Five-spot problem

Fig. 14. A model of the five-spot problem with one production well in the center and four injection wells at the corners.

Now, let us consider the standard test case of the five-spot problem. Here, one production well P in the center of the computational domain ii is surrounded by four injection wells Ij, j = 1,. ..,4, located at the corners of the model as displayed in Figure 14. Assume a thin oil bearing layer trapped between two sealing layers as shown in Figure 14, which allows us to reduce the problem to two-dimensions by taking a (rri-a 2)-slice of the model at the depth of the oil bearing layer. The distances between a particular location X = (xi, X2) in the computational domain Q and the wells are given by... 

Fig. 15. The total velocity field is displayed as streamlines (a) and as velocity vectors (b) for the five-spot problem. The plots in (c) and (d) show the zoomed section of the top left quarter.

Fig. 16. [Reproduced in colour in Plate 31 on page 443.] Five-spot problem computed with ADERl. Color plots indicating the injection of water during the simulation at six different times, (a) t = to (b) t = tioo (c) t = toio] (d) t = fseo (e) t = trso and (f) f = feoo-...

In order to confirm the good performance of the proposed ADER schemes, we compare their numerical results, on the five-spot problem, with two different reservoir simulators, whose results we use as reference solutions. The chosen simulators ECLIPSE and FrontSim are two commercial software packages used by the majority of reservoir simulation groups in oil industry. We remark, that these simulators solve the coupled system of the pressure and saturation equations (29)-(27) and therefore consider the effect of pressure changes due to changes in saturation. 

This in turn typically causes changes in the total velocity field, but as confirmed by the following results, these effects can be neglected for the homogeneous five-spot problem. The reference solutions are both computed on a two-dimensional Cartesian mesh consisting of 100 x 100 rectangular cells, which are fixed throughout the simulation. 

This study was performed to estimate the additional oil recovery by pattern polymer flooding using Kelzan in a California viscous-oil reservoir. Laboratory experiments were performed to estimate polymer solution viscosity, adsorption and dispersion upon displacement by normal injection water. Waterflood and polymer flood oil recovery curves were obtained for a laboratory core packed with sand representative of the reservoir. A numerical model was developed to simulate polymer floods in linear or five-spot patterns in single-layer or stratified reservoirs- An analytical solution to the linear polymer flood problem was developed to provide a quick estimate of incremental oil obtainable by polymer flooding and to provide a check on the accuracy of the numerical model. 

Figure 18 shows the results (on the five-spot model problem) obtained by an adaptive ADER4 scheme. The shocks are resolved sharper in this case compared to the ADERl results of Figure 16. Moreover, the rarefaction appears to be much smoother, there is no structure of an underl3ung triangular mesh visible. However, if we look at the corresponding adaptive mesh in Figure 19, we see that there is even a coarser mesh in the areas of the rarefactions. In fact, behind the shocks the error indicator allows the recoarsening of the mesh to its coarsest level. This is due to the increased approximation quality of the higher order ADER4 scheme, which uses piecewise cubic polynomials instead of piecewise constant functions to reconstruct the water saturation...

In order to ensure that the slides are equally affected by the heat treatment, it is important that only a few (five to six) slides are microwaved at a time. This is because the microwaves within the oven become unevenly distributed, leading to areas of low and high intensity. Those areas with a high intensity of micro-waves are referred to as hot spots. Tissue that is unevenly microwaved may produce variable or poor results Microwave ovens with a turntable help to reduce this problem. 

Two cases of Henoch-Schonlein purpura (a disease that involves purple spots on the skin, joint pain, gastrointestinal problems, and glomerulonephritis) have been reported in individuals that were using aloe. One patient had taken "some juice extracted from four to five leaflets of Aloe vera" (he had taken the same remedy several months prior with no reaction) 24 hours prior to developing a rash, abdominal pain followed by rapid deterioration of renal function and abnormal levels of serum creatinine (Cholongitas et al. 2005). The second patient had taken "extracts of Aloe vera" (parts consumed and dose consumed unspecified) for approximately 1 week. He developed general edema and palpable purpura that eventually resolved (Cao et al. 2005). 

In the United States, the Occupational Safety and Health Administration (OSHA) is responsible for ensuring the safety of the environment within an industrial plant. Environmental problems caused by a facility that reaches beyond the fences of the facility become the concern of the EPA. Environmental problems can become lethal to human life if left uncontrolled. This is what happened in Donora, Pennsylvania in 1948 that resulted in the deaths of 20 people. Donora was situated in a low spot of the surrounding terrain. A meteorological temperature inversion blanketed the area for five days. Near Donora, a steel mill, sulfuric acid plant, and zinc smelter poured sulfur dioxide into the atmosphere. The weather inversion kept the sulfur dioxide close to the ground and people sickened and died. 

There is one other extremely clever method of setting up the tempo of a project. Say you ve composed the perfect background music for a thirty-second commercial spot. The only problem is that the music runs for thirty-five seconds at 120 bpm. Speeding up the tempo will make it run faster, but how much do you need to increase the tempo Guessing and experimenting will eventually get you the right answer, but ACID has a feature that lets you automatically adjust the tempo using just one marker. 

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