## 1 Introduction  
History matching is defined as the act of adjusting a model of a reservoir until it closely reproduces the past behavior of a reservoir [1]. Using decline curve analysis (DCA) to history match and estimate future production with high accuracy is challenging. The empirical approach of DCA has many limitations, specifically the need for several years of production to be able to estimate ultimate recovery with high certainty. The objective of any model that estimates reserves is basically to mitigate the risks in estimating future production and help build a sound economic analysis for a given reservoir. Achieving an accurate reserve estimation without the need of a significant period of production data is truly challenging. The common goal of all these production history matching and forecasting methods is to estimate reserves available for production by utilizing and understanding the available data. DCA has been used to forecast production since its development by Arps in 1945 [2]. In unconventional reservoirs (ultra-low permeability reservoirs), using the classic Arpsexponential, harmonic, and hyperbolic DCA equations is unrealistic and leads to substantial risk of overstating reserves. The main reason behind this is the fact that the Arps model is developed for reservoirs undergoing boundary-dominated flow (BDF), which is nearly unreachable as a flow behavior in ultra-low permeability reservoirs. The dominant flow behavior throughout much of--and possibly the entire life of--an ultra-low permeability well is transient flow. Several decline curve models have been introduced to accommodate and fit the production trends from unconventional wells. A very common method is the Arps hyperbolic decline with a best-fit b-value. When attempting to find the best-fit b-value, the results sometimes require b to be greater than 1. The hyperbolic decline exponent, b, in the Arps decline curve equation has a physical meaning in reservoir engineering [3] and should be between 0 and 1. In general, b is believed to be constant but in the transient flow regime period, b tends to change with time. The Arps hyperbolic decline model is a typical Arps decline model with a b-value estimated using a \"best fit\" to accommodate the changing value of b during the transient period and find an overall best fit that usually results in a b-value more than 1. [PERSON] and [PERSON] (2010) showed that b > 1 gives physically impossible results when the Arps cumulative production equation is evaluated at infinite time. Nonetheless, the Arps cumulative production equation gives acceptable results when finite values of producing time or economic rate limits are used. In applying this decline equation, there has to be an economic or time limit in order to observe physically possible results of cumulative production [4]. A commonly used decline model is the stretched exponential production decline (SEPD) model introduced by [PERSON] in 2009. For the physical basis of the SEPD, it can be described as the integral of all exponential decays, with a \"fat-tailed\" probability distribution of the time constants. The model attempts to mimic heterogeneity by describing the production-decline behavior of the reservoir as a group of exponential declines from a number of contributing volumes, with a specific distribution of time constraints and where discharge of fluid from a tank against a fixed backpressure results in an exponential decline of flow rate versus time. Oil or gas reservoirs are regarded as a collection of connected tanks (cells) discharging against different backpressures and with different resistances or \"time constants\". Natural interpretation of stretched exponential decay is generated by a sum (integral) of pure exponential decays, and the stretched exponential is the resulting model [5]. The main advantage of this decline model is its ability to fit the transient period of the production very well compared to other methods [6].