Summary A matrix/fracture exchange model for a fractured reservoir simulator isdescribed. Oil/water imbibition is obtained from a diffusion equation withwater saturation as the dependent variable. Gas/oil gravity drainage andimbibition are calculated by taking into account the vertical saturationdistribution in the matrix blocks. Introduction In most simulators intended for naturally fractured reservoirs, the fractureand matrix systems are considered to be two overlapping media. Flow between thetwo is described in various ways by means of source and sink terms. Thedescription of the matrix/fracture interaction is a key point in the modeling of dual-porosity systems. In this paper, the modeling of oil/water imbibition is based on thediffusion equation approach of Beckner et al. The effect of gravity isincorporated through a modification of the boundary conditions imposed. Analytical and numerical solutions are presented, and computed results arecompared with experimental data. presented, and computed results are comparedwith experimental data. Gas/oil gravity drainage and imbibition are calculatedby taking into consideration the vertical saturation distribution in thematrix. The principles for the implementation of the proposed methods in areservoir simulator are described. The following limitations and assumptions apply.The models presented are valid only for two-phase oil/water and gas/oilsystems.Matrix blocks within a grid cell are identical and box-shaped withdimensions L, L, and L.For oil/water systems, capillary continuity exists inside a grid cellbetween vertically stacked matrix blocks.The two phases in the fracture system are gravity segregated.Analytical solutions can be obtained only in the oil/water case and onlyif the water level in the fracture system rises with a constant velocity andthe diffusion coefficient is constant.The matrix-block gas and oil are at capillary/gravitationalequilibrium. Flow Equations Dual-porosity reservoirs are modeled by the continuum approach, where thefracture and matrix systems are considered to be two overlapping continuousmedia. The basic equations for isothermal fluid flow in porous media aretransformed to a system of ordinary differential equations by means of theintegral finite-difference method (see Pruess and Bodvarsson). In case of adual-porosity, single-permeability reservoir composed of a continuous fracturesystem containing discontinuous matrix blocks, the following equations areobtained for each component (1 =o, g, or w) and the kth grid cell in thereservoir. Fracture equation: (1) Matrix equation: (2) where mi = (3) (4) The summation is over all phases--i.e., =o, g, and w. The sum over Index isover all grid cells adjacent to grid-cell number k. Hence Index k refers to theboundary between the grid cells k and . The individual terms of Eqs. 1 and 2 describe the transport of Component ithrough Phase by various mechanisms. For further details regarding theequations and their derivation, see Bech. Water/Oil Imbibition The formulations of the matrix/fracture exchange term in manydouble-porosity simulators suffer from two limitations:imbibition from newly contacted matrix-block face as the fracture water level advances is not described andsaturation gradients within the matrix blocks are notmodeled. These factors can be taken into consideration by modeling theimbibition as a diffusion process. The matrix-block water saturationdistribution is determined from (5) (6) In the derivation of Eqs. 5 and 6, it is assumed that the flow is 2D andthat the fluid and the rock are incompressible. It is also assumed thatoil-phase pressure gradients and gravity terms are negligible. The diffusion equation (Eq. 5) has been solved with the boundary condition S= S at the part of the matrix-block surface that is submerged in water and Sw =S, elsewhere on the surface (see Fig. 1). Initially, the matrix-block watersaturation is S everywhere. The boundary condition (7) where S corresponds to zero capillary pressure at the surface, implies thatinstantaneous imbibition occurs at the matrix/fracture in-terface. A delayedimbibition can be introduced by the boundar conditions: (8) (9) and S = S (z) is the ultimate matrix-block water saturation at the height z, which may be equal to or less than 1 - Sor. Here t = t (z) denotes the time when zwf = z; i.e., t (z) is the time whenimbibition starts at the height z. is an inverse time constant. The problem as presented in Eqs. 5 through 9 must be solved numerically. This is done by using central differences and applymg a Newton-Raphsonalgorithm in solving the nonlinear algebraic equations.
Hydraulic fracturing of horizontal wells is a relatively novel stimulation treatment. For most depths of petroleum interest fractures away from the well are vertical and normal to the minimum horizontal stress direction. If a horizontal well is drilled at a trajectory other than the expected fracture direction a complicated fracture-to-well connection is likely to occur. Past literature has suggested a longitudinal fracture initiation followed by a tortuous turning path towards the final fracture direction, or multiple fractures. This phenomenon happens within a distance that is a few times the well diameter. The consequences are increased treatment pressures and a potentially severe fracture width reduction at the turning point. Post-treatment production performance is likely to be affected substantially. Reservoir simulation studies were carried out demonstrating the dependence of production rate on the angle between the horizontal well and the final fracture trajectory, the length of the perforated interval and the dimensionless fracture conductivity. under an assumption of a choke effect at the turning point. An angle threshold was identified beyond which the well production decreases significantly. Introduction Inducing hydraulic fractures is an additional, and relatively novel, technique of stimulating the performance of horizontal wells. In general, horizontal wells are attractive alternating s to vertical wells, where the formation thickness is relatively small and the ratio of horizontal to vertical permeability is small. Furthermore, Deimbacher et al. showed that in formations that are highly anisotropic in the horizontal plane, the direction of drilling the horizontal well (with respect to the maximum horizontal permeability axis) is crucial. Unfractured horizontal wells drilled normal to the maximum horizontal permeability (which usually coincides with the maximum horizontal stress direction) can outperform both fractured vertical wells and longitudinally fractured horizontal wells. It was also shown that in thin formations a small vertical-to-horizontal permeability anisotropy variable,, is not as important. With regard to hydraulic fracturing, as shown by Muxherjee and Economides and Economides et al., two mutually exclusive scenarios for drilling a horizontal well can be distinguished: the well may be drilled either along the direction of the maximum horizontal stress,, and, thus accept a longitudinal fracture, or may be drilled along the minimum horizontal stress,, leading to the possibility of transverse fractures. However, these limiting cases may not lend themselves in offshore locations where the well trajectories may be dictated by the logistics of drilling from a platform. All angles between horizontal well and minimum horizontal stress direction will result in a more complicated fracture geometry. When the horizontal well direction does not coincide with one of the principal stress axes, the hydraulically induced fracture will experience both shear and tensile failure. P. 891^
Convection-diff usion equations are difficult to solve when the convection term dominates because most solution methods give solutions which oscillate in space. Previous criteria b a e d on the one-dimensional convection-diffusion equation have shown that finite difference and Galerkin (linear or quadratic basis functions) will not give oscillatory solutions provided the Peclet number times the mesh size (Pe A x ) is below a critical value. These criteria are based on the solution at the nodes, and ensure that the nodal values are monotone. Similar criteria are developed here for other methods: quadratic Galerkin with upwind weighting, cubic Galerkin, orthogonal collocation on finite elements with quadratic, cubic or quartic polynomials using Lagrangian interpolation, cubic or quartic polynomials using Hermite interpolation, and the method of moments. The nodal values do not oscillate for collocation or moments methods with Hermite cubic polynomials regardless of the value of Pe Ax.A new criterion is developed for all methods based on the monotonicity of the solutions throughout the domain. This criterion is more restrictive than one based only on the nodal values. All methods that are second order (Ax') or better in truncation error give oscillatory solutions (based on the entire domain) unless Pe Ax is below a critical value. This value ranges from 2 for finite difference methods to 4.6 for Hermite, quartic, collocation methods.PeAx =z 2 Christie er a1.' provided a criterion for the one-dimensional steady-state equation with the Galerkin finite element method and linear or quadratic trial functions. Linear functions give the criterion (l), whereas quadratic functions give the criterion Pehx s 4(2)
Summary A numerical front-tracking technique, based on a transformation of the governing equations into a moving coordinate system (MCS), is applied to a finite-element reservoir simulator. The method is especially suited for studies of tertiary oil-recovery pilots with chemical flooding and other miscible displacement processes. The new front-tracking technique is compared to conventional finite-element formulations with a uniform grid over the entire domain. Comparisons with other methods show that computer time can be greatly reduced with the same accuracy. Introduction Accurate simulation of EOR processes (such as chemical and polymer flooding) is very important for the design and appraisal of pilots and the prediction of full-field performance. The most prediction of full-field performance. The most serious limitation on making predictions is the inability to simulate these processes with a large enough number of gridblocks to reduce the numerical dispersion to an acceptable level. Lake et al. studied the Benton stage chemical flood project with two-dimensional (2D) simulation (cross section) and a stream tube program to account for areal and confinement effects. With this procedure the effect of dispersion is not accounted for properly. The MCS is a numerical method that performs three-dimensional (3D) simulation without numerical dispersion in a symmetric flooding pattern. The method can be used in EOR studies pattern. The method can be used in EOR studies as shown in Fig. 1. Coreflood and cross-sectional models are used to study the physical process and reservoir layering, as well as other components crucial for the design of the flood. The streamtube simulator and the 3D symmetric pattern simulator (where the MCS-is applied) study simultaneously the sensitivity to parameter variation and full-field performance. The symmetric pattern simulation performance. The symmetric pattern simulation can provide response functions of individual streamtubes through the detailed simulation of a small area. Many studies were conducted to develop methods for reducing numerical dispersion. The basic numerical methods-finite-difference and weighted-residual methods-show either oscillations close to the sharp front(s) or a smearing of the front.
The hydraulic behaviour of the fractures in a fractured carbonate reservoir is a function of fracture intensity, aperture, intrinsic permeability, length, height and orientation, all of which influence the scale of connectivity and ultimately storage, productivity and reserves. If a geologically realistic fracture model is not appropriately incorporated into upscaled fracture properties for a dynamic simulation, it may still be possible to match a short production history, but calculations of field-wide fracture pore volumes and forecasts of future reservoir development may be poor and uncertain. To accurately represent the fractures, discrete fracture network (DFN) models were built and used to constrain fracture geometries and their hydraulic properties for use in forecasting, field development options and uncertainty characterization. The workflow illustrated in this paper shows how a DFN may be validated and calibrated through the simulation of transient bottom hole pressures from individual drill stem tests and pressure interference data, followed by upscaling to a full-field dynamic simulation model. This DFN-to-simulation workflow, applicable to most conventional fractured reservoirs, successfully matched reservoir pressure history for the field as a whole and for individual wells without having to locally modify any of the upscaled fracture properties around the wells. Sensitivity analysis identified key fracture drivers having the greatest impact upon the history match, and these were combined to produce history matched Low and High Case models. Production forecasts for the Low, Base and High Cases were used to predict reserves, manage risk and optimize the field development plan.Supplementary material: Supplementary figures are available at https://doi.org/10.6084/m9.figshare.c.5001203Thematic collection: This article is part of the The Geology of Fractured Reservoirs collection available at: https://www.lyellcollection.org/cc/the-geology-of-fractured-reservoirs
A deterministic model is described which enables the stress conditions to be calculated based on the structural growth history. The method provides a framework for estimating natural fracture distribution and directions. The method is developed for two dimensional cross sections and can be extended to three dimensions. A depth converted seismic profile is used for dividing the geological section into a finite number of time defined formations. The individual formations can be intersected by fault traces. The paleo-appearance of the cross section is constructed through steps in geological time by stripping off formations from the top. In each step the remaining section is decompacted under the decreased load based on the compaction trend of the individual formations. For structural restoration, unfolding and movement along fault traces are made. During this backstripping the displacements are recorded for the subsequent stress analysis. Stresses are calculated by finite element modelling. An elastic/plastic strain hardening model is proposed. It includes a time dependent creep model making it possible to predict formation properties through geological time. An elastic/plastic finite element model has been designed to calculate stresses by use of these displacements. The results of the stress analysis are predictions of the stress state being either elastic or plastic: compaction, compressible shear or tension fracture orientations is given too for each point in the reservoir at the steps in geologic time. The level of stress can be "history matched" with well data, well test analysis, fracture identification logs, in situ stress measurements etc.. Illustration of the method is presented for two oil bearing chalk reservoirs overlying salt diapirs in the North Sea. One example illustrates the stress state in a heavily fractured reservoir and the other, a less fractured reservoir.
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