Journal of Petroleum Engineering

Volume 2015, Article ID 214084, 16 pages

http://dx.doi.org/10.1155/2015/214084

## A New Approach in Pressure Transient Analysis: Using Numerical Density Derivatives to Improve Diagnosis of Flow Regimes and Estimation of Reservoir Properties for Multiple Phase Flow

London South Bank University, UK

Received 5 April 2015; Accepted 7 June 2015

Academic Editor: Alireza Bahadori

Copyright © 2015 Victor Torkiowei Biu and Shi-Yi Zheng. 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

This paper presents the numerical density derivative approach (another phase of numerical welltesting) in which each fluid’s densities around the wellbore are measured and used to generate pressure equivalent for each phase using simplified pressure-density correlation, as well as new statistical derivative methods to determine each fluid phase’s permeabilities, and the average effective permeability for the system with a new empirical model. Also density related radial flow equations for each fluid phase are derived and semilog specialised plot of density versus Horner time is used to estimate relative to each phase. Results from 2 examples of oil and gas condensate reservoirs show that the derivatives of the fluid phase pressure-densities equivalent display the same wellbore and reservoir fingerprint as the conventional bottom-hole pressure BPR method. It also indicates that the average effective ranges between 43 and 57 mD for scenarios (a) to (d) in Example 1.0 and 404 mD for scenarios (a) to (b) in Example 2.0 using the new fluid phase empirical model for estimation. This is within the value used in the simulation model and likewise that estimated from the conventional BPR method. Results also discovered that in all six scenarios investigated, the heavier fluid such as water and the weighted average pressure-density equivalent of all fluid gives exact effective as the conventional BPR method. This approach provides an estimate of the possible fluid phase permeabilities and the % of each phase contribution to flow at a given point. Hence, at several stabilisation points, the relative can be generated.

#### 1. Introduction

Several sets of well and reservoir models have been generated with pressure derivatives with different boundary conditions. Likewise, several type curves, which account for different combinations of wellbore, reservoir characteristics, and boundary effects with associated flow regimes for computation of well and reservoir parameters, have been used to simplify well test interpretation. This demonstrates that the log-log plot of the pressure derivative is a powerful tool for reservoir model identification in pressure transient analysis.

However, in practice, each current method of transient data analysis has its own strengths and limitations with no single pressure and production data analysis method capable of handling all types of data and reservoir types with clear reliable results [1]. The log derivative and derivative type curve, which have remained reference flow regime’s diagnostic tools for over four decades, are the only unified approach for welltest interpretation and are applicable in a wide range of situations.

The derivative method, which is the greatest breakthrough in welltest analysis, was first introduced by Tiab in 1976 [2, 3] and developed by French mathematician Dominique Bourdet in 1983 [4]. It has remained the reference solution for identifying flow regime, boundary response, and use for diagnosing complex reservoir features till date. This approach has helped to reduce the uncertainties surrounding the interpretation of welltest data because key regions of radial flow and boundary features have been adequately diagnosed. However, due to the nonunique solution of the mathematical fluid flow equation, mostly in heterogeneous reservoir, most engineers in the industry are compelled to use analytical model and type curve solutions to match complex model, which is oftentimes not realistic. Assumptions made are ignored while pursuing a perfect match and results obtained from this approach are often misleading [5].

This marked the beginning of numerical well testing in the industry by Zheng, 2006 [5], although the approach started from the early 1990s [6–10]. Zheng made more advances in 2006, providing more solutions to the nonunique problems mostly in heterogeneous reservoirs through numerical welltesting, thereby promoting its application. More papers have been published by researchers on the subject, thereby reflecting the advancement of numerical welltesting and its application in solving various reservoir engineering practical problems.

One of the main limitations of the pressure derivative is that the measured pressure data must be constructed into derivative data, by means of numerical differentiation. Oftentimes derivative data from real field are very noisy and difficult to interpret, resulting in various smoothing techniques developed by researchers on this subject. It is practically believed that smoothing of pressure derivative data often alters the characteristics of the data. Also, it is difficult to distinguish between fluid and reservoir fingerprints in critical saturated reservoirs.

Another limitation of these derivatives is diagnosing flow regimes in complex reservoir structures such as complex faulted systems and high permeability streak with interbedded shales, which is common in deep water turbidite systems, channel-levee, lobe, and channelized deposits. Also, in situations of multiphase flow around the wellbore, the derivative data are always noisy and difficult to interpret, resulting in the application of deconvolution and various smoothing techniques to obtain a perceived representative model which often might not be. Additionally, the analytical solution for transient pressure analysis is limited to single phase flow, which in real case is never the situation. Presently there are few literatures or research on multiphase transient pressure analysis. However, the combination of the new statistical approach [11] and the density derivative approach serves as a support tool for better interpretation and estimation of reservoir properties in these conditions.

The diagnosis of flow, which appears as distinctive patterns in the pressure derivative curve, is a vital point in welltest interpretations since each flow regime reflects the geometry of the flow streamlines in the tested formation. Hence, for each flow regime identified, a set of well and/or reservoir parameters can be estimated using the region of the transient data that exhibits the characteristic pattern behaviour [11]. In the study, the pressure derivative formulation from Horne (1995) [12] and the new statistical approach by Biu and Zheng (2015) [11] would be used throughout the analysis.

The mathematical formulation for pressure derivative by Horne (1995) [12] is given asAlso the mathematical formulation for the new statistical derivative approach by Biu and Zheng (2015) [11] is given as follows.

*Model 1. *Consider the following:

*Model 2 (the exponential function). *Consider the following:

*Model 3. *Consider the following:

*Model 4 (the time function). *Consider the following:Equations (1) to (5) are the derivative and statistical models used for flow regime diagnosis, behaviours, and estimation of wellbore and reservoir properties using the log-log derivative plot.

#### 2. Theoretical Concept of the Density Derivatives

The basic concepts involved in the derivation of fluid flow equation include(i)conservation of mass equation,(ii)transport rate equation (e.g., Darcy’s law),(iii)equation of state.Consider flow in a cylindrical coordinate with flow but with flow in angular and -directions neglected as shown in Figure 1; the equations are given as follows:Equation (6) represents the conservation of mass. Since the fluid is moving, the equationis applied. By conserving mass in an elemental control volume as shown in Figure 1 and applying transport rate equation, the following equation is obtained:Expand the equation using Taylor series:Equations (6) to (9) apply to both liquid and gas. Equation (9) is known as the general diffusivity equation and for each fluid; the density or pressure term in (9) can be replaced by the correct expression in terms of density or pressure.