Impacts of proppant distribution on development of tight oil reservoirs with threshold pressure gradient

Field evidence indicates that proppant distribution and threshold pressure gradient have great impacts on well productivity.Aiming at the development of unconventional oil reservoirs in Triassic Chang-7 Unit,Ordos Basin of China,we presented an integrated workflow to investigate how(1)proppant placement in induced fracture and(2)non-linear flow in reservoir matrix would affect well productivity and fluid flow in the reservoir.Compared with our research before(Yue et al.,2020),here we extended this study into the development of multi-stage fractured horizontal wells(MFHWs)with large-scale complicated fracture geometry.The integrated workflow is based on the finite element method and consists of simulation models for proppant-laden fluid flow,fracture flow,and non-linear seepage flow,respectively.Simulation results indicate that the distribution of proppant inside the induced cracks significantly affects the productivity of the MFHW.When we assign an idealized proppant distribution instead of the real distribution,there will be an overestimation of 44.98%in daily oil rate and 30.63%in cumulative oil production after continuous development of 1000 days.Besides,threshold pressure gradient(TPG)also significantly affects the well performance in tight oil reservoirs.If we simply apply linear Darcy’s law to the reservoir matrix,the overall cumulative oil production can be overrated by 77%after 1000 days of development.In general,this research provides new insights into the development of tight oil reservoirs with TPG and meanwhile reveals the significance of proppant distribution and non-linear fluid flow in the production scenario design.

Characteristics of unsteady flow in porous media while considering threshold pressure gradient with Green＇s function

The flow behavior in porous media with threshold pressure gradient(TPG) is more complex than Darcy flow and the equations of motion, and outer boundary and inner boundary with TPG are also different from Darcy flow for unsteady flow of a producing well in a reservoir. An analytic method to solve this kind of problem is in a need of reestablishment. The classical method of Green's function and Newman product principle in a new way are used to solve the unsteady state flow problems of various shapes of well and reservoir while considering the TPG. Four Green's functions of point, line, band and circle while considering the TPG are achieved. Then, two well models of vertical well and horizontal well are built and simultaneously the function to calculate the moving boundary of each well model is provided. The results show that when considering TPG the pressure field is much different, which has a sudden pressure change, with a moving boundary in it. And the moving boundary of each well model increases with time but slows down rapidly, especially when the TGP is large.

Numerical investigation of a coupled moving boundary model of radial flow in low-permeable stress-sensitive reservoir with threshold pressure gradient

The threshold pressure gradient and formation stress-sensitive effect as the two prominent physical phenomena in the development of a low-permeable reservoir are both considered here for building a new coupled moving boundary model of radial flow in porous medium. Moreover, the wellbore storage and skin effect are both incorporated into the inner boundary conditions in the model. It is known that the new coupled moving boundary model has strong nonlinearity. A coordinate transformation based fully implicit finite difference method is adopted to obtain its numerical solutions. The involved coordinate transformation can equivalently transform the dynamic flow region for the moving boundary model into a fixed region as a unit circle, which is very convenient for the model computation by the finite difference method on fixed spatial grids. By comparing the numerical solution obtained from other different numerical method in the existing literature, its validity can be verified. Eventually, the effects of permeability modulus, threshold pressure gradient, wellbore storage coefficient, and skin factor on the transient wellbore pressure, the derivative, and the formation pressure distribution are analyzed respectively.

TRANSIENT PRESSURE OF PERCOLATION THROUGH ONE DIMENSION POROUS MEDIA WITH THRESHOLD PRESSURE GRADIENT

This paper studies the transient pressure of percolation during one production and one shutting in one dimension porous media with threshold pressure gradient, the differential equations are derived and solved with numerical computation. Basing on numerical solution, it is analyzed that: 1. the relation between the steady pressure at well bore (or at endpoint) and threshold pressure gradient, shut-in time, and the corresponding formulae are derived; 2, the regulation of transient pressure peak. The result is very useful and will help experiments and applications in the development of low permeability reservoirs with threshold pressure gradient.

The establishment of a new deliverability equation considering threshold pressure gradient

The flowing mechanism of a low permeability gas reservoir is different from a conventional gas reservoir,especially for that with higher irreducible water saturation the threshold pressure gradient exists. At present,in all the deliverability equation,the additional pressure drop caused by the threshold pressure gradient is viewed as constant,but this method has big error in the practical application. Based on the non-Darcy steady flow equation,the limited integral of the additional pressure drop is solved in this paper and it is realized that the additional pressure drop is not a constant but has something to do with production data,and a new deliverability equation is derived,with the relevant processing method for modified isochronal test data. The new deliverability equation turns out to be practical through onsite application.

PROJECTED GRADIENT DESCENT BASED ON SOFT THRESHOLDING IN MATRIX COMPLETION

Matrix completion is the extension of compressed sensing.In compressed sensing,we solve the underdetermined equations using sparsity prior of the unknown signals.However,in matrix completion,we solve the underdetermined equations based on sparsity prior in singular values set of the unknown matrix,which also calls low-rank prior of the unknown matrix.This paper firstly introduces basic concept of matrix completion,analyses the matrix suitably used in matrix completion,and shows that such matrix should satisfy two conditions:low rank and incoherence property.Then the paper provides three reconstruction algorithms commonly used in matrix completion:singular value thresholding algorithm,singular value projection,and atomic decomposition for minimum rank approximation,puts forward their shortcoming to know the rank of original matrix.The Projected Gradient Descent based on Soft Thresholding(STPGD),proposed in this paper predicts the rank of unknown matrix using soft thresholding,and iteratives based on projected gradient descent,thus it could estimate the rank of unknown matrix exactly with low computational complexity,this is verified by numerical experiments.We also analyze the convergence and computational complexity of the STPGD algorithm,point out this algorithm is guaranteed to converge,and analyse the number of iterations needed to reach reconstruction error.Compared the computational complexity of the STPGD algorithm to other algorithms,we draw the conclusion that the STPGD algorithm not only reduces the computational complexity,but also improves the precision of the reconstruction solution.

Evaluation of Well Spacing for Primary Development of Fractured Horizontal Wells in Tight Sandstone Gas Reservoirs

Methods for horizontal well spacing calculation in tight gas reservoirs are still adversely affected by the complexity of related control factors,such as strong reservoir heterogeneity and seepage mechanisms.In this study,the stress sensitivity and threshold pressure gradient of various types of reservoirs are quantitatively evaluated through reservoir seepage experiments.On the basis of these experiments,a numerical simulation model(based on the special seepage mechanism)and an inverse dynamic reserve algorithm(with different equivalent drainage areas)were developed.The well spacing ranges of Classes I,II,and III wells in the Q gas field are determined to be 802–1,000,600–662,and 285–400 m,respectively,with their average ranges as 901,631,and 342.5 m,respectively.By considering both the pairs of parallel well groups and series well groups as examples,the reliability of the calculation results is verified.It is shown that the combination of the two models can reduce errors and provide accurate results.

A fractal approach to low velocity non-Darcy flow in a low permeability porous medium

In this paper, the mechanism for fluid flow at low velocity in a porous medium is analyzed based on plastic flow of oil in a reservoir and the fractal approach. The analytical expressions for flow rate and velocity of non-Newtonian fluid flow in the low permeability porous medium are derived, and the threshold pressure gradient （TPG） is also obtained. It is notable that the TPG （J） and permeability （K） of the porous medium analytically exhibit the scaling behavior J ~ K-D＇r/（l＋Or）, where DT is the fractal dimension for tortuous capillaries. The fractal characteristics of tortuosity for capillaries should be considered in analysis of non-Darcy flow in a low permeability porous medium. The model predictions of TPG show good agreement with those obtained by the available expression and experimental data. The proposed model may be conducible to a better understanding of the mechanism for nonlinear flow in the low permeability porous medium.

Tight gas production model considering TPG as a function of pore pressure,permeability and water saturation

Threshold pressure gradient has great importance in efficient tight gas field development as well as for research and laboratory experiments.This experimental study is carried out to investigate the threshold pressure gradient in detail.Experiments are carried out with and without back pressure so that the effect of pore pressure on threshold pressure gradient may be observed.The trend of increasing or decreasing the threshold pressure gradient is totally opposite in the cases of considering and not considering the pore pressure.The results demonstrate that the pore pressure of tight gas reservoirs has great influence on threshold pressure gradient.The effects of other parameters like permeability and water saturation,in the presence of pore pressure,on threshold pressure gradient are also examined which show that the threshold pressure gradient increases with either a decrease in permeability or an increase in water saturation.Two new correlations of threshold pressure gradient on the basis of pore pressure and permeability,and pore pressure and water saturation,are also introduced.Based on these equations,new models for tight gas production are proposed.The gas slip correction factor is also considered during derivation of this proposed tight gas production models.Inflow performance relationship curves based on these proposed models show that production rates and absolute open flow potential are always be overestimated while ignoring the threshold pressure gradients.

Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.

A noise removal algorithm for DR Images based on adaptive estimation of threshold

When DR （Digital Radiography） images are filtered, it is necessary to preserve the edges and key details. But the existing methods may inevitably take fine details mistaken for noise to remove. In order to solve the problem an improved anisotropic diffu- sion filtering model is proposed. Firstly, a novel diffusion function is introduced based on Perona and Malik model, which well overcomes the high rate of convergence. Secondly, the gradient threshold is modified to an adaptive estimation function, so it is bet- ter at adaptive threshold regulations according to the pixels and iteration times. Finally, the edges are extracted from the restored im- ages and the results are evaluated quantificationally. It is shown from the experiments that the proposed method is effective not only in noise reduction but also in details preserved.

A multi-scale scheme for image segmentation using neuro-fuzzy classification and curve evolution

In this paper, we present a new scheme to segment a given image. This scheme utilizes neuro-fuzzy system to derive a proper set of contour pixels based on multi-scale images. We use these fuzzy derivatives to develop a new curve evolution model. The model automatically detect smooth boundaries, scaling the energy term, and change of topology according to the extracted con- tour pixels set. We present the numerical implementation and the experimental results based on the semi-implicit method. Experi- mental results show that one can obtains a high clualitv edge contour.

Water Gauge Image Denoising Model Based on Improved Adaptive Total Variation

As an important part of water level warning in water conservancy projects,often due to the influence of environmental factors such as light and stains,the acquired water gauge images have sticky,broken and bright spot conditions,which affect the identification of water gauges.To solve this problem,a water gauge image denoising model based on improved adaptive total variation is proposed.Firstly,the regular term exponent in the adaptive total variational equation is changed to an inverse cosine function;secondly,the differential curvature is used to distinguish the image noise points and increase the smoothing strength at the noise points;finally,according to the characteristics of the gradient mode and adaptive gradient threshold after Gaussian filtering,the New model can adaptively denoise in the smooth area and protect the edge area,so as to have the characteristics of both edge-preserving denoising.The experimental results show that the new model has a great improvement in image vision,higher iteration efficiency and an average increase of 1.6 dB in peak signal-to-noise ratio,and an average increase of 9%in structural similarity,which is more beneficial to practical applications.

Study of the Seepage Mechanism in Thick Heterogeneous Gas Reservoirs

The seepage mechanism plays a crucial role in low-permeability gas reservoirs.Compared with conventional gas reservoirs,low-permeability sandstone gas reservoirs are characterized by low porosity,low permeability,strong heterogeneity,and high water saturation.Moreover,their percolation mechanisms are more complex.The present work describes a series of experiments conducted considering low-permeability sandstone cores under pressuredepletion conditions(from the Xihu Depression in the East China Sea Basin).It is shown that the threshold pressure gradient of a low-permeability gas reservoir in thick layers is positively correlated with water saturation and negatively correlated with permeability and porosity.The reservoir stress sensitivity is related to permeability and rock composition.Stress sensitivity is generally low when permeability is high or in the early stage of gas reservoir development.It is also shown that in sand conglomerates,especially the more sparsely filled parts,the interstitial materials among the conglomerates can be rapidly dislodged from the skeleton particles under stress.This material can therefore disperse,migrate,and block the pore throat producing serious,stress-sensitive damage.

Threshold pressure gradient of fluid flow through multi-porous media in low and extra-low permeability reservoirs

After analyzing many studies of fluid flow theory of multi-porous media in low and extra-low permeability reservoirs and the numerical simulation of non-Darcy flow, we found that a negative flow rate occurs in the existing non-Darcy flow equation, which is unreasonable. We believe that the existing equation can only be considered as a discriminant to judging Darcy flow or non-Darcy flow, and cannot be taken as a fluid flow governing equation of multi-porous media. Our analysis of the experimental results shows that the threshold pressure gradient(TPG) of low and extra-low permeability reservoirs is excessively high, and does not conform to fluid flow through multi-porous media in the actual reservoir situation. Therefore, we present a reasonable TPG ranging from 0.006 to 0.04 MPa/m at the well depth of 1500 m and oil drainage distance of 500 m. The results of our study also indicate that the non-Darcy flow phenomenon will disappear when the TPG reaches a certain value. In addition, the TPG or non-Darcy flow in low and extra-low permeability reservoirs does not need to be considered in the productivity prediction and reservoir numerical simulation. At present, the black oil model or dual-porous media is suitable for simulating low and extra-low permeability reservoirs.

WELL TESTING ANALYSIS FOR HORIZONTAL WELL WITH CONSIDERATION OF THRESHOLD PRESSURE GRADIENT IN TIGHT GAS RESERVOIRS

A fundamental solution for homogeneous reservoir in infinite space is derived by using the point source function with the consideration of the threshold pressure gradient. The fundamental solution of the continuous point source function is then derived based on the Green function. Various boundary conditions of the reservoirs are considered for this case and the corresponding solutions are obtained through the mirror image reflection and the principle of superimposition. The line source solution is obtained by integration. Subsequently, the horizontal-well bottom hole pressure response function for a non-linear gas flow in the homogeneous gas reservoir is obtained, and the response curve of the dimensionless bottom hole pressure and the derivative for a horizontal well in the homogeneous gas reservoir are obtained. In the end, the sensitivities of the relevant parameters are analyzed, The well test model presented in this paper can be used as the basis of the horizontal well test analysis for tight gas reservoirs.

Dual-porosity model of rate transient analysis for horizontal well in tight gas reservoirs with consideration of threshold pressure gradient

Most researches of the threshold pressure gradient in tight gas reservoirs are experimental and mainly focus on the transient pressure response, without paying much attention to the transient rate decline. This paper establishes a dual-porosity rate transient decline model for the horizontal well with consideration of the threshold pressure gradient, which represents the non-Darcy flow in a fracture system. The solution is obtained by employing the Laplace transform and the orthogonal transform. The bi-logarithmic type curves of the dimensionless production rate and derivative are plotted by the Stehfest numerical inversion method. Seven different flow regimes are identified and the effects of the influence factors such as the threshold pressure gradient, the elastic storativity ratio, and the cross flow coefficient are discussed. The presented research could interpret the production behavior more accurately and effectively for tight gas reservoirs.

Discussion of liquid threshold pressure gradient

Some authors believe that a minimum pressure gradient(called threshold pressure gradient(TPG))is required before a liquid starts to flow in a porous medium.In a tight or shale oil formation,this TPG phenomenon becomes more important,as it is more difficult for a fluid to flow.In this paper,experimental data on TPG published in the literature are carefully reviewed.What we found is that a very low flow velocity corresponding to a very low pressure gradient cannot be measured in the experiments.Experiments can only be done above some measurable flow velocities.If these flow velocities and their corresponding pressure gradients are plotted in an XY plot and extrapolated to zero velocity,a non-zero pressure gradient corresponds to this zero velocity.This non-zero pressure gradient is called threshold pressure gradient in the literature.However,in the regime of very low velocity and very low pressure gradient,the data gradually approach to the origin of the plot,demonstrating a non-linear relationship between the pressure gradient and the velocity.But the data do not approach to a point of zero velocity and a threshold pressure gradient.Therefore,the concept of threshold pressure gradient is a result of data misinterpretation of available experimental data.The correct interpretation is that there are two flow regimes:nonlinear flow regime(non-Darcy flow regime)when the pressure gradients are low,and linear flow regime(Darcy flow regime)when the pressure gradient is intermediate or high.The nonlinear flow regime starts from the origin point.As the pressure gradient is increased,the curve becomes a straight line demonstrating the linear flow regime.We have verified our views by first analyzing the causes of non-Darcy flow,and then systematically analyzed typical experimental data and correlations in the literature.We conclude that TPG does not exist.We also use several counter examples to support our conclusion.

Exact analytical solutions for moving boundary problems of one-dimensional flow in semi-infinite porous media with consideration of threshold pressure gradient

By defining new dimensionless variables, nonlinear mathematical models for one-dimensional flow with unknown moving boundaries in semi-infinite porous media are modified to be solved analytically. The exact analytical solutions for both constant-rate and constant-pressure inner boundary constraint problems are obtained by applying the Green＇s function. Two transcendental equations for moving boundary problems are obtained and solved using the Newton-Raphson iteration. The exact analytical solutions are then compared with the approximate solutions. The Pascal＇s approximate formula in reference is fairly accurate for the moving boundary development under the constant-rate condition. But another Pascal＇s approximate formula given in reference is not very robust for constant-pressure condition problems during the early production period, and could lead to false results at the maximum moving boundary distance. Our results also show that, in presence of larger TPG, more pressure drop is required to maintain a constant-rate production. Under the constant-pressure producing condition, the flow rate may decline dramatically due to a large TPG. What＇s more, there exists a maximum distance for a given TPG, beyond which the porous media is not disturbed.

NUMERICAL SIMULATION STUDY ON SURFACTANT FLOODING FOR LOW PERMEABILITY OILFIELD IN THE CONDITION OF THRESHOLD PRESSURE

Based on the non-Darcy flow characteristics of surfactant flooding in the low permeability oilfield, considering the changes of threshold pressure and influence of surfactant on convection, diffusion, adsorption and retention, a mathematical model is established for a three-dimensional, two-phase, three-component surfactant flooding. A new treatment for the changes of threshold pressure and a novel correction method for the relative permeability curve in the process of surfactant flooding are derived, which enhances the matching degree between the mathematical model and field practice. The mathematical model was used to perform the numerical simulation study for a pilot test of surfactant flooding in Chao 45 Block of Daqing Oilfield, a proper injection plan was optimized. After the optimized plan was carried out in oilfield, the desirable effects, like pressure-reducing, injection rate increase, and the increase of oil recovery, were achieved. The average oil increase for single well reaches 37%, the ratio of cost to revenue is above 1：4, so the economic effect of scale is promising.