This paper presents a computer method to obtain the shape factors and equal cell volumes of the channels for any well spacing pattern from a potentiometric model. By using this program the authors have processed the darn for the seven-spot, direct line-drive and The staggered line-drive patterns. The data for the five-spot pattern had been previously processed by a non-computer method and are included for completeness. The shape factors and volumes for the channels are presented in tables for those who want to use them to process data using their own permeability relationships and viscosities of their reservoir oils. The authors have used the data and sets of representative permeability curves to process sample calculations of water flood performances. The comparison of the calculated results shows that the influence of well spacing is small. The permeabilities of the reservoir rock to oil and water had a greater influence on oil recovery for a given pore- volume through put of water than the well spacing pattern. The more water-wet the reservoir rock, the better the possibility of permeabilities which are conducive to food recover. The viscosity of the reservoir oil also influences the recovery more than the well spacing pattern. The reduction in the percentage recovery of oil with increase in viscosity of the reservoir oils is small when oil viscosities are in the range of 0.1 to 3. Above this range the reductions in recoveries are extensive. Sample comparisons of the time required for different patterns to recover the oil are presented. Results of an examples calculation are given to show the effect of the permeability profile on recovery.
The effect of well spacing pattern on the recovery of oil when flooding with either gas or water has been studied by many investigators. Muskat et al. presented an analysis using conductivity, sweep efficiency and unit mobility to the time of breakthrough. Dyes et al. used experimental techniques (X-ray shadowgraphs) and different mobility ratios. They presented quantitatively the relationship between mobility and sweep efficiency at and after breakthrough. Hauber presented a method to predict waterflood performance for arbitrary well spacing patterns and mobility ratios. Craig et al., using techniques similar to Dyes et al. to determine sweep efficiency foil a five- spot pattern. added the use of relative permeability curves at breakthrough and thereafter. Douglas et al. used relative permeabilities and continuously changing saturations throughout the entire five-spot flood pattern. In obtaining their solutions they used finite-difference equations. Higgins and Leighton also used relative permeabilities and continuously changing saturations throughout the pattern before and after breakthrough. They employed techniques that process a flood-pattern calculation on the computer in about one minute. The methods of Douglas et al., and Higgins and Leighton both checked closely the laboratory results for a wide range of mobility ratios. This paper presents some sample performances calculated by the Higgins and Leighton method that show the effect on recovery of different permeabilities and viscosities using the seven-spot, the line-drive and the staggered line-drive, as well as the five-spot flood pattern. No previous paper has presented these data using different permeability curves and continuously changing saturations throughout the flood patterns. The paper also presents
the results and analyses of the flood-pattern prediction,
the computer techniques for determining the shape factors and volumes from the potentiometric models for the foregoing flood patterns, and
the shape factors and volumes of the channels of the flood pattern in the event reservoir engineers may like to process waterflood calculations using their own permeability curves and reservoir oils.
The use of channels taken from a potentiometric model (see Fig. 1) to aid in calculating the performances of water floods of nonlinear patterns has been thoroughly explained in the literature. Therefore, very little theory. discussion, or proof regarding this phase will be repeated in this paper. The computer method presented in this paper to calculate the volumes and the shape factors of the channels of potentiometric models employs the trapezoidal rule for the volumes and the Pythagorean theorem (the hypotenuse equals the square root of the sum of the squares of the two sides of a right triangle) for the shape factors. In calculating the volume of a channel, the area of each cell in the channel is determined and then multiplied by the thickness to obtain the volume. In determining the areas of the cells, trapezoids are constructed whose vertical sides are spaced delta x apart, as shown in Fig. 2. The length of the sides is the difference between an ordinate cut off by the top and bottom of the cell - usually equipotentials. The coordinates of the points along an equipotential or streamline are obtained by Lagrange's equation of interpolation for which the constants are coordinate points at the intersections.