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Summary Physics-based and empirical rate-time models inherently assume constant bottomhole flowing pressure (BHP), an assumption that may not hold for many unconventional wells. Hence, applying these models without accounting for BHP variations might lead to inaccurate (a) flow regime identification, (b) estimation of the parameters of these models, and (c) estimated ultimate recovery (EUR) and drainage volumes. This study evaluates and compares the predictions of rate-time relations including and ignoring corrections for time-varying BHP for both synthetic and shale gas wells. We generate a real gas synthetic case with errors in the time-varying BHP. First, we convert pressures into pseudopressures. Second, we deconvolve the pseudopressure history by applying the regularized exponential basis function inverse scheme to obtain an equivalent rate—the unit-pseudopressure-drop rate at standard conditions—at constant BHP. Third, we history match the production using the scaled single-phase compressible fluid physics-based model for three different approaches: (a) using rate-time-pressure data with rate-pseudopressure deconvolution, (b) using rate-time-pressure data using just rate-pressure deconvolution, and (c) using only rate-time data. Finally, we compare the results in terms of their history matches and estimated reservoir parameters. We conclude by illustrating the application of this procedure to shale gas wells. For the synthetic case, the fit of the single-phase compressible fluid rate-time model using rate-pseudopressure deconvolution can accurately estimate the original gas in place, characteristic time, gas permeability, and fracture half-length. In contrast, considerable errors are noted when either using rate-pressure deconvolution or failing to account for variable BHP. Regarding the shale gas examples, the rate-pseudopressure deconvolution scheme accurately identifies the flow regimes present in the well, which can be difficult to detect by only analyzing rate-time data. For this reason, the fits of the scaled single-phase compressible fluid model using only rate-time result in unreasonably large estimates of the reservoir parameters and EUR. In contrast, the application of rate-pseudopressure deconvolution constrains the fits of the single-phase compressible fluid model yielding more realistic estimates of the time of end of transient flow, and EUR. This paper illustrates the application of a workflow that accounts for variable BHP by estimating an equivalent constant unit-pseudopressure-drop gas rate (at standard conditions). We illustrate the workflow for a particular decline-curve model, but the workflow is general and can be applied to any rate-time model. The approach history matches and forecasts the production of unconventional gas reservoirs using rate-time models more accurately than assuming constant BHP.

Summary Extrapolation of history matched single-phase flow solutions is a common practice to forecast production from tight-oil reservoirs. Nonetheless, this approach (a) omits multiphase flow effects that exist below bubblepoint conditions and (b) has not included the quantification of the uncertainty of the estimated ultimate recovery (EUR). This work combines a new two-phase (oil and gas) flow solution within a Bayesian framework to address the uncertainty in the EUR. We illustrate the application of the procedure to tight-oil wells of west Texas. First, we combine the oil and the gas flow equations into a single dimensionless two-phase flow equation. The solution is a dimensionless flow rate model that can be easily scaled using two parameters: hydrocarbon pore volume and characteristic time. Second, this study generates the probabilistic production forecasts using a Bayesian approach in which the parameters of the model are treated as random variables. We construct parallel Markov chains of the parameters using an adaptative Metropolis-Hastings (M-H) Markov chain Monte Carlo (MCMC) for this purpose. Third, we evaluate the robustness of our inferences using posterior predictive checks (PPCs). Finally, we quantify the uncertainty in the EUR percentiles using the bootstrap method. The results of this research are as follows. First, this work shows that EUR estimates based on single-phase flow solutions will consistently underestimate the ultimate oil recovery factors in solution-gas drives where the reservoir pressure is less than the bubblepoint. The degree of underestimation will depend on the reservoir and flowing conditions as well as the fluid properties. Second, the application of parallel Markov chains using an adaptative M-H MCMC scheme that addresses the correlation between the model’s parameters solves the issues of mixing and autocorrelation of Markov chains and, thus, it speeds up speeding up the convergence of the Markov chains. Third, we generate replicated data from our posterior distributions to assess the robustness of our inferences (PPCs). Finally, we use hindcasting to calibrate and strengthen our inferences. To our knowledge, all these approaches are novel in EUR forecasting. Using a Bayesian framework with a low-dimensional (two-parameter) physics-based model provides a fast and reliable technique to quantify the uncertainty in production forecasts. In addition, the use of parallel chains with an adaptative M-H MCMC accelerates the rate of convergence and increases the robustness of the method.

This paper describes scenario generators and portfolio models for oil and gas exploration and production (E&P) planning. The problem is how to allocate funds to a set of oil recovery projects to achieve a desired tradeoff between risk and return. A mix of development and exploration projects is assumed, further stratified by reservoir depth and uncertainty level of reserves. The scenario generator incorporates distributions for seven uncertain reservoir parameters, obtained from engineering and geologic data in the Tertiary Oil Recovery Information System (TORIS) database (http://www.netl.doe.gov/scngo/Petroleum/Software/database.html). Capital and operating expenditure data comes from the IHS Energy Discovery Module (www.ihsenergy.com). A slightly compressible-liquid tank model is used to model the yearly oil production from each well in each project. Oil price trajectories are generated using a mean reverting model. A complex income and tax calculation evaluates the net present value (NPV) of free cash flow, and a Monte Carlo simulation generates NPV scenarios for each project. We describe and compare optimization models that minimize several risk measures, including variance, semivariance, expected loss, and loss probability, as well as models that maximize expected utility. Risk measure values, optimal project weights (assuming any level of fractional participation is allowed in all projects), and portfolio NPV distributions arising from the various risk measures are compared, and contrasted with results of widely used ranking procedures that ignore risk.

The first part of this paper presents a new procedure for constructing analytical solutions of multicomponent two-phase flow problems in one-dimensional homogeneous penneable media, provided the corresponding simpler single-phase problem can be solved. An explanation of its mathematical basis is given.

Considering flooding of stratified reservoirs, this paper focus on the case with significant gravity effects. Assuming the fluids are incompressible, we develop a new VerticaI Equilibrium (VE) theory, and introduce pseudo-functions to describe the displacement problem. From the form of the pseudorized equations we conclude that previously proposed analytical methods for assessing flood performance may lead to erroneous results due to incorrect treatment of gravity. We derive travelling wave solutions for the pseudorized problem and compare an example case with numerical simulations. We show that the travelling wave solution can be formulated as a new generalized Dietz solution which accounts for reservoir stratifications and capillary effects. Previously published interface models are obtained in the limit of zero capillary pressure. We derive several results for stationary displacement, and introduce a graphical method for assessing the vertical sweep efficiency under these circumstances.