Geofluids

Volume 2019, Article ID 1309042, 9 pages

https://doi.org/10.1155/2019/1309042

## A Method of Differentiating the Early-Time and Late-Time Behavior in Pressure-Pulse Decay Permeametry

^{1}School of Civil Engineering, Guizhou University, Guiyang 550025, Guizhou, China^{2}School of Civil Engineering, Chongqing University, Chongqing 400045, China

Correspondence should be addressed to Chaolin Wang; nc.ude.uqc@50010615102

Received 22 October 2018; Accepted 18 February 2019; Published 9 April 2019

Academic Editor: Craig T. Simmons

Copyright © 2019 Yu Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The pressure-pulse decay is a preferred technique for determining permeability of unconventional gas reservoir rocks. The pressure-pulse decay often shows quite different characteristics during the early time and the later time. Most approaches for estimating the permeability proposed in the literature are required to use the later-time pressure-pulse decay measurements. However, the later-time data are often selected subjectively, lacking a universal criterion. In this paper, a method of differentiating the early-time and late-time behavior for pressure-pulse decay test is proposed. The analytical results show that the critical time (dimensionless time) of early-/late-time decay characteristics mainly depends on the volume ratios, and it increases first and then decreases with the volume ratios. The critical time for cases with same chamber sizes is much less than that for cases with unequal chamber sizes. Applicability of the proposed methods is examined using a numerical simulator, TOUGH+REALGASBRINE. The numerical results show that the pressure gradient along the sample varies nonlinearly at the early time and becomes a constant at the late time. Then, the proposed method is applied to real data for permeability estimation. It is found that the early-time behavior is negligible as the volume ratio takes on small values. As the volume ratios increase, the deviation becomes significant and considerable permeability errors will be produced if these early-time data are used.

#### 1. Introduction

Permeability is typically considered the critical parameter for commercial gas production [1–3], geological storage facilities for CO_{2} [4], and radioactive waste storage [5]. Quantitative description and modeling of single-phase flow or multiphase flow in porous media require accurate estimation of permeability and the relationship between capillary pressure, saturation, and permeability [6–8]. The pressure-pulse decay is a widely used method for permeability measurement of tight rocks. It is initially proposed by Brace et al. [9] for the determination of permeability in tight rocks. This technique is based on the analysis of the differential pressure between the upstream and downstream circuits within a sample. Note that Brace’s solution assumed constant pressure gradient in the sample and ignored compressive storage effect. Hsieh et al. [10] and Dicker and Smits [11] extended Brace’s model to include compressive storage effect in their analytical solutions of the problem. Jones [12] introduced a factor “” to simplify Dicker and Smits’s solution, which was further developed by Cui et al. [13] by taking consideration of sorption effects. Moreover, multiple optimized methodologies have been developed based on the conventional pulse decay technique. Lasseux et al. [14] suggested a step decay method that enables accurate, simultaneous characterization of the Klinkenberg-corrected permeability, Klinkenberg coefficient, and porosity from gas permeability experiments. Yang et al. [15] presented a modified pressure-pulse decay method for determining permeabilities of tight reservoir cores. In their method, only one gas chamber is applied at one end of the test core sample, while the other end of the core is sealed. Feng et al. [16] proposed a new experimental design by creating dual pressure pulses to avoid the pressure disturbance due to compressive storage and adsoprtion/desorption effect during the course of measurement. Hannon [17] provided a bidirectional model of pressure-pulse decay permeametry, which reaches equilibrium more than 7.5 times faster than the standard pressure-pulse decay. The pressure decay will deviate from the exponential behavior in the early-time period, and considerable error may result if early-time data are used for permeability calculation [18, 19]. As a result, most of the authors [12, 15, 19–21] used the late-time measurements for permeability calculation. On the other hand, the early-time measurements may contain important information about the rock samples. For example, Kamath et al. [22] employed early-time data to estimate heterogeneity in the sample and Zhao et al. [23] utilized early-time data to analyze the nonlinear flow behavior. However, the time point differentiating the early-time and late-time behavior was not clearly defined and was often somewhat arbitrary in literature. The objective of this study is to develop a physically sound and mathematically accurate method for differentiating the early-time and late-time behavior.

#### 2. Method for Differentiating the Early-Time and Late-Time Behavior

##### 2.1. Early-Time and Late-Time Behavior

The governing equation for fluid flow in the core sample using the pressure-pulse decay method can be specified as follows [9, 11]:with initial and boundary conditions:where denotes the distance along the sample; is real time; refers to sample length; , , , and refer to fluid compressibility, fluid viscosity, porosity, and permeability, respectively; , , and refer to the volume of sample pore, upstream chamber, and downstream chamber, respectively; and and are the pressure of the upstream chamber and downstream chamber, respectively.

Brace et al. [9] assumed that the pressure gradient was constant along the length of the sample (i.e., ) and presented a simplified solution asin whichwhere is the final pressure when .

The general analytical solution of equation (1) in terms of dimensionless differential pressure,

was presented by Dicker and Smits [11]:where are the roots of equation (7); and are the ratios of pore volume to upstream and downstream, respectively; and is dimensionless time, defined in equation (8):

The analytical solution of equation (6) includes an early-time behavior and a late-time behavior. In the early-time period, the effect of the downstream boundary condition is not felt. Although the early-time data can be used to investigate heterogeneity in core samples [22], more information than can be obtained from flow tests along is required. Therefore, the late-time solution, dominated by the first term in equation (6), is recommended for routine permeability calculation [12, 19].

Jones [12] simplified equation (6) into a similar form as Brace et al.’s by introducing a factor “” as

Therefore, the late-time solution for predicting permeability was then given as

Brace’s solution of equation (4) can be treated as a particular late-time solution of Dicker and Smits’s for .

Dicker and Smits [11] present convincing reasons that the volume of upstream and downstream chambers should be equal. On one hand, equal volumes could keep the mean pore pressure constant. A desirable situation of symmetry can be created by making upstream and downstream volumes identical, which indicates that any pressure decreases in the upstream chamber can be offset by an equal downstream pressure increase. On the other hand, equal chamber volumes make equation (6) become a single exponential quickly, since all even terms of equation (6) cancel when . Figure 1 presents the relationship between dimensionless pressure and time obtained from the general analytical solution (equation (6)) for . The dashed lines and the solid lines refer to the first term and all terms in the summation (equation (6)), respectively. The first term (i.e., the late-time solution), representing the single-exponential behavior of pulse decay curve, overlaps the general solution for almost the entire portion of pressure decay when is small. However, with the increase of volume ratios , the contributions of higher terms to the first term increase, leading to significant deviation of pressure decay curve from single-exponential behavior at the early time. Consequently, Jones [12] and Feng [19] recommend that the late-time pressure decay should be used for permeability calculation. However, neither of them figured out the time point differentiating the early-time and late-time behavior.