Mixed-type curves and the lightcone frame in Minkowski 3-space

2020 ◽  
Vol 17 (06) ◽  
pp. 2050088 ◽  
Author(s):  
Tongchang Liu ◽  
Donghe Pei

In this paper, we study mixed-type curves in Minkowski 3-space. Mixed-type curves are regular curves, and there are both non-lightlike points and lightlike points in a mixed-type curve. For non-lightlike curves and null curves in Minkowski 3-space, we can study them by a Frenet frame or a Cartan frame, respectively. But for mixed-type curves, the two frames will not work. As far as we know, no one has yet given a frame to study them in Minkowski 3-space. So, we give the lightcone frame in order to provide a tool for studying this type curves in mathematical and physical research. As an application of the lightcone frame, we define an evolute of a mixed-type curve. We also give some examples to show the evolutes.

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Xin Zhao ◽  
Donghe Pei

The evolutoid of a regular curve in the Lorentz-Minkowski plane ℝ 1 2 is the envelope of the lines between tangents and normals of the curve. It is regarded as the generalized caustic (evolute) of the curve. The evolutoid of a mixed-type curve has not been considered since the definition of the evolutoid at lightlike point can not be given naturally. In this paper, we devote ourselves to consider the evolutoids of the regular mixed-type curves in ℝ 1 2 . As the angle of lightlike vector and nonlightlike vector can not be defined, we introduce the evolutoids of the nonlightlike regular curves in ℝ 1 2 and give the conception of the σ -transform first. On this basis, we define the evolutoids of the regular mixed-type curves by using a lightcone frame. Then, we study when does the evolutoid of a mixed-type curve have singular points and discuss the relationship of the type of the points of the mixed-type curve and the type of the points of its evolutoid.


Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2852
Author(s):  
Xin Zhao ◽  
Donghe Pei

In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz–Minkowski plane R12. The pedal curve is always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curve. For a mixed-type curve, the pedal curve at lightlike points cannot always be defined. Herein, we investigate when the pedal curves of a mixed-type curve can be defined and define the pedal curves of the mixed-type curve using the lightcone frame. Then, we consider when the pedal curves of the mixed-type curve have singular points. We also investigate the relationship of the type of the points on the pedal curves and the type of the points on the base curve.


Author(s):  
P. Noverri

Delta Mahakam is a giant hydrocarbon block which is comprised two oil fields and five gas fields. The giant block has been considered mature after production for more than 40 years. More than 2,000 wells have been drilled to optimize hydrocarbon recovery. From those wells, a huge amount of production data is available and documented in a well-structured manner. Gaining insight from this data is highly beneficial to understand fields behavior and their characteristics. The fields production characterization is analyzed with Production Type-Curve method. In this case, type curves were generated from production data ratio such as CGR, WGR and GOR to field recovery factor. Type curve is considered as a simple approach to find patterns and capture a helicopter view from a huge volume of production data. Utilization of business intelligence enables efficient data gathering from different data sources, data preparation and data visualization through dashboards. Each dashboard provides a different perspective which consists of field view, zone view, sector view and POD view. Dashboards allow users to perform comprehensive analysis in describing production behavior. Production type-curve analysis through dashboards show that fields in the Mahakam Delta can be grouped based on their production behavior and effectively provide global field understanding Discovery of production key information from proposed methods can be used as reference for prospective and existing fields development in the Mahakam Delta. This paper demonstrates an example of production type-curve as a simple yet efficient method in characterizing field production behaviors which is realized by a Business Intelligent application


2021 ◽  
Vol 73 (09) ◽  
pp. 8-10
Author(s):  
Justin Hayes

If you talk to a typical subsurface professional working on unconventionals today (e.g., a reservoir engineer, completion engineer, geologist, petrophysicist, etc.) as I have in person and through media such as LinkedIn, you will find that many lament one key thing: Our sophisticated models have been reduced too much. Of course, I am generalizing and those are not the words they use; the lamentations come in many forms. The dissatisfaction with oversimplification is most easily observed as dis-taste for the type curve, the simplified model we use to predict upcoming new drills. (Yes, I know many of you will want to refer to them by their “proper” name: type well curve; I will be sticking with the colloquial version.) A simple meme posted on LinkedIn about type curves garnered one of the most engaged conversations I have seen amongst technical staff. The responses varied from something like “Thank God someone finally said this out loud” to comments such as “I don’t know anything better than type curves.” Most comments were closer to the former than the latter. What is even more remarkable is that our investors feel the same. In personal conversations, many of them refer to our type curves simply as “lies.” This perception, coupled with the historical lack of corporate returns, led investors away from our industry in droves. Many within the industry see it differently and want to blame the exodus on other factors such as oil and gas prices, climate change, competition from renewables, other environmental, social, and governance (ESG) issues, the pandemic, or OPEC’s unwillingness to “hold the bag” any longer. If you ask them, though, investors will tell you a simple answer: The unconventional business destroyed way too much capital and lied too much through the type curves. Why is it that both investors and technical staff are unhappy with our ability to accurately model future performance? Why can’t we deliver returns? The typical unconventional-focused oil and gas company has two models that are critical to the business. First is the subsurface model, with which we are all intimately familiar in its various forms, and the second is the corporate financial model, which is focused on cash flows, income, and assets/liabilities. It is unfortunate that the two models are separate. It means we must simplify one or both so they can communicate with each other. How can you observe this oversimplification while it is happening? It is happening when the finance staff say, “Please just give me a simple type curve and well count; I need to model, optimize, and account for debt/leverage, equity, and cash flows.” Meanwhile, the technical staff say, “Please just give me a CAPEX budget or a well count; I need to model, optimize, and account for well spacing, completion design, land constraints, and operational constraints.” Looking back, we know that the winner in this tug-of-war of competing needs was the type curve.


2019 ◽  
Vol 9 (1) ◽  
pp. 206
Author(s):  
Guofeng Han ◽  
Yuewu Liu ◽  
Wenchao Liu ◽  
Dapeng Gao

Pressure communication between adjacent wells is frequently encountered in multi-stage hydraulic fractured shale gas reservoirs. An interference test is one of the most popular methods for testing the connectivity of a reservoir. Currently, there is no practical analysis model of an interference test for wells connected by large fractures. A one-dimensional equation of flow in porous media is established, and an analytical solution under the constant production rate is obtained using a similarity transformation. Based on this solution, the extremum equation of the interference test for wells connected by a large fracture is derived. The type-curve of pressure and the pressure derivative of an interference test of wells connected by a large fracture are plotted, and verified against interference test data. The extremum equation of wells connected by a large fracture differs from that for homogeneous reservoirs by a factor 2. Considering the difference of the flowing distance, it can be concluded that the pressure conductivity coefficient computed by the extremum equation of homogeneous reservoirs is accurate in the order of magnitude. On the double logarithmic type-curve, as time increases, the curves of pressure and the pressure derivative tend to be parallel straight lines with a slope of 0.5. When the crossflow of the reservoir matrix to the large fracture cannot be ignored, the slope of the parallel straight lines is 0.25. They are different from the type-curves of homogeneous and double porosity reservoirs. Therefore, the pressure derivative curve is proposed to diagnose the connection form of wells.


SPE Journal ◽  
2012 ◽  
Vol 18 (01) ◽  
pp. 97-113 ◽  
Author(s):  
Ayala H Luis F. ◽  
Peng Ye

Summary Rate-time decline-curve analysis is the technique most extensively used by engineers in the evaluation of well performance, production forecasting, and prediction of original fluids in place. Results from this analysis have key implications for economic decisions surrounding asset acquisition and investment planning in hydrocarbon production. State-of-the-art natural gas decline-curve analysis heavily relies on the use of liquid (oil) type curves combined with the concepts of pseudopressure and pseudotime and/or empirical curve fitting of rate-time production data using the Arps hyperbolic decline model. In this study, we present the analytical decline equation that models production from gas wells producing at constant pressure under boundary-dominated flow (BDF) which neither employs empirical concepts from Arps decline models nor necessitates explicit calculations of pseudofunctions. New-generation analytical decline equations for BDF are presented for gas wells producing at (1) full production potential under true wide-open decline and (2) partial production potential under less than wide-open decline. The proposed analytical model enables the generation of type-curves for the analysis of natural gas reservoirs producing at constant pressure and under BDF for both full and partial production potential. A universal, single-line gas type curve is shown to be straightforwardly derived for any gas well producing at its full potential under radial BDF. The resulting type curves can be used to forecast boundary-dominated performance and predict original gas in place without (1) iterative procedures, (2) prior knowledge of reservoir storage properties or geological data, and (3) pseudopressure or pseudotime transformations of production data obtained in the field.


2011 ◽  
Vol 105-107 ◽  
pp. 1539-1546
Author(s):  
Chun Yan Zhao ◽  
Guo Yong Zheng

Based on dynamic triaxial test results of saturated soft clay, influence factors and variations on accumulated deformation were analyzed. The Parr’s equation on accumulated deformation was modified to create an attenuation-type curve model on accumulated deformation of saturated normal consolidation clay. In this model, dynamic strength was introduced and a new parameter called equivalent dynamic stress level was added. Besides, based on comparative analysis on variations between failure-type and attenuation-type curves, a failure-type curve model was created on accumulated deformation of saturated normal consolidation clay. The two models can take cycle number, coupling of static and dynamic deviator stress, consolidation way into consideration. The models were verified by test results. The correlation coefficients are more than 0.96 for optimization of test results based on the two models, and there is good agreement between optimized and test curves, which show that the two models are suitable to predict variations of accumulated deformation under different loading cases and consolidation ways. In order to improve prediction accuracy, it is suggested that loading cases and consolidation ways should be consistent with in-situ conditions when dynamic triaxial tests being used to determine the constants in the models.


2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Nemat Abazari ◽  
Martin Bohner ◽  
Ilgin Sağer ◽  
Alireza Sedaghatdoost ◽  
Yusuf Yayli

Abstract In this paper, we investigate the representation of curves on the lightlike cone $\mathbb {Q}^{3}_{2}$ Q 2 3 in Minkowski space $\mathbb {R}^{4}_{2}$ R 2 4 by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone $\mathbb {Q}^{3}_{2}$ Q 2 3 in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function $\kappa _{2}=0$ κ 2 = 0 , and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.


2017 ◽  
Vol 21 (1) ◽  
pp. 17 ◽  
Author(s):  
Wangang Chen ◽  
Yu Yang ◽  
Hansen Sun ◽  
Chengwei Zhang ◽  
Qin Wen ◽  
...  

To analyze the effects of the leakage recharge of the aquifer on the initial dewatering of coalbed methane wells, the mathematical seepage model of water in the coalbed considering the aquifer leakage was established by using the leakage coefficient according to the unsteady seepage theory. The model was solved after Laplace transform and the Stehfest numerical reverse inversion was used to obtain the solution in right space. Then, the log-log type curves of pressure and pressure derivative were created with new combinations of parameters. Based on the natural seepage mechanism, the influence of aquifer leakage on curve shape was judged. It is found that the radial flow ends earlier as the leakage coefficient increases. Moreover, it was proposed to obtain reservoir permeability, skin factor, and leakage coefficient by using type curve matching. The type curves are useful for quantitatively evaluating the level of leakage, thereby guiding the adjustment of the following production system for CBM wells. Curvas de solución y tipo para el modelo de filtración de capas carboníferas acuíferas con recarga de fugasResumenEste estudio estableció el modelo matemático de filtración de agua en una capa carbonífera al estimar la salida acuífera con el uso del coeficiente de fuga, de acuerdo con la teoría de filtración inestable, para analizar los efectos en la recarga de pérdida de fluidos de un acuífero en el drenado inicial para pozos de gas metano.  El modelo se resolvió tras usar la transformación Laplace y la inversión numérica Stehfest para encontrar la respuesta en el lugar indicado. Luego, se creó la representación algorítmica de la presión y la presión derivativa con nuevas combinaciones de parámetros. Se evaluó la influencia de la pérdida de fluido del acuífero en la forma de la curva con base al mecanismo físico de filtración. Se estableció que el flujo radial finaliza antes de que el coeficiente de pérdida de fluido se incremente. Además, se propone el uso de la curva tipo correspondiente para obtener la permeabilidad del reservorio, el factor de daño y el coeficiente de pérdida de fluido. Las curvas tipo son útiles para evaluar cuantitativamente el nivel de la pérdida de fluido, y de esta manera guiar el ajuste de un sistema de producción consecuente para pozos de gas metano de carbón.


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