Methods of Decline Curve Analysis for Shale Gas Reservoirs

## Abstract:

With help from horizontal wells and hydraulic fracturing, shale gas has made a significant contribution to the energy supply. However, due to complex fracture networks and complicated mechanisms such as gas desorption and gas slippage in shale, forecasting shale gas production is a challenging task. Despite the versatility of many simulation methods including analytical models, semi-analytical models, and numerical simulation, Decline Curve Analysis has the advantages of simplicity and efficiency for hydrocarbon production forecasting.

#### Authors:

#### Lei Tan^{1}, Lihua Zuo^{2}, and Binbin Wang^{3}

^{1}State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China. ^{2}Department of Petroleum Engineering, Texas A&M University, College Station, TX 77843, USA. ^{3}Geochemical & Environmental Research Group, Texas A&M University, College Station, TX 77845, USA

These authors contributed equally to this work.

Received: 2 January 2018 / Accepted: 28 February 2018 / Published: 5 March 2018

In this article, the eight most popular deterministic decline curve methods are reviewed: Arps, Logistic Growth Model, Power Law Exponential Model, Stretched Exponential Model, Duong Model, Extended Exponential Decline Model, and Fractural Decline Curve model. This review article is dedicated to summarizing the origins, derivations, assumptions, and limitations of these eight decline curve models. This review article also describes the current status of decline curve analysis methods, which provides a comprehensive and up-to-date list of Decline Curve Analysis models for petroleum engineers in analysis of shale gas reservoirs. This work could serve as a guideline for petroleum engineers to determine which Decline Curve models should be applied to different shale gas fields and production periods.

## Introduction

Shale gas has become an important energy production component due to the availability of horizontal well drilling and hydraulic fracturing techniques since the late 1980s. Figure 1a shows that currently natural gas accounts for one-quarter of the world energy supply , and Figure 1b illustrates that shale gas accounts for about half of the natural gas produced in the USA and could increase to 70% in 2040, as predicted by the Energy Information Agency of the United States.

**Figure 1.** The percentage of shale gas in total world energy consumption. (a) World energy consumption map; (b) natural gas production map (Modified after [1,2]).

Shale formations are characterized by natural fractures (Figure 2a), low-permeability rocks, and nanopores associated with organic matter, clay minerals, carbonate, and various silts (Figure 2b). In shale, gas content is mainly composed of free gas in the nanopores and adsorbed gas in the rocks and other components such as carbon dioxide, nitrogen, and hydrogen sulfide. The co-existence of these components with different size and properties has posed a major challenge to investigating shale.

**Figure 2.** Shale samples. (a) Outcrop of natural fractures in Woodford Shale, Oklahoma, USA [3]; (b) High-resolution images of Montery shale, California, USA [4]. A, B, C/D are the scan images under different resolutions for the same shale core sample. A, B, and C are for the same core sample, under different resolutions, 10μm, 3 μm and 2 μm respectively. D is for another core sample under 2 μm resolution. Arrows in B and C mark pores less than 80 nm. Arrows in D mark carbonate micrite.

To simulate gas production in shale, complex gas transport mechanisms and the presence of natural and hydraulic fractures need to be considered, which makes the simulation of shale gas reservoirs a challenging task. There are many gas transport mechanisms coexisting during the gas production process, such as gas diffusion, gas slippage, and gas desorption. The complexity of these mechanisms makes classical free gas diffusivity equations inadequate to model the gas transport in shale. Existing simulation models for gas transport in shale have not fully captured the complex mechanism, and a new comprehensive and reliable gas transport model still needs to be constructed and studied [5,6,7].

With meticulously measured petrophysical information [8,9,10,11], the analytical, semi-analytical, or numerical solutions based on an accurate reservoir model could lead to reasonable production forecast. However, it is expensive and time-consuming to perform petrophysical experiments in order to build such a sophisticated reservoir model, because a lot of geological and petrophysics data need to be gathered and analyzed before being implemented into the reservoir model.

For instance, test wells need to be drilled, and multiple well logging processes need to be performed in order to obtain the necessary input (e.g., formation location, depth, resistivity, permeability, fracture properties, gas properties) for the reservoir model. These processes are expensive and time-consuming, and some logging techniques are not sensitive enough to measure the multi-scale properties in shale. In contrast, DCA only requires the production data, and little reservoir information is needed. DCA is efficient with adequate accuracy to meet industrial needs. In the past few decades, several DCA models have been proposed and benchmarked with commercial reservoir simulators or shale gas production data before being applied to more shale gas reservoirs.

In this work, eight most popular DCA models for shale gas reservoirs will be reviewed. These models start from the classical Arps model and end with the new Fractional Decline Curve model. There have been previous review works or comparison works for different DCA models, such as Kanfar and Wattenbarger [12], Joshi and Lee [13], Lee [14], and Hu et al. [15], but there are a few new models proposed during the past several years and there is a need to include new applications of these DCA models in an updated review. This work will provide the most current theory and applications of DCA models in shale gas reservoirs.

## DCA Models and Applications

### Origins of DCA

In essence, DCA models are regressions for historical production data. When the production is controlled, the application of DCA is limited because the production rates are mostly constant. As the exploration and production techniques are further developed, there have been more capacity wells drilled; the production rates of these wells usually show a rapid drop at the early stage, then the decreasing rate of the production slows down at late production time. At late production time, these wells require more sophisticated DCA models to simulate the production decline trend.

Ralph Arnold is one of the pioneers who first applied data analysis techniques to estimate the oil reserves. The earliest reference to DCA was made by Arnold and Anderson [16], who assumed that the production rate at any time is a constant fraction of the rates at the preceding time. In Arnold [17], the author states that “Twenty years ago the estimate of oil reserves was computed generally by calculating the contents of the supposed reservoir rock from data regarding thickness, extent, etc., guessing at the saturation and percentage of recoverable oil, and finally arriving at a very rough approximation of the desired information.

Nowadays, thanks to the great mass of data available and to the perfection of methods of computation, more accurate results are obtained. If the production of a given well over a reasonable period (even a period of days in some instances) is known, the future production of the well by years and in totality can be computed with remarkable accuracy, and thus arrive at the recoverable reserve for this well and its surrounding area.” Arnold did not specifically use the terminology “Decline Curve Analysis,” but “great mass of data” and “methods of computation” are the exact input data for DCA. The main idea introduced in Arnold [17] is exactly what DCA does to forecast the production. Since then, there have been more applications of DCA models for production forecasting [18,19]. For a complete review of early DCA work, refer to [20], where the author reviewed all the major DCA models developed before 1945.

### Arps Decline Model

The classical DCA model was proposed in Arps [20], where the author proposed a hyperbolic function with three parameters to simulate the decline of flow rate. In the Arps model, bottomhole pressure is fixed, the skin factor is constant, and the flow regime is boundary dominated flow. To derive the DCA model, the concept of loss ratio was first introduced. The loss ratio is defined as the ratio between the production drop of the current time step and that of the previous time step. Based on observations, Arps proposed two different scenarios of loss ratio. The first scenario is to assume that the loss ratio is a constant,

where b is a positive constant. Integrating Equation (1), we get the exponential decline functions as follows:

where q i is the initial rate, in bbl/day.

The second scenario is to assume that “first differences of the loss ratios are approximately constant”, i.e.,

The double integration of Equation (3) allows us to obtain the rate–time relationship for hyperbolic decline:

where a i is the initial loss ratio.

Based on this decline model, the curve has a slope − 1/b on a log–log paper. During the fitting process, these parameters can be determined by calculating the derivatives of production data with respect to time.

Although Arps model is simple and fast, it often fails to accurately fit the decline curve of unconventional reservoirs and predict the estimated ultimate recovery (EUR) [21]. The Arps model often tends to overestimate the EUR for shale gas wells because it assumes that a boundary-dominated flow regime prevails [22,23]. Since most shale gas wells rarely reach the boundary-dominated flow regime, the Arps model cannot be applied directly to shale gas reservoirs without significant modifications [24]. Following the Arps decline curve, various new models were proposed to better model the boundary-dominated flow phase, and will be presented below.

### Modified Hyperbolic Decline Model

Arps model was originally designed for pseudo-radial boundary-dominated flows (BDF), which occur in medium–high-permeability formations. However, for tight-gas shale reservoirs, the fracture-dominated flow regimes dominate and the late-time-flow regime is rarely reached. Meanwhile, the Arps decline model leads to an unbounded and unrealistic estimate of EUR for b ≥ 1 [25].

To resolve this issue, Equations (2) and (4) need to be applied in a piecewise fashion, which means using different DCA curves for different production time intervals. The hyperbolic decline function, Equation (4), is applied in the early stage; after the decline rate reaches a certain value, the exponential decline function, Equation (2), is used. This process could be achieved by applying computer programs to determine the switch point, which is the point when the decline rate is smaller than a certain limit (5% is often used).

To generalize Equations (2) and (4) in one expression so that there is no need to change formulas for different time intervals, Robertson [26] proposed the following Modified Hyperbolic Decline (MHD) model:

where a , β , and N are all positive constant. Note that when β → 1 , Equation (5) is equivalent to Equation (4); when N → ∞ , Equation (5) is equivalent to Equation (2) ([26]).

Another equivalent way to express the modified hyperbolic model was proposed in Seshadri and Mattar [27], where hyperbolic decline in the early life of a well is shifted to exponential decline in the late life by imposing a predetermined decline rate, D * . Once the decline rate reaches D*, the decline curve switches from a hyperbolic function to an exponential function. The equivalent MHD model is written as

where q_{i} is the initial rate, in bbl/day, and D_{1} and D_{2} are decline rates.

Meyet et al. [28] pointed out that the selection of the decline rate D * can be empirically determined and has no physical basis. The authors also showed that the EUR forecasts by MHD are often greater than those of other DCA methods. Paryani et al. [29] also showed that the MHD model provides over-optimistic estimates of reserves and longer remaining life of shale oil wells.

### Power Law Exponential Decline Model

As mentioned in Section 2.2, Arps [20] introduced the so-called “loss-ratio” as in Equation (1). When b is a constant, the exponential decline, Equation (2), is obtained. In Ilk et al. [30], the authors proposed another approach to formulate parameter b :

where D_{∞} , D_{1} are the decline constant at infinite time and initial time, and n^{^} is the time exponent.

The physical interpretation of Equation (7) is that the loss ratio can be approximated by a decaying power function with constant behavior at late production time. By substituting Equation (7) into Equation (1) and integrating, the following production rate is derived:

where q^{^}_{i} is the rate “intercept”, D^{^}_{i} is the initial decline constant, D ∞ is the decline constant at infinite time, and n^{^} is the time exponent. In addition, the parameters D and D^{^}_{i} are defined as follows:

The model defined by Equation (8) is called the Power Law Exponential (PLE) Decline Model. It is based on the Arps decline curves and uses the power law to approximate the production rate. This model is developed specifically for shale gas wells.

The PLE model has extra variables to account for both transient and boundary-dominated flows. However, four unknowns in this model, q^{^}_{i} , D^{^}_{i}, D_{∞} , and n^{^} , cause many degrees of freedom while fitting real-world data, which can be impractical. There are many methods to solve the PLE model, and multiple solutions are acceptable. For example, the interpolation algorithms could be written using regression techniques like a classical least squares method.

If D_{∞} = 0 , Equation (8) usually overestimates the flow rate. Equation (8) is reduced to the following form:

However, Equation (10) only holds true during transient flows [31], which could be determined by using the derivate plot of the flow rate where the slope is −1. Therefore, attention is needed to select appropriate data when using Equation (10).