Interface Velocity Research Articles

Dissolution of Ti–N inclusions in liquid titanium during electron beam melting

The dissolution of Ti–N solids in liquid titanium was investigated using experimental and numerical methods. To characterize the dissolution rate and the transport phenomena associated with the dissolution process, cylindrical Ti–N rods were immersed into a molten pool of commercially pure titanium produced using an electron beam button furnace. Tests were conducted for various immersion times to observe the evolution in the rod tip profile and measure the volume loss. Generally, the volume fraction of dissolved solid increases with increasing immersion time. A numerical model has been developed to describe the experimental process and aid in understanding the dissolution process. Overall, the predicted volume loss for different immersion times agrees well with the experimental measurements. An effective mass transfer coefficient, in the range of 4.2 × 10−5 to 4.9 × 10−5 m/s, has been derived based on the model results for the experimental conditions examined (e.g. interface velocities and temperatures in the range of 0.09 – 0.11 m/s, and 1766 – 1778 °C, respectively).

Transient Taylor\u2013Dean flow in a composite annulus with porous walls partially filled with porous material

The sole aim of this article is to examine the relative contribution of suction/injection parameter on Taylor–Dean flow in a composite annular gap partially filled with porous material. In the present setup, the Newtonian fluid flow is induced by the circumferential motion of both cylinders and pressure gradient imposed in the Azimuthal direction. The mathematical model governing the flow is rendered dimensionless using appropriate dimensionless quantities transformed using the Laplace transform technique. Using suitable Ansatz, the equation is reduced to the Bessel differential equations and solved. The solution of converted to the time domain using a well-known numerical scheme known as the Riemann-sum approximation. The variation of the Newtonian fluid for different flow parameters is presented graphically. The solution method is validated by obtaining the steady-state solution and also using the implicit finite different approach (IFD); comparison of the methods is depicted in tabular form (see Tables 1, 2). It is deduced generally that the Newtonian fluid is higher when injection at the outer cylinder except when Da is small also higher interfacial velocity can be achieved by taking positive value of beta.

Estimating bubble interfacial heat transfer coefficient in pool boiling

Interfacial heat transfer coefficient (IHTC) is one the most deterministic factors of nucleate boiling characteristics in pool boiling. IHTC and bubble dynamics have a reciprocal effect on each other, which makes accurate estimations of IHTC even more challenging, but important for nucleate boiling studies. In this study, by considering interfacial heat resistance, micro-layer heat transfer and conjugate heat transfer, a comprehensive model that is capable of simulating phase change phenomena and IHTC has been developed and a modified form of Volume of Fluid method (VOF) is used to simulate pool boiling phenomena. In this study, three major challenges in simulating bubble dynamics and related phase change heat transfer using a VOF method have been successfully addressed to achieve more accurate simulations. First, spurious currents resulting from approximating curvature in VOF and interface diffusion have been minimized by introducing curvature smoothing and employing a simplified coupled level set volume of fluid method (S-CLSVOF). Second, simulation instability due to the concentration of source terms on the interface has been addressed by smearing source terms around the interface. Third, micro-layer thickness, which is noticeably smaller than computational cells, has been calculated based on a linear depletable micro-layer method. Effects of three-phase contact line were adopted by introducing a dynamic contact angle model and temperature variation inside the heater. The model shows great agreement with available experimental and numerical results in nucleate boiling of water and R113 in terms of bubble growth and departure rate, frequency and temperature distribution, and micro-layer thickness. The new model developed in this study simulates IHTC in two water cases and its relationship to interfacial relative velocity.

Evolution of the mixed-mode character of solid-state phase transformations in metals involving solute partitioning

Abstract Partitioning phase transformations in the solid state are principally subjected to two processes that take place: the redistribution, through long-range diffusion, of the partitioning element, and the lattice transformation taking place at the interface. Consequently, the usual approximation to consider one of these two processes as controlling the rate of the phase transformation is of limited accuracy. For a more accurate description, the so-called mixed-mode character of partitioning phase transformations is to be taken into account. In the present study, it is shown that the mixed-mode character can be quantified and that it has a significant effect on the kinetics. By means of examples involving either substitutional (Mo in Ti) or interstitial (C in Fe) partitioning elements, it is shown that a gradual change of the character of the transformation occurs during the phase transformation, shifting from initially interface-controlled (which implies the largest interface velocity) towards more diffusion-controlled.

An analytical theory for the forced convection condensation of shear-thinning fluids onto isothermal horizontal surfaces

We present an analytical theory for laminar forced convection condensation of saturated vapor on horizontal surfaces. The condensation produces shear-thinning film moving downstream due to the viscous-drag occurring at the vapor-liquid interface. The mathematical model is built based on a few input parameters, viz. power-law index (n), nondimensional film-thickness (ηδ,l), Prandtl number (Pr), and inertia number (Mc). A set of output parameters is used to analyze the distinct characteristics of the fluid-flow and condensation, viz. the condensate’s nondimensional mass flow rate (m^), Nusselt number, specific enthalpy ratio (Rh), thermal retention coefficient (Θ), and nondimensional wall-shear stress (τ^w). We have identified the subtlety of shear-thinning film flow when liquid’s thermophysical properties vary according to the changes in wall-shear and the interfacial drag. Contextually, we illustrate that a rise in shear-thinning effect (obtained by decreasing n), keeping Mc and ηδ,l fixed, results in a decrease of m^, Rh, τ^w, and (1/Θ). We have demonstrated that for a fixed Rh, a shear-thinning film, compared to the Newtonian film, would exhibit a greater ηδ,l but a smaller interfacial velocity (fi′). Furthermore, a greater film thickness is required for low Pr liquids to attain the same degree of subcooling compared to high Pr liquids. We perform systematic investigation over a wide range of ηδ,l. We observe that for small ηδ,l values, the vapor boundary-layer moving onto the liquid-film exhibits similar flow-features as found in the well-known Blasius boundary-layer. Conversely, at large ηδ,l, the present solution would remarkably differ from the Blasius solution. Finally, we establish an approximate theory for small ηδ,l motivated by the linearity in the cross-stream variations of velocity and temperature within the thin-film. This approximate theory gives rise to analytical correlations for Rh, Θ, τ^w, and m^, which would be useful for engineers.

Induced side-branching in smooth and faceted dendrites: theory and phase-field simulations.

The present work is devoted to the phenomenon of induced side branching stemming from the disruption of free dendrite growth. We postulate that the secondary branching instability can be triggered by the departure of the morphology of the dendrite from its steady state shape. Thence, the instability results from the thermodynamic trade-off between non monotonic variations of interface temperature, surface energy, kinetic anisotropy and interface velocity within the Gibbs-Thomson equation. For the purposes of illustration, the toy model of capillary anisotropy modulation is prospected both analytically and numerically by means of phase-field simulations. It is evidenced that side branching can befall both smooth and faceted dendrites, at a normal angle from the front tip which is specific to the nature of the capillary anisotropy shift applied. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Temporal reverse osmotic salt filtration mechanism of multi-layered porous graphene

Reverse osmosis (RO) technology is currently the most progressive, energy-saving and efficient membrane separation technology . Meanwhile, graphene becomes a promising candidate for fabricating the RO membranes in water desalination due to its high salt rejection and water flux. The concept of “temporal selectivity” is first proposed in our previous work in terms of the time difference between the penetration time of an ion passing through the pore and the tangential slipping time for the ion sliding across the pore. Nevertheless, the temporal selectivity mechanism of multilayered graphene membrane remains ambiguous. In this paper, the RO process of saltwater through porous graphene column RO membrane is studied by using molecular dynamics (MD) simulations method, and the effects of rotating angular velocity and the thickness of RO membrane on desalination performance of seawater are considered first. The MD results show that the salt rejection increases with the rotation speed of porous membrane increasing while the water flux initially increases and then decreases . Meanwhile, the interfacial slip velocity increases linearly with angular velocity increasing. On the other hand, the increasing thickness of porous graphene membrane can enhance the selectivity and reduce the permeability of water molecules. As expected, the tri-layered porous graphene RO membrane can achieve high salt rejection at low interfacial slip velocity. In order to ensure high selectivity and energy conservation and efficient, the pore structure of the porous graphene RO membrane is optimized. The results show that the optimized nanopores can increase the water flux significantly, whereas the salt rejection is not changed appreciably. It is found that the pore size of the innermost layer membrane near the feed region has the most significant effect on the water flux. The water flux increases sharply with the increase of pore diameter and the salt rejection remains totally higher than 80%. Moreover, the RO membrane with a special Type 3 structure exhibits excellent performance in seawater desalination, specifically, the ultrahigh water flux reaches 20029 L·cm<sup>–2</sup>·d<sup>–1</sup> and the super salt rejection arrives at 94%. The research results further clarify and verify the mechanism of the temporal selectivity in RO process, and improve the water flux under the condition of the same membrane thickness by designing gradient hole. The findings can conduce to the in-depth theoretical understanding of porous graphene-based membranes and designing and developing the large-scale seawater desalination devices and water filtration equipment.

THERMO CAPILLARY CONVECTION WITH DIFFUSIVE-TYPE EVAPORATION IN A THREE-DIMENSIONAL CHANNEL UNDER THE CONDITIONS OF COMBINED THERMAL LOAD

Three-dimensional generalization of the Ostroumov-Birikh solution of the classical convection equations has been obtained in order to model two-layer flows under the conditions of liquid-vapor phase transition of the diffusive type at the interface. A liquid and gas-vapor mixture fills an endless rectangular duct subjected to the action of a longitudinal temperature gradient and a transversely directed gravity field. The exact solution is interpreted as a solution describing the flows on the working section in a sufficiently long cuvette. The distinguishing characteristics of evaporative convection in the working systems like the ethanol-nitrogen and HFE7100-nitrogen are investigated in the case of different types of the boundary temperature regimes imposed on channel walls. Here, the thermal load is applied on a substrate, according to the linear law with respect to the longitudinal coordinate, whereas the upper and lateral walls are thermally insulated. The longitudinal temperature gradients and transverse temperature drops in the fluid systems, caused both by external thermal load and evaporation processes at the interface, lead to the formation of vortex structures with a complex symmetry and a peculiar vapor concentration field. The flow topology essentially differs with respect to the intensity of the thermal load and type of the liquid coolant. The characteristics of evaporative convection modes, including the evaporative mass flow rate and maximum interfacial temperature and velocity, are analyzed. The study of the characteristic features appearing upon boundary heating allows one to determine flow control mechanisms and to forecast possible forms of instability.

A semi-analytical model based on unsteady conductive heat transfer in steam-assisted gravity drainage (SAGD)

Steam-assisted gravity drainage (SAGD) is one of the most successful in-situ recovery methods for bitumen reservoirs. Seeking the function of oil production with parameters of steam injection is significant for rapid performance evaluation of the SAGD process in fields. In this study, a new semi-analytical model based on unsteady conductive heat transfer is developed to estimate SAGD performance and describe the evolution of the steam chamber synchronously. These are implemented by interface velocity of steam chamber, as the function of the parameters of steam with the exponent-shape assumption and energy balance to link the oil output and chamber growth. It is worth mentioning that the equation of oil production is obtained based on the assumption of unsteady temperature distribution along with the interface, which is close to the real condition of the reservoir in fields. Then, the model is validated by using the Phase B, UTF project of the Athabasca reservoir. Finally, the impacts of steam injection parameters on SAGD oil production are discussed in this paper. Initial higher rate of heat injection to reach the production peak quickly and the subsequent lower rate for maintaining peak should be adopted to achieve the efficient target of oil production.

Prediction of the interfacial disturbance wave velocity in vertical upward gas-liquid annular flow via ensemble learning

Interfacial disturbance wave velocity is an important parameter for the study of momentum transfer between the gas core and the liquid film at the two-phase interface, which directly affects the calculation of frictional pressure drop. Since an exact analytical solution of the interfacial disturbance velocity cannot be derived by the two-phase flow theory equations, an ensemble learning framework for the disturbance wave velocity is constructed and a new model is proposed, which is appropriate for predicting different flow conditions in vertical two-phase flow. The dimensionless velocity-related parameters of interfacial disturbance waves are obtained by feature selection based on the interfacial shear force model, and the grid search method is equipped to tune the important parameters. By comparing the current and literature data prediction results, the extrapolation and applicability of the ensemble learning model are further verified. For the Extra Tree model, the Mean Absolute Percentage Error of the optimized Extra Tree model is less than 20%, and the relative measurement uncertainty is within ±25% for 95.67% of the results. It shows that the proposed ensemble learning framework provides a novel approach in the study of interfacial wave spatiotemporal parameters.

Remarkable enhancement and electronic mechanism for hydrogen storage kinetics of Mg nano-composite by a multi-valence Co-based catalyst

Transition metals are the traditional catalysts to improve the hydrogen storage performance of Mg. In the present article, the multi-valence Co catalyst (Co@CNTs) is prepared to improve the kinetics of Mg. The dehydrogenation temperature decreases, and desorption kinetics of MgH2 is significantly enhanced by the addition of the catalyst. The dehydrogenation of MgH2–Co@CNTs composite is a random bulk nucleation/surface and three-dimensional growth with constant interface velocity reaction. Although the dehydrogenation activation energy of MgH2–Co@CNTs composite is reduced, the enthalpy change and entropy change of the composite are similar to the raw MgH2, which means the dehydrogenation is mainly dissociation of MgH2. Besides, the cycle performance of MgH2–Co@CNTs composite is stable. The damping of the performance for the composite is negligible after 50 cycles, which benefits from the in-site formed Co3MgC0.5 in the composite. The Co3MgC0.5 can act as a ‘hydrogen pump’ to adjust the hydrogen storage performance of Mg. Theoretical studies show that multi-valence Co can trap H2 molecules and weaken their σ-bonds. In the first cycle, the bridging effect of Co plays an important role in the improvement of H2 dissociation and release. In the subsequent cycles, Co6C1 becomes the active site to assist the hydrogenation and dehydrogenation of the composite.

The effect of surfactant on the drag and wall correction factor of a drop in a bounded medium

Abstract In the present article, the analytical solution for creeping motion of a drop/bubble characterized by insoluble surfactant is examined at the instant it passes the center of a spherical container filled with Newtonian fluid at low Reynolds number. The presence of surfactant characterizes the interfacial region by an axisymmetric interfacial tension gradient and coefficient of surface dilatational viscosity. Under the assumption of the small capillary number, the deformation of spherical phase interface is not taken into account. The computations not only yield information on drag force and wall correction factor, but also on interfacial velocity and flow field for different values of surface tension gradient and surface dilatational viscosity. In the limiting cases, the analytical solutions describing the drag force and wall correction factor for a drop in a bounded medium reduces to expressions previously stated by other authors in literature. The results reveal the strong influence of the surface dilatational viscosity and surface tension gradient on the motion of drop/bubble. Increasing the surface tension gradient and surface dilatational viscosity, results in linear variation of drag force. When the surface tension gradient increases, the drag force for unbounded medium increases more as compared to the bounded medium hence wall correction factor decreases with increase in surface tension gradient whereas it increases with increase in surface dilatational viscosity.

Dipping into a new pool: The interface dynamics of drops impacting onto a different liquid.

When a drop impacts onto a pool of another liquid, the common interface will move down at a well-defined speed for the first few milliseconds. While simple mechanistic models and experiments with the same fluid used for the drop and pool have predicted this speed to be half the impacting drop speed, this is only one small part in a rich and intricate behavior landscape. Factors such as viscosity and density ratios greatly affect the penetration speed. By using a combination of high-speed photography, high-resolution numerical simulations, and physical modeling, we disentangle the different roles that physical fluid properties play in determining the true value of the postimpact interfacial velocity.

Transdimensional simultaneous inversion of velocity structure and event locations in downhole microseismic monitoring

In downhole microseismic monitoring, the velocity model plays a vital role in accurate mapping of the hydraulic fracturing image. For velocity model uncertainties in the number of layers or interface depths, the conventional velocity calibration method has been shown to effectively locate the perforation shots; however, it introduces nonnegligible location errors for microseismic events, especially for complex geologic formations with inclinations. To improve the event location accuracy, we have exploited the advantages of the reversible jump Markov chain Monte Carlo approach in generating different dimensions of velocity models and adopted a transdimensional Bayesian simultaneous inversion framework for obtaining the effective velocity structure and event locations simultaneously. The transdimensional inversion changes the number of layers during the inversion process and selects the optimal interface depths and velocity values to improve the event location accuracy. The confidence intervals of the simultaneous inversion event locations estimated by Bayesian inference enable us to evaluate the location uncertainties in the horizontal and vertical directions. Two synthetic examples and a field test are presented to illustrate the performance of our methodology, and the event location accuracy is shown to be higher than that obtained using the conventional methods. With less dependence on prior information, our transdimensional simultaneous inversion method can be used to obtain an effective velocity structure for facilitating highly accurate hydraulic fracturing mapping.

Web-Based Open-Source Tool for Isotachophoresis.

We present the development of a client-side web-based simulator for complex electrophoresis phenomena, including isotachophoresis. The simulation tool is called Client-based Application for Fast Electrophoresis Simulation (CAFES). CAFES uses the broad cross-browser compatibility of JavaScript to provide a rapid and easy-to-use tool for coupled unsteady electromigration, diffusion, and equilibrium electrolyte reactions among multiple weak electrolytes. The code uses a stationary grid (for simplicity) and an adaptive time step to provide reliable estimates of ion concentration dynamics (including pH profile evolution), requiring no prior installation nor compilation. CAFES also offers a large database of commonly used species and their relevant physicochemical properties. We present a validation of predictions from CAFES by comparing them to experimental data of peak- and plateau-mode isotachophoresis experiments. The code yields accurate estimates of interface velocity, plateau length and relative intensity, and pH variations while significantly reducing the computation time compared to existing codes. The tool is open-source and available for free at https://microfluidics.stanford.edu/cafes.

Two-Dimensional Transient Heat Transfer Model for High-Temperature Laser-Scanning Confocal Microscopy

Abstract High-temperature laser-scanning confocal microscopy (HT-LSCM) has proven to be an excellent experimental technique through in situ observations of high temperature phase transformation to study kinetics and morphology using thin disk steel specimens. A 1.0 kW halogen lamp within the elliptical cavity of the HT-LSCM furnace radiates heat and imposes a nonlinear temperature profile across the radius of the steel sample. When exposed at the solid/liquid interface, this local temperature profile determines the kinetics of solidification and phase transformation morphology. A two-dimensional numerical heat transfer model for both isothermal and transient conditions is developed for a concentrically solidifying sample. The model can accommodate solid/liquid interface velocity as an input parameter under concentric solidification with cooling rates up to 100 K/min. The model is validated against a commercial finite element analysis software package, strand7, and optimized with experimental data obtained under near-to equilibrium conditions. The validated model can then be used to define the temperature landscape under transient heat transfer conditions.

Effects of carbon segregation and interface roughness on the mobility of solid-liquid interface in Fe-C alloy: A molecular dynamics study

The kinetic coefficients (μ) for the fcc Fe-(0–0.5 wt%)C alloy solidification and melting were investigated using molecular dynamics simulations. The activation energy for the diffusion of C atoms (QC) at the solid-liquid interface was calculated using the Debye-Waller factor <u2> to reveal the effect of C atoms dragging on the interface mobility. The influence of C segregation and interface roughness on interface mobility was also investigated. Simulation results show that for melting, the C content has a minor effect on the μ, and the μ is mainly ranged 18.1–19.4 cm/s•K, while for solidification, the μ linearly decreases with the increasing C content, and the μ is ranged 9.6–17.9 cm/s•K. In addition, a ‘platform’ zone was observed under low undercooling, in which the interface velocity is close to zero and suggests weak interface mobility, resulting from the segregation and dragging of C atoms at the interface. The ‘platform’ zone size linearly increases with the increasing C content. The QC increases with the increasing C content, which is ranged 0.71–0.91 eV for the fcc Fe-(0.1–0.4 wt%)C alloy solidification, indicating the C atomic motion is weakened and the C atoms dragging is reinforced due to C content increasing. Interface roughness and C atoms distribution analyses show that for the fcc Fe-C alloy solidification, a smooth solid-liquid interface is unfavorable for interface mobility, and the smooth interface is usually accompanied by the non-uniform distribution of C atoms. Therefore, increasing the interface roughness may be helpful for improving interface mobility and segregation for alloy solidification.

Intrusive and non-intrusive two-phase air-water measurements on stepped spillways: A physical study

Self-aerated free-surface flow studies have a more recent history compared to classical fluid dynamics. Traditional velocimetry techniques are adversely affected by the presence of gas–liquid interfaces. In the present study, detailed air-water flow measurements were performed in a highly turbulent free-surface flow, and three velocimetry approaches were applied: (a) centreline dual-tip phase-detection needle probe measurements at all step edges downstream of the inception location of free-surface aeration; (b) Optical Flow (OF) data based upon ultra-high-speed video movies through the left sidewall; and (c) Optical Flow (OF) data based upon ultra-high-speed video movies overlooking the self-aerated flow, in a direction normal to the pseudo-bottom formed by the staircase profile. The study was conducted in a steep channel and three stepped invert geometries were tested. The results highlighted both advantages and limitations of the three complementary metrologies in free-surface flows with strong turbulence. The dual-tip phase-detection probe delivered reliable interfacial velocity data, but in the form of point measurements. The sideview OF technique provided a great level of details of the cavity recirculation and shear zone between mainstream and cavity, but the data were unreliable for void fractions>0.30 and only limited to the sidewall region. The top view OF technique characterised the surface velocity field across the entire chute width, highlighting the occurrence of three-dimensional air-water surface patterns. One observed limitation of the OF was the requirements of a high frame rate (i.e. 10,000 fps or more) and high-quality light source. Overall, these measurement techniques provided complementary results for a better understanding of the physical behaviour of highly turbulent multiphase flows on stepped channels.

High-Velocity Air–Water Flow Measurements in a Prototype Tunnel Chute: Scaling of Void Fraction and Interfacial Velocity

Aeration occurs in many natural and human-made flows and must be considered in engineering design. In water infrastructure, air–water flows can be violent and of very high velocity. To date, most fundamental research and engineering design guidelines involving air–water flows have been based upon laboratory scale measurements with limited validation at prototype scale with larger Reynolds numbers. Herein, unique measurements were conducted in high-velocity air–water flows in the tunnel chute of the 225-m-high Luzzone arch Dam in Switzerland. For each of the two test series, an array of 16 double-tip conductivity probes was installed in the circular tunnel chute of 3 m diameter and slope of ≈37° measuring void fraction, bubble count rate, interfacial velocity, and droplet sizes for four different discharges of up to 15.9 m3/s corresponding to Reynolds numbers of up to 2.4 × 107 and mean flow velocities of up to 38 m/s. Void fraction and interfacial velocity distributions, as well as design parameters such as depth-averaged void fractions and flow resistance, compared well with previous laboratory studies and empirical equations. The droplet chord sizes exhibited scale effects, and care must be taken if air–water mass transfer and droplet momentum exchange processes are assessed at the laboratory scale.

A Case Study on the Water-Oil Interface of Shunbei Oilfield Based on Dynamic Data

Shunbei Oilfield is characterized by substantial heterogeneity and a complex oil–water relationship. The water-oil interface is dynamically changing, and it is a crucial parameter for reserve calculation and evaluation. The main purpose is to analyze the effect of fluid flow in multi-scale media on the water-oil interface. It is well known that the fracture-cavity reservoirs have well-developed fractures and karst caves, and their distribution is complex in Shunbei Oilfield. This paper presents a way to simplify the fracture-cavity system first, then uses a unit of oil wells as a system to study the water-oil interface, which avoids impact on the water-oil interface due to oil production. A detailed step by step procedure for solving the semi-analytical solution of water-oil interface in a fracture-cavity reservoir by using an explicit algorithm and a successive steady-state method is presented. The solution can be used to investigate water-oil interface behavior. In this paper, we validated this method with the actual data for a relatively similar actual reservoir. Sensitivity analyses about the effects of the main parameters including production rates, cave volume and initial oil–water volume ratio on interfacial migration velocity are also presented in detail. The water breaking time of oil wells is fully investigated. The water-oil interface movement chart under different development conditions is established to predict the water-oil interface in the late stage of oil well production and extend the waterless developing period. Being based on this chart, a water breakthrough warning can be realized, and oil recovery can be improved. The findings of the research have led to the conclusion that the rising speed of water-oil interface is proportional to the production rate, on the contrary, it is inversely proportional to cave volume and initial oil–water volume ratio. As well production goes on, the water-oil interface rises at different rates. After the well is put into production for one year, the water-oil interface rises by 16.38%, 12.56% and 4.24% according to the condition that production rate is 10%, the initial oil–water volume ratio is 0.7, and the cave volume is 100 × 104 m3. This method is not only suitable for any period and any well type in the development of Shunbei Oilfield; it also has the function of calculating the real-time water-oil interface of a single well and multi-wells. This new method has the characteristics of easy calculation and high accuracy. The method in this paper can be further developed as it has great applicability in fracture-cavity reservoirs.