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Threshold pressure gradient (TPG) is a key parameter determining the pore-scale fluid dynamics. In tight gas reservoirs, both gas and water exist in the porous rock, and the existing water can be divided into irreducible and movable water. However, how movable water saturation will influence TPG has not yet been investigated. Therefore herein, nuclear magnetic resonance (NMR) and high-pressure mercury intrusion (HPMI) experiments were performed to determine pore-scale water distribution, movable water saturation, and pore throat distribution in the core plugs. Subsequently, the air bubble method was used to measure TPG as a function of movable water saturation and permeability inside tight gas core plugs, finding that TPG increased from 0.01 MPa/m to 0.25 MPa/m with the movable saturation increased from 2% to 35%. Finally, a semi-empirical model was derived to describe the correlation between TPG, movable water saturation, and permeability, which performed better than previous models in the literature. These insights will advance the fundamental understanding of TPG in tight gas reservoirs and provide useful guidance on tight gas reservoirs development. [ABSTRACT FROM AUTHOR]

• A moving boundary model for flow in a multilayered low-permeability reservoir is built. • The number of layers can be arbitrary in the model. • The exact similarity solution of the nonlinear model is obtained. • The exact analytical solution can reduce to the solution for Darcy flow. • A formula for the relative error of the transient pressure caused by ignoring the moving boundary is presented. A nonlinear generalized multiple moving boundary model of one-dimensional non-Darcy flow in heterogeneous multilayered low-permeability porous media with a threshold pressure gradient is constructed, in which the total rate of fluid injection into the porous media remains constant. The number of layers in the model can be arbitrary, and thus the generalized model will be very suitable for describing the one-dimensional non-Darcy flow characteristics in low-permeability reservoirs with strong heterogeneity. Through the similarity transformation method, the exact analytical solution of the multiple moving boundary model is obtained, and the formula for the subrate of fluid injection into every layer is provided. Moreover, it is strictly proved that the exact analytical solution can reduce to the solution of Darcy flow as the threshold pressure gradient in different layers simultaneously tends to zero. Through the exact analytical solution, the effects of the layer threshold pressure gradient, the layer permeability ratio, and the layer elastic storage ratio on the moving boundaries, the spatial pressure distributions, the transient pressure, and the layer subrate in low-permeability porous media are discussed. Through comparison of the exact analytical solutions, it is also demonstrated that incorporation of the multiple moving boundary conditions is very necessary in the modeling of non-Darcy flow in heterogeneous multilayered porous media with a threshold pressure gradient, especially when the threshold pressure gradient is large. In particular, an explicit formula is presented for estimating the relative error of the transient pressure introduced by ignoring the moving boundaries in the modeling. All in all, solid theoretical foundations are provided for non-Darcy flow problems in stratified reservoirs with a threshold pressure gradient. They can be very useful for strictly verifying numerical simulation results, and for giving some guidance for project design and optimization of layer production or injection during the development of heterogeneous low-permeability reservoirs and heavy oil reservoirs so as to enhance oil recovery. [ABSTRACT FROM AUTHOR]

A stimulation technology of multi-stage fractured horizontal well is commonly used for enhancing the productivity in the development of low-permeable oil reservoirs. However, it is well known that the fluid flow in low-permeable oil reservoirs doesn't obey conventional Darcy's law, and the threshold pressure gradient exists; as a result, in fact, the model of multi-stage fractured horizontal well in low-permeable oil reservoirs belongs to a nonlinear moving boundary problem. And realization of the numerical simulation for the model is very complicated and difficult. It is significant to figure out the threshold pressure gradient effect on the numerical simulation results for the solution of actual engineering problems. In view of the concerns, based on the subsurface Darcy's flow module in COMSOL Multiphysics, an effective numerical simulation method is presented here: The non-Darcy kinematic equation at the full pressure gradient range is expressed at the interface of the gravitational acceleration vector in Darcy's law. Furthermore, the simulation method is strictly verified through the comparison with two analytical solutions for a one-dimensional case and a two-dimensional case (radial flow) respectively. Consequently, according to a constructed model, which can cover non-Darcy flow in the unstimulated reservoir area with threshold pressure gradient, Darcy's flow in the stimulated reservoir area and Darcy's flow in the main hydraulic fractures, the two-dimensional numerical simulation of the multi-stage fractured horizontal well in the low-permeable oil reservoir is realized just through the subsurface Darcy's flow module. Finally, through analyses and discussions on the numerical results for several designed simulation scenarios, some significant conclusions are obtained such as due to the threshold pressure gradient effect, there exists large undeveloped areas in low-permeable oil reservoirs, which don't change with the production time for the constant well production pressure condition; SRV fracturing can largely improve the utilization degree of the low-permeable reservoir area between the adjacent main hydraulic fractures at the early production period; it is very necessary to incorporate the threshold pressure gradient effect in the simulation for the optimal well spacing design in the development of low-permeable oil reservoirs. • Simulation of multi-stage fractured horizontal well is realized by COMSOL. • The threshold pressure gradient (TPG) in low-permeable reservoir is considered. • The moving boundary introduced by the TPG effect can be simulated effectively. • The simulation method is verified through the comparison with analytical solutions. • Some significant conclusions for petroleum engineering applications are obtained. [ABSTRACT FROM AUTHOR]

It is well known that the Non-Newtonian Bingham fluid flow in porous media does not obey the conventional linear Darcy's law due to the yield stress for the Bingham fluid: There exists a threshold pressure gradient, which means that the seepage flow only happens when the threshold pressure gradient is overcome. The principle of non-Darcy seepage flow with the threshold pressure gradient is also applicable into the situation of the fluid flow in the low-permeable porous media. Here, a nonlinear moving-boundary mathematical model is built for the semispherical centripetal non-Darcy seepage flow with the threshold pressure gradient in a three-dimensional infinite heavy oil reservoir with the type of Bingham fluid; wherein the moving boundary conditions are incorporated for describing the effect of the threshold pressure gradient. In consideration of the strong nonlinearity of the model, the similarity transformation method is applied into obtaining the exact analytical solution of the model. In order to keep full self-similarity for the model, the inner boundary condition is set as variable flow rate that increases linearly with the time. As a result, an exact analytical solution for the nonlinear moving-boundary mathematical model of semispherical centripetal non-Darcy seepage flow with the threshold pressure gradient is obtained. The existence and the uniqueness of the exact analytical solution are also strictly proved. It is also theoretically proved that as the threshold pressure gradient tends to zero, the exact analytical solution can be reduced to that of a mathematical model of semispherical centripetal Darcy's seepage flow. The presented exact analytical solution can be used for strictly verifying the validity of the numerical methods for solving the three-dimensional moving boundary models of non-Darcy seepage flow with the threshold pressure gradient in the actual engineering problems. From the exact analytical solution, it is also revealed that when the threshold pressure gradient exists, the spatial pressure distribution exhibits an instructive feature of compact support; as the threshold pressure gradient tends to zero, the sensitivity of its effect on the transient distance of the moving boundary and the transient pressure will grow, which reveals the difficulty in accurately determining the position of the moving boundary by the numerical methods and the serious uncertainty problem in the interpretation of the threshold pressure gradient by the pressure transient analysis method in engineering as the threshold pressure gradient is rather small. Through the comparison of the two different exact analytical solutions that corresponds to the two different models with and without incorporating the moving boundary conditions for describing the effect of the threshold pressure gradient, it is demonstrated that when the moving boundary conditions are not incorporated in the modeling, the effect of the threshold pressure gradient on the spatial pressure distribution, the transient pressure and the productivity index can be overestimated largely. Therefore, it is very necessary to incorporate the moving boundary conditions in the modeling of non-Darcy seepage flow with the threshold pressure gradient. The study in the paper definitely provides solid theoretical basis of fluid mechanics for the relevant engineering applications in the development of heavy oil reservoirs and low-permeable reservoirs in petroleum engineering and in the development of water resources in low-permeable formations in hydraulic engineering. • A moving boundary model of centripetal seepage flow of Bingham fluid is built. • Exact analytical solution of the model is obtained by similarity transformation. • It can be used for verifying the validity of three-dimensional flow simulation. • Existence and uniqueness of the exact analytical solution are proved strictly. • Moving boundary must be incorporated in modeling this Bingham fluid flow problem. [ABSTRACT FROM AUTHOR]

Abstract Many engineering technologies involve the moving boundary problems for the radial seepage flow with a threshold pressure gradient, such as the well testing in the development of low-permeability reservoirs, heavy oil reservoirs and groundwater resources. However, as a result of the strong nonlinearity, an exact analytical solution of the moving boundary problems for the radial seepage flow with a threshold pressure gradient has not been obtained yet. Here, a dimensionless moving boundary mathematical model for the radial fluid flow in an infinite low-permeability reservoir with a threshold pressure gradient is developed first. The setting of a variable well production rate for an inner boundary condition can make the mathematical model exhibit a full self-similarity property. Second, by introducing some similarity transformations, the nonlinear system of PDEs of the model can be equivalently transformed into a closed pseudo-linear system of ODEs, whose exact analytical solution can be easily obtained. What's more, the existence and the uniqueness of the exact analytical solution to the moving boundary model are also proved strictly through the mathematical analysis. Third, it is also strictly proved that the exact analytical solution can be degenerated to that of the model of the Darcy's radial fluid flow as the threshold pressure gradient approaches to zero. Finally, by a comparison of model analytical solutions, it is demonstrated that the moving boundary conditions must be incorporated in the modeling of the radial seepage flow with a threshold pressure gradient; otherwise, the effect of the threshold pressure gradient on the radial seepage flow can be overestimated largely. Highlights • A moving boundary model of radial flow with threshold pressure gradient is built. • Moving boundary must be incorporated in modeling this threshold problem. • Exact analytical solution of the model is obtained by similarity transformations. • Existence and uniqueness of the exact analytical solution are proved strictly. • It can be used for verifying the validity of two-dimensional flow simulation. [ABSTRACT FROM AUTHOR]

The non-Darcy seepage flow in low-permeable coalbed and the strong adsorption of coalbed methane are the two key characteristics in the process of coalbed methane development in China. Based on these concerns, a model of unsteady seepage flow in low-permeable coalbed with moving boundary is built, which incorporates both the existence of threshold pressure gradient and the stable and unstable adsorption of coalbed methane simultaneously. Moreover, the effects of wellbore storage and skin effect are both considered, which can be incorporated in the inner boundary conditions. Due to the strong nonlinearity and complexity of the model with moving boundary, a verified numerical method of spatial coordinate transformation based fully implicit finite difference method is adopted to obtain its numerical solutions. By using these numerical solutions, the effects of the dimensionless coefficient of wellbore storage, the skin factor, the dimensionless threshold pressure gradient and the dimensionless coefficient of unstable desorption on the formation pressure distribution are analyzed, and some significant conclusions are obtained. [ABSTRACT FROM AUTHOR]

Nowadays, development of heavy oil usually demands massive hot water injection and meanwhile produces considerable carbon footprint. Radiofrequency (RF) heating, as a "Non-aqueous" method, has potentials to be a cleaner and efficient way for heavy oil development in near future. In this paper, we presented a novel numerical model for simulating non-Newtonian Bingham fluid flow in heavy oil reservoirs based on RF heating. Compared to previous studies, this paper focuses on issues including: (1) Application of RF heating in heavy oil reservoir; (2) Non-Darcy flow induced by threshold pressure gradient (TPG) of Bingham fluid; (3) Temperature dependent TPG and viscosity; (4) Two-way coupling between non-Darcy flow, transient heat transfer and electromagnetic (EM) field. To incorporate TPG in numerical analyses, an effective method was utilized by modifying the gravitational acceleration vector in Darcy's law. This method was verified with analytical solution. By doing this, the induced moving boundary by TPG can be simulated, and pressure-disturbed area can be determined. Results show that RF heating significantly mitigates the impediment induced by TPG and high viscosity near well. The moving boundary stops motion at around 1000th day of production and the extreme disturbed distance is 63 m. Besides, it is found that no matter whether TPG is considered or not, RF heating has a positive impact on the reservoir development. However, for the production time less than 1000 days, RF heating is more effective when TPG is considered. Finally, we also found it very necessary to incorporate TPG in the simulations to determine optimal well spacing. If TPG is neglected in kinematic equation, the optimal well spacing would be overestimated by over 50%. At the same time, the change of TPG value with temperature cannot be ignored. Otherwise, the optimal well spacing would be underestimated by 15%. This paper reveals the potentials of RF heating method for non-Newtonian Bingham fluid production and provides an alternative way for clean, environmental, and highly efficient development of heavy oil reservoir. • Simulations of Bingham fluid flow in heavy oil reservoir using RF heating are carried out. • TPG dependent on temperature is considered to represent yield stress of Bingham fluid. • Electromagnetic field, temperature field and flow field are coupled based on FEM methods. • Optimal well spacing with RF heating, TPG effect and moving boundary is analyzed. [ABSTRACT FROM AUTHOR]

Abstract: The dimensionless mathematical models of one-dimensional flow in the semi-infinite long porous media with threshold pressure gradient are built for the two cases of constant flow rate and constant production pressure on the inner boundaries. Through formula deduction, it is found that the velocity of the moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on the moving boundary, which is very different from the classical heat-conduction Stefan problems. However, by introducing some similarity transformation from Stefan problems, the exact analytical solutions of the dimensionless mathematical models are obtained, which can be used for strict validation of approximate analytical solutions, numerical solutions and pore-scale network modeling for the flow in porous media with threshold pressure gradient. Comparison curves of the dimensionless pressure distributions and the transient dimensionless production pressure under different values of dimensionless threshold pressure gradient are plotted from the exact analytical solutions of problems of the flow in semi-infinite long porous media with and without threshold pressure gradient. It is shown that for the case of constant flow rate the effect of the dimensionless threshold pressure gradient on the dimensionless pressure distributions and the transient dimensionless production pressure is not very obvious; in contrast, for the case of constant production pressure the effect on the dimensionless pressure distributions is more obvious especially at the larger dimensionless distance near the moving boundary; and for the case of constant production pressure, the smaller the dimensionless threshold pressure gradient is, the larger the dimensionless pressure is, and the further the pressure disturbance area reaches. [Copyright &y& Elsevier]

In the paper, the study on the exact analytical solution of the moving boundary problem of one-dimensional Bingham fluid seepage flow is extended from the infinite long porous media (Liu et al., 2012) to the finite long porous media. Two exact analytical solutions are presented by appropriately relying on some methods of mathematical physics and mathematical techniques. One is for the model with finite closed outer boundary condition; the other is for the model with finite constant pressure outer boundary condition. The existence and the uniqueness of the exact analytical solutions are also strictly proved theoretically. In addition, the numerical solutions of the two models by the finite difference method are also provided. Through the comparison, it is found that these exact analytical solutions have very excellent agreement with the numerical solutions although few terms of the infinite function series existent in the exact analytical solutions have to be retained for the calculation. Furthermore, for the two models, the effect of the threshold pressure gradient on the transient pressure and the transient pressure derivative at the inner boundary for the whole flow process is analyzed through the analytical solutions. Finally, through the comparison of the relevant model solutions, it is concluded that it is very necessary to incorporate the process of moving boundary for the modeling of non-Darcy Bingham fluid flow in finite long porous media with threshold pressure gradient; otherwise, large errors can be introduced in predicting the transient pressure and the transient pressure derivative in the porous media. The presented work can support solid theoretical foundations for the experiment design of measuring the threshold pressure gradient and the pressure transient analysis in the field of inverse problems in the petroleum engineering, which have been widely involved in the development of low-permeable oil reservoirs and heavy oil reservoirs. • Two typical models of 1D Bingham fluid flow in finite long porous media are built. • Moving boundary process caused by threshold pressure gradient is incorporated. • Their exact analytical solutions are both obtained by mathematical physics method. • Existence and the uniqueness of the exact analytical solutions are strictly proved. • Solid theory for experiment design and pressure transient analysis is provided. [ABSTRACT FROM AUTHOR]

In this paper, numerical simulations of fractured vertical well in low-permeable oil reservoir are carried out and two issues: threshold pressure gradient (TPG) in reservoir matrix and proppant distribution in hydraulic fractures are focused on. A physical definition of effective fracture length (EFL) is presented to evaluate the fracture effectiveness due to uneven proppant distribution. In terms of numerical processing for TPG, based on the subsurface Darcy's flow module in COMSOL Multiphysics, an effective numerical simulation method is adopted by substituting the gravitational acceleration vector in Darcy's law (Liu et al., 2019b). The simulation results show that, proppant distribution has great influence on fracture effectiveness. Unappealing proppant distribution can reduce effective fracture length by 49.6%, which is negative for low-permeable oil reservoir development. TPG in reservoir matrix mainly affects the moving boundary and pressure-disturbed area in reservoir. When TPG in matrix reduces from 0.02 MPa/m to 0 MPa/m, the distance of moving boundary at 300 days could increase for almost double and pressure-disturbed area enlarged drastically for around 74%. Furthermore, a heterogenous low-permeable oil with secondary fracture networks is designed to study the joint effect of TPG and proppant distribution. If we neglect these two factors, cumulative oil rate will be greatly overestimated by 38.6% at 1000 days. At the same time, the existence of natural fractures could also affect the streamline distribution in low-permeable oil reservoir and contribute to fluid flow in reservoir. • Simulation of fractured vertical well in low-permeable oil reservoir is realized by COMSOL. • Proppant distribution in fracture and TPG in matrix are considered in simulation model. • Effect of proppant distribution on effective fracture length is analyzed quantitatively. • Effect of TPG on moving boundary and pressure-disturbed area are analyzed quantitatively. [ABSTRACT FROM AUTHOR]