Analytical Modeling of Linear Flow in Pressure-Sensitive Formations

Abstract

Summary The mathematical models commonly used to describe fluid flow through porous media are dependent on various simplifying assumptions, one of them being that the permeability is independent of pore pressure. However, during fluid withdrawal, reservoir permeability may be reduced because of deformation of the porous rock. The pressure dependence of permeability is more pronounced in tight formations. When dealing with pressure-sensitive formations, the assumption of a constant permeability is inappropriate. In this study, analytical models are developed to model production from hydraulically fractured tight formations with pressure-dependent permeability, for both constant-pressure- and constant-rate production scenarios. By deriving an explicit relationship between time and pseudotime, it is shown that the analytical liquid solutions can be directly applied to pressure-dependent permeability reservoirs. This paper develops the appropriate transformation, and discusses its application by comparing the numerical solution of the nonlinear problem with the analytical solution proposed here. The close agreement between these solutions demonstrates the accuracy of the proposed methodology in forecasting the behavior of pressure-sensitive formations. We used the models developed in this work to address the following question: Is it possible that the permeability is decreased so much that a reduction in rate results when drawdown is increased? We show that the answer to the question is no. Depending on the strength of the nonlinearity, there could be a point beyond which the rate will not improve measurably as the flowing pressure is lowered. However, for a particular reservoir with a constant-permeability modulus, it is not possible to reduce the production rate by increasing the drawdown. This is contrary to previous publications that suggest that in a reservoir with pressure-dependent permeability, there is an optimum drawdown for maximum production. In a companion paper, we explore conditions where such an optimum drawdown could exist.

Rate Transient Analysis of Heterogeneous Unconventional Reservoirs Exhibiting Deteriorated Induced-Fractures with Exponential Permeability Profile

“…In those cases, the solutions in this work are still applicable by introducing pseudopressure or pseudotime concepts. For instance, to incorporate the impact of pressure-sensitive permeability and linearize the temporal portion of PDE, we can use a pseudotime transformation to replace the real time t a = 1 k normali ∫ 0 t k ( p ) normald t …”

“…One of the commonly used correlations to relate permeability to pore pressure is the following exponential function. , k ( p ) = k i e σ ( p − p normali ) …”

“…One of the commonly used correlations to relate permeability to pore pressure is the following exponential function. 27,28 = k p k ( ) e p p i ( ) i (23) in which k and k i are the current and initial permeability, respectively; σ is the permeability modulus. As commonly accepted, the pseudotime should be evaluated based on the average pressure within the region of investigation.…”

Novel Mathematical Model for Transient Pressure Analysis of Multifractured Horizontal Wells in Naturally Fractured Oil Reservoirs

“…The equation captures the feature of pressure sensitivity of hydraulic fractures, which has been widely used in the field. − The permeability modulus, independent of pressure or pressure difference, reflects the effect of pressure sensitivity, ranging from 10 –4 to 10 –3 psi –1 . Accordingly, this work focusses on forecasting the permeability of hydraulic fractures with the pressure change and analyze typical flow behaviors on the basis of the permeability modulus of hydraulic fractures.…”

Perturbation Solution for One-Dimensional Flow to a Constant-Pressure Boundary in a Stress-Sensitive Reservoir

“…Unfortunately, accounting for the variation of permeability with pore fluid pressure renders the governing pressure diffusion equation nonlinear and thus not amenable to exact analytical solutions. This type of nonlinear pressure diffusion problem is often attacked by using the pseudopressure (Raghavan et al 1972;Tabatabaie et al 2017), which is essentially a Kirchhoff transformation. This has the effect of slightly "lessening" the effect of the nonlinearity, by moving the nonlinearity from the second-order space-derivative term, to the first-order time-derivative term.…”