S U M M A R YWe compare different finite-difference schemes for two-dimensional (2-D) acoustic frequencydomain forward modelling. The schemes are based on staggered-grid stencils of various accuracy and grid rotation strategies to discretize the derivatives of the wave equation. A combination of two O( x 2 ) staggered-grid stencils on the classical Cartesian coordinate system and the 45 • rotated grid is the basis of the so-called mixed-grid stencil. This method is compared with a parsimonious staggered-grid method based on a fourth-order approximation of the first derivative operator. Averaging of the mass acceleration can be incorporated in the two stencils. Sponge-like perfectly matched layer absorbing boundary conditions are also examined for each stencil and shown to be effective.The deduced numerical stencils are examined for both the wavelength content and azimuthal variation. The accuracy of the fourth-order staggered-grid stencil is slightly superior in terms of phase velocity dispersion to that of the mixed-grid stencil when averaging of the mass acceleration term is applied to the staggered-grid stencil.For fourth-order derivative approximations, the classical staggered-grid geometry leads to a stencil that incorporates 13 grid nodes. The mixed-grid approach combines only nine grid nodes. In both cases, wavefield solutions are computed using a direct matrix solver based on an optimized multifrontal method. For this 2-D geometry, the staggered-grid strategy is significantly less efficient in terms of memory and CPU time requirements because of the enlarged bandwidth of the impedance matrix and increased number of coefficients in the discrete stencil.Therefore, the mixed-grid approach should be suggested as the routine scheme for 2-D acoustic wave propagation modelling in the frequency domain.Modelling seismic wave propagation is essential for understanding complex wave phenomena in a realistic heterogeneous medium. Numerical results from finite-difference (FD) modelling are particularly useful since they provide the complete wavefield response. Frequency-domain forward modelling is of special interest for multisource experiments, such as tomographic experiments, because of its computational efficiency (Pratt & Worthington 1990;Štekl & Pratt 1998). Moreover, realistic rheology is easily incorporated into the modelling scheme by introducing complex velocities. The key step in frequency-domain finite-difference (FDFD) modelling that controls computational efficiency is the numerical inversion of a massive matrix equation. The matrix structure depends on the spatial derivative approximations. We shall discuss what the useful features of this matrix are for wave modelling accuracy and for computational efficiency.Elastodynamic finite-difference time-domain (FDTD) techniques moved from second-order approximations of spatial derivatives (Madariaga 1976;Virieux 1984Virieux , 1986 to higher-order approximations (Dablain 1986;Levander 1988) using staggered-grid stencils with a good trade-off between modelling acc...

Summary Water-injection-induced fractures are key factors influencing successful waterflooding projects. Controlling dynamic fracture growth can lead to largely improved water-management strategies and, potentially, to increased oil recovery and reduced operational costs (well-count and water-treatment-facilities reduction), thereby enhancing the project economics. The primary tool that reservoir engineers require to guarantee an optimal waterflood field implementation is an appropriate modeling tool, which is capable of handling the dynamic fracturing process in complex reservoir grids. We have developed a new modeling strategy that combines fluid flow and fracture growth in one reservoir simulation. Dynamic fractures are free to propagate in length and height-direction with respect to poro- and thermoelastic stresses acting on the fracture. A prototype simulator for contained fractures was tested successfully. We have extended the coupled simulator to incorporate noncontained fractures. The new simulator, called FRAC-IT, handles fracture-length and -height growth by evaluating a fracture-propagation criterion on the basis of a Barenblatt (1962) condition. The solution of the 5D problem is computed by use of a tuned Broyden (1965) approach. We demonstrate the capabilities of the coupled simulator by showing its application to a complex reservoir-simulation model. The fracture modeling is used to history match an injectivity test in a five-spot injection pattern using produced water. The coupled-simulation results and the field-data interpretation show a very good match. The outcome of the injection test led to an appropriate waterflood-management strategy adapted to the specific reservoir conditions and, in terms of production, to a net oil-production increase of 50 to 100%. The field example shows how the coupled-simulator technology can be used to achieve optimized waterflood-management strategies and increased oil recovery. Introduction Waterflooding is often applied to increase the recovery of oil in mature reservoirs or to maintain the reservoir pressure above bubblepoint in the case of green fields. Even though often unnoticed, water injection frequently is taking place under induced-fracturing conditions. The rock fracturing has a strong influence on the water injectivity and the areal distribution of the fluids in the reservoir. A qualitative example of the impact of the fracture orientation on the areal sweep is demonstrated in Fig. 1. We show streamlines in two different water-injection-pattern configurations for two fracture orientations (i.e., line-drive and five-spot geometry, and fracture oriented toward the producer and away from the producer. The density of the streamlines indicates that the fracture orientation changes the areal sweep. In order to achieve optimized water-injection management, dynamic fracture propagation needs to be estimated properly before the injection, controlled during operations, and monitored to ensure predictions and reality do not deviate significantly. The tools commonly used to study fracture growth numerically are analytical fracture simulators, which often are based on a single-well model in a simplified reservoir formation. Generally, reservoir heterogeneity is reduced to a number of horizontal layers with homogeneous properties and a laterally infinite extent. Fracture propagation is described using a pseudo-3D description (van den Hoek et al. 1999). For many field developments under waterflooding, fracture propagation is estimated with acceptable error bars using these or similar tools. The major drawbacks areAreal reservoir heterogeneity is not accounted for.Varying poro- and thermoelastic stresses along the fracture are neglected.Injection pressures have large error bars because the reservoir response is not properly captured.Nearby well's influences (e.g., pattern flood) are not captured. In the past, many attempts have been made to address these issues. Common approaches can be grouped into fully implicit simulators (Tran et al. 2002), where both fluid-flow and geomechanical equations are solved simultaneously on the same numerical grid, and coupled simulators (Clifford et al. 1991), where a standard, finite-volume reservoir simulator is coupled to a boundary-element-based fracture-propagation simulator. To our knowledge, both approaches are not standard and currently not used in the industry becauseModels need to be purpose built (i.e., reservoir models from standard reservoir simulator cannot be used).Fracture propagation is oversimplified.Numerical stability is questionable. We have developed an extension to an existing reservoir simulator to circumvent these shortcomings. We use a coupled-simulator approach based on a two-way communication strategy between the fully numerical reservoir simulator and the half-analytical geomechnical-modeling part. The new simulator enables the modeling of fluid flow and dynamic fracture propagation in a combined way. We have applied the tool to field applications for waterflooding projects in which injector/producer shortcuts are a potential risk (pattern floods) and also to environments in which fracture containment and estimating accurate injection pressures are the main concerns. In this paper, we briefly review the coupled-simulator approach and discuss the application to a waterflooding field example.

Geophysical Journal International, v. 148, n. 3, p. 476-498, 2002. http://dx.doi.org/10.1046/j.1365-246x.2002.01573.xInternational audienc

The performance of many waterfloods [and enhanced-oil-recovery (EOR) schemes] is characterized by fluid injection under fracturing conditions. Especially when the geology is complex and the mobility of the reservoir is low, induced fractures can be of the same order as the well spacing, which has a significant (in general undesired) impact on both areal sweep and vertical conformance. Therefore, fluid injection needs to be actively managed and surveyed in order to design an appropriate injection strategy over time.We have analyzed historical injection/production-test, injection step-rate-test, and falloff (FO) test (FOT) data of an existing complex waterflood in the Pierce field, North Sea. The mental subsurface model that emerged from this data analysis was developed further through a series of dynamic fracture-propagation simulations. While the data analysis was a relatively standard procedure, the fracture-modeling part was far from trivial and included simulations using a standalone fracture modeling tool and a more sophisticated coupled dynamic fracture-propagation reservoir simulator, both being in-house software tools.The combined analysis was used to develop a better understanding of the waterflood performance. The main improvement compared to previous work was the integration of the data analysis and the dynamic modeling work rather than looking at each data source individually. In combination, a consistent explanation of the observed reservoir behavior was achieved. This has resulted in changes in the day-to-day water injection management and is expected to play a key role in longer-term development strategies Raw-Data AnalysisThe first step toward a better understanding of the Pierce waterflood response was a detailed analysis of all the raw data available such as daily injection/production rates and THP/BHP, step-rate injection test(s), buildup and FOT in producers and injectors, respectively. In the following, we are presenting an example of each of these data groups, while the stress data and pressure-transient data are discussed in separate paragraphs that follow.

Water-injection induced fractures are key factors influencing successful waterflooding projects. Controlling dynamic fracture growth can lead to largely improved water management strategies and potentially to increased oil recovery and reduced operational costs (well count and water treatment facilities reduction), thereby enhancing the project economics. The primary tool that reservoir engineers require to guarantee an optimal waterflood field implementation is an appropriate modeling tool that is capable of handling the dynamic fracturing process in complex reservoir grids. We have developed a new modeling strategy that combines fluid-flow and fracture-growth in one reservoir simulation. Dynamic fractures are free to propagate in length- and height-direction with respect to poro- and thermo-elastic stresses acting on the fracture. A prototype simulator for contained fractures was successfully tested. We have extended the coupled simulator to incorporate non-contained fractures. The new simulator handles fracture length and height growth by evaluating a fracture propagation criterion that is based on a Barenblatt condition. The solution of the five-dimensional problem is computed using a tuned Broyden approach. We demonstrate the capabilities of the coupled simulator by showing its application to a complex reservoir simulation model. The fracture modeling is used to history match an injectivity test in a five-spot injection pattern using produced water. The coupled simulation results and the field data interpretation show a very good match. The outcome of the injection test led to an appropriate waterflood management strategy adapted to the specific reservoir conditions and, in terms of production, in a net oil production increase of 50–100%. The field example shows how the coupled simulator technology can be used to achieve optimized waterflood management strategies and increased oil recovery. Introduction Waterflooding is often applied to increase the recovery of oil in mature reservoirs or to maintain the reservoir pressure above bubbelpoint in the case of green fields. Even though often unnoticed, water injection is frequently taking place under induced fracturing conditions. The rock fracturing has a strong influence on the water injectivity and the areal distribution of the fluids in the reservoir. A qualitative example of the impact of the fracture orientation on the areal sweep is demonstrated in Figure 1. We show streamlines in two different water-injection pattern configurations for two fracture orientations i.e., line-drive and five-spot geometry, and fracture orientated towards the producer and away from the producer. The density of the streamlines indicates that the fracture orientation changes the areal sweep. In order to achieve optimized water injection management, dynamic fracture propagation needs to be properly estimated prior to the injection, controlled during operations, and monitored to ensure predictions and reality do not deviate significantly. The common tool used to study fracture growth numerically are analytical fracture simulators that often are based on a single well model in a simplified reservoir formation. Generally, reservoir heterogeneity is reduced to a number of horizontal layers with homogeneous properties and laterally infinite extent. Fracture propagation is described using a pseudo-three dimensional description (van den Hoek et al.1). For many field developments under waterflooding fracture propagation is estimated with acceptable error bars using these or similar tools. The major drawbacks are that:Areal reservoir heterogeneity is not accounted forVarying poro- and thermo-elastic stresses along the fracture are neglectedInjection pressures have large error bars since the reservoir response is not properly capturedNearby wells influences in e.g. patternfloods are not captured.