Ambastha, A.K., and Ramey Jr., H.J., Stanford U. (USA)
Reservoirs with a fluid bank, or a burning front, reservoirs with a reduced or an increased permeability region around the wellbore, and geothermal reservoirs are often modeled as composite reservoirs. Reservoirs with a fluid bank include reservoirs undergoing waterflood, chemical flood, polymer flood, CO2 flood, and steam injection. Eight well tests reported in the literature exhibiting composite reservoir behavior have been analyzed using the deviation time method. The dimensionless deviation times obtained from pressure and pressure derivative responses for a well in a composite reservoir have been used for analyzing the well tests. Analysis shows the estimate of discontinuity radius to be sensitive to both the real and the dimensionless deviation times used. The estimated discontinuity radius from the deviation time method may represent a lower bound for discontinuity radius, if the swept region is not cylindrical. Also, obtaining an accurate deviation time for small mobility contrasts may be difficult. Limitations on the deviation time method due to wellbore storage effects have been quantified indicating a need to minimize wellbore storage in composite reservoir well tests. The effects of a transition region on the dimensionless deviation time have been studied using an analytical solution for a three-region reservoir.
A two-region composite reservoir is made up of two concentric regions of differing hydraulic diffusivities. Figure 1 shows a schematic diagram of a two-region composite reservoir. The inner and outer regions of a composite reservoir have different, but uniform, rock and fluid properties, and are separated by a discontinuity. Reservoirs with a fluid bank, or a burning front, reservoirs with a reduced or an increased permeability region around the wellbore, and geothermal reservoirs with thermal discontinuities or boiling regions are often modeled as composite systems. Reservoirs with a fluid band include reservoirs undergoing waterflood, chemical flood, polymer flood, CO2 miscible flood, and steam injection.
A well test in a composite reservoir exhibits wellbore storage effects, a first semi-log line corresponding to the inner region mobility, a long transition, and a second semi-log line corresponding to the outer region mobility, in sequence. The first and the second semi-log lines may be masked by wellbore storage and outer boundary effects, respectively. However, if the first semi-log line develops, a deviation time,, from the first semi-log line may be obtained. This deviation time can be used to estimate discontinuity radius by:
where t is in hours, and t is the dimensionless deviation time based on discontinuity radius. The variable R is the discontinuity radius in feet. The term, , is the hydraulic diffusivity of the inner region in md-psi/cp. Equation (1) is the basis of the deviation time method to estimate discontinuity radius. Other methods to estimate discontinuity radius or swept volume are: the intersection time method, the pseudosteady state method, and the type-curve matching method. The deviation time method requires the least amount of data for analysis.
Previous investigators have proposed a number of values for dimensionless deviation time. Dimensionless deviation time values were derived by either the drainage-radius concept, or a graphical analysis of numerical or analytical pressure responses from composite reservoirs. A summary of dimensionless deviation times proposed by several authors is presented in Table 1. Sosa et al. used the average dimensionless deviation time of 0.389 proposed by Merrill et al. (Table 1) to analyze simulated falloff tests in water injection wells.
Satman et al. and Tang used the Eggenschwiler et al. analytical solution, and graphed in () as a function of to correlate the pressure responses for all discontinuity radii to the response for RD = 500. The choice RD = 500 is arbitrary, and is given by:
Ambastha and Ramey correlated the pressure derivative responses for all discontinuity radii by graphing a semi-log pressure pressure derivative as a function of. Figure 2 shows the semi-log pressure derivative responses from the Eggenschwiler et al. solution for several values of mobility and storativity ratios, and, respectively, given by (3)
Wellbore storage is neglected in generating the pressure derivative responses shown of Fig. 2. Figure 2 applies for non-zero skin values, as the pressure derivative is independent of skin, if wellbore storage is zero.