A unified approach to the nonlinearity of the diffusivity equation and assessment of pseudotime

We present a unified approach that quantifies the degree of nonlinearity for the hydraulic-diffusivity equation used for single-phase-fluid (gas, oil, or water) production from an infinite-actingpressure-sensitive formation subject to constant bottomhole pressure (BHP). Using the unified approach, we assess pseudotime validity for transient linear flow and quantify the associated error in the estimation of formation properties from rate-transient analysis (RTA). Using the experimental and correlation-generated data, a set of exponential relations for rock and fluid properties are obtained as a function of pressure. As a result of the chosen exponential relations, all the properties causing nonlinearity are lumped into diffusivity coefficient g and transmissibility T, which are exponential functions of pressure with four pressure-dependent exponents (i.e., g ~ a₀, a₁ and T ~ b₀, b₁). The degree of the nonlinearity (NL) is defined as the root mean square of the diffusivity-coefficient deviation from its initial value with respect to pseudopressure, where the limit NL ¼ 0 recovers the linear hydraulic-diffusivity equation. We generate numerous synthetic cases with a wide range of rock and fluid properties at different pressure drawdowns to explain the effects of each property on a₀, a₁, b₀, and b₁, and consequently on NL. The results reveal that the cases with a highly pressure-dependent permeability or porosity tend to exhibit high nonlinearity. We use synthetic cases to demonstrate the relationship between NL and the pseudotime error in the estimation of the contact area and initial permeability using RTA. The results show that the cases with compressible gas may not exhibit high nonlinearity when a well produces from a pressure-sensitive formation. In fact, the pseudotime error might become less significant in the compressible-gas cases with high-pressure-sensitive rocks than in the slightly compressible cases. The analysis of RTA results indicates that the pseudotime error increases with the nonlinearity, and the pseudotime error is less than 10%, when NL < 0.65 and < 1.15 for the cases with the positive and negative values of a₀, respectively. Finally, we apply the proposed unified nonlinearity approach to five field cases, including three gas, one oil, and one water production from different formations to assess pseudotime. The unified approach provides a reliable avenue to assess the accuracy of the pseudotime and to identify the cases where the use of pseudotime can cause significant errors.