Current models used to forecast production in unconventional oil and gas formations are often not producing valid results. When traditional decline curve analysis models are used in shale formations, Arps b-values greater than 1 are commonly obtained, and these values yield infinite cumulative production, which is non-physical.. Additional methods have been developed to prevent the unrealistic values produced, like truncating hyperbolic declines with exponential declines when a minimum production rate is reached. Truncating a hyperbolic decline with an exponential decline solves some of the problems associated with decline curve analysis, but it is not an ideal solution. The exponential decline rate used is arbitrary, and the value picked greatly effects the results of the forecast. A new empirical model has been developed and used as an alternative to traditional decline curve analysis with the Arps equation. The new model is based on the concept of logistic growth models. Logistic growth models were originally developed in the 1830s by Belgian mathematician, Pierre Verhulst, to model population growth. The new logistic model for production forecasting in ultra-tight reservoirs uses the concept of a carrying capacity. The carrying capacity provides the maximum recoverable oil or gas from a single well, and it causes all forecasts produced with this model to be within a reasonable range of known volumetrically available oil. Additionally the carrying capacity causes the production rate forecast to eventually terminate as the cumulative production approaches the carrying capacity. The new model provides a more realistic method for forecasting reserves in unconventional formations than the traditional Arps model. The typical problems encountered when using conventional decline curve analysis are not present when using the logistic model. Predictions of the future are always difficult and often subject to factors such as operating conditions, which can never be predicted. The logistic growth model is well established, robust, and flexible. It provides a method to forecast reserves, which has been shown to accurately trend to existing production data and provide a realistic forecast based on known hydrocarbon volumes. / text

Neste trabalho de tese, estudamos o modelo de crescimento logístico de populações biológicas utilizando a abordagem de espaço de estados. Os estados não observados são as biomassas anuais, a equação de observação é linear e a equação de estado é não linear. As distribuições de probabilidade utilizadas para os termos de erro de observação aditivos são: Normal, t-student, Skew-normal e Skew-t. As distribuições Log-normal, Log-t, Log-skew-normal e Log-skew-t são consideradas para os erros de observação multiplicativos. A inferência nos modelos é realizada considerando-se métodos Bayesianos e as distribuições a posterior de interesse são aproximadas utilizando-se algoritmos MCMC e a aproximação de Laplace. Apresentamos duas aplicações, a primeira referente a pesca de camarão marinho na costa do Chile, na qual a variável observável é o rendimento médio anual de pesca (captura por unidade de esforço média). Na segunda é considerada a pesca de lagostim vermelho na costa de Chile, na qual além do rendimento médio anual da pesca, observa-se as estimativas anuais de biomassa vulnerável, obtidas através de estudos de área varrida. Para o primeiro conjunto de dados, os modelos com erros de observação multiplicativos têm melhor performance, particularmente os modelos Log-skew-normal e Log-skew-t. Considerando estes resultados, no segundo caso utilizamos somente erros multiplicativos e a distribuição a posteriori preditiva mostra que cada variável observável parece ter sua própria família de distribuição de probabilidades. Além disso, os resultados também revelam uma crescente complexidade do modelo ao incorporar a classe mais geral de distribuições assimétricas. / We study the logistic population growth model using a state-space approach. The non observable states are the annual biomass of the population with a linear observation equation and a non-linear state equation. The probability distribution used for the additives observation error terms are Normal, Student-t, Skew-normal and Skew-t, and Log-normal, Log-t, Log-skew-normal and Log-skew-t for multiplicative observation errors terms. The inference about the parameters of the models is performed using Bayesian methods, with MCMC algorithms and Laplace approximations. We present two applications to real data sets. The first in marine shrimp population off the coast of Chile, where observable variable is the average annual fishing yield. The second application is for the population of the red squat lobster off Chile, where in addition to the average annual fishing yield, a second observable variable was included. In the first case, the multiplicative observational errors models presented the best results. Particularly the Log-skew-normal and Log-skew-t models has the better performances. Considering these results, in the second application we use only multiplicative observation errors models.

Neste trabalho de tese, estudamos o modelo de crescimento logístico de populações biológicas utilizando a abordagem de espaço de estados. Os estados não observados são as biomassas anuais, a equação de observação é linear e a equação de estado é não linear. As distribuições de probabilidade utilizadas para os termos de erro de observação aditivos são: Normal, t-student, Skew-normal e Skew-t. As distribuições Log-normal, Log-t, Log-skew-normal e Log-skew-t são consideradas para os erros de observação multiplicativos. A inferência nos modelos é realizada considerando-se métodos Bayesianos e as distribuições a posterior de interesse são aproximadas utilizando-se algoritmos MCMC e a aproximação de Laplace. Apresentamos duas aplicações, a primeira referente a pesca de camarão marinho na costa do Chile, na qual a variável observável é o rendimento médio anual de pesca (captura por unidade de esforço média). Na segunda é considerada a pesca de lagostim vermelho na costa de Chile, na qual além do rendimento médio anual da pesca, observa-se as estimativas anuais de biomassa vulnerável, obtidas através de estudos de área varrida. Para o primeiro conjunto de dados, os modelos com erros de observação multiplicativos têm melhor performance, particularmente os modelos Log-skew-normal e Log-skew-t. Considerando estes resultados, no segundo caso utilizamos somente erros multiplicativos e a distribuição a posteriori preditiva mostra que cada variável observável parece ter sua própria família de distribuição de probabilidades. Além disso, os resultados também revelam uma crescente complexidade do modelo ao incorporar a classe mais geral de distribuições assimétricas. / We study the logistic population growth model using a state-space approach. The non observable states are the annual biomass of the population with a linear observation equation and a non-linear state equation. The probability distribution used for the additives observation error terms are Normal, Student-t, Skew-normal and Skew-t, and Log-normal, Log-t, Log-skew-normal and Log-skew-t for multiplicative observation errors terms. The inference about the parameters of the models is performed using Bayesian methods, with MCMC algorithms and Laplace approximations. We present two applications to real data sets. The first in marine shrimp population off the coast of Chile, where observable variable is the average annual fishing yield. The second application is for the population of the red squat lobster off Chile, where in addition to the average annual fishing yield, a second observable variable was included. In the first case, the multiplicative observational errors models presented the best results. Particularly the Log-skew-normal and Log-skew-t models has the better performances. Considering these results, in the second application we use only multiplicative observation errors models.

This study compares four recently developed decline curve methods and the traditional Arps or Fetkovich approach. The four methods which are empirically formulated for shale and tight gas wells are: 1. Power Law Exponential Decline (PLE). 2. Stretched Exponential Decline (SEPD). 3. Duong Method. 4. Logistic Growth Model (LGM). Each method has different tuning parameters and equation forms. The main objective of this work is to determine the best method(s) in terms of Estimated Ultimate Recovery (EUR) accuracy, goodness of fit, and ease of matching. In addition, these methods are compared against each other at different production times in order to understand the effect of production time on forecasts. As a part of validation process, all methods are benchmarked against simulation. This study compares the decline methods to four simulation cases which represent the common shale declines observed in the field. Shale wells, which are completed with horizontal wells and multiple traverse highly-conductive hydraulic fractures, exhibit long transient linear flow. Based on certain models, linear flow is preceded by bilinear flow if natural fractures are present. In addition to this, linear flow is succeeded by Boundary Dominated Flow (BDF) decline when pressure wave reaches boundary. This means four declines are possible, hence four simulation cases are required for comparison. To facilitate automatic data fitting, a non-linear regression program was developed using excel VBA. The program optimizes the Least-Square (LS) objective function to find the best fit. The used optimization algorithm is the Levenberg-Marquardt Algorithm (LMA) and it is used because of its robustness and ease of use. This work shows that all methods forecast different EURs and some fit certain simulation cases better than others. In addition, no method can forecast EUR accurately without reaching BDF. Using this work, engineers can choose the best method to forecast EUR after identifying the simulation case that is most analogous to their field wells. The VBA program and the matching procedure presented here can help engineers automate these methods into their forecasting sheets.