“ AÑO AÑO DEL BUEN SERVICIO AL CIUDADANO”
UNIVERSIDAD NACIONAL NACIONAL DE PIURA FACULTAD DE INGENIERIA DE MINAS ESCUELA DE INGENIERIA DE MINAS
LIMITE FINAL DE UNA EXPLOTACION EXPLOT ACION SUPERFICIAL EMPLEANDO EL CONO FLOTANTE Y EL ALGORITMO DE LERCHS GROSSMAN PROFESOR: GUILLERMO RAMIREZ GARCIA CURSO:
MINERIA SUPERFICIAL
CICLO:
VIII
ALUMNO: JUAN QUEVEDO GUERRERO GUERRERO
2019 1
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INDICE
OBJETIVOS…………………………………………………………………………… OBJETIVOS……………………………………………………………………………3 3 INTRODUCCIÓN…………………………………………………………………….. INTRODUCCIÓN……………………………………………………………………..4 4 1 ALGORITMOS DE OPTIMIZACIÓN OPTIMIZACIÓN DEL PIT FINAL……………………6 1.1.
CONO FLOTANTE…………………………………………………... FLOTANTE…………………………………………………...6 6
1.2.
CASO PRÁCTICO…………………………………………………… PRÁCTICO……………………………………………………11 11
2 MÉTODO DE OPTIMIZACIÓN DE TAJO DE LERCHS-GROSSMAN…17 LERCHS-GROSSMAN…17 1.3.
LERCHS GROSSMAN……………………………………………… GROSSMAN………………………………………………17 17
1.4.
PARAMETROS ECONOMICOS…………………………………… ECONOMICOS……………………………………19 19
1.5.
PARAMETROS GEOTECNICOS………………………………….. GEOTECNICOS…………………………………..21 21
1.6.
METODO LERCHS Y GROSSMAN (PROCESO PARA EL DISEÑO)……………………………………………………………… DISEÑO)………………………………………………………………22 22
1.7.
OPTIMIZACIÓN DE UN PIT………………………………………... PIT………………………………………...23 23
3 ANÁLISIS DE LOS ALGORITMOS DE OPTIMIZACIÓN UTILIZANDO GEMCOM SURPAC Y UPL………………………………………………… UPL …………………………………………………24 24 a. METODOLOGÍA……………………………………………………... METODOLOGÍA……………………………………………………...24 24 4 CASOS PRÁCTICOS……………………………………………………….. PRÁCTICOS………………………………………………………..33 33 Ejercicio 1…………………………………………………………………….. 1……………………………………………………………………..34 34 Ejercicio 2…………………………………………………………………….. 2……………………………………………………………………..38 38
CONCLUSIONES…………………………………………………………………… CONCLUSIONES……………………………………………………………………41 41
BIBLIOGRAFIA……………………………………………………………………… BIBLIOGRAFIA………………………………………………………………………42 42
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OBJETIVOS
Determinar las ventajas de optimización optimización que tiene el método de Lerchs Grossman respecto al cono flotante. Analizar el método de Lerchs Lerchs Grossman Grossman para establecer los bloques bloques que que se explotan y los que no. Determinar los límites del método de Lerchs Grossman en el punto en el cual se maximicen los ingresos y con ello definir la secuencia de extracción dentro de los límites del pit. Definir los procesos que desarrollan desarrollan el algoritmo algoritmo del cono flotante para para la optimización del pit final y analizar todos los cálculos que congloba el desarrollo del método. Desarrollar ejemplos prácticos que ayuden con el entendimiento del procedimiento de cada algoritmo y poder solucionar posteriormente ejercicios similares.
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INTRODUCCIÓN La evaluación económica de un macizo rocoso suele ser una tarea bastante engorrosa a la que se ven enfrentados la mayoría de Los ingenieros de Minas, a la hora de realizar un planeamiento adecuado, en la secuencia de extracción y límites de explotación del material de un proyecto minero que maximice los ingresos obtenidos, puesto que se hace necesario conocer la mayoría de las variables posibles, para poder realizar una predicción del beneficio de la extracción con un buen nivel de confianza. Existen distintos métodos para evaluar estas variables; este artículo se centrará en la aplicación del algoritmo de Lerchs-Grossmann y el del cono flotante a un depósito hipotético, en el cual se genera el pit óptimo estableciendo los límites de este en el punto en el cual se maximizan los ingresos, y definiendo la secuencia de extracción del material contenido dentro de los límites de este pit. Estos métodos son ampliamente usados en depósitos masivos y diseminados, y en menor medida en depósitos estratificados siempre y cuando sean horizontales y sub horizontales. Debido a que uno de los parámetros importantes dentro del diseño de una explotación a cielo abierto es la definición de los límites físicos f ísicos de la explotación, estos métodos son usados como base en el software de simulación y optimización minera de mayor distribución en el mundo, y finalmente llevados al manejo de estos algoritmos en Microsoft Excel. El algoritmo de Lerchs-Grossmann las herramientas (Surpac y UPL) no tienen grandes diferencias cuantitativas, en cuanto a cantidad de bloques a minar y ganancia neta se refiere, pero sí difieren dif ieren gráficamente en la configuración del pit, esto principalmente debido a condicionamientos en la operación del UPL. Por otra parte, para el método del cono flotante sucede lo contrario, pues la geometría final del pit dada por las dos herramientas es bastante similar, pero
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El notable incremento que han sufrido todos los costos asociados al desarrollo de una explotación minera (maquinaria, salarios, etc.) junto con la explotación de yacimientos que poseen cada vez leyes más bajas, ha hecho que el diseño final f inal de la explotación a cielo abierto tenga que llevarse a cabo con criterios económicos, de tal forma que dicho diseño no comprometa, en ningún caso, la futura viabilidad económica de la explotación. Esta filosofía de trabajo ha permitido desarrollar, en las últimas décadas, diferentes algoritmos que tienen como objetivo optimizar la explotación. Generalmente trabajan sobre un modelo de la mineralización constituido por un bloque tridimensional regular. Existen distintos métodos para evaluar las variables como el de lerchs Grossman y el cómo flotante a un depósito hipotético, en el cual se genera un pit óptimo estableciendo los límites de este en el punto en el cual se maximizan los ingresos y definiendo def iniendo la secuencia de extracción del material contenido dentro de los límites del pit. Estos, métodos m étodos son ampliamente usados en depósitos masivos y diseminados.
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1. ALGORITMOS DE OPTIMIZACIÓN DEL PIT FINAL
1.1.
CONO FLOTANTE
El método del cono flotante es una de las técnicas más ampliamente aceptada en el diseño de los límites finales de un pit, pit , porque es de rápida ejecución, veloz y de fácil conceptualización. La programación lógica incluida es empleada después del método convencional de secciones-transversales, se procede a utilizar un llamado punto equilibrio en el radio de área a descapotar como criterio básico de optimización. La principal diferencia con técnicas manuales, es que esta utiliza el concepto de cono flotante en tres dimensiones(alto, ancho, profundo) para remover incrementos en lugar de secciones verticales para generar la geometría final del pit. Un pit es generado y analizado mediante la construcción de una forma cónica en dirección inversa y moviendo su vórtice de un bloque de mena a otro. La forma del cono es definida de manera tal que esté conforme con las restricciones de pendiente de diseño del piten las distintas áreas del depósito. El computador es utilizado en la generación de las configuraciones cónicas 3D y en el cálculo del valor neto de cada cono, mediante la suma de los valores de todos los bloques de mena y estéril encerrados dentro del cono. Finalmente, un límite 3D del pit es obtenido mediante la remoción del frustrum de todos los conos con valor neto positivo. El método del cono flotante es significativamente restrictivo, debido a la gran cantidad de esfuerzo computacional requerido. Ciertamente, la determinación de un soporte mutuo entre conos solapados, no es usualmente requerida debido a que usan más implementaciones del algoritmo del cono flotante. El método
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Por esta razón, el método heurístico del cono flotante muchas veces falla para generar verdaderos límites finales del pit La teoría de los conos flotantes para determinar los límites económicos del Rajo, data de los años 60. La técnica consiste en una rutina que pregunta por la conveniencia de extraer un bloque y su respectiva sobrecarga. Para esto el algoritmo tradicional se posiciona sobre cada bloque de valor económico positivo del modelo de bloques y genera un cono invertido, donde la superficie lateral del cono representa el ángulo de talud. Si el beneficio neto del cono es mayor o igual que un beneficio deseado dicho cono se extrae, de lo contrario se deja en su lugar. En el siguiente esquema se presenta un perfil de un modelo de bloques sometido al algoritmo del cono móvil optimizante, donde cada bloque está definido por un valor económico, es decir lo que significa económicamente su extracción. Es así que los bloques con valor negativo representan a los bloques de estéril con su costo de extracción asociado (-10) y los bloques de mineral son representados por el beneficio global que reporta su extracción (Beneficio Global = Ingresos Costos = 810 - 10 = 800). En un intento por considerar el soporte mutuo entre conos, técnicas más eficientes aplican sofisticadas programaciones matemáticas. Además de la aplicación de sistemas conceptuales de análisis que han sido desarrollados.
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En el ejemplo anterior podemos observar que el extraer el bloque de valor positivo (+800) y sus 15 bloques de estéril asociado (-10 cada uno), genera un beneficio final de +650, correspondiente al beneficio de extraer dicho bloque con su sobre carga asociada
1.1.1. BONDADES DEL DEL CONO FLOTANTE. El cono flotante tiene esa denominación ya que es una versión mejorada de la tradicional rutina del cono flotante. El creador fue el ingeniero Marc Lemieux, quién detectó una serie de deficiencias y mermas económicas producidas por el método convencional de conos flotantes y en 1979 publicó el artículo “Moving Cone Optimizing Algorythm”, y Algorythm”, y como resultado se obtuvo diseños muy superiores en el aspecto económico, que aquellos obtenidos con el algoritmo convencional. Las principales mejoras de la rutina del cono flotante con respecto al método tradicional fueron:
1.1.2. SECUENCIAS DE DE EXTRACCIÓN DE CONOS:
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En la figura se puede apreciar el beneficio que reporta la extracción de cada bloque. Los bloques con beneficio positivo ya se les han descontado lo que cuesta extraer dicho bloque o costo mina (-10). Si el primer cono se construye en el bloque (1) y suponiendo un ángulo de talud, entonces dicho bloque no puede ser extraído (Beneficio = -10). Al no ser factible la extracción del bloque (1), el segundo cono se construye en el bloque (2), donde el beneficio neto del cono es de +10, siendo en consecuencia ventajosa su extracción, quedando la figura de la siguiente forma:
Continuando con la secuencia, el tercer cono se construye en el bloque (3), resultando un beneficio de +30
De este análisis se concluye que los tres bloques con valor económico mayor que cero son extraídos con un beneficio económico de +40, sin embargo un correcto análisis debiera obtener un pit con valor de +60, dejando en su lugar el
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De lo anterior se desprende que la incorrecta secuencia con que se analizan los conos, produce pérdidas económicas cuya magnitud, obviamente, depende de la complejidad de la mineralización, de la variabilidad de las leyes, etc. El problema antes descrito es resuelto por el nuevo algoritmo introduciendo el concepto del “cono negativo”, algoritmo que consiste en extraer todos los bloques con beneficio positivo, para posteriormente devolverlos al rajo con su respectiva sobrecarga y así analizar la conveniencia de extraerlos o bien eliminarlos. En el ejemplo presentado anteriormente, se aprecia que al devolver el bloque (3) con su respectiva sobrecarga, se produce un beneficio económico pues se libera un valor de +20, esto indica que dicho bloque al no extraerse en su condición más favorable debe ser eliminado del análisis. En la práctica la técnica del cono negativo presenta deficiencias similares a las obtenidas mediante lo que se podría llamar el cono positivo, sin embargo un análisis simultáneo de ambas técnicas (cono positivo y negativo) produce resultados satisfactorios.
1.1.3. CONOS CON SOBRECARGA SOBRECARGA RELACIONADA Este es el principal aporte del método del cono móvil optimizante, consiste en analizar conos que tengan sobrecarga compartida, por ejemplo:
Los bloques (1) y (2) tienen un beneficio de +70 (incluido el costo mina). Al
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No obstante si se analiza en su conjunto se ve que es ventajosa su extracción, pues esta trae consigo un beneficio de +40. + 40.
1.2.
CASO PRÁCTICO
En el siguiente gráfico tenemos un modelo de bloques en dos dimensiones de la sección X - X´ y se necesita definir los límites del pit final en base a los siguientes parámetros definidos a continuación. continuación.
7 bloques cuyas leyes son son mayores mayores a la ley de corte de mina (0.319 % Cu)
2 bloques cuyas leyes leyes son son mayores mayores a la ley de corte de planta planta (0.230 % Cu)
El resto de los bloques con leyes menores a la ley de corte de planta planta
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1.2.1. DESARROLLO Orden de evaluación del cono.
1.2.2. PARÁMETROS DEL DISEÑO DISEÑO
Tamaño del Bloque: 50 x 50 x 50 Factor de Tonelaje: 12.5 ft3/ton
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Bloques entre Ley de corte de Mina y Planta
Ventas = (% Cu x 20 )lb/tn x 0.51 $/lb x 10,000 10,000 tn = 10,200 x % Cu
Costo = 10,000 ton x 3.25 $/tn = $ 32,500
Ganancia = (10,200 x % Cu ) -32,500
Bloques mayores que la Ley de corte de Mina
Ventas = (% Cu x 20 )lb/tn x 0.51 $/lb x 10,000 10,000 tn = 10,200 x % Cu
Costo = 10,000 10,000 ton x 3.25 $/tn = $ 32,500 32,500
Ganancia = (10,200 x % Cu ) -32,500
Evaluación Cono # 1
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Solo bloques mayores a la Ley de corte de Mina serán usados como bases de Conos, bloques superiores son evaluados antes que los bloques profundos.
2. MÉTODO DE OPTIMIZACIÓN DE TAJO DE LERCHS-GROSSMAN LERCHS-GROSSMAN 2.1.
LERCHS GROSSMAN
Este algoritmo de programación dinámica original fue demostrado en el diseño de la configuración óptima de bloques para ser removidos en una sección transversal bi-dimensional (2D) (Lerchs-Grossmann, 1965). Tomando una sección transversal 2D de un modelo de bloques y las restricciones de ángulo de pared final definidas. El algoritmo procede calculando la suma de la columna de valores originales de bloques para cada bloque. Este valor de la suma de la
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por el ensamble de estas secciones transversales 2D, y generalmente encuentra que el límite final puede también afectar el valor del ángulo final de pit. La aplicación de la teoría gráfica para el diseño de límites de pit fue creativamente introducido por (Lerchs-Grossmann.1965) El modelo del algoritmo integra problemas zero-one dentro de la red de ramificaciones. Los vértices son equivalentes a bloques minados, y los arcos direccionados impuestos representan las limitaciones de la pendiente del pit. Estos arcos direccionados indican la relación entre bloques de estéril que deben ser removidos antes de minar un bloque de mena en particular. Entonces cualquier contorno de pit factible es obtenido mediante el cierre de la gráfica. (Lerchs y Grossmann) reconocieron que el pit final es un problema de determinación del cierre de una gráfica con unamasa total máxima.
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2.2. PARAMETROS ECONOMICOS 2.2.1. PRECIOS El precio es una de las variables más importantes de la optimización un precio alto o bajo pueden fácilmente determinar la vialidad o no de un determinado proyecto, el precio de la mayoría de metales depende de la oferta y la demanda.
2.2.2. COSTOS
2.2.2.1 Costos directos de minado Incluye los gastos de perforación, voladura, carguío y al costo base de acarreo
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2.2.2.3 Costos de Procesamiento por Tonelada de Mineral Para una Concentradora el costo puede ser por tonelada de mineral molido o por tonelada de concentrado y para una operación de lixiviación con el tankhouse lleno a través de la vida de la mina, el costo será asignado por libra de cobre vendido. Si el desmonte es chancado y acarreado por f ajas a los botaderos, este costo debe de ser asignado solo al desmonte y no a las toneladas del mineral.
2.2.2.4 Costos de Transporte y Tratamiento fuera de la Propiedad Esta categoría de cargos para una operación incluye aquellos gastos relativos al secado de concentrados, manipuleo y trasporte, fundición y refinería. Estos costos se cargan por libra de cobre vendible y típicamente están en el rango de
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2.3.
PARAMETROS GEOTECNICOS
2.3.1 ALTURA DE BANCO. Es la distancia vertical entre cada nivel horizontal del Tajo, los elementos del banco se ven en la figura adjunta, en general todos los bancos tienen la misma altura, a menos que las condiciones geológicas muestren lo contrario. La altura de banco depende de las características físicas del depósito, el grado de selectividad requerido en la separación del mineral y el desmonte con el equipo de carguío; el ratio de producción, el tamaño del equipo y el tipo del equipo para satisfacer los requerimientos de producción y de las condiciones climáticas.
2.3.2 TALUDES DEL TAJO
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2.4.
METODO LERCHS Y GROSSMAN (PROCESO PARA EL DISEÑO)
El método permite diseñar el contorno de una explotación a cielo abierto de tal forma que se maximice la diferencia entre el valor total de la m ineralización explotada y el costo total de la extracción del mineral y estéril. Proceso para Lerchs y Grossman en 2-D:
Se requiere requiere de una sección sección con con bloques bloques de ley conocida conocida y su
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Finalmente en la fila superior un bloque bloque hacia hacia el lado derecho derecho mostrara el mayor valor derivado en la fila. Desde este bloque se sigue la recta de las flechas a fin de obtener la pared final óptima de l a sección.
2.5.
OPTIMIZACIÓN DE UN PIT
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3. ANÁLISIS DE LOS LOS ALGORITMOS ALGORITMOS DE OPTIMIZACIÓN OPTIMIZACIÓN UTILIZANDO GEMCOM SURPAC Y UPL
3.1. METODOLOGÍA 3.1.1. Modelo de Bloques Se diseñó un modelo de bloques hipotético de un material calcáreo partiendo
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bloque con dimensiones 20m x 20m x 20m y un total de 32.500 bloques dentro del modelo. Las listas generadas se unieron en una hoja de cálculo de Microsoft Excel, a tres columnas X, Y, Z, a esta lista se añadió una cuarta columna a la que se le asigno él % de contenido de carbonatos de cada bloque de acuerdo a l os rangos establecidos utilizando la función aleatoria de Microsoft Excel.
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g. Densidad del mineral: Se considera que la densidad del mineral es de 2.1 ton/m3 , ya que los materiales calcáreos como la caliza tiene un rango de densidades entre (1.8 y 2.2 ton/m3 )
3.2. RESULTADOS Y ANÁLISIS Luego de la definición del modelo de bloques y de la asignación de los parámetros técnicos y económicos, se muestran a continuación los resultados
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cuadro de texto con el número de bloques que entran al pit final y el valor total
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Los resultados de la optimización obtenidos mediante UPL, se presentan a
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3.2.4. Comparación de resultados
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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