油藏数值模拟习题

中国石油大学（北京） 石油工程本科教学课程
油藏数值模拟上机作业题
设有一均质等厚油藏，共有两层，顶深2100米。

第一层厚30米，Kx=Ky=50mD, φ=0.25；第二层厚20米，Kx=Ky=40mD ，φ=0.20；垂向不渗透。

其它油藏物性参数及流体性质跟示范算例完全相同。

本油藏采用正五点井网注水开采。

取其中两个相邻注采单元区块，长800米，宽400米，如图所示。

给定生产井单井产液量60m 3/d ，注水井单井注入速度60m 3/d ，注采井都在两层完井。

(1)预测第10年末此区块总体生产指标、各井井底流压及油藏内压力、饱和度分布。

(2)若实际油藏的油井投产前进行了压裂，使得第五年末的生产井底流压比未压裂时即
情况(1)高2.0MPa ，请对模拟油藏内近井区渗透率和井的采油指数进行修正，完成历史拟合，再重新计算第十年末的上述指标。

作业要求：
上交数值模拟软件两次运算结果数据电子文件：
姓名学号_1.res ，姓名学号_2.res
生产井 注水井。

第一章储层流体的物理性质二.计算题1.(1)该天然气的视分子量M=18.39该天然气的比重传=0.634(2)Imol该天然气在此温度压力下所占体积:V~2.76xl0'4(m3)2.(1) m=69.73xl03(g)(2)pnO.0180* 106(g/m3)=0.0180(g/cm3)3.Z=0.864.Bg=0.005235.Ng=21048.85 xl04(m3)6.(1) Cg=0.125(l/Mpa)(2) Cg=0.0335(l/Mpa)7.Z=0.848.Vg地面=26.273 (标准米3)9.p g=0.2333(g/cm3)10.p g=0.249(g/cm3)11.Ppc=3.87344 (MPa)Pci > Ppc > Pc212.(1) Z-0.82(2)Bg=0.0103(3)Vg 地下=103(m3)(4)Cg=0.1364(l/Mpa)(5)|ig=0.0138 (mpa • s)13.R S CO2=65 (标准米3/米3)RSCH4=19 (标准米3/米3)RSN2=4.4 (标准米3/米3)14.Rs=106.86 (标准米3/米3)15.(1) Rsi=100 (标准米3/米3)(2)Pb=20 (MPa)(3)Rs=60 (标准米3/米3)析出气ARs=40 （标准米3/米3）16.V/Vb=0.976217.yo=0.704(g/cm3)18.yo=0.675(g/cm3)19.Bo=1.29520.Bt=1.28321.Rs=71.3 (Nm3/m3)Bo=1.317Bg=0.00785Bt=1.457Z=0.85422.P=20.684Mpa 下:Co=1.422xl。

—3 (1/Mpa)Bo=1.383P=17.237MpaT：Bo=1.390Bt=1.390Rs=89.068 (Nm3/m3)P=13.790MpaT：Bo=1.315Bt=1.458Rs=71.186 (Nm3/m3)Bg=7.962xl。

CHINA UNIVERSITY OF PETROLEUM油藏数值模拟编程大作业结果综合报告院系名称：石油工程学院专业班级：研开发班考生姓名：学号：任课老师：刘月田完成日期：2017 年5 月10 日一、题目要求要求编制计算机程序完成该油藏渗流问题的模拟求解。

具体要求： （一）使用显式和隐式两种方法。

（二）输出油藏投产后如下时刻的压力分布，1w 井底流压和2w 井产油量：初始时刻、10天、1个月、2个月、1个季度、半年、1年、2年、5年；给出压力刚好达到稳定的时刻及压力分布、1w 井底流压和2w 井产油量。

（三）对照（二）中内容，比较分析显式和隐式两种方法的计算过程及结果有何不同。

（四）建议计算过程用国际单位制，输出的结果中压力用MPa ，产量用m 3/d 。

二、物性参数厚度分布=初始参数/60，单位：m渗透率分布=初始参数+50，单位：10-3μm2孔隙度分布=（初始参数×0.025+15），%三、物理特征上述油藏具有如下物理特征：（1） 二维平面（薄层）油藏，厚度(,)H x y ； （2） 单相、微可压缩流体；（3） 岩石不可压缩，岩石及地层具有非均质性：(,)K x y ，(,)x y φ； （4） 达西渗流，恒等温，不考虑重力影响； （5） 南部边界为定压边界，其他边界为封闭边界； （6） 油藏开发过程中压力始终保持在饱和压力以上。

四、数学模型首先取地层平面为二维坐标平面、以正东和正北方向分别作x 和y 轴，建立平面直角坐标系。

1. 渗流控制方程单相微可压缩流体流体渗流方程为：0[()]Vr K pp g D q C tαρααφμ∂∇⋅∇-∇+=∂ 忽略重力影响，并考虑维数因子(,)H x y α=，且μ为常数，代入上式得1()Vr pHK p Hq H Ctφμ∂∇∇+=∂ 写成直角坐标形式，得11()()Vr p p pHK HK Hq H C x x y y tφμμ∂∂∂∂∂++=∂∂∂∂∂ 2. 初始条件(,,0)20ini p x y p Mpa ==3. 边界条件南部边界（定压边界）：20ini p p Mpa == 其他边界（封闭边界）：0ps∂=∂，其中s 为边界法线方向。

油藏数值一、选择题1．双模包括物理模拟和（B数学模拟）2．随着计算机的迅速发展，求解数学方程组常用（A数值模拟）3．油藏模拟的基础在于油藏描述和（A生产动态）4．模拟是用（C油藏模型）来研究油藏的各种物理性质和流体在其中的流动规律5．数学模型来研究某个物理过程变化规律是通过求解某一物理过程的（A数学方程组）6．数值模型用离散化方法将偏微分方程组转化为（D有限差分方程组）7．计算机模型是将各种数学模型的计算方法编制成（C计算机程序）8．油藏数值模拟用（B数值）方法求解油藏数学方程组9．离散化就是把整体分割为若干（D单元）10．有限差分法是对（C网格范围内的各点）求解。

11．有限差分法使偏微方程被（B代数方程组）所取代12．块中心网格是用（A网格块中心）来表示小块坐标的13．点是心网格是用（B结点）的位臵来确定小块的中心14．函数P（x,t）的一阶向前差商为（A）15．函数p(x,t)的一阶后差商为（D）16．函数p(x,t)的一阶中心差商为（C）17．函数p(x,t)二阶中心差商为（C）18．当网格中有一口井时，均质地层的网格等值供给半径为（D）19．半隐式方法是（D以上都对）20．根据每一组份的质量守恒建立的渗流数学模型称为（A 组份）模型21．动态预测是在历史拟合的基础上对（C未来的开发指标）进行计算22．IMPES方法是（C隐式求解压力方程，显式求解饱和度）的23．半隐式方法是（D同时求出压力和饱和度）的24．适定问题是指一个问题的解（D存在、唯一且稳定）25．认识油田的主要方法有直接观察法和（B模拟法）26．差分方程数值解的方法有（D上述ABC）27．数值模拟输入的数据为（D上述都不对）28．下列哪一个为一维显式差分格式（A29．下列一维差分格式中哪一个截断误差最小（C）30．下列哪一个为二维显式差分格式（A）31．下列哪一个是标准排列格式（A）32．下列哪一个是对角排列格式（B）33．下列哪一个是点交替排列格式（C ）34．下列哪一个是交替对角排列格式（D ）35．一阶向前差商的截断误差的阶是（C 1）36．一阶向后差商的截断误差的阶是（C 1）37．二阶差商的截断误差的阶是（A 2）38．对数学模型命名时按先后顺序最先出现的顺序是（B 维数）39．建立数学模型常用的物理原理包括（D 上述ABC ）40．二阶微分方程的基本类型为（D 都是）41．下述表达式表示定压外边界条件的是（A ),,,(|1t z y x f P ab =）42．下列表达式表示定流量外边界条件的是(B),,,(|2t z y x f n p =∂∂) 43．下述表达式表示封闭外边界条件的是(C),,,(|2t z y x f n p ab =∂∂) 44．下述表达示混合外边界条件的是(D {),,,(|}{3t z y x f ap N P ab =+∂∂}45．下述表达示表示定井底压力内边界条件(A P(rw,t)=常数))46．下述表达示表示定产量内边界条件的是(B r rw r p |∂∂=常数)47．下述表达示表示变井底压力内边界条件(C P(rw,t)=φ1(t)))48．下述表达示表示变产量内边界条件的是(D rrw r p |∂∂=φ2(t))49．块中心网格和点中心网格的差分方程是(A 一样的)50．块中心网格的点中心网格的离散点数是(B 不相同的)51．p(x.t)的一阶向前差商的截断误差为 (A O (△X)) 52．p(x.t)的一阶向后前差商的截断误差为 (A O (△X)) 53．p(x.t)的一阶中心差商的截断误差为 (B O (△X2)) 54．p(x.t)的二阶中心差商的截断误差为 (B O (△X2)) 55．加权六点格式中,要使之成为显式差分格式,必须满足(A θ=056．加权六点格式中,要使之成为隐式差分格式,必须满足(C θ=1/2 )57．加权六点格式中,要使之成为克兰克.尼克森差分格式,必须满足(Bθ=1/2 )58．加权六点格式中,要使之成为克兰克.尼克森差分格式,必须满足(Bθ=1/2 )59．显式差分格式是( A有条件)收敛的.60．隐式差分格式是( B无条件)收敛的61．显示差分格式的稳定条件是(A△t/△x2<=1/2)62．对于各向同性的正方形网格的等值供给半径为(C 0.208△x)63．差分方程组的直接解法适用于处理(B系数矩阵阶数不太高)的问题.64．差分方程组的迭代解法主要用于处理(A系数矩阵阶数较高)的问题65．传导系数的处理方法是(D以上都是)66．以下哪个是渗透率的平均取值方法(D 以上都是)67．对渗透率取值一般取(A 上游权)的处理方法.68．X 方向的传导系数为(A i j i x h y K x ∆∆][T j x ,μ)69．Y 方向的传导系数为(B yjxi h K yj ∆∆][T j y,μ) 70．将含水饱和度归一化的公式为(C Sor Sw c Sw c Sw S W ---=1)71．差分方程组的直接解法的特点为(D 都对)72．差分方程组的迭代解法的特点是(C A 和B)73．差分方程组的系数的处理方法有(D A,B,C 都对)74．对数学模型按相态分类有(D A,B,C 都对)75．对数学模型按空间维数分有无零维模型(A 有)76．下列哪个是按模型的使用功能及特点进行分类的(D A,B,C 都对)77．历史拟合的对象是(D A,B,C 都对)78．历史拟合在含水拟合时,主要是对(C 相对渗透率曲线)的修改.79．历史拟合在压力拟合时,主要调整内容为(D A,B,C 都对)80．对含水率和油气比的影响因素有(D A 和B)81．正确应用数值模拟技术的重要环节是( C A 和B)一、解释概念1. 模拟:模拟是利用模型来模拟物理过程。

秋季学期《油藏描述》在线考试补考（适用于4月份考试）
油藏静态描述的核心是对储集体（）进行研究和评价。

A:空间展布
B:构造特征
C:沉积相变化
D:非均质性
参考选项：D
（）是控制成岩作用的关键因素。

A:岩性
B:流体
C:温度
D:压力
参考选项：D
精细构造描述的重点是要描述对油气田开发有重要影响的（）。

A:构造演化机制
B:断裂体系
C:地应力
D:低级序构造
参考选项：D
地震相分析的关键就是根据（），并结合其它资料将地震相转为相应的沉积相。

A:地震相参数
B:沉积特征
C:测井响应特征
D:地震相特征
参考选项：D
（）是降低孔隙利用系数的主要因素。

A:孔道非均质
B:层内非均质
C:孔间非均质
D:表面非均质
参考选项：C
（）模拟即是要模拟物体点及其性质在三维空间的联合分布。

A:示性点过程
B:布尔
C:高斯
D:指示
参考选项：A
1。

PAPER NO. 28902B HERIOT WATT UNIVERSITYDEPARTMENT OF PETROLEUM ENGINEERINGExamination for the Degree ofMeng in Petroleum EngineeringReservoir SimulationMonday 17th April 199509.30- 12.30( 70% of Total marks )NOTES FOR CANDIDATES(1)Answer ALL the questions and try to confine your answer to the space provided on thepaper.(2)The amount of space and the relative mark for the question will give you some idea of thedetail that is required in your answer.(3)If you need more space in order to answer a question then continue on the back of thesame page indicating clearly (by PTO) that you have done so.(4)The total number of marks in this examination is 262; this will be rescaled to give anappropriate weighted percentage for the exam. The marks are relative and, together withthe space available, should give an approximate guide to the level of detail required.(5)There is a compulsory 15 minute reading time on this paper during which you must notwrite anything.(6)You will be allocated 3 hours to complete this paper.1Q1.List two uses of numerical reservoir simulation for each of the following stages of field development.(i)At the appraisal/early field development stage1.(2)2.(2)(ii)At a stage well beyond the maximum oil production in a large North Sea field: 1.(2)2.(2)Q2.At any stage in a reservoir development by waterflooding, the engineer may use material balance calculations and/or numerical reservoir simulation. Under which particular circumstances would you use each of these approaches?(i) Material Balance?(3)(ii) Reservoir simulation?(3)Q3.Given all the problems and inaccuracies, which are known to exist in the application of reservoir simulation, why do engineers still use it?(2)2Q4WaterReservoir X is a light oil reservoir (35º API) being developed by waterflooding. The reservoir comprises a high average permeability massive sand overlying lower average permeability laminated sands. The thickness of each sand is approximately equal and there are shales at the interface of these two sands. However, the operator is uncertain if these shales are continuous or discontinuous.The following questions refer to Figure 1 above.(i)What would the main differences be between the cases where the shales in the abovereservoir were continuous and where they were discontinuous?(4) (ii)Say briefly how you would go about investigating this using reservoir simulation.(4)3(iii)Suppose a reservoir engineer took 10 vertical grid blocks (NZ=10) in a simulation model of this system. Would you expect the local (kv.kh) values in each of the main reservoirsands to be similar or different? Explain your answer briefly.Similar/different(1)Explain(4)(iv)If this reservoir well had a gas cap, then gas coning might be a problem.What is gas coning? Draw a rough sketch.(4)How would you use reservoir simulation to investigate this problem?(4)4(v)In which two ways would the grid used to investigate gas coning be different from that which was used in the full field waterflooding simulation?1.(2) 2.(2)Q5.Two of the main numerical problems/errors that arise in reservoir simulators are due to numerical dispersion and grid orientation.Explain each of these terms briefly saying - what it means, its origin and how we might get round it. Draw a simple hand drawn sketch illustrating each.(i)Numerical Dispersion: Sketch(5) (ii)Numerical Dispersion - Explanation?(5)56(iii)Grid Orientation - sketch(4)(iv)Grid orientation - Explanation?(4)Q6.A very simple single phase pressure equation is given by Eq. 1 below.∂∂ÊËÁˆ¯˜=∂∂ÊËÁˆ¯˜P t P x 22Eq. 1(i)Write down how this equation is discretised in an explicit finite difference scheme - briefly explain your notation.(4)(ii)If an implicit finite difference scheme was used to solve Eq. 1, then a set of linear equations would arise which could be solved using either a direct or an iterative linear equation solution technique. Briefly explain each of the bold terms above:• Implicit finite difference scheme(4)• Set of linear equations(4)• Direct linear equation solution technique(4)• Iterative linear equation solution technique(4)78Q7.Statement: “The Equations of Two Phase Flow can be derived easily simply by using Material Balance and Darcy’s Law”.Explain this statement with reference to two phase flow - you do not need to actually derive the equations and, indeed, you may not use any equations other than Darcy’s law.(8)Q8.(i)Draw a schematic sketch of a single grid block of size D x by D y by D z, showing the porosity f , the oil and water saturations S o and S w (only 2 phases present).(2)9(ii)Using the sketch in part (i) above, derive expressions for (a) the volume of oil in the grid block, V o ; (b) the mass of oil in the grid block, M o , introducing the formation volume factor, B o .(a)V o ?(3)(b)M o ?(3)(iii)Write an expression for the oil flux, J o , saying briefly what it is, any units it might be expressed in and explaining any terms you introduce.(6)10Q9.(i)Name three ways in which a Black Oil reservoir simulation differs from a Compositional simulation model.1.2.3.(6)(ii)Draw a simple sketch of a single grid block showing what is meant by a component and a phase.(4)(iii)Using the notation C IJ to denote the mass composition of component I per unit volume of phase J (dimensions of C IJ are mass/volume), derive an expression - based on thequantities labelled in your sketch in (ii) above - for the mass of component I in the grid block.(6)(iv)Give one example of (a) where you would use a Black Oil model and (b) where you would use a Compositional simulation model.(a)Use Black Oil model?(3)(b)Use Compositional model?(3)Q10.Figure 2: The figure below shows the basic idea of “upscaling” or “Pseudo-isation”.With reference to Figure 2 above, answer the following:11(i)What is meant by “Upscaling” with reference to the modelling of say a waterflood.(2)(ii)What is the difference between “rock” relative permeabilities and pseudo-relative permeabilities?(4)(iii)In order to perform upscaling in reservoir simulation, we need both an Upscaling Methodology and Upscaling Mathematical Techniques. Explain very briefly themeaning of the bold terms.• Methodology(4)• Techniques?(4) 12Q11.(i)Sketch (roughly) a semi variogram for each of the following permeability models:(a) a correlated random field with a range of 100m and a sill of 10,000 mD2; and(b) a laminated system where the laminae are of constant width of 1cm and where the highpermeability = 2D and the low permeability = 1D. Label your sketches clearly.(a)(6)(b)(6) (ii)What can you deduce about the standard deviation of the correlated random field in (i)(a) above.(3)13(iii)Sketch the correlogram for case (i) above.(5)Q12.Figure 3 below shows the sketch of simple 3 layer model.2 cmx5 cm1 cm y(i) Calculate the effective permeability of the above model in the x-direction; show your working.(5)14(ii)Calculate the effective permeability of the above model in the y-direction, show your working.(6)(ii) Suppose we put a very find grid (say of size 0.1 cm x 0.1 cm) on the 3 layer model in Figure 3 above. If we jumbled up all the grid blocks randomly so that the new model had no discernable structure, would the new effective permeability be: greater than that in (i) and (ii)?; less than that in (i) and (ii)?; in between these values? Explain your answer.(6)15Q13.In miscible flow in a random correlated field, explain how the mixing zone grows with time in each of the following cases (illustrate your answers with simple sketches):(i) dispersive flow(4)(ii) fingering flow(4)(iii) channelling flow(4)(iv) On the same diagram below, sketch the expected type of fractional flow curve f(c) vs c) you would expect for each type of flow.(6)16Q14.In the Kyte and Berry pseudo-isation (upscaling) method, describe briefly (using a diagram) how numerical dispersion is taken into account (no detailed mathematics isrequired).(10)17Q15.(i)List the main categories in the hierarchy of stratal sedimentary elements - give one short sentence explaining each of these.(8)(iii)Describe which of the above length scales of sedimentary heterogeneity are likely to have most significance for the following reservoir flow phenomena:* Reservoir pay-zone connectivity:(3)* Gravity slumping or water over-ride:(3)* Vertical sweep efficiency:(3)* Residual/Remaining oil saturation(3)18Q16.You have been given the following distribution of core-plug permeabilities in a particular reservoir unit which includes a higher permeability a fluvial channel sand overlying alower permeability deltaic sand:With reference to Figure 4 above: (a) explain the probable reason that the permeability distribution has the above form; (b) sketch the sort of permeability models (laminar, bed and formation scale) you might use for the flow simulation of this unit.(a)(3)(b)(continue on the back of this sheet if necessary)(10)19Q17.(i)Draw a sketch of water displacement of oil across the laminae in a water-wet laminated system at (a) “low” flow rate (capillary dominated) and (b) “high” flow rate(viscous dominated); in this sketch show where the residual remaining oil is andgive a sentence or two of explanation.(a)(8)(b)(8) 20(ii)Is the effective water permeability at residual (i) remaining oil saturation (across the laminae) higher in case (a) or (b) in part (i) above? Explain.(6) (iii)What are the implications of the results in (i) and (ii) above for upscaling in reservoir simulation?(6)21。