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Albert C. Reynolds

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Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*26 (04): 1590–1613.

Paper Number: SPE-204236-PA

Published: 11 August 2021

Abstract

Summary In the context of production optimization, we consider the general problem of finding the well controls that maximize the net present value (NPV) of life-cycle production, where the well controls are either the bottomhole pressure (BHP) or a rate (oil, gas, water, or total liquid) at each well on a set of specified control steps (time intervals), with the limitations on surface facility considered as nonlinear-state constraints [e.g., field-liquid-production rates (FLRs), field-water-production rates (FWRs), and/or field-gas-production rates]. If the reservoir simulation used for reservoir management has sufficient adjoint capability to compute gradients of the objective function and all state constraints, we show that one can develop a significantly more computationally efficient procedure by replacing the adjoint-enhanced reservoir simulator by a proxy model and optimizing the proxy. Our methodology achieves computational efficiency by generating a set of output values of the cost and constraint functions and their associated derivative values by running the reservoir simulator for a broad set of input design variables (well controls) and then using the set of input/output data to train a proxy model to replace the reservoir simulator when computing values of cost and constraint functions and their derivatives during iterations of sequential quadratic programming (SQP). The derivation of the equations for computing the proxy-based model that uses both function and gradient information is similar to that of least-squares support vector regression (LS-SVR). However, this method is referred to as gradient-enhanced support vector regression (GE-SVR) because, unlike LS-SVR, the method uses derivative information, not just function values, to train the proxy. Similar to LS-SVR, improved (higher) estimated optimal NPV values can be obtained by using iterative resampling (IR). With IR, after each proxy-based optimization, one evaluates the cost and constraint functions and their derivatives at the estimated optimal controls using reservoir-simulator output, and then adds this new input/output information to the training set to update the proxy models for predicting NPV and constraints. Using the updated proxies, one applies SQP optimization again. IR continues until the simulator and proxy evaluated at the latest estimate of the optimal well controls give the same value of NPV within a specified percentage tolerance and the constraints evaluated by reservoir simulator at the latest optimal well controls are such that the constraints are satisfied within some small specified tolerance. Our results indicate that proxy-based optimization with iterative resampling might require up to an order of magnitude less computational time than pure reservoir-simulator-based optimization. By comparing the results generated with an LS-SVR proxy with the GE-SVR results, we find that GE-SVR is roughly an order of magnitude more computationally efficient than LS-SVR but also provides a better approximation of a complex cost-function surface so that it is possible to locate multiple optima in cases where LS-SVR fails to identify the multiple optima.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Improved Oil Recovery Conference, August 31–September 4, 2020

Paper Number: SPE-200360-MS

Abstract

The main objective of this work is to investigate efficient estimation of the optimal design variables that maximize net present value (NPV) for the life-cycle production optimization during a single-well CO 2 huff-n-puff (HnP) process in unconventional oil reservoirs. During optimization, the NPV is calculated by a machine learning (ML) proxy model trained to accurately approximate the NPV that would be calculated from a reservoir simulator run. The ML proxy model can be obtained with either least-squares support vector regression (LS-SVR) or Gaussian process regression (GPR). Given forward simulation results with a commercial compositional simulator that simulates miscible CO 2 HnP process in a simple hydraulically fractured unconventional reservoir model with a set of design variables, a proxy is built based on the ML method chosen. Then, the optimal design variables are found by maximizing the NPV based on using the proxy as a forward model to calculate NPV in an iterative optimization and training process. The sequential quadratic programming (SQP) method is used to optimize design variables. Design variables considered in this process are CO 2 injection rate, production BHP, duration of injection time period, and duration of production time period for each cycle. We apply proxy-based optimization methods to and compare their performance on several synthetic single-well hydraulically fractured horizontal well models based on Bakken oil-shale fluid composition. Our results show that the LS-SVR and GPR based proxy models prove to be accurate and useful in approximating NPV in optimization of the CO 2 HnP process. The results also indicate that both the GPR and LS-SVR methods exhibit very similar convergence rates and require similar computational time for optimization. Both ML based methods prove to be quite efficient in production optimization, saving significant computational times (at least 5 times more efficient) than using a stochastic gradient computed from a high fidelity compositional simulator directly in a gradient ascent algorithm. The novelty in this work is the use of optimization techniques to find optimum design variables, and to apply optimization process fast and efficient for the complex CO 2 HnP EOR process which requires compositional flow simulation in hydraulically fractured unconventional oil reservoirs.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*25 (04): 1938–1963.

Paper Number: SPE-193925-PA

Published: 13 August 2020

Abstract

Summary Solving a large‐scale optimization problem with nonlinear state constraints is challenging when adjoint gradients are not available for computing the derivatives needed in the basic optimization algorithm used. Here, we present a methodology for the solution of an optimization problem with nonlinear and linear constraints, where the true gradients that cannot be computed analytically are approximated by ensemble‐based stochastic gradients using an improved stochastic simplex approximate gradient (StoSAG). Our discussion is focused on the application of our procedure to waterflooding optimization where the optimization variables are the well controls and the cost function is the life‐cycle net present value (NPV) of production. The optimization algorithm used for solving the constrained‐optimization problem is sequential quadratic programming (SQP) with constraints enforced using the filter method. We introduce modifications to StoSAG that improve its fidelity [i.e., the improvements give a more accurate approximation to the true gradient (assumed here to equal the gradient computed with the adjoint method) than the approximation obtained using the original StoSAG algorithm]. The modifications to StoSAG vastly improve the performance of the optimization algorithm; in fact, we show that if the basic StoSAG is applied without the improvements, then the SQP might yield a highly suboptimal result for optimization problems with nonlinear state constraints. For robust optimization, each constraint should be satisfied for every reservoir model, which is highly computationally intensive. However, the computationally viable alternative of letting the reservoir simulation enforce the nonlinear state constraints using its internal heuristics yields significantly inferior results. Thus, we develop an alternative procedure for handling nonlinear state constraints, which avoids explicit enforcement of nonlinear constraints for each reservoir model yet yields results where any constraint violation for any model is extremely small.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*25 (01): 139–161.

Paper Number: SPE-191480-PA

Published: 17 February 2020

Abstract

Summary A common pitfall in probabilistic history matching is omitting the local variation of spatial uncertainties and falsely generalizing the learning from local data to the entire field. This can lead to radical overestimation of uncertainty reduction and bad reservoir‐management decisions. In this paper, we propose a methodology to quantify and correct for the error that arises from the omission of local variation in probabilistic history matching. Most performance metrics in an oil field, such as the original oil in place (OOIP) and the estimated ultimate recovery (EUR), are field‐scale objective functions that depend on properties (e.g., porosity) over the entire field. On the other hand, many measurement data from wells [e.g., bottomhole pressure (BHP)] are mainly sensitive to the reservoir properties near the locations where they are measured, and thus they are susceptible to local variations of reservoir properties. Calibrating field‐scale objective functions to local well data without properly characterizing the local variation can overestimate the uncertainty reduction of field‐scale objective functions. In this paper, we derived formulas to quantify errors in the posterior cumulative distribution functions (CDFs) of the objective functions resulting from the omission of local variation. We also provide a way to correct for the error and to recover the true posterior CDFs. Through theoretical derivation, we show that the modeling error that arises from the omission of local variation is dependent on the magnitude of the global and local variations of the uncertain properties (e.g., porosity). The larger the local variation relative to the global variation, the larger the error in the estimated posterior distributions. The error also depends on the variogram of the local variation and the detection range of the data. The error is larger for cases with a long variogram for the local variation and a short data‐detection range. In addition, the modeling errors for different measurement data points can be highly correlated even when the measurement errors for these data are independent. To correct for this modeling error, analytical and empirical formulas are proposed that have been shown to greatly improve the accuracy of the posterior distributions in a number of cases. To the best of our knowledge, this is the first time that the modeling error from the omission of local variation in the probabilistic history‐matching process has been quantified and corrected. The methodology proposed could help improve the reliability of the result from probabilistic history matching.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*25 (01): 001–036.

Paper Number: SPE-182684-PA

Published: 17 February 2020

Abstract

Summary Generating an estimate of uncertainty in production forecasts has become nearly standard in the oil industry, but is often performed with procedures that yield at best a highly approximate uncertainty quantification. Formally, the uncertainty quantification of a production forecast can be achieved by generating a correct characterization of the posterior probability‐density function (PDF) of reservoir‐model parameters conditional to dynamic data and then sampling this PDF correctly. Although Markov‐chain Monte Carlo (MCMC) provides a theoretically rigorous method for sampling any target PDF that is known up to a normalizing constant, in reservoir‐engineering applications, researchers have found that it might require extraordinarily long chains containing millions to hundreds of millions of states to obtain a correct characterization of the target PDF. When the target PDF has a single mode or has multiple modes concentrated in a small region, it might be possible to implement a proposal distribution dependent on a random walk so that the resulting MCMC algorithm derived from the Metropolis‐Hastings acceptance probability can yield a good characterization of the posterior PDF with a computationally feasible chain length. However, for a high‐dimensional multimodal PDF with modes separated by large regions of low or zero probability, characterizing the PDF with MCMC using a random walk is not computationally feasible. Although methods such as population MCMC exist for characterizing a multimodal PDF, their computational cost generally makes the application of these algorithms far too costly for field application. In this paper, we design a new proposal distribution using a Gaussian mixture PDF for use in MCMC where the posterior PDF can be multimodal with the modes spread far apart. Simply put, the method generates modes using a gradient‐based optimization method and constructs a Gaussian mixture model (GMM) to use as the basic proposal distribution. Tests on three simple problems are presented to establish the validity of the method. The performance of the new MCMC algorithm is compared with that of random‐walk MCMC and is also compared with that of population MCMC for a target PDF that is multimodal.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Europec featured at 81st EAGE Conference and Exhibition, June 3–6, 2019

Paper Number: SPE-195471-MS

Abstract

In the petroleum industry, well testing is a common practice that consists of wellbore pressure, temperature and flow rates data acquisition to estimate parameters that govern the flow in porous media. Injection-falloff testing is particularly important for offshore reservoirs, especially for the oil reserves that contain high carbon dioxide and sulfur content. In this environment, a conventional well test in an exploratory well should not be run in order to avoid discarding high concentrations of these gases to the atmosphere. Therefore, there is a need for developing techniques for analyzing pressure data from injection-falloff tests. In this work, we have developed an approximate semi-analytical solution for wellbore pressure response during gas injection and falloff well tests in reservoirs containing oil and gas with complex composition by applying the Thompson and Reynolds steady-state theory. For the injection period, we first determine the overall concentrations distributions from a system of hyperbolic conservation equations using the method of characteristics (MOC), assuming a one-dimensional homogeneous reservoir with incompressible fluids and constant molar density, and neglecting capillary, gravity effects, volume changing on mixing and diffusion. During the falloff stage, it is assumed that there is no phase nor concentration movement in the reservoir, which is reasonable as we neglect capillary pressure, diffusion, gravity force and fluid compressibilities. Once we have the concentration profiles in the reservoir, we can calculate the total mobility distributions and then integrate the pressure gradient given by Darcy's law to find the wellbore pressure response. The semi-analytical approximate solution obtained was validated against the commercial numerical simulator STARS from CMG. After validation, the developed model was used as a forward model to estimate absolute permeability and skin factor by history matching noisy data obtained from the numerical simulator mentioned.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Reservoir Simulation Conference, April 10–11, 2019

Paper Number: SPE-193841-MS

Abstract

We previously published a two-dimensional data-driven model (INSIM-FT) for history matching waterflooding production data and to identify flow barriers and regions of high connectivity between injector-producer pairs. This two-dimensional INSIM model assumed vertical wells. The history-matched models can be used for prediction of waterflooding performance and life-cycle waterflooding optimization. The INSIM-FT-3D model presented here extends INSIM-FT to three dimensions, considers gravity and enables the use of arbitrary well trajectories. INSIM-FT-3D places nodes at each well perforation and then adds nodes throughout the reservoir. Flow occurs through "streamtubes" between each pair of connected nodes. Mitchell's best candidate algorithm is used to place nodes and a three-dimensional (3D) connection map is generated with Delaunay triangulation. Pressures and saturations at nodes, respectively, are obtained from IMPES-like pressure equations and a Riemann solver that include gravity effects. With history-matched model(s) as the forward model(s), we estimate the optimal well controls (pressure or rates at control steps) that maximize the life-cycle net-present-value (NPV) of production under waterflooding using a gradient-based method that employs a stochastic gradient. Two 3D reservoirs are considered to establish the viability of using INSIM-FT-3D history-matched models for waterflooding optimization, a channelized reservoir and the Brugge reservoir. Unlike history-matching and waterflooding optimization based on reservoir simulation models, INSIM-FT-3D is not a detailed geological model. Moreover, the time required to run INSIM-FT-3D is more than one order of magnitude less the cost of running a comparable reservoir simulation model.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Reservoir Simulation Conference, April 10–11, 2019

Paper Number: SPE-193925-MS

Abstract

Solving a large-scale optimization problem with nonlinear state constraints has proven to be challenging when adjoint gradients are not available for computing the derivatives needed in the basic optimization algorithm employed. Here, we present a methodology for the solution of an optimization problem with nonlinear and linear constraints where the true gradients that cannot be computed analytically are approximated by ensemble-based stochastic gradients based on an improved stochastic simplex approximate gradient (StoSAG). For the most part, our discussion is focused on the application of our procedure to waterflooding optimization where the optimization variables are the well controls and the cost function is the life-cycle net present value (NPV) of production. The optimization algorithm used for solving the constrained optimization problem is sequential quadratic programming (SQP) with constraints enforced using the filter method. We introduce modifications to StoSAG that improve its fidelity, i.e., the improvements give a more accurate approximation to the true gradient (assumed here to equal the gradient computed with the adjoint method) than the approximation obtained using the original StoSAG algorithm. The improvements to the basic StoSAG vastly improve the performance of the optimization algorithm; in fact, we show that if the basic StoSAG is applied without the improvements, then SQP may yield a highly suboptimal result for optimization problems than many nonlinear state constraints involve.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Reservoir Simulation Conference, April 10–11, 2019

Paper Number: SPE-193921-MS

Abstract

Multistage hydraulic fracturing of a horizontal well in an unconventional reservoir tends to induce a complex fracture network (CFN) which is challenging to characterize by conventional methods. In this work, we develop a fracture characterization workflow to estimate the geometric configuration and fracture properties of a CFN by assimilating microseismic event data and production data, sequentially. A novel stochastic fractal model, that is consistent with rock physics and outcrop observations, is developed in order to generate realizations of the complex fracture network. In the first stage of the two-stage assisted history matching workflow, we estimate the parameters of the stochastic fractal model (fracture intensity, average fracture length, orientation and fracture distribution) by using a genetic algorithm to history match data for the locations of microseismic events. In the second stage, the production data from the shale reservoir are assimilated by the ES-MDA algorithm to estimate the stimulated reservoir volume (SRV) and its average permeability, fracture permeability, aperture and porosity. In the unconventional shale gas reservoir simulator used as the forward model, large-scale fractures are modeled via the embedded discrete fracture model (EDFM) and a dual-porosity, dual-permeability (DP-DK) model is used for modeling the SRV and small scale fractures. The simulator includes Knudsen diffusion and the Langmuir adsorption/desorption model. For validation, we consider a synthetic shale gas reservoir with a horizontal well that has been stimulated by multistage hydraulic fracturing. A particular realization of the variables that describe the reservoir model is used to generate observed data for microseismic events and production rates. The parameters to be adjusted to match the observed microseismic events are the expected values of the length, orientation and intensity of the distribution of the natural fractures and the fractal pattern. Results show that we obtain good estimates of the expected value of natural fracture length, orientation, intensity and fracture distribution by history matching observations of locations of microseismic events. These estimates provide an updated stochastic fractal model for the configuration of CFN. The history-matched fractal model is used to generate an ensemble of fracture distributions consistent with microseismic data as candidate fracture configurations when estimating fracture properties by matching production data. We obtain much better history matches, future performance predictions, estimates of stimulated reservoir volume and its average permeability and estimates of fracture permeability, porosity and aperture when we match both microseismic and production data than we only match production data. When both seismic and production data are matched for synthetic cases and parameters are properly scaled, the true values of parameters and reservoir performance predictions are within the P25-P75 confidence intervals calculated from the ensemble of history matched models in virtually all cases. In practice, the proper characterization of the CFN and reservoir properties should be useful for placing new wells and designing fracture treatments.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Reservoir Simulation Conference, April 10–11, 2019

Paper Number: SPE-193918-MS

Abstract

Important decisions in the oil industry rely on reservoir simulation predictions. Unfortunately, most of the information available to build the necessary reservoir simulation models are uncertain, and one must quantify how this uncertainty propagates to the reservoir predictions. Recently, ensemble methods based on the Kalman filter have become very popular due to its relatively easy implementation and computational efficiency. However, ensemble methods based on the Kalman filter are developed based on an assumption of a linear relationship between reservoir parameters and reservoir simulation predictions as well as the assumption that the reservoir parameters follows a Gaussian distribution, and these assumptions do not hold for most practical applications. When these assumptions do not hold, ensemble methods only provide a rough approximation of the posterior probability density functions (pdf 's) for model parameters and predictions of future reservoir performance. However, in cases where the posterior pdf for the reservoir model parameters conditioned to dynamic observed data can be constructed from Bayes’ theorem, uncertainty quantification can be accomplished by sampling the posterior pdf. The Markov chain Monte Carlos (MCMC) method provides the means to sample the posterior pdf, although with an extremely high computational cost because, for each new state proposed in the Markov chain, the evaluation of the acceptance probability requires one reservoir simulation run. The primary objective of this work is to obtain a reliable least-squares support vector regression (LS-SVR) proxy to replace the reservoir simulator as the forward model when MCMC is used for sampling the posterior pdf of reservoir model parameters in order to characterize the uncertainty in reservoir parameters and future reservoir performance predictions using a practically feasible number of reservoir simulation runs. Application of LS-SVR to history-matching is also investigated.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*23 (06): 2409–2427.

Paper Number: SPE-191378-PA

Published: 08 October 2018

Abstract

Summary We design a new and general work flow for efficient estimation of the optimal well controls for the robust production-optimization problem using support-vector regression (SVR), where the cost function is the net present value (NPV). Given a set of simulation results, an SVR model is built as a proxy to approximate a reservoir-simulation model, and then the estimated optimal controls are found by maximizing NPV using the SVR proxy as the forward model. The gradient of the SVR model can be computed analytically so the steepest-ascent algorithm can easily and efficiently be applied to maximize NPV. Then, the well-control optimization is performed using an SVR model as the forward model with a steepest-ascent algorithm. To the best of our knowledge, this is the first SVR application to the optimal well-control problem. We provide insight and information on proper training of the SVR proxy for life-cycle production optimization. In particular, we develop and implement a new iterative-sampling-refinement algorithm that is designed specifically to promote the accuracy of the SVR model for robust production optimization. One key observation that is important for reservoir optimization is that SVR produces a high-fidelity model near an optimal point, but at points far away, we only need SVR to produce reasonable approximations of the predicting output from the reservoir-simulation model. Because running an SVR model is computationally more efficient than running a full-scale reservoir-simulation model, the large computational cost spent on multiple forward-reservoir-simulation runs for robust optimization is significantly reduced by applying the proposed method. We compare the performance of the proposed method using the SVR runs with the popular stochastic simplex approximate gradient (StoSAG) and reservoir-simulations runs for three synthetic examples, including one field-scale example. We also compare the optimization performance of our proposed method with that obtained from a linear-response-surface model and multiple SVR proxies that are built for each of the geological models.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 24–26, 2018

Paper Number: SPE-191480-MS

Abstract

A common pitfall in history matching is to falsely generalize the learning from local data to the entire field, which can lead to radical over-estimation of uncertainty reduction and bad reservoir management decisions. This problem is referred to as the local-global problem and in this paper a methodology is proposed to quantify and correct for the error arise from this problem. Most performance metrics in an oil field, such as estimated ultimate recovery (EUR), are field-level ob jective functions that depend on properties (e.g., porosity) over the entire field. On the other hand, most measurement data (e.g., BHP) are sensitive only to a local area around the wells and are thus susceptible to local variation of geological properties. Calibrating field-level objective functions and multipliers of global properties (e.g., porosity) to local well data over-estimates the reduction of global uncertainties. In this paper, we derived the formula to quantify error in the calibrated posterior distribution (S-Curve) resulted from the local-global problem, as well as a correction factor to recover the true posterior S-Curve. Through theoretical derivation, it is shown that the model error arise from the local-global problem is dependent on the magnitude of the global and local variation of the uncertain properties (e.g., porosity). The larger the local variation relative to the global variation, the larger the error is in the estimated posterior S-Curve. The error also depends on the variogram of the local variation, and the detection range of the data. The error is larger for case with long variogram for the local variation and short data detection range. In addition, this model error can be highly correlated for different measurement data points even when the measurement error for these data were independent. To address this local-global modeling error, a series of analytical and empirical formula is proposed, which has successfully corrected the error and greatly improve the posterior S-Curve for a series of cases. To the best of our knowledge, this is the first time the error from the local-global problem is quantified and corrected. The methodology proposed could help improve the reliability of the result from probabilistic history matching.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*23 (03): 919–936.

Paper Number: SPE-185824-PA

Published: 04 January 2018

Abstract

Summary We provide analytical solutions for the wellbore pressure during an injection/falloff-test problem under radial-flow conditions in homogeneous porous media where the injected fluid is carbonated water. For both the injection and falloff periods, we assume an isothermal process with thermodynamic equilibrium, a linear adsorption isotherm, and viscosities that depend only on the carbon dioxide (CO 2 ) concentration. We also neglect CO 2 diffusion, gravity effects, and capillarity effects. For the injection period, we first determine the saturation and concentration distributions with time in the reservoir by applying the method of characteristics to solve the appropriate system of hyperbolic conservation equations, where we assume incompressible fluids. In solving for water saturation and CO 2 concentration in water, we neglect the change in water volume caused by the variation of the CO 2 concentration in water. After solving for the saturation and concentration profiles, the pressure solution can be obtained by integrating Darcy's law, from the wellbore radius to infinity, while assuming an infinite-acting reservoir and invoking the Thompson-Reynolds steady-state theory (Thompson and Reynolds 1997b ). Because Darcy's law does not assume incompressible flow, the pressure solution generated does not assume incompressible flow. To obtain an analytical expression for the wellbore pressure, however, we do assume that for injection and falloff, the total flow-rate profile in the reservoir is constant in a region from the wellbore to a radius greater than the radius of the flood front. The region within this radius increases with time and it is referred to as the steady-state region or zone (Thompson and Reynolds 1997b ). During the falloff stage, it is assumed that there is no change in saturation in the reservoir, which is reasonable because we neglect capillary pressure, the gravity force, and fluid compressibilities when determining the saturation profile. Using these assumptions, we generate analytical solutions for a carbonated-water-injection (CWI)/falloff test and compare these solutions with those obtained with a commercial reservoir simulator using very fine spatial grids and very small timesteps. This comparison suggests that the analytical solutions presented can be used reliably to analyze pressure data obtained during CWI/falloff tests.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*23 (02): 367–395.

Paper Number: SPE-182660-PA

Published: 19 December 2017

Abstract

Summary We develop and use a new data-driven model for assisted history matching of production data from a reservoir under waterflood and apply the history-matched model to predict future reservoir performance. Although the model is developed from production data and requires no prior knowledge of rock-property fields, it incorporates far more fundamental physics than that of the popular capacitance–resistance model (CRM). The new model also represents a substantial improvement on an interwell-numerical-simulation model (INSIM) that was presented previously in a paper coauthored by the latter two authors of the current paper. The new model, which is referred to as INSIM-FT, eliminates the three deficiencies of the original data-driven INSIM. The new model uses more interwell connections than INSIM to increase the fidelity of history matching and predictions and replaces the ad hoc computation procedure for computing saturation that is used in INSIM by a theoretically sound front-tracking procedure. Because of the introduction of a front-tracking method for the calculation of saturation, the new model is referred to as INSIM-FT. We compare the performance of CRM, INSIM, and INSIM-FT in two synthetic examples. INSIM-FT is also tested in a field example.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Europec featured at 79th EAGE Conference and Exhibition, June 12–15, 2017

Paper Number: SPE-185824-MS

Abstract

The injection-falloff test problem can be considered as two independent stages, injection and falloff. For the injection period, we first determine the saturation and concentration distributions with time in the reservoir from the appropriate system of hyperbolic conservation equations by assuming a one-dimensional homogeneous medium containing incompressible fluids and applying the method of characteristics (MOC). We assume an isothermal process with thermodynamic equilibrium and a linear adsorption isotherm, neglecting the CO 2 diffusion.Then the pressure solution can be obtained by integrating the expression for the pressure gradient by Darcy's law, from the wellbore radius to infinity while assuming an infinite-acting reservoir. Because Darcy's law does not assume incompressible flow, the pressure solution does not need to assume incompressible flow. In order to obtain an analytical expression for the wellbore pressure, one must assume that for injection and falloff, the total flow rate profile in the reservoir is constant in a region from the wellbore to a radius greater than the radius of the flood front. The region within this radius increases with time and it is referred to as the steady-state region or zone ( Thompson and Reynolds, 1997b ). During the falloff stage, it is assumed that there is no change in saturation in the reservoir, which is reasonable because we neglect capillary pressure, the gravity force and fluid compressibilities when determining the saturation profile. We assume an isothermal process with thermodynamic equilibrium and a linear adsorption isotherm and that viscosities depend only on the CO 2 concentration. We neglect diffusion effects and the volume that CO 2 occupies in the water and oil phases.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Reservoir Simulation Conference, February 20–22, 2017

Paper Number: SPE-182684-MS

Abstract

Generating an estimate of uncertainty in production forecasts has become almost standard in the oil industry but is often done with procedures that yield at best a highly approximate uncertainty quantification. Formally, the uncertainty quantification of a production forecast can be achieved by generating a correct characterization of the posterior probability density function (pdf) of reservoir model parameters conditional to dynamic data and sampling this pdf correctly. While Markov chain Monte Carlo (MCMC) provides a theoretically rigorous method for sampling any target pdf that is known up to a normalizing constant, in reservoir engineering applications, researchers have found that it may require extraordinarily long chains containing millions to hundreds of million of states to obtain a a correct characterization of the target pdf. When the target pdf has a single mode or has multiple modes concentrated in a small region, it is possible to implement a proposal distribution based essentially on random walk so that the resulting Markov chain Monte Carlo algorithm based on the Metropolis-Hastings acceptance probability can yield a good characterization of the posterior pdf. However, such a method may still require the generation of millions of states in the chain in order to obtain a proper sampling of the posterior pdf. For a high-dimensional multimodal pdf with modes separated by large regions of low or zero probability, characterizing the pdf with MCMC based on random walk is not feasible. While methods such as population MCMC exist for characterizing a multimodal pdf, their computational cost generally makes the application of these algorithm far too costly for field application. In this paper, we design a new proposal distribution based on a Gaussian mixture pdf for use in MCMC where the posterior pdf can be a multimodal pdf, possibly with the modes spread far apart. Simply put, the method generates modes using a gradient-based optimization method and constructs a Gaussian mixture model to use as the basic proposal distribution. Tests on three simple problems are presented to establish the validity of the method. The performance of the new MCMC algorithm is compared with random walk MCMC and is also compared with population MCMC for a target pdf which is multimodal.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Reservoir Simulation Conference, February 20–22, 2017

Paper Number: SPE-182660-MS

Abstract

We develop a new data-driven model for the assisted history matching of production data from a reservoir under waterflood. Although the model is developed from production data and requires no prior knowledge of rock property fields, it incorporates far more fundamental physics than that of the popular capacitance-resistance model (CRM). The new model also represents a substantial improvement on an interwell numerical simulation model (INSIM) which was presented previously in a paper co-authored by the last two authors of the current paper. The new model, which is referred to as INSIM-FT, eliminates the three deficiencies of the original INSIM data-driven model. (1) For some complex cases, e.g., when a producer is converted to an injector or when injected water from more than one injector passes through an intermediate well node, the INSIM procedure for calculation of water saturation degrades to an ad hoc calculation which introduces inaccuracies. Our new model uses an accurate front-tracking procedure to calculate water saturation, hence the name INSIM-FT. (2) The original INSIM formulation assumes relative permeabilities are known a priori which defeats the objective of finding a model without requiring knowledge of petrophysical properties; INSIM-FT estimates relative permeabilities by historymatching. (3) Unlike CRM, the original INSIM model does not provide a reasonable characterization of how water from an injector is allocated among producers and thus does not reliably identify large-scale geological features such as faults. INSIM-FT remedies this INSIM deficiency. The reliability of INSIM-FT for history-matching, future reservoir performance prediction and reservoir characterization is validated with two synthetic models, and its performance is compared with that of CRM. Finally, INSIM-FT is applied to a field case.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*21 (06): 2195–2207.

Paper Number: SPE-173214-PA

Published: 14 December 2016

Abstract

Summary Recently, Emerick and Reynolds ( 2012 ) introduced the ensemble smoother with multiple data assimilations (ES-MDA) for assisted history matching. With computational examples, they demonstrated that ES-MDA provides both a better data match and a better quantification of uncertainty than is obtained with the ensemble Kalman filter (EnKF). However, similar to EnKF, ES-MDA can experience near ensemble collapse and results in too many extreme values of rock-property fields for complex problems. These negative effects can be avoided by a judicious choice of the ES-MDA inflation factors, but, before this work, the optimal inflation factors could only be determined by trial and error. Here, we provide two automatic procedures for choosing the inflation factor for the next data-assimilation step adaptively as the history match proceeds. Both methods are motivated by knowledge of regularization procedures—the first is intuitive and heuristical; the second is motivated by existing theory on the regularization of least-squares inverse problems. We illustrate that the adaptive ES-MDA algorithms are superior to the original ES-MDA algorithm by history matching three-phase-flow production data for a complicated synthetic problem in which the reservoir-model parameters include the porosity, horizontal and vertical permeability fields, depths of the initial fluid contacts, and the parameters of power-law permeability curves.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*21 (03): 0786–0798.

Paper Number: SPE-173217-PA

Published: 15 June 2016

Abstract

Summary CO 2 -water-alternating-gas (CO 2 -WAG) flooding generally leads to higher recovery than either continuous CO 2 flooding or waterflooding. Although CO 2 injection increases microscopic displacement efficiency, unless complete miscibility is achieved, suboptimal sweep efficiency may be obtained because of gravity segregation and the channeling of CO 2 through high-permeability zones or by viscous fingering. Alternating water injection with CO 2 injection results in better mobility control and increases sweep efficiency. Water injection also increases pressure that promotes miscibility. However, poorly designed WAG parameters can result in suboptimal WAG performance. In this work, given the number of WAG cycles and the duration of each WAG cycle, we apply a modification of a standard ensemble-based optimization technique to estimate the optimal well controls that maximize life-cycle net present value (NPV). By optimizing the well controls, we implicitly optimize the WAG ratio (volume of water injected divided by the volume of gas injected). We apply the optimization methodology to a synthetic, channelized reservoir. The performances of optimized WAG flooding, optimized waterflooding, and optimized continuous CO 2 flooding are compared. Because of the similarity between WAG and surfactant alternating gas (SAG foam), we also optimize the SAG process and provide a more computationally efficient way to optimize the SAG process with the optimal well controls obtained from WAG as the initial guesses for the optimal controls for SAG.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*21 (06): 2175–2194.

Paper Number: SPE-173213-PA

Published: 02 June 2016

Abstract

Summary We derive and implement an interwell-numerical-simulation model (INSIM), which can be used as a calculation tool to approximate the performance of a reservoir under waterflooding. In INSIM, the reservoir is characterized as a coarse model consisting of a number of interwell control units, where each unit has two specific parameters: transmissibility and control pore volume (PV). By solving the mass-material-balance and front-tracking equations for the control units, the interwell fluid rates and saturations are obtained so that phase-producing rates can be predicted. The ability of INSIM to predict water cut and phase rates is the most important innovation included in INSIM. INSIM is applied to perform history matching and to infer the interwell connectivity and geological characteristics. INSIM has a number of advantages. First, the model parameters estimated from history matching provide a relative characterization of interwell-formation properties. The model can handle changes in the flow directions caused by changing well rates, including shutting in wells or converting producers to injectors, whereas with the common correlation-based interwell-connectivity method, the well interactions are assumed to be fixed. Second, the previous methods, which have similar computational complexity to INSIM, can only provide the total liquid-production rate, whereas with our procedure, we can calculate the oil- and water-flow rates and hence history match water-cut data. Third, because we can calculate the oil- and water-flow rates, our method can be used for waterflooding optimization but with far-less computational effort than with the traditional method by use of a reservoir simulator.

**Includes:**Supplementary Content

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