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英文原文The new three-dimensional subsidence influencefunction denoted by nkgC. Gonza lez Niciezaa, M.I. A lvarez Ferna ndezaA. Mene ndez D azb, A.E. A lvarez VigilcAbstract：This study presents a three-dimensional development of the n k g influence function with the aim of predicting subsidence phenomena and characterizing the shape and dimensions of the corresponding trough. The parameters “n” and “k”characterize the ground and “g” is related to the gravity. This function depends on two physical concepts: the first is gravity, which characterizes the forces acting on the ground, and the second, the convergence of the roof and floor of the mine workings due to the stress state of the ground. Caving in of the roof generates direct subsidence, and the swelling of the floor, indirect subsidence, which allow us to establish the shape of the trough.The physical concepts introduced are fundamental in the mathematical implementation of the n k g influence function, allowing a more intuitive interpretation of the subsidence trough and notably facilitating the work of calibration, validation and sensitivity analysis. These concepts likewise allow the scope of application of influence functions to be extended to non-horizontal seams, also taking into account the quality of the rock mass and the presence of preferential sliding directions, in both the roof and the floor of the seam.In the development of this paper, we shall first see the physical concepts considered, to then present the three-dimensional implementation of our n k g influence function. We shall see the results obtained when calibrating the proposed numerical model with real data obtained from subsidence measurements in a coalmine in the Coal Basin of Asturias, situated in the North of Spain.Keywords：Subsidence; Influence functions; Underground mining1. IntroductionGround subsidence is the movement of the ground due to the loss in sustaining capacity of the sub-soil. In the majority of cases, this is directly related to human activities that either alter the water table or when carrying out deeppit mining of ore. The ground that is above the excavated area is altered, collapses, is partially compacted and progressively subsides as the work of extraction advances.We may speak of two types of mining subsidence depending on its extension. On the one hand, there is localized subsidence, which is concentrated in areas in the proximity of the excavated area and takes the form of a highly localized abrupt depression that is limited in extension. Extensive subsidence, on the other hand, results in the formation of a topographic depression on the surface (subsidence trough) that is more or less regular in shape and which is directly related to the depth of the excavated areas being exploited in the mine. In these cases, the subsidence is large in the central area and decreases progressively towards the sides, said subsidence being accompanied by horizontal displacement.The movements of the ground produced by these two types of subsidence may produce damage to buildings,roads, railway lines, oil pipelines or any other infrastructure in the surrounding areas. To estimate,quantify and prevent this damage, models must beestablished that allow us to determine a priori issues such as what the greatest vertical displacement of the surface of the ground will be and where it will be produced, or what the lateral extension of said subsidence will be.To answer these questions, different numerical methods have been developed in recent years 1 that include methods based on influence functions 2, cross section functions 3, empirical formulas 4, or numerical models based on finite elements or finite differences 5,6.The application of any of these numerical methods to real cases has to be accompanied by the development of three processes: calibration of real data, validation andsensitivity analysis.Calibration is the process of configuring a numerical model on the basis of the input data on the subsidence registered in the field. It implies executing several models, changing the parameters that characterize the subsidence until the model simulates the real behavior of the ground with the appropriate degree of reliability.Validation is the process of confirming the goodness of the calibration, considering an independent data set to that used as input, which has been obtained subsequent to this calibration.Finally, sensitivity analysis consists in varying the parameters of the proposed mathematical models, with the aim of estimating how this affects the results. This allows us to determine the most significant parameters and to predict the subsidence that will be produced in the future.Having established these steps, we shall now go on to develop the approach that we have adopted in our work, initially describing the theoretical basis of our method and subsequently applying it to a series of real cases.2. Approach and scope of the studyThe study presented here includes the introduction of a series of physical concepts that enable ground subsidence to be studied on the basis of its real behavior and of how the excavation carried out underground is transmitted to the topographic surface in the form of subsidence. Taking the methods that use n k influence functions as a reference 3, we generalize said functions with the aim of obtaining a mathematical model thatdescribes the subsidence trough as precisely as possible.To do so, the effect of two concepts that have not been taken into account in the traditional analysis of influence functions will be introduced. The first is gravity, which characterizes the forces acting on the ground, and the second is the convergence of the mine workings due to the stress state of the ground, a geometrical concept that when applied to the roof and floor of the mine workings allow us to define the shape of the expected subsidence trough by breaking the total subsidence down into two types of subsidence: direct subsidence (due to the convergence of the roof) and indirect subsidence (due to the convergence of the floor).These two concepts are fundamental, as they enable the phenomenon of subsidence to be interpreted in a more intuitive way. This will distinctly facilitate the calibration, validation and sensitivity analysis of the phenomenon of subsidence under study.On the other hand, the method developed in this paper enables the subsidence of seams of any slope whatsoever to be explained, even totally vertical seams, thus avoiding the limitation of n k influence functions. Previously, these had only been used to predict the subsidence generated by the mining of horizontal or semi-horizontal mine workings (with slopes of below 301).We shall call this new function the generalized n k g influence function (“g”of gravity), in the sense that it justifies the formation of a double subsidence trough, at the same time as being applicable to both horizontal and vertical slopes.Firstly, we shall introduce the physical concepts on which our interpretation of subsidence is based to then go on to formulate these concepts mathematically in a new generalized influence function. We shall then see its application to the simulation of sub-vertical seams, and finally we shall present the results of our model in a particular case corresponding to the Coal Basin of Asturias (situated in Northern Spain).3. The physical concept of subsidence caused by excavationWhen carrying out underground excavation, the stress state of the ground is modified and a series of movements appear in the rock mass, directed towards the interior of the excavated cavity. Said movements have a preferential direction that depends on the free faces and on the direction of gravity.When speaking of longwall coal mining, the geometry of said cavities, in which the thickness of the seams is small in relation to their length and width, the displacements of the wall rock, though important, are not found to be representative with respect to thephenomena of roof and floor convergence. Taking a horizontal coal seam of this type as a reference, the alteration of the stress state will cause movements of the ground in a vertical direction, downwards from the roof and upwards from the floor.The direction of these movements depends on two effects:A stress effect in the perimeter of the excavated cavity, the direction of which is determined by the normal unit vector ns of the free face of the excavation and directed towards the cavity.A gravitational effect, the direction of which is determined by a vertical unit vector ng; parallel to gravity.Let nsr be the normal vector ns to the free face in the roof, and nsf the normal vector ns in the floor. From here on, we shall use the subscripts r and f to refer to scalar or vectorial values in the roof and floor of the seam.As can be seen in Fig. 1, both vectors coincide in their direction and sense in the roof of the seam. We may thus say that their effects are summed, and therefore the movement of the ground situated in the roof of the cavity will be vertical and downward. In the floor of the cavity, however, the vector nsf and the vector ng have the same direction but the opposite sense,and hence their effect is counteracted, both effects possibly canceling each other out if the gravitational and stress effects are similar. Thus, the magnitude of thedisplacements will necessarily be much lower in the floor than in the roof.The caving in of the roof results in a new cavity, though now above the previous one. Similar phenomena to those described above will again be produced in the new cavity, such that the caving in of the roof of this cavity will in turn once more generate another cavity situated above it, which will be progressively transmitted upward, possibly reaching the topographic surface, where subsidence appears. This situation is analogousto the movement of a bubble (the cavity) in a fluid (the ground), in which an upward push acts on the bubble.The process that triggers subsidence may thus be described by means of two interrelated phenomena:The movement of the ground towards the cavity, with a downward trajectory due to the combination of the effect of stresses perpendicular to the cavity and the action of gravity.The migration of the cavity towards the topographic surface, which follows an identical direction to the movement of the ground, though in the opposite sense.Throughout this upward migration, the original cavity suffers a decrease in size because of swelling processes in the ground (see Fig. 1). This decrease in size may be sufficiently significant for the cavity to stabilize at some point during its upward migration and stop ascending, in the same way that the viscosity of a fluid checks the ascent of a bubble.Therefore, once the excavated cavity is produced, the ground surrounding it collapses and caving in progresses vertically until possibly reaching the topographic surface (see Fig. 2), in which case subsidence of the ground is produced.The cavity is transmitted both vertically and horizontally in such a way that the maximum subsidence occurs at point Q, while it will decrease progressively at the adjacent points, thus forming the subsidence trough shown in Fig. 2.In the case of the movements of the floor of the mine workings, there initially exists a tendency for the cavity to progress downward, as a result of the stress state inthe surrounding area. However, this effect is counteracted by gravity, which impedes the development of this tendency, rapidly checking it. Let us now consider a non-horizontal seam such as the one represented in Figs. 3a and b (the representation of the subsidence in this figure is magnified so as to be able to appreciate its shape, the same criterion being applicable to the other figures). The aforementionedeffects intervene in the caving in of the roof: on the one hand, the tendency of the ground to move in a perpendicular direction to the roof (stress effect), and on the other, the tendency to do so vertically (gravitational effect). Consequently, the resulting trajectory is no longer vertical, but is a result of the combination of the vectors that represent the two movements, as represented in Fig. 3a. The vector that marks said trajectory has been denominated -sr and is given by:= + . , rsnrg(1)where the subscript r refers to the roof of the seam and mr is a gravitational weighting factor constant that represents the relative importance of both effects on the roof of the seam.To explain what occurs in terms of the progression of cavities, it must be borne in mind that the ascending cavity follows the direction marked by the vector sr;though in the opposite sense. The caving in of the roof into the original cavity (cavity 0) originates a new cavity(cavity 1), situated in the direction marked by -sr: Likewise, as the roof of cavity 1 collapses, a new cavity 2 will appear in the direction s-1r : This new cavity will be situated closer to the surface and in a new direction n-1sr that becomes progressively more parallel to ng: The upward progression of the cavity is no longer a straight trajectory, its direction being corrected as it ascends. The effect of gravity makes the ascent vector s-ir tend progressively to a vertical orientation. Thus, the cavity will follow a parabolic trajectory such as that represented in Fig. 3b.This trajectory is similar to the one that a bubble would follow in a fluid when a horizontal component is initially given to its movement and corresponds to a parabola whose axis is parallel to ng: At least initially, the effect of this trajectory of non-vertical ascent of the cavity is that the surface subsidence originated by the caving in of the cavity roof is not situated over the vertical of the seam, but is laterally displaced.Once the subsidence reaches the surface, the displacements that it generates are distributed throughout the affected area, as represented in Figs. 3a and b, in the form of a trough. The maximum displacements are produced at a point that will be denominated Qr, which is the point of intersection between the parabola that defines the trajectory of the cavity and the topographic surface. The subsidence decreases, along all the directions of the topographic surface, as the distance to the point Qr increases, tending to zero when this distance is sufficiently large.Said point on the surface, Qr, will therefore suffer a displacement whose magnitude will be determined by the maximum of the distribution of subsidence, the value w(Qr) represented in Fig. 3b, and whose direction is defined by the tangent vector to the upward parabolic trajectory of the cavity at the point Qr.Moreover, as a consequence of the position of the seam, the possibility exists of the cavity progressing from the floor, i.e. caving in of the floor may be produced in a similar way to that of the roof. This situation is schematically represented in Fig. 4a, inwhich the trajectory of migration of the cavity is defined by -sf := + . fsfnfg(2)Therefore, the mining of an inclined seam may originate two subsidence troughs: one corresponding to the movements induced in the roof of the seam, which will be denominated from here on direct subsidence or roof subsidence; and the other, corresponding to themovements that take place in the floor, which will be called indirect subsidence or floor subsidence.It can be seen in Fig. 4b that if point A represents the center of a small cavity being excavated in the mine workings and point Q is the vertical projection of A on the topographic surface, said cavity produces a maximum direct subsidence at point Qr and a maximum indirect subsidence of the floor at point Qf, distributed asymmetrically with respect to Q. What is more, as the roof cavity and the floor cavity have different lengths,they may be checked differently and hence the magnitude of both types of subsidence may differ markedly. In general, floor subsidence will be greater than that of the roof, since the length of the cavity is greater.As well as the length of the trajectory, which will condition the degree to which the cavity will be checked in its ascent, the magnitude of the vectors sr and sf also influence the magnitude of the subsidence. In effect, in the roof of the seam, the vectors ng and nsr present the same sign, which means that the total vector will have a greater magnitude that in the floor, where ng and the normal vector to the seam, nsf ; present vertical components of the opposite sign, thus being subtractive the total vector (see Fig. 4). The extreme case arises in a horizontal seam, in which the two vectors in the floor, which are vertical and opposite in sign, totally counteract each other. In contrast, in a vertical seam, the composition ng and ns in the roof and floor gives rise to two symmetrical vectors, of equal magnitude.The relative positions of ng and ns in the roof and floor of the excavated mine workings not only condition the magnitude of the vectors sr and sf ; and therefore the maximum value of surface subsidence, but also where this is produced. As the vertical component of sf is smaller than that of sr; the parabolic trajectory of the cavity in the floor is more horizontal than that in the roof, the tendency to a vertical orientation of sf is slower, and the parabola cuts the topographic surface at a point Qf further from point Q than point Qr.When takes the value zero, the gravitational effect is eliminated, and hence the rupward trajectory of the cavity would now not be a parabola, but rather a straight line perpendicular to the seam. If mf is zero, the trajectory will never reach the topographic surface and hence no subsidence is generated.Values of mr and mf greater than zero permit this straight trajectory to be modified towards a parabola and mark the speed at which said parabola tends to a vertical orientation. As a result, the greater the gravitational parameters, the closer the point of maximum subsidence will be to the vertical projection of the center of the seam.If the ground is homogeneous and isotropic in the roof and floor of the excavated cavity, then we shall have = : However, these hypotheses are not fulfilled in the rfmajority of real cases.中文译文被 nkg 指示的新的三维下沉影响功能摘要：这种学习提出一种关于 n k g 影响功能的三维发展，其目的在于预言下沉现象和使相应的水槽的形状和尺度的特征。具有特征地面参数“n”和“k”和 “g”与重力有关。这种功能依赖于两个物理的概念： 第一是重力，这使作用于地面的力量具有的特征。第二，开采工作面的顶板和底板的变形取决于地面的压力状态。在中开掘巷道的顶板产生直接的下沉，和底板的增大，间接的下沉，这允许我们建立水槽的形状。物理的概念介绍在数学的执行n k g影响功能中基本原则，允许更多对下沉水槽进行直觉的解释和显著地促进校准，有效和灵敏性分析的工作。这些概念同样地允许应用被延伸到多水平裂隙的影响功能的范围，也考虑在裂隙的顶板和底板中的大块岩石的质量优惠的滑行方向。在本文的发展中，我们将首先看见物理的概念然后考虑提出对我们的 n k g 影响功能进行三维的陈述。我们将看见从位于西班牙的北部 Asturias 的煤盆地中的一种煤矿中的下沉测量方法获得的真实的数据，通过数字模型校准获得的结果。关键词： 下沉； 影响发挥作用； 地下的采矿1、简介地面下沉在丧失次土壤的支撑能力时发生的地面运动。在大多数情况下，这直接与任一个改变地下水有关的人类活动或者实行深部采矿。在开采空间之上的地面被改变，崩溃，部分地被压实，并且随着工作面的推进而显著。我们可能谈论依赖于其延伸的两类采矿下沉。一方面，在那里被定位下沉，这在开采空间附近区域上被集中或在延伸时的一种高度定位、突然和有限的下沉形式。广泛的下沉，另一方面，正常的的地表（下沉水槽）下沉的形成或多或少直接与在矿上所使用的开采深度有关。在这些情况下，中心的区域下沉最大并且朝着四周减少，说明下沉陪伴水平移动。被这两类下沉产生的地面移动可能产生对周围区域的建筑物，道路，铁路，输油管或者任何其它基础设施的损坏。为量化并防止这种损坏，建立模型并允许我们确定一个诸如 priori 问题，比如最大的下沉是多少？在哪里？或者下沉将如何向四周延伸，。回答这些问题，不同的数字方法在最近几年得到发展 1 包括基于影响功能的方法 2 ，通过部分功能 3 ，经验性的公式 4 ，或者基于有限的要素和差异的数字模型 5,6 。这些数字的方法都陪伴三个过程的发展： 真实的数据，有效分析和灵敏性的校准。校准是输入在登记的领域中下沉时的数据的基础上构成一个数字模型的过程。它意味着执行几个模型，使下沉具有特征的参数，直到模型以可靠性、适当的模拟地面的真实的行为。有效是确认校准的过程，在获得校准之后考虑对被用作输入的一个独立的数据组。最后，灵敏性分析在于不同数学模型的参数，目的在于如何估计这种影响结果。这允许我们确定最重要的参数并且预言将在未来产生的下沉。已建立这些步骤，我们现在将继续发展在我们的工作中已采取的门径，描述一系列的真实的情况，在我们的理论基础上运用它。2论文的概括和范围在这里论文提出在真实的情况下，学习如何控制地铁下开采的下沉和传送到地表的下沉的一系列概念。采取参考使用 n k 影响作用的方法 3 ，我们总结说，功能目的在于获得尽可能精确描述下沉水槽的一个数学的模型。这样做，被考虑尚未影响功能结果的传统的分析中的两个概念将加以介绍。第一是重力，这具有作用于地面的力量，第二地面的压力状态是因为开采工作面的压缩， 一个几何学的概念运用于开采工作面的顶板和底板时，允许我们通过打破整个的下沉，把期待的下沉水槽的形状定义到两类的下沉中： 直接下沉 (因为顶板 )的压缩和间接下沉 (因为底板) 的压缩。当它们实现按照一种更多直觉的方式被解释的下沉的现象时，这两个概念是基本的，这将清楚促进学习下面的下沉的现象的校准，有效分析和灵敏性。另一方面，在本文中被解释实现无论什么任何斜坡，甚至完全垂直的裂隙的下沉方法，从而避免 n k 影响功能的限制。以前，这些仅仅用来预言被水平的采矿产生的下沉
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