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Summary Finding multiple posterior realizations through a reservoir history-matching procedure is of significance for uncertainty quantification, risk analysis, and decision making in the course of reservoir closed-loop management. Previously, an efficient, distributed, optimization algorithm—global linear-regression (GLR) with distributed Gauss-Newton (GLR-DGN)—has been proposed in the literature to iteratively minimize multiple objective functions by performing Gauss-Newton (GN) optimizations concurrently while dynamically sharing information between dispersed regions in a reduced parameter space. However, theoretically, the required number of initial training models should be larger than the number of parameters to guarantee a unique solution for the GLR equation for sensitivity-matrix estimation. This limitation makes the large-scale reservoir history-matching problem with a large amount of parameters almost intractable. We enrich the previous history-matching framework by integrating our recently proposed smooth local parameterization (SLP) and DGN for the sensitivity-matrix calculation. Motivated by the fact that one specific flow response mainly depends on a few influential or local parameters, which can be generally identified by the physical position of the wells (e.g., the parameters in the zone surrounding the wells), which is particularly true for large-scale reservoir models, this paper presents a new integration of subdomain linear-regression (SLR) with DGN, referred to as SLR-DGN. This SLP allows us to independently represent the globally spatial parameter field within low-order parameter subspaces in each subdomain. On the basis of the SLP procedure, only a few training models are required to compute local sensitivity of the response functions using a subdomain linear regression. SLP is a linear transformation with smoothness and differentiability, which makes it particularly compatible with Newton-like gradient-based optimization algorithms. Furthermore, we also introduce an adaptive scheme, named weighting smooth local parameterization (WSLP), in which the minimization algorithm adaptively determines the weighting coefficients and the optimal domain decomposition (DD) correspondingly, to mitigate the negative effects of an inappropriate DD strategy. We support our framework with numerical experiments for a four-variable toy model and a modified version of the sensitivity analysis of the impact of geological uncertainties on production (SAIGUP) model with spatially dependent parameters. Comparisons with previous GLR-DGN have shown that our new framework can generate comparable and even better results with significantly reduced computational cost. This SLP has high scalability, because the number of training models depends primarily on the number of local parameters in each subdomain and not on the dimension of the underlying full-order model. Activating more subdomains results in fewer local parameter patterns and enables us to run fewer training models. For a large-scale case study in this work, to optimize 412 global parameters, SLR-DGN needs only 100 initial-model simulations. In comparison to GLR-DGN where the parameters are defined over the entire domain, the central-processing-unit cost is reduced by a factor of several orders of magnitude, while retaining reasonable accuracy.

Based on fractal theory (FT) and fractional calculus (FC), a new fractally fractional diffusion model (FFDM) of composite dual-porosity has been developed to evaluate performance of multiple fractured horizontal wells (MFHWs) with stimulated reservoir volume (SRV) in tight gas reservoirs (TGRs). More specifically, FT is used to characterize the complex and heterogeneous fracture network (FN) both inside and outside of SRV, while anomalous behavior of diffusion processes both inside and outside of SRV is quantified by applying the temporal fractional derivatives. The FFDM is then solved by the Laplace transformation, line source function, the numerical discrete method, and superposition principle. The transient pressure responses are then inversely converted from Laplace domain into real time domain with the Stehfest algorithm, and the FFDM is also validated, and type curves are generated as well. Flow stages are subsequently identified together with analysis on characteristics of the type curves, especially the anomalous features different with those generated from the conventional Euclidean model. Sensitivity analyses of some related parameters have also been discussed as well. And the FFDM is then also matched with the real field well-testing data of a MFHW with SRV in a TGR. The proposed FFDM provides a new understanding of the performance of MFHWs with SRV in TGRs, which can be used to interpret the field pressure data more accurately and appropriately.

In this paper, a generalized and pragmatic framework has been developed to optimize the vector well pattern in tight sandstone reservoirs. More specifically, the maximum principal formation stress, natural fracture orientation, and principal permeability orientation are identified as the three key factors associated with well productivity, while both micro-elements of fractures are summed up and coefficients of the same fracture are combined according to the angle between the principal permeability orientation and direction of the hydraulic fractures to form an equation matrix. The sensitivity analysis is then conducted to examine effect of the maximum horizontal principal stress and the principal permeability on horizontal well production while determining the optimal orientation of the horizontal wellbore. Subsequently, theoretical calculation is performed to design and optimize the well pattern as well as fracture properties. It is found that the orientation of a horizontal wellbore should be perpendicular to the direction of the maximum horizontal principal stress if the angle (θ) between the orientation of the principal permeability k x and the direction of the maximum horizontal principal stress is less than 30 ° . If 30 ° ≤ θ ≤ 60 ° , the orientation of the horizontal wellbore should be perpendicular to the direction of the maximum horizontal principal stress for a weak anisotropic reservoir k x / k y 2 , while it should be perpendicular to the orientation of principal permeability k x for a strong anisotropic reservoir k x / k y ≥ 2 . The orientation of a horizontal wellbore needs to be perpendicular to the orientation of the principal permeability (kx) if 60 ° ≤ θ ≤ 90 ° . In addition to improving the recovery factor compared with that of the conventional well spacing, it is found from a field application that the optimized vector well pattern can effectively extend the stable production period, reduce water-cut, increase oil recovery up to 4.5%, and improve the overall reservoir performance.

Carbon dioxide miscible flooding has been proven to be one of the most effective enhanced oil recovery (EOR) technologies, particularly for light and medium oil reservoirs. However, specific effects of pore structure on CO2 miscible flooding recovery in low permeability reservoir lack in-depth understandings. In this paper, pore structures are specifically studied by means of the molecular mechanics to evaluate their effects on the CO2 EOR in the low permeability reservoir. First, a series lab experiments are performed for the pore and fluid characterization. More specifically, the pore throat size and distribution frequency are measured from the high-pressure mercury injection and nuclear magnetic resonance. The minimum miscibility pressure is determined from the slim-tube tests with known oil compositions tested from gas chromatography analysis. Second, the regularity of CO2 extraction is explored on the basis of molecular mechanics and the thickness of raffinate is calculated. Finally, the raffinate volume and recovery ratio in the pores are calculated after the CO2 miscible flooding. The results show that a raffinate-layer with thickness of 0.13 μm remains on the surface of the pore after the CO2 miscible flooding, which would cause the oil to be immobile since the throat could be blocked when the throat radius is smaller than 0.26 μm. The recoveries of cores C-1 and C-2 are 70.1 % and 61.4 % from calculations and 68.4 % and 59.8 % from experiments, whose errors are 2.5 % and 2.7 %, respectively. This study would be beneficial to analyze the CO2 miscible flooding in reservoirs with different pore structures and provide technical support for improving CO2 utilization efficiency.

Summary Recently, a multiwell-pad-production (MWPP) scheme has been the center of attention as a promising technology to improve shale-gas (SG) recovery. However, the increasing possibility of multiwell pressure interference (MWPI) in the MWPP scheme severely distorts flow regimes, which strongly challenges the traditional pressure-transient analysis methods that focus on single multifractured horizontal wells (SMFHWs) without MWPI. Therefore, a methodology to identify pressure-transient response of the MWPP scheme with and without MWPI is urgent. To fill this gap, a new semianalytical pressure-transient model of the MWPP scheme is established by use of superposition theory, Gauss elimination, and the Stehfest numerical algorithm. Type curves are generated, and flow regimes are identified by considering MWPI. Finally, a sensitivity analysis is conducted. Our results show that there are good agreements between our proposed model and numerical simulation; moreover, our semianalytical approach also demonstrates a promising calculation speed compared with numerical simulation. Some expected flow regimes are significantly distorted by MWPI. In addition, well rate determines the distortion of pressure curves, whereas fracture length, well spacing, and fracture spacing determine when the MWPI occurs. The smaller the gas rate, the more severely flow regimes are distorted. As the well spacing increases, fracture length decreases, fracture spacing decreases, and the occurrence of MWPI occurs later. The stress-sensitivity coefficient has little to no influence on the occurrence of MWPI. Similar to the concept of the dual-porosity model, three new flow regimes—the single-well flow regime, MWPI flow regime, and MWPP flow regime—are artificially defined to systematically characterize the flow regimes of the MWPP scheme. This work offers some additional insights on pressure-transient response for the MWPP scheme in the SG reservoir, which can provide considerable guidance for fracture-properties estimation and well-pattern optimization for the MWPP scheme.

Summary The detection of interacting behavior between the hydraulic fracture (HF) and the natural fracture (NF) is of significance to accurately and efficiently characterize an underground complex-fracture system induced by hydraulic-fracturing technology. This work develops a semianalytical pressure-transient model in the Laplace domain to detect interacting behavior between HF and NF depending on pressure-transient characteristics. Our results have shown that no matter what the flow state (compressible or incompressible flow) within a hydraulically induced fracture system, we can easily detect interacting behavior between HF and NF depending on whether the “dip” shape occurs at the formation radial-flow regime. Referring to sensitivity analysis, distance between NF and well, horizontal distance between NF and HF, and NF length are the three most sensitive factors to detect fracture-interacting behavior. Depending on interference analysis, although the pressure-transient characteristics of a pseudosteady-state dual-porosity model can interfere with our proposed methodology, the difference between our model and a pseudosteady-state dual-porosity system lies in whether the value of the horizontal line of dimensionless pressure derivative is equal to 0.5 at the formation radial-flow regime. Our work obtains some innovative insights into detecting fracture-interacting behavior, and the valuable results can provide significant guidance for refracturing operations and fracture detection in an underground fracture system.

Abandonment pressure of shale gas reservoirs has been in the center of attention to evaluate the CO 2 storage in such reservoirs. However, the evaluation of abandonment pressure in shale gas reservoirs is not clearly defined. To quantitatively determine the abandonment pressure for CO 2 storage in depleted shale gas reservoirs, we introduced two effective and reasonable semi-analytical models, named pressure transient analysis (PTA) and rate transient analysis (RTA), based on a free-simulator approach. Firstly, by line source function, CO 2 seepage models for an injection well with constant injection rate or constant injection pressure were simultaneously proposed in consideration of gas adsorption, diffusion flow, stress sensitivity and non-Darcy flow effects. Then, we solved seepage models using Laplace transformation, perturbation method, superposition principle and numerical discretization method, thus obtained transient-pressure and transient-rate solutions of the injection well. Further, we compared the proposed mathematical models with existing models. Furthermore, we systematically examined the sensitivity of the parameters, thereby constraining the intrinsic uncertainty to evaluate the abandonment pressure. Finally, we applied the proposed method to study a real case from Sichuan Basin in China, thus demonstrating the promising prosperity of this new quantitative determination of abandonment pressure for CO 2 storage in depleted shale gas reservoirs. Our results showed that the maximum difference between our proposed method and numerical method was lower than 5%, suggesting a good agreement between the two methods. Moreover, our results also showed that free-simulator approach gave much faster calculation speed compared to numerical method. As a result, we provided a framework for the quantitative determination of abandonment pressure, by coupling pressure transient analysis (PTA) and rate transient analysis (RTA) during CO 2 injection at constant injection pressure and injection rate. Moreover, our results also showed that the abandonment pressure has power law relationship with the injection pressure using PTA method, and the abandonment pressure has logarithm relationship with CO 2 injection volume using RTA method.

Shale gas reservoir is typical unconventional reservoir, it's necessary to take advantage of multi-stage fractured horizontal well so as to develop those kinds of reservoirs, which can form high conductivity hydraulic fractures and activate natural fractures. Due to the existence of concentration gap between matrix and fractures, desorption gas can simultaneously diffuse into the natural fractures and hydraulic fractures. This process can be called dual diffusion. Based on the triple-porosity cubic model, this paper establishes a new well testing model of multi-stage fractured horizontal well in shale gas reservoir with consideration of the unique mechanisms of desorption and dual diffusion in matrix. Laplace transformation is employed to solve this new model. The pseudo pressure transient responses are inverted into real time space with stehfest numerical inversion algorithm. Type curves are plotted, and different flow regimes in shale gas reservoirs are identified and the effects of relevant parameters are analyzed as well. Considering the mechanism of dual diffusion in matrix, the flow can be divided into five regimes: early linear flow; pseudo-steady state inter-porosity flow; the diffusion from matrix into micro-fractures; the diffusion from matrix into hydraulic fractures and boundary-dominated flow. There are large distinctions of pressure response between pseudo steady state diffusion and unsteady state diffusion under different value of pore volume ratio. It's similar to the feature of pseudo-steady state inter-porosity flow, diffusion coefficient and Langmuir parameters reflect the characters of pseudo-steady state diffusion. The numbers of stage of hydraulic fractures have certain impact on the shape factor of matrix and the inter-porosity coefficient. This new model is validated compared with some existing models. Finally, coupled with an application, this mew model can be approximately reliable and make some more precise productivity prediction.

Currently, most models for multiple fractured horizontal wells (MFHWs) in naturally fractured unconventional reservoirs (NFURs) are based on classical Euclidean models which implicitly assume a uniform distribution of natural fractures and that all fractures are homogeneous. While fractal theory provides a powerful method to describe the disorder, heterogeneity, uncertainty and complexity of the NFURs. In this paper, a fractally fractional diffusion model (FFDM) for MFHWs in NFURs is established based on fractal theory and fractional calculus. Particularly, fractal theory is used to describe the heterogeneous, complex fracture network, with consideration of anomalous behavior of diffusion process in NFURs by employing fractional calculus. The Laplace transformation, line source function, dispersion method, and superposition principle are used to solve this new model. The pressure responses in the real time domain are obtained with Stehfest numerical inversion algorithms. The type curves of MFHW with three different outer boundaries are plotted. Sensitivity analysis of some related parameters are discussed as well. This new model provides the relatively more accurate and appropriate evaluation results for pressure transient analysis for MFHWs in NFURs, which could be applied to accurately interpret the real pressure data of an MFHW in field.