基于影响函数法的倾斜煤层长臂开采地表沉陷预测外文文献翻译

英文原文An influence function method based subsidence predictionprogram for longwall mining operations in inclined coal seamsLUO Yi, CHENG Jian-weiDepartment of Mining Engineering, West Virginia University, Morgantown WV, USA Abstract: The distribution of the final surface subsidence basin induced by longwall operations in inclined coal seam could besignificantly different from that in flat coal seam and demands special prediction methods. Though many empirical prediction methods have been developed, these methods are inflexible for varying geological and mining conditions. An influence function method has been developed to take the advantage of its fundamentally sound nature and flexibility. In developing this method, significant modifications have been made to the original Knothe function to produce an asymmetrical influence function. The empirical equations for final subsidence parameters derived from US subsidence data and Chinese empirical values have been incorporated into the mathematical models to improve the prediction accuracy. A corresponding computer program is developed. A number of subsidence cases for longwall mining operations in coal seams with varying inclination angles have been used to demonstrate the applicability of the developed subsidence prediction model.Keywords: subsidence prediction; influence function method; inclined coal seam; longwall mining+ IntroductionLongwall mining in inclined coal seams has beena common practice in some major coal mining countries. It has long been reported that the characteristics of the final subsidence basin induced by long-wall operation in inclined coal seam are differentfrom that caused by mining in a level coal seam.Though many different methods have been proposedfor predicting final subsidence for inclined coal seam,most of them are empirical types including graphicaland profile function methods. The empiricalmethods are site specific and a method developed forsubcritical condition would not work for supercritical case and vise versa. In comparison, influencefunction methods are more flexible to adapt for thevariation from subcritical to supercritical underground extraction conditions. Influence functionmethods in the form of zone area methods have beenused in predicting final subsidence over inclined coalseam. In this method, the major influence zone ofsubsidence due to extraction of an element of coalseam, in circular shape in a flat coal seam, is transformed into an elliptical shape and is systematicallydivided into a finite number of subzones. Theamount of the subsidence influence due to extractionin each of the subzones is predetermined. The finalsubsidence at the surface point is the summation ofthe influences from those subzones that are locatedin the mined area.However, the zone area method isvery tedious to apply and lack of sufficient accuracy.In this paper, an influence function method is developed for the prediction of the finalsurface movement and deformations over a longwall gob extracted in an inclined coal seam.The Germanexperience and the findings from subsidence research in the US have been combined in developing this method.In general, the longitudinal direction oflongwall mining panel is laid along the seam strikedirection while its transverse direction follows theseam dipping direction. Since the characteristics offinal subsidence along a longitudinal major cross-section is the same as that over a flat coal seam, thisproposed method only provide method to predictfinal surface movements and deformations along amajor transverse cross-section.+ Proposed influence functionThe two fundamental steps of employing influence function methods in subsidence prediction are:1) definition of the influence function that describethe distribution of subsidence influence on theground surface caused by the extraction of one element of the coal seam, and 2) integration of the influence function over the mine area. A proper formof influence function should be carefully defined sothat it can well represent the mechanism involved inthe subsidence process.In this paper, the original form of the influencefunction for subsidence in the Knothes Theory ismodified to represent the asymmetrical subsidenceinfluence along the seam dipping direction。 Fig. 1shows the scheme of applying influence functionmethod in predicting final subsidence due to miningin an inclined coal seam. A convention is followed toestablish the global coordinate system (O-X) inwhich the origin (O) is located directly (vertically)above the left (lower) panel edge and its positivedirection points to the right (upper) panel edge side.The surface point where the final surface subsidenceis to be predicted is located xp distance from the origin of the global coordinate system. A number ofimportant parameters involved in defining the influence function are: The limit angles on the lower and upper sides of the panel ( H and L ), respectively.When they are plotted upwards from the lower and upper edges of the panel, they specify the edges of the final subsidence basin, respectively. When they are plotted from a surface point downward, they delineate the zone within which the extraction in the coal seam would influence the surface point to subside. These two lines are the lower and upper influence boundary lines.Generally, the limit angles depend on the angle of the seam inclination, .Rom developed a4graph to determine H and L based on German subsidence experience.Chinese also derivedtheir own empirical formulae for the limit angles for a number of mining districts. Nadir angle ( ) shows the spatial relation between a specified surface point (point P inFig. 1) and the extraction point in the coal seam that influences surface point P to subside the most (point Z). It should be noted that the P-Z line equally divides the angle formed between the lower and upper influence boundary lines drawn downwards from the surface point P. The nadir angle is determined by = 1 (4 )2HL(1)The effective radii of influence function (R L and RH ) on lower and upper sides of the point of maximum extraction influence (Z) are shown in Fig. 1. They can be determined by finding the intersection points between the inclined coal seam and the lower and upper influence boundary lines, respectively. The line equations for the lower and upper influence boundaries (RH andRL ) when the prediction point is at xp from the left panel edge are shown in Eqs.(2) and (3), respectively.Fig. 1 Diagram showing relations for using influence function method to predict final surface subsidence over a longwall gob in inclined coal seamy = -(xp - x)tan Ly = -(x - xp )tan HThe line equation for the inclined coal seam isy = -h1 +x tan x xpx xp（2）（3）（4）The coordinates of the intersection point between the lower influence boundary and the coal seam are determined from Eqs.(2) and (4) ash - x tan Lx = 1pLtan - tan L（5）h - x tan LLy = 1pL tan tan - tan L（6）The coordinates of the intersection point between upper influence boundary and the coal seam are determined asxHy= xh + x tan = 1pHtan + tan Hh + x tan + 1pHtan （7）H ptan + tan LH （8）The location of the maximum extraction influence point (Z) is determined by finding the intersection point between the line equation of P-Z and that of the inclined coal seam. The coordinates of the maximum extraction influence point areh + xpxZ =1tan 1tan 3 tan （9）yz = h (xp) = 1 h1 tan + xptan 1+ tan tan + xp （10）The effective radii of major influence on the lower and upper sides can be simply determined using Eqs.(11) and (12), respectively. These radii of major influence are dependent on xp , L , H and overburden depth.RL = xZ - xLRH = xH - xZ（11）（12）The other parameter to define the influence function is the maximum possible subsidence, Smax . In general, Smax decreases as the overburden depth increases. Therefore, Smax along a transverse cross- section varies with xp . It is reasonable to assume that for an inclined coal seam Smax at a prediction point is a function of the effective distance (he ) instead of the true depth. The effective distance is simply calculated from the coordinates of the prediction point (xp , 0) and the point of maximum extraction influence (xZ , yZ ).()22he =xZ - xp yZ（13）The true subsidence factor can be estimated by substituting he into the empirical equation derived previously by the author from the collected US and Australian longwall subsidence dataa = 1.9381(he+ 23.4185)-0.1884（14）The maximum possible subsidence at surface point xp for an inclined coal seam is determined by Eq.(15).Smax = am cos （15）In the first step in defining the influence function, the original form of the influence function for subsidence in Knothes theory is modified for the extraction in an inclined coal seam as shown in Eq.(16) 8 . Unlike the influence function for the flat coal seam, the influence function for inclined coal seam (Eq.(16) changes its magnitude and distribution at different locations due to different effective distance. It should be noted that a local coordinate system is used in expressing this influence function. The local coordinate system places its origin at the prediction point (P) and aligns its positive direction to the right side. In Eq.(16), the local coordinate x is obtained as xxp . x -s Smax e f (x ) = RL RL x x 0-eS max RH RH x 0The influence function in Eq.(16) becomes twopieces with the center point at x= 0. The influence function for mining a 7-ft (2.1 m) thick coal seam at an inclination of 30 is shown in Fig. 2. It is plotted for a point that is about 827 ft (252 m) directly above the mined coal seam. At this location, the point of the maximum extraction influence (Z) moves to the right side of the prediction point for a horizontal distance of about 192 ft (59 m). The effective distance (he ) is about 741 ft (226 m) which is smaller than the actual depth. The radius of the major influence on the lower side is about 599 ft (183 m) while that on the upper side is about 404 ft (123 m). It should be noted that the influence function in Eq.(16) is not continuous at the center point as shown in the figure. The maximum value of the influence function for the lower side at x=0 is Smax /RL while that on the upper side is Smax /RH . Since RH is smaller than RL , the difference between the two maximums is Smax = (1/RH RL ). Due to such discontinuity nature of the influence function, the irregularities will surely appear in the resulting profiles of the final surface movements and deformations.Fig. 2 Original and adjusted influence function fora 30 inclined coal seamIn order to make the influence function (Eq.(16) continuous at the center point, it is necessary to adjust the influence function in Eq.(16) on both the lower (x0) and the upper (x0) sides. The adjustment method should meet the following two fundamental requirements: The maximum value of the adjusted influence function, f s ( x), on the lower side at x=0should be equal to that on the upper side, orf s ( x = 0) Lower = f s ( x = 0)Upper（17） On each side of the center point, the total contribution of influences to the final subsidence at the surface prediction point should remain the same before and after the adjustments.00ss - RLf (x )dx =- RLf (x )dxH Rf (x )dx =RH f(x )dx（18） ss 00In doing so, the values of the influence function for both lower and upper sides at thecenter point (x = 0)are forced to beSmax 1 +1 to meet the first requirement. In order to2 RLRH meet second requirement, the influence function on each side is multiplied by a linear adjustment function as shown in Fig. 3. In each of the adjustment functions, AL and AH are the coefficients to make Eq.(17) possible and BL and BH are the slopes of the adjustment functions for the lower and upper side of the influence function, respectively. These coefficients for the linear equations are defined asFig. 3 Adjustment functions for the influence function on the lower and upper sides A = 1 RL +1 L2 RH（19） A = 1 RH +1 H2 RL1 RL2 R+1(1+ fL () - 2 fL()BL= H RL（20）1 RH2 R+1(1+ fH () - 2 fH( )BL= L RHIn Eq.(20), fL () and fH () are the adjustments for varying inclination angle. Their values at each inclination angle are determined individually using Eq.(21) and regression studies are performed and the resulting empirical equations are:0.9808 + 0.0041 - 0.00062 38.2 fL () = 0 fH ()=1+ 0.0063 38.2（21）The adjusted influence function after applying the adjustments in Eqs.(19), (20) and (21) is shown in Eq.(22). The distribution of the adjusted influence function is plotted back to Fig. 2. After the adjustments, both sides of the influence function meet perfectly at the center point but the influence function is skewed toward the upper side. This asymmetry becomes more apparent as the inclination angle ( ) increases. It should also be pointed out that the current adjustment method would not ensure the continuity of the first derivative at the center point. x - Smax e RL ( A+ B x )x 0sf (x ) = RLLL x （22）-eSR max RH ( A+ B x )x 0HHH+ Final surface movements and deformations Determination of final subsidenceBased on the concept of the influence function method, the final subsidence at a surface point is the summation of the all influences received at this point caused by the extraction in the coal seam. Mathematically, it is the integral of the influence function over the mined area. After considering the over- hanging overburden strata over the panel edges and an equivalent transformation of the coordinate system, the final surface subsidence at the prediction point can be determined by integrating the adjusted influence function between the left and right inflection points, O1 and O2 (Fig. 1). The offsets of the inflection points at the coal seam level (d1 and d2 ) are first determined using the empirical formula (Eq.(23) derived from the collected US subsidence data 9 . The actual overburden depths on the left and right edges of the panel (h1 and h2 ) are used in Eq.(23) to obtain the offsets of inflection points on the left and right sides of the panel, respectively.(h12 )d1,2 = h1,2 0.382075 0.999253（23）It is reasonable to project the two inflection points from the coal seam to the surface with the nadir anglesince this angle signifies the line of the maximum influence on the ground surface caused by the underground extraction (Fig. 1). Through such projection, the coordinates of the left and right in- flection points in the global coordinate system, x1and x2, are determined as: x1 = d1 cos - (h1 - d1 sin ) tan x2 = (W - d2 )cos - (h2 + d2 sin ) tan （24）In performing the integration of the influence function between the inflection points, the inflection points should be expressed in local coordinates. x1 = x1 - xpx2 = x2 - xp（25）Since the influence function (Eq.(22) is defined as two pieces, the integration should be performed intwo segments, one on the left side of the prediction point and another on its right as2shown in Eq.(26). The final subsidence at prediction point, xp , is shown in Fig. 1 as the shaded area and is determined as：2 x x b1-b2-S (x ) = Smax ( A- B x )e RL dx + Smax ( A- B x )e RH dxp RLL a1 L RHHa2 H（26）Depending on where the prediction point is, there are the following three possibilities involved in the integration of the influence function.- Both inflection points are located on the left side of the prediction points, or x2xp . In this case, the lower and upper bounds of the two integrals are：a1 = x1 , b1 = x2, a2 = 0, b2 = 0- The prediction point is located between the left and right inflection points, or x1 xpxp a1 = 0, b1 = 0, a2 = x1 , b2 = x2 Final movement and deformationsThe other final surface movement (i.e., horizontal displacement) and deformations (i.e., slope, strain and curvature) are directly related to the final surface subsidence. The final surface slope is defined as:i (xp) = dS (xp )dxp（27）Based on the subsidence theories, the final horizontal displacement is proportional to the final slope. For a flat coal seam, the proportionality coefficient is defined as R2 /h where R is the radius of major influence and h is the overburden depth. For the inclined coal seam, the average of RL and RH should be substituted for R and the effective distance he for h in the determination of the proportionality coefficient. Therefore, the horizontal displacement at the prediction point is defined as:U (xp) = (RL + RH )24he i (xp )（28）The final surface s