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The pressure response during a well test provides important information about reservoir properties, well productivity or reservoir performance. In order to have an option to replace traditional production tests, avoiding the gas flaring and increasing operational safety, injectivity test began to play an important role in the management of reservoirs. Although analytical models are able to describe injection and falloff periods, multiple rate models still need to be developed. This work presents a new formulation to compute pressure response when performing multiple rate injectivity test in a water injection well operating in an oil zone. First, the proposed formulation is compared to injection/falloff models and after a multiple rate scheme is performed. This formulation extends prior models presented in the literature. The accuracy of the proposed solution was assessed by comparison with a numerical simulator. The proposed formulation is also used to determine the reservoir equivalent permeability in any specific period of injection or shut-in. Keywords Well-testing • Injectivity test • Partial differential equation • Reservoir evaluation List of symbols B Phase formation volume factor c to Total compressibility c r Rock compressibility f w Fractional flow of water h Thickness (m) k l Absolute permeability (= o,) k rl Relative permeability (= ,) M Mobility ratio p i Initial pressure p l Pressure (= , w) p wf Well bottom-hole pressure q inj Injection rate q o Oil rate q w Water rate r f Waterfront radius r w Wellbore radius S w Water saturation t Time t

Completion of water injection wells directly into production oil zone has become a common practice on offshore development projects. Hence, injectivity tests started to play an important role for reservoir management of these fields. In an injectivity test a single phase (water) is injected continuously for a certain period of time in an oil saturated zone followed by a falloff period. Though single-phase flow of a slightly compressible fluid analysis methods are well documented in literature, transient two-phase solutions and analysis techniques need further development. Analytical solutions that model pressure behavior during a single layer injectivity test were recently presented in literature. This paper presents a new analytical solution for a vertical water injection well completed in a layered oil reservoir of infinite extent. To the best authors’ knowledge, the proposed solution was not presented before. Two phase immiscible displacement of oil by water in a multilayer reservoir is modeled by a system of partial differential equations. This system is weakly non-linear and can be decoupled by assuming steady-state conditions in the two-phase region. The solution is then obtained by the method of characteristics on the two-phase region coupled with the infinite acting single phase radial flow solution valid for the oil region. The solution derived in this work can be used to compute the wellbore pressure, injection rates and the water front for each individual layer without resorting of a numerical flow simulator. Parameter estimation from field data can be obtained from the proposed solution if a production logging rate profile is available. Two synthetic examples are used to verify the solution derived in this work. The first example considers the behavior of a two-layer reservoir whereas in the second case a reservoir with seven layers is analyzed. The wellbore pressure is compared to a finite difference simulator. An excellent agreement is obtained for both examples.