Applications of Rate Normalization for Well Performance Analysis (III)
If Mother Nature can lie, it will (Anonymous)
💠Introduction
Part I and Part II of this series introduced the concept of rate normalization as a means to approximate the convolution integral, resulting in a practical plotting function to analyze variable-rate performance and obtain well and reservoir parameters. The focus was on analyzing the PSS regime, which is characterized by a straight line formed by the rate-normalized pressure drop when plotted against the material balance time (tmb). However, while it is rigorously true that PSS will yield a straight line in a Cartesian plot, the converse is not guaranteed. This straight line is a necessary, but not a sufficient, condition❗
Thus, a judicious and systematic approach is required to unambiguously identify not only boundary-dominated conditions but also any other flow regime of interest present in the well response. This article delves into the aspects that are central to tackling this problem, such as
✅Model Identification
✅Flow Regimes
✅ Pressure Derivative
💠Model Identification
The goal of production performance analysis is to determine the properties that describe a system (well + reservoir), denoted as S, by measuring the response or output O of such a system to an input signal I applied to it. The input signal I may be a rate change, in which the resulting response O is a pressure change or vice versa (Figure 1)
A forward model describes how a system S responds to a signal I. Symbolically, I * S = O. However, the case is that only I and O are available, and the objective of the analysis is to uncover S. This is the core of an inverse problem where the parameters of the system are unknown and are to be determined. Unfortunately, often, the solution to an inverse problem is non-unique, with different models producing a response similar to the observed output. The analyst should pin down the most suitable model by analyzing the response in conjunction with geophysical, geological, production, and operational information.
💠Flow Regimes
The selection of a suitable model is linked to the identification of the specific flow regimes associated with it. The shape of the streamlines defines a flow regime and can change according to the prevailing flow geometry, and follows a chronological sequence. These sequences span through early, middle, and late time regions. Well geometry and near-wellbore conditions control the response during the early time. The middle time region relates to reservoir properties. The late time region describes the effect of boundaries on the response, which can be physical, defined by the geological setting, or artificially created by interfering wells. The flow behaviors of interest in practical applications are the following:
▶️Steady State: The pressure does not vary with time, typical of a constant pressure situation triggered by an active aquifer or a gas cap.
▶️Semi Steady State: This flow behavior is observed in a closed or bounded reservoir, absent of any external source of energy, where the pressure disturbance has reached all boundaries, and declines at the same rate independently of position.
▶️Transient State: This condition, also known as infinite or semi-infinite acting, is observed before the semi-steady state develops. The pressure changes with time and location. Among the flow regimes of interest within the transient condition are the radial, spherical, and linear regimes. Figure 2 depicts the radial flow geometry where the streamlines converge at the wellbore; flow is two-dimensional, symmetrical around an axis (the well). The streamlines are straight in a plane, with the cross-section exposed to flow decreasing towards the center of the system, while the isopotentials are concentric. A vertical well, where the pressure disturbance brought about a rate change hasn't reached the boundaries, will exhibit radial flow. Radial flow is synonymous with infinite acting conditions.
As described by Figure 3, in the linear geometry, the streamlines are parallel, with the cross-section exposed to flow staying constant. Likewise, the isopotentials are also parallel to each other. This flow regime is characteristic of fractured vertical wells, multi-fractured horizontal wells, a well in a channel, or in between two faults.
💠Pressure Derivative
The pressure derivative is one of the most relevant concepts in well testing analysis and its counterpart rate-transient analysis, in terms of model identification. Tiab (1980) first introduced the use of pressure derivative into well testing analysis, as an aid in the interpretation of interference testing. Bourdet et. al. (1983, 1989) extended the idea of the application of the pressure derivative to flow regime identification by defining the logarithmic pressure derivative function as
In real variables,
.For infinite acting conditions, the solution of the diffusivity equation for a vertical well producing at a constant rate in a homogeneous reservoir is given by the line-source solution (Matthews, 1967, Earlougher, R., 1977):
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The application of the pressure derivative results in the following,
Thus, during infinite acting conditions, a plot of the pressure derivative against time results in a plateau. Note that the skin factor disappears. Likewise, for boundary conditions, the pressure derivative is given by (Horne, 1990)
The previous expression implies that a log-log plot of the pressure derivative will result in a unit-slope straight line: this is the distinctive identifying feature of the PSS regime. Figure 4 shows a log-log plot of the pressure response for a vertical well at the center of a circle for different reds (dimensionless external radius). The logarithmic derivative is flat during infinite-acting conditions, which last until tda = 0.1 after which it develops a unit-slope line corresponding to boundary-dominated conditions. The Cartesian derivative, also known as the primary pressure derivative (dPd/dtd), shows a negative unit slope during transient conditions and a plateau during PSS regime. Note that the pressure derivative transitions from radial flow into PSS in less than 0.1 log-cycles, as opposed to the pressure response (Pd), which takes about two log-cycles to reach a unit-slope trend.
In summary, the geometrical shape, or fingerprint, of the pressure derivative function is flow-regime specific. Once a flow regime is recognized, specialized analyses in the form of semi-log and cartesian analysis are conducted with the purpose of estimating relevant well and reservoir parameters (Ehligh-Economides, 1988).
💠Derivative Analysis and Normalization
So far, the derivative analysis has been based on the constant rate solution. The concept and application of the former can be readily extended to a variable rate analysis by noting that a generalized superposition derivative can be defined for multi-rate schedule (Stewart, 2011)
The application of the derivative definition to the rate-normalized pressure drop expression for boundary-dominated conditions, with Δte as tmb, results in
Thus, using tmb as the superposition time function, a log-log plot of ΔP yields a unit-slope straight line. The corollary is that data falling on the unit-slope line will form a straight line in a Cartesian plot. Moreover, during infinite acting radial flow, ΔP is flat. Note that tmb is not the rigorous superposition time function for radial flow; however, it is approximately correct. Figure 5 shows the log-log representation of the rate-normalized pressure drop along with its corresponding derivative; the analysis is analogous to that shown in Figure 4 for the constant rate scenario
💠Application
The log-log analysis of the field case presented in Part I is shown in Figure 6. The derivative is noisy, typical of field data. A unit-slope line (red) can be drawn through the derivative (red squares) after 400 hrs, approximately: this data corresponds to boundary-dominated conditions or PSS; a Cartesian plot of ΔP/q against tmb should produce a straight line. The Cartesian or primary derivative shows a plateau for the data that follows the unit-slope line. Also note that the ΔP/q and its derivative merge after 1000 hrs. Before 300 hrs, the data is in transient regime; an apparent plateau forms between 100 and 200 hrs.
💠Takeaways
✅ Production performance analysis is achieved by defining the proper well-reservoir model, based on the observations (pressures and rates); this is an inverse model.
✅ Model recognition is based on identifying prevailing flow regimes; this is achieved by the pressure derivative
✅ The definitive fingerprint of PSS is a unit-slope line on the tmb-derivative plot. This, not the Cartesian plot, should be the primary diagnostic check
The next article in the series will delve into different schemes that can be used to perform the differentiation process, as well as other useful derivative definitions
💠References