Fracture Flow Research Articles

Numerical analysis of seepage law for radial fluid flow in a single fracture: Comparison between smooth and rough fractures

This paper investigated the differences for the hydraulic characteristics in a single fracture between using the Navier–Stokes (N–S) equation and Darcy's law, which would be benefit to understand the seepage mechanism in the fracture. A numerical model of the radial flow was established considering the aperture size and water injection flow rate. Some conclusion could be given. First, the Darcy's law only described the seepage characteristics when the flow rate was small when the flow rate and pressure response have a linear relationship. While the N–S equation could describe the linear and nonlinear seepage characteristics, resulting in a better model of the actual fracture seepage flow. Second, the aperture size had a limited influence on the water pressure and seepage velocity inside the fracture when the flow rate was small. It began to have a significant impact influence on the seepage characteristics inside the fracture with the aperture increased. Third, the flow–pressure response conformed to the Forchheimer equation in the fracture. The critical Reynold number would decrease from 1.2 to 0.0116 when fracture aperture decreased from 3 to 0.5 mm using the N–S equation. The degree of nonlinearity of the fluid flow increases with fracture roughness increasing. This work gave a guidance to the difference in the two seepage theories and correction for the result by Darcy law, which was widely used in the engineering calculation.

Changes in Water Age During Dry‐Down of a Non‐Perennial Stream

AbstractNon‐perennial streams, which lack year‐round flow, are widespread globally. Identifying the sources of water that sustain flow in non‐perennial streams is necessary to understand their potential impacts on downstream water resources, and guide water policy and management. Here, we used water isotopes (δ18O and δ2H) and two different modeling approaches to investigate the spatiotemporal dynamics of young water fractions (Fyw) in a non‐perennial stream network at Konza Prairie (KS, USA) during the 2021 summer dry‐down season, as well as over several years with varying hydrometeorological conditions. Using a Bayesian model, we found a substantial amount of young water (Fyw: 39.1–62.6%) sustained flows in the headwaters and at the catchment outlet during the 2021 water year, while 2015–2022 young water contributions estimated using sinusoidal models indicated smaller Fyw amounts (15.3% ± 5.7). Both modeling approaches indicate young water releases are highly sensitive to hydrological conditions, with stream water shifting to older sources as the network dries. The shift in water age suggests a shift away from rapid fracture flow toward slower matrix flow that creates a sustained but localized surface water presence during late summer and is reflected in the annual dynamics of water age at the catchment outlet. The substantial proportion of young water highlights the vulnerability of non‐perennial streams to short‐term hydroclimatic change, while the late summer shift to older water reveals a sensitivity to longer‐term changes in groundwater dynamics. Combined, this suggests that local changes may propagate through non‐perennial stream networks to influence downstream water availability and quality.

Monitoring Subsurface Fracture Flow Using Unsupervised Deep Learning of Borehole Microseismic Waveform Data

Monitoring Subsurface Fracture Flow Using Unsupervised Deep Learning of Borehole Microseismic Waveform Data

Establishment and application of propped hydraulic fracture conductivity theoretical model based on fracturing efficiency index

Hydraulic fracturing technology is a key technology for the development of unconventional oil and gas reservoirs, and the conductivity of hydraulic fractures has a great impact on their production. The paper introduces the "fracturing efficiency index" as means of characterising the dynamic relationship between hydraulic fracture expansion and proppant filling. It also suggests a way to figure out the proppant fracture inflow capacity. This approach addresses issues such as uneven fracturing fluid dosage, a wide range of injection discharges, and inconsistent sand spreading concentration encountered during fracturing operations. Based on this model, the influencing factors of the flow-conducting capacity such as proppant particle size, sand spreading concentration, liquid injection displacement, and proppant crushing rate, were analyzed, and the calculated results of this model are closer to the experimental results of the API flow-conducting capacity test compared with the calculated results of the reference model. Finally, this model was used to optimize the fluid volume, sand volume, and displacement of hydraulic fracturing in the Jiping 1H shale gas field. The research results show that: (1) the fracturing efficiency index is negatively correlated with the conductivity of hydraulic fractures, whereas is positively correlated with the enhancement of conductivity; (2) our model is applicable to the calculation of supported fracture inflow capacity for high sand concentration, large particle size, intermediate fracture flow and low crushing rate, and the results are close to more than 98.58%, 98.29%, 95.68% and 95.08% of the experimental results, respectively; (3) On-site hydraulic fracturing does not result in higher production capacity with more fracturing fluid dosage, higher displacement and more sand addition; there are optimal fluid, optimal displacement and optimal sand intervals, which are optimized to increase production by 26.92%. It can be seen that the hydraulic fracture conductivity model can optimize various parameters of on-site fracturing construction.

Particle transport during asymmetric net-zero cyclic fluid flow: Flow in tubes and radial flow in fractures

Cyclic fluid flow enhances transport in porous media. We study particle transport during asymmetric net-zero cyclic fluid flow where the fluid injection rate is higher than the withdrawal rate. Inside a tube, the threshold fluid velocity required to mobilize a particle depends on the relative particle size: larger particles obstruct flow more effectively than small particles and require a smaller velocity to be transported. Then, flow rates can be selected so particles move forward during the injection phase and remain stationary during withdrawal. Similarly, asymmetric cyclic fluid flow transports particles radially along fractures despite the net-zero fluid flow. The flow velocity decreases away from the injection point and particles migrate to a characteristic “terminal radial distance”. Unlike tube flow, a grain has a diminishing obstructing effect on the radial flow field, and the threshold flow rate required to mobilize grains increases with increasing particle size during radial flow.

A new numerical model for simulation of flow on rough fracture

Hydraulic fracturing is an important means to develop unconventional oil and gas. The fractures produced by hydraulic fracturing have the characteristics of a rough surface, high tortuosity, and strong randomness. The fluid migration law in rough fractures is more complicated, which leads to the change of the law of conductivity in fractures becoming more complex. Meanwhile, it is difficult to get certain regularity for the fracture surface obtained by direct fracturing. As a result, it is an arduous task to explore the influence of fracture roughness on conductivity by experimental means. Therefore, it is a feasible idea to use simulation to study the flow conductivity in rough fractures, which can be calculated by Darcy's seepage equation. In previous studies, quantitative characterization of rough fractures was rarely studied, and the three-dimensional Darcy seepage model of rough fractures was lacking too. In this paper, the rough surface profile and waveform are analyzed by analogy, and the height of the rough fracture surface, as well as the phase and wavelength of the waveform, were taken as the control parameters to generate the rough fracture surface. Then, the fractal dimension was used to characterize the roughness of the fracture surface. Coupled with the rough fracture model and the Darcy flow model, a three-dimensional Darcy flow model of rough fracture was established. Next, a new parameter, equivalent permeability, was defined to quantify and characterize the fracture permeability affected by roughness. Subsequently, the simulation results were verified by the experimental data. Finally, the effects of fracture surface fractal dimension, fracture width, proppant distribution, shear slip, and tortuous angle on the flow field were studied. This study provides a new model for the simulation of a rough fracture flow field, which provides important insight for understanding the properties of rough fracture flow fields.

Blasingame production decline curve analysis for fractured tight sand gas wells based on embedded discrete fracture model

Blasingame production decline curve analysis (PDCA) is crucial for extracting natural gas from fractured tight sandstone reservoirs (FTSRs). The natural gas flow in FTSRs is complex due to high-velocity non-Darcy (hvnD) flow, threshold pressure gradient (TPG), and stress sensitivity of reservoirs. These factors pose a challenge in solving nonlinear equations. In this paper, we establish a mathematical model for the flow of natural gas in FTSRs. The fracture is characterized by a hvnD flow coupled using an embedded discrete fracture model (EDFM). The gas flow in the matrix is coupled with the TPG. Stress sensitivity has been considered in both fractures and matrix. The finite volume method (FVM) was employed to discretize and solve the nonlinear equations in the model. The pressure-normalized rate algorithm and gas flow model were used to present a Blasingame production decline curve for fractured horizontal wells in FTSRs. The Blasingame production decline curve can be used to identify natural gas flow regimes, matrix petrophysical properties, and fracture parameters. The effect of the hvnD flow in fractures, the lvnD flow in the matrix (TPG), stress sensitivity, fracture parameters, and production schedule on the Blasingame production decline curve and pressure profile are discussed in detail. This study provides scientific and theoretical support for predicting gas wells performance and the gas production decline law in FTSRs.

Generalized Analytical Solutions of Vertically Fractured Wells in Commingled Reservoirs: Field Case Study

Summary Accurate identification of the individual-layer parameters for vertically fractured wells in commingled reservoirs is essential for development plan design, reservoir numerical simulation, and stimulation measure selection. Different semi-analytical and numerical models are generally applied in multilayer transient testing (MLT) analysis to determine the properties of individual layer. However, these approaches require numerous computations and are complicated to program due to the fracture and reservoir discretization. This work thus presents the generalized analytical solutions of vertically fractured wells in infinite, closed, or constant-pressure commingled reservoirs with both computational and functional simplicity. The fully analytical solutions are derived based on the early-time approximate solutions of infinite-conductivity fracture and trilinear flow models, infinite-conductivity fracture solutions, pressure superposition principle, and Duhamel principle. A systematic verification by employing a standardized well testing software and trilinear flow model is conducted to ensure the general application accuracy of the presented solutions. The results show that the developed analytical solutions are valid when the dimensionless fracture conductivity is more than 2 (FcD > 2) with an average absolute percent deviation (AAD) of ~2% for pressure and that is ~4% for pressure derivative. The developed analytical solutions also exhibit improvements in early-time pressure and derivative calculation. Finally, a field case of a four-layer fractured well is interpreted by the developed solutions and well testing software to illustrate the feasibility. The interpretation results of two methods are nearly identical, with only a minor difference. The developed analytical solutions are computationally accurate while maintaining functional simplicity and can be considered as an alternative to the current semi-analytical and numerical approaches in MLT analysis.

A matrix solver approach for fracture flow simulation by Analytic Element Method

The manuscript presents a study on 2D groundwater flow through porous and fractured media employing Analytic Element Method. The approximate solution based on a series expansion is presented for the problem with multiple fractures. Influence of each fracture can be expressed in a series satisfying Laplace’s equation perfectly. The unknown coefficients in series expansion are determined by using the discharge potentials of all the elements linked to them. Dimension, location and conductivities were randomly selected for the inhomogeneities. Furthermore, a matrix method as a direct solver for AEM, obtained by implementing the intern boundary conditions as well as convergence analysis of iterative method of Gauss–Seidel type are also discussed.

A Novel Deformation Analytical Solution and Constitutive Model for Fractured Rock Masses

In order to study the deformation and stability of a fractured rock mass, existing research suggests that fracture deformation, which is usually obtained by evaluating the equivalent deformation modulus, dominates the deformation of a fractured rock mass. However, this parameter is difficult to obtain in theory and practice, which limits the application of rock mass deformation analysis methods. In order to calculate the deformation of fractured rock masses, the mass of the rock is regarded as a sponge-like material, and it is assumed that the deformation of a water-saturated fractured rock mass under external force load is approximately equal to the net flow of fracture water. Based on this assumption, firstly, the relationship between rock mass deformation and fracture flow is studied through a single-fracture rock mass model. The hydraulic properties of the fracture are characterized by the permeability coefficient, and the fracture deformation in the rock mass is equivalent to the fracture flow. The fracture deformation calculation formula is derived from the fracture hydraulics calculation formula, and this is compared with the measured data. The rationality of the calculation formula was verified. On this basis, the calculation formula for rock mass deformation, including multiple groups of fracture surfaces, is proposed, and the stress-strain constitutive relationship of a complex rock mass is established. The correctness of the calculation method was verified by comparing it with other theoretical calculation results.

Experimental Study of Flow Boiling Regimes Occurring in a Microfluidic T-Junction.

Microchannel flow boiling is an efficient cooling method for high-heat-flux electronic devices. To understand the flow boiling regime in a T-shaped microchannel, this paper prepared T-shaped microchannels of different sizes and designed an experimental platform for the visualization of flow boiling in microchannels, and aimed to study the evolution characteristics of two-phase flow patterns in T-shaped microchannels. The influences of the flow rate and channel size on the boiling flow pattern inside a T-shaped microchannel were experimentally observed and quantitatively described. The results indicate that the occurrence position of the vaporization core gradually migrates from branch channel to main channel as the wall temperature increases. The flow boiling at the bifurcation of the T-shaped microchannel mainly includes the extrusion fracture flow, bubble flow, plug-annular alternating flow and annular flow, in which the annular flow can be further divided into the intermittent annular flow and the stable annular flow. In addition, a high flow rate and small channel size can lead to the disappearance of the bubble flow, and the presence of the bubble flow delays the appearance of the annular flow.

A Physics-Informed Spatial-Temporal Neural Network for Reservoir Simulation and Uncertainty Quantification

Summary Surrogate models play a vital role in reducing computational complexity and time burden for reservoir simulations. However, traditional surrogate models suffer from limitations in autonomous temporal information learning and restrictions in generalization potential, which is due to a lack of integration with physical knowledge. In response to these challenges, a physics-informed spatial-temporal neural network (PI-STNN) is proposed in this work, which incorporates flow theory into the loss function and uniquely integrates a deep convolutional encoder-decoder (DCED) with a convolutional long short-term memory (ConvLSTM) network. To demonstrate the robustness and generalization capabilities of the PI-STNN model, its performance was compared against both a purely data-driven model with the same neural network architecture and the renowned Fourier neural operator (FNO) in a comprehensive analysis. Besides, by adopting a transfer learning strategy, the trained PI-STNN model was adapted to the fractured flow fields to investigate the impact of natural fractures on its prediction accuracy. The results indicate that the PI-STNN not only excels in comparison with the purely data-driven model but also demonstrates a competitive edge over the FNO in reservoir simulation. Especially in strongly heterogeneous flow fields with fractures, the PI-STNN can still maintain high prediction accuracy. Building on this prediction accuracy, the PI-STNN model further offers a distinct advantage in efficiently performing uncertainty quantification, enabling rapid and comprehensive analysis of investment decisions in oil and gas development.

Effect of fracture characteristics on groundwater flow adjacent to high-level radioactive waste repository

In the study, a 2-D numerical model-delineating the high-level radioactive waste repository surrounded by hosting granite-rock was designed to evaluate pressure build-up, elevated temperature, and fracture flow caused by heat generation at the repository. During the early stage (0.1 years), build-up pressure at the repository induced radial groundwater flow. On the other hand, buoyancy-driven vertical flow was dominant during the late stage (2,000 years). The pressure build-up and elevated temperature varied significantly in bentonite, excavation-damaged zone and granite, resulting in different groundwater velocity. Fracture flow in both single and intersecting fractures were also influenced by the pressure and temperature, but their impact varied depending on the geometrical properties (e.g. orientation, elevation, lateral position) of the fracture. In complex discrete fracture network models, build-up pressure in the early stage mainly generated preferential fracture flow through few pathways, while in the late stage, convection-circulating flow induced by the buoyancy was distinct. In the early stage, higher density and connectivity of fracture network did not lead to an increase in average fracture flow, whereas the circulating flow in the late stage became more frequent and larger. Especially, long fractures served as both preferential flow pathways in the early stage and conduits for circulating flow in the late stage. The circulating flow was highly dependent on the spatial distribution of the long fractures, causing a significant uncertainty of average fracture flow in the late stage. The low angle of intersection between fractures resulted in low connectivity that decelerated the average fracture flow in both stages.

Infill well deployment optimization based on a novel multi-well productivity model considering inter-well interference for shale gas reservoir

Optimizing the deployment of infill wells is significant for improving the recovery of shale gas reservoir. A multi-well productivity prediction model considering complex fracture morphology and flow interference between multiple-staged fractured horizontal wells is established in order to optimize the deployment of infill well. The cumulative production curves of three simultaneous wells in a shale gas area in Sichuan Basin are accurately predicted with a relative error less than 5.5%. The optimal deployment location and timing of infill wells were determined quantitatively to maximize the 20-year cumulative production of infilling well and parent wells. The results show that: The later the infill wells were deployed, the more production the two parent wells produced, and the less the infill well produced; When the spacing d 1,3 between parent well 1 and infill well is larger, the two wells produce more and the parent well 2 produces less; The optimal deployment time for infill well is 2.85 years; The optimal well spacing d 1,3 between parent well 1 and infill well is 450 m, and the optimal well spacing d 2,3 between parent well 2 and infill well is 350 m. The findings of this study can help for better understanding of improving shale gas EUR.

Heat transfer characterizations of water flow through a single fracture with integrating water dissolution effects for the Enhanced Geothermal Systems

The Enhanced Geothermal Systems (EGS) is the technology that generates connected fracture networks in the Hot Dry Rock (HDR) reservoir with the application of water as the working fluid for hydraulic fracturing and geothermal energy production. A good understanding of water flow behaviors in the single fracture is critically important because connected fracture networks are composed of individual fractures. It should be noticed that the dissolution reaction between the reservoir rock and the working fluid (water) will change the fracture surface directly and consequently affect the mass and heat transfer process of water through the rock fracture in the HDR reservoir. In this study, the heat transfer process of water flow in the single fracture by considering dissolution reactions between water and fracture surfaces of reservoir rock have been investigated under high pressure and temperature conditions. The experiments of dissolution reaction and convective heat transfer have been performed separately for different temperature conditions with the rock sample of 0.2 mm fracture aperture. The influences of reaction duration, mass flow rates and initial temperature have been presented and analyzed. It is found that mass flow rates and temperature have significant effects on measured temperature of fracture surfaces. The effects of dissolution reaction on fracture surface morphology exists but it is extremely small for heat transfer. Based on experimental and simulation data, a modified correlation that integrates reaction duration, mass flow rates and changes of thermophysical properties has been proposed and its maximum error that compares with experimental results is less than 7%. The results of this study provided an accurate correlation for simulations and characterizations of the Hot Dry Rock reservoirs, which is of great importance for geothermal energy production and optimization.

Emergence of Unstable Focused Flow Induced by Variable‐Density Flows in Vertical Fractures

AbstractFluids with different densities often coexist in subsurface fractures and lead to variable‐density flows that control subsurface processes such as seawater intrusion, contaminant transport, and geologic carbon sequestration. In nature, fractures have dip angles relative to gravity, and density effects are maximized in vertical fractures. However, most studies on flow and transport through fractures are often limited to horizontal fractures. Here, we study the mixing and transport of variable‐density fluids in vertical fractures by combining three‐dimensional (3D) pore‐scale numerical simulations and visual laboratory experiments. Two miscible fluids with different densities are injected through two inlets at the bottom of a fracture and exit from an outlet at the top of the fracture. Laboratory experiments show the emergence of an unstable focused flow path, which we term a “runlet.” We successfully reproduce the unstable runlet using 3D numerical simulations and elucidate the underlying mechanisms triggering the runlet. Dimensionless number analysis shows that the runlet instability arises due to the Rayleigh‐Taylor instability (RTI), and flow topology analysis is applied to identify 3D vortices that are caused by the RTI. Even under laminar flow regimes, fluid inertia is shown to control the runlet instability by affecting the size and movement of vortices. Finally, we confirm the emergence of a runlet in rough‐walled fractures. Since a runlet dramatically affects fluid distribution, residence time, and mixing, the findings in this study have direct implications for the management of groundwater resources and subsurface applications.

An efficient computational framework for height-contained growing and intersecting hydraulic fracturing simulation via SGBEM–FEM

An efficient computational framework to model the growth of intersecting height-contained hydraulic fractures is proposed. The fracture in solids is modeled by the classical theory of linear elastic fracture mechanics, whereas the injection flow in hydraulic fractures is treated as channel flow. The governing equations of fracture mechanics are formulated in terms of weakly-singular weak-form boundary integral equations, with the knowledge of the fracture surface geometry exploited to reduce the integral dimension. The fluid flow within the fracture is treated using finite element method as one dimensional flow in an arbitrarily curved channel. The symmetric Galerkin boundary element method and finite element method are coupled with the traction continuity condition. The model is then formulated in a variational form and augmented with Lagrange multipliers to prescribe displacement constraints on intersections of hydraulic fractures, while fluid constraints are enforced by a consolidation of associated pressure degrees of freedom. Pertinent numerical implementation techniques, including spatial and temporal discretizations, special crack tip element, crack propagation criteria and remeshing strategy, are presented. The proposed framework is verified by comparing to the numerical results from a three dimensional hydraulic fracturing simulator and then verified by comparing the stress intensity factors to the XFEM and Murakami’s results for both star shaped cracks and symmetric branched cracks. The verification shows the utility properly enforces the physical constraints on the intersections of fractures. Three numerical examples are shown to demonstrate the capability of the proposed computational framework modeling the intersecting hydraulic fracturing process in a single or separated height-contained pay zones.

Experimental and numerical investigation of Non-Darcy flow in propped hydraulic fractures: Identification and characterization

Non-Darcy flow occurs in all porous media. The phenomenon of non-Darcy flow is particularly evident in propped fractures. It is crucial to understand the causes of non-Darcy flow, identify its onset, and characterize it quantitatively. We first observed the phenomenon of non-Darcy flow in propped fractures through laboratory experiments. Subsequently, 75 pore-scale propped fracture models with different porosities, apertures, and proppant particle sizes were established, and 2850 simulations at different pressure gradients were performed to identify and quantitatively characterize the non-Darcy flow in propped fractures. Calculation of the non-Darcy coefficients of various regions in the numerical model revealed that non-Darcy flow arises due to flow resistance. A model for predicting the crucial Reynolds number based on the apparent permeability (Rec=ehkD−1β−1/99) was proposed to enable more accurate identification of the onset of non-Darcy flow, and the predicted results were in strong accord with the observed data. A more constitutive non-Darcy coefficient prediction model was developed to quantify the non-Darcy effect, which is expressed as β=408af−0.34s−0.76φ−4.44. This study explains the root cause of non-Darcy flow in propped fractures, identifies the occurrence of non-Darcy flow more efficiently, and accurately characterizes non-Darcy flow.

Pore-scale insight into the effect of porous walls on fluid flow in rough-walled fractures

This study investigates the influence of porous walls on fluid flow behavior in single rough-walled fractures. A series of direct numerical simulations (DNS) were conducted to examine fluid flow through rough-walled fractures surrounded by porous walls under various hydraulic gradients. The simulation results clearly demonstrate the progressively significant impact of porous walls on flow characteristics as porosity increases. Regression analyses indicate that the Forchheimer's law, which characterizes the nonlinear flow of rock fractures, remains valid in the presence of porous walls. Moreover, increased porosity enhances fluid exchange and leads to an increased ability for single fractures to transmit fluid. Two double-parameter equations were established to quantitatively characterize the effects of porous walls on hydraulic conductivity of the fracture and fluid exchange between porous walls and the fracture. Further analysis revealed that the angle between the fracture profile and main flow direction profoundly influences the inflow/outflow exchange between the fracture and its adjacent porous walls. Moreover, the pore-scale results indicate that it is unreasonable to equate the effect of porous walls with slip boundaries as suggested in previous studies. The results and findings of this study deepen our understanding of flow behavior at the interface between the rough-walled fracture and adjacent porous walls, and they serve as a basis for accurately simulating large-scale fracture networks that consider the hydraulic contribution of fracture walls.

Mechanism of strength deterioration of red sandstone on reservoir bank slopes under the action of dry–wet cycles

To investigate the micro-scale mechanism of strength deterioration under different times of dry–wet cycles, laboratory tests of physical properties, triaxial compression, X-ray diffraction (XRD) and scanning electron microscopy (SEM) were conducted on red sandstone on reservoir bank slopes. The research results showed that when the 5th dry–wet cycle ended, the dry mass and peak strength started to rapidly decline, while the porosity and saturated mass increased rapidly. In general, all of these behaviours become relatively stable when the number of cycles increased. Similarly, cohesion and internal friction angle changed most significantly from 0 to 10 cycles and then became stable. In addition, the physical expansion of the adsorbed water film and the dissolution and expansion of mineral particles increased the number of internal microcracks and pores and the porosity and saturated mass of the sample. In addition, the pore fluid effect and fracture flow effect made the microcracks in the red sandstone grow and connect; thus, the peak strength of the sample decreased. Moreover, during the dry–wet cycles, the change in the relative content of mineral particles and the pore fluid effect weakened the clay cementation, and then the dry mass and cohesion of the samples decreased. The research reported in this paper will play a very significant role in the scientific analysis of slope stability in the Three Gorges Reservoir area.