Brent formation in the North Sea. A detailed description of the geology of the cliff-face studied and of the approach used in this project have been presented by Ravenne et al. Here we shall only consider the problem of modelling random sets by using a probabilistic method for representing the spatial distribution of the facies (sandstone, shaly sandstone, shale) in a heterogeneous reservoir. Before presenting the method, we review the main procedures for modelling random sets, that are used in the petroleum industry. REVIEW OF THE EXISTING METHODS Boolean Sets A simple way is to consider a heterogeneous medium as consisting of sandstone lenses in a shale matrix (or vice versa). Boolean sets (Matheron, Serra, Jeulin) are a mathematical way for modelling this type of deposit, that has been used for many years in other fields (Fig. 2). This method consists of putting lenses of a predetermined shape (e.g. ellipses or rectangles) at random points in the domain under study (i.e. the points are statistically uniformly distributed in space). Lenses are not correlated. The advantage of this approach is that it is easy to use in 2D or 3D spaces. It only depends on a few parameters: the number of seed points per unit space (called density), the shape of the lenses (fixed or variable), their size and orientation. The model is very flexible. The parameters can be modified locally in order to reproduce the real phenomenon more accurately. Clearly the more complicated the model is, the more parameters there are to fit but this can be overcome by fitting them by trial and error. P. 591^

Summary The simulation of multiphase flow presents several difficulties, includingthe occurrence of sharp moving fronts when convection is dominating,the need for a good approximation of velocities to calculate the convective terms of the equation, andflow singularities around wells. To handle the first difficulty, we propose a Godunov-type higher-order scheme based on a piecewise linear approximation of the saturation associated with a multidimensional slope limiter. With respect to the second, the pressure equation is approximated by means of a mixed-hybrid formulation equivalent to the classic mixed formulation but yielding a positive-definite linear system. To solve the third difficulty, we introduce macroelements around wells. Numerical experiments illustrate the capabilities of the method. Introduction In a previous paper,1 we proposed a finite-element method (FEM) for 2D diphasic simulations. In addition to the global pressure formulation,1,2 the basic tools for this method were a higher-order Godunov-type3 discontinuous FEM to approximate the saturation equation with little numerical diffusion and a mixed finite-element approximation of the pressure equation to reduce grid-orientation effects through better coupling of the pressure and saturation equa-tions. In this paper, we present new developments added to this method concerning the approximation of the saturation equation, the resolution of the pressure equation, and the representation of wells. The discontinuous FEM consists of a discontinuous, piecewise, linear or bilinear approximation of the saturation that allows us to design a higher-order scheme. Inside each cell, the water-conservation equation is multiplied by test functions and integrated by parts. Numerical fluxes approximating water flow rates on the edges are calculated at Gauss integration points on the edges by use of Godunov equations solving a 1D Riemann problem and involving the two traces of the saturation at the Gauss points. Extending ideas developed by Van Leer,4 we added a multidimensional slope limiter that greatly enhances the scheme's stability and that prevents overshoots and oscillations2 to the discontinuous FEM (see Ref. 5 for a similar approach). For the pressure equation, the usual mixed finite-element formulation2,6,7 yields a linear system that is difficult to solve because it is not positive-definite. To overcome this drawback, we use the equivalent mixed-hybrid formulation.2,8 In addition to the pressure inside the cells and the total flow rates across the edges, this formulation introduces as new unknowns the pressure values on the edges. The original degrees of freedom can be eliminated to obtain a linear system in the new unknowns that is symmetric, positive-definite, and easy to solve. Finally, well models called macroelements9 are presented. A union of cells around a well is divided into several sectors inside which the flow is assumed to be radial and is calculated through a simplified 1D simulator in radial coordinates. For each well, a finite number of 1D radial problems (the macroelement) links the finite-element variables to the well variables, allowing for direct calculation of the bottomhole pressure (BHP), for example, and for precise calculation of water breakthrough. With such a method, one can simulate in particular nonradial flow around producing wells. All these features are incorporated in the BIDIMIX simulator for tw0-phase incompressible flow developed jointly by INRIA, Inst. Français du Pétrole (IFP), and Elf Aquitaine. Performance of the simulator is demonstrated on four different numerical tests: a comparison with a first-order finite-difference method on a simulation of a quarter of a five-spot pattern, an edgedrive coning problem, an imbibition problem, and a field-scale problem. Saturation Equation Ref. 1 describes the saturation equation in detail. The water-conservation law isEquation 1 where the water volumetric flow vector, uw, can be written asEquation 2 Here, the usual terms have been rearranged in such a way that each physical effect is clearly identified: r=-?(x)Pcmax(x)?[a(Swr)] is the capillary diffusion term; b0(Swr) ut, is the standard transport term, with the fractional flow, b0, and the total water/oil flow vector, ut; b1(Swr)q1 describes the differential action of gravity on the two fluids, with q1 proportional to ?D(x); and b2(Swr)q2 describes the differential action of capillary pressure heterogeneity on the two fluids with q2 proportional to ?Pcmax(x). The last three terms will be approximated in the same way; therefore, it is convenient to introduce the transport termEquation 3 Typical boundary conditions for these equations areEquation 4 on the outer boundary, ? l, of the field,Equation 5 (given water injection rate) on the boundary, ?e, of an injection well; andEquation 6 (capillary effects neglected) on the boundary, ?s, of a production well. The initial condition isEquation 7 Discontinuous Finite-Element Approximation of Saturation. The reduced saturation, S wr, is approximated by Swra?M1, which is linear on every Triangle (bilinear on every Parallelogram) K of the finite-element mesh covering the field domain (see Fig. 1). By use of forward differencing in time, the saturation is obtained at each timestep through a two-step calculation. The first step is a finite-element calculation, and the second step limits the slope of the saturation calculated in the first step. Discontinuous Finite-Element Approximation of Saturation. The reduced saturation, S wr, is approximated by Swra?M1, which is linear on every Triangle (bilinear on every Parallelogram) K of the finite-element mesh covering the field domain (see Fig. 1). By use of forward differencing in time, the saturation is obtained at each timestep through a two-step calculation. The first step is a finite-element calculation, and the second step limits the slope of the saturation calculated in the first step.

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The fluid injection in sedimentary formations may generate geochemical interactions between the fluids and the rock minerals, e.g., CO 2 storage in a depleted reservoir or a saline aquifer. To simulate such reactive transfer processes, geochemical equations (equilibrium and kinetics equations) are coupled with compositional flows in porous media in order to represent, for example, precipitation/dissolution phenomena. The aim of the decoupled approach proposed consists in replacing the geochemical equilibrium solver with a substitute method to bypass the huge consuming time required to balance the geochemical system while keeping an accurate equilibrium calculation. This paper focuses on the use of artificial neural networks (ANN) to determine the geochemical equilibrium instead of solving geochemical equations system. To illustrate the proposed workflow, a 3D case study of CO 2 storage in geological formation is presented.

The emergence of liquid-rich and gas shale reservoirs presents major strategic opportunities and challenges for the oil and gas industry. Accurate estimation of Stock Tank Oil and Gas Initially In Place (STOIIP & GIIP) is one of the priority tasks before defining the reserves. An accurate method is proposed to calculate hydrocarbon volumes using high-resolution geological models taking advantage of huge improvements made during last decade in the field of characterization and geological modeling of unconventional reservoirs. This exact method provides fluids in place in reservoir and surface conditions with an extended black-oil formulation including condensates. The physic including the equilibrium between gravity and eventually capillary forces and adsorbed gas is fully respected using, for each lithofacies, the most accurate available geological description with 3D porosity distributions, Langmuir isotherms (Langmuir, 1918), capillary pressure curves, and thermodynamic data. Adsorbed and liquid-rich gases are considered. This method calculating hydrocarbons in place is the natural endpoint of any workflow devoted to the geological modeling of newly discovered reservoirs, particularly suited to heterogeneous reservoirs. The knowledge generated by this calculation has significant impact on fracturation programs to increase the recovery rate and field development planning.

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Conventional gradient methods have already been applied to reservoir engineering for matching the history of former field performances. The key point of these methods is to select the best areal reservoir zoning for reduction of the amount of reservoir parameters to be identified. In this paper we propose a zoning based on reservoir lithofacies, thus making a more natural than geographical choiee. By introducing gradients to numrmze an objective function that measures the difference between observed and simulated well pressure responses, we can effectively achieve the inversion of petrophysical lithofacies parameters. By fixing the value of petrophysical parameters, the influence of the geometry can be studied by varying the geologie body's dimensions. Several examples of inversion are given at the end of the artiele to illustrate the effectiveness of this gradient method.

More and more computerized models provide reservoir descriptions of million ceIls, which correspond to the complexity of the heterogeneities met in natural rocks. Huid flow simulations within these media need upscaling techniques. The Dual Mesh Method considering specific discretisation for each unknown allows to solve this upscaling problem by doing adaptive homogeneisation. In this paper, non linear problems are addressed and time steps different for each resolution are considered. Applications in heterogeneous media with different mobility ratios are given and show that the dual mesh method could be applied to full field simulations with more accurate solutions than with an a priori upscaling.