CN1235062C - 对存在多相液流的物理系统形成表示的方法 - Google Patents
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Abstract
使用计算机系统对其中存在多相液流的物理系统形成表示的方法,该方法包括:使用计算机系统生成一个包含多个系数的初始矩阵,所述多个系数中的第一组系数包含该矩阵的对角线上的元素,所述多个系数中的第二组系数包含该矩阵的不在对角线上的元素,所述多个系数中的每个系数都对应于并且代表液体容器的一个位置,并与容器中的液流相关联;调节所述初始矩阵的系数,将矩阵元素限定为可在物理上实现的值,并通过调节所述多个系数中的每个系数来跟踪物理系统的行为,以根据液流穿越每个元素的方向对每一元素的结点进行加权,使得每个对角线上的元素最小为零,并且每个不在对角线上的元素小于或等于零;将调节后的系数存储在所述计算机系统的计算机系统存储器中;并且使用该计算机系统利用调节后的系数生成对该物理系统的表示。
Description
相关申请
本申请按35U.S.C.§-119(e)要求给予美国专利申请No.60/215,697,标题为“储油层仿真与建模的方法和系统”的优先权,该申请由Stephen R.Kennon,Kok Thye Lim,Scott A.Canaan,Steven B.Ward,Stuart W.Pond,Jr.及Edward J.Barragy于2000年6月29日提交,该文献按其提交的整体在此结合以资对比。
技术领域
本发明一般涉及使用计算机系统对其中存在多相流的物理系统形成表示的方法,还涉及对液体容器中的液流进行预测的方法。
背景技术
在一定条件下能够在数学上对物理系统建模,以便仿真其行为。有各种各样的从非常简单的极为复杂的方法对这种系统建模。物理系统建模一种比较精致的方法是通过使用有限元分析。顾名思义,有限元分析涉及以数学模型表示物理系统的各个有限元,以及在提供一组预定的边界条件下对这模型的求解。另一种相当的对物理系统建模的方法是通过使用有限差分分析。有限差分分析涉及对被建模的空间内各点进行建模,并计算这些点之间的差。有限差分分析常用于仿真流体的动态行为。
使用流线,逆流向方法的传统的有限差分技术和有限元技术一般用来仿真储油层中石油生产。虽然这些方法的每一种有其优点,但也有各自的缺点。一般来说,有限差分技术产生物理上现实的结果,但是它们不很精确。另一方面,有限元技术则更为精确,但是它们产生物理上不实际的结果。
作为它们各自的不足的结果,通常所使用的有限差分技术和有限元技术这两者一般要求大量的计算资源。在有限差分技术的情形下,能够达到合理的精确性,但是这需要比在有限元模型中所需要的更多的结点。这增加了计算精确的解所需要的存储器和CPU的时间量。在有限元技术的情形下,使用较少的结点能够达到相当的精确性,但是解可能是不实际的。例如,浓缩性可能大于百分之100,或渗透性可能是负的。于是,可能存在解不收敛或收敛很慢这样的问题。因而这些技术可靠性较低并在达到可接受的精度之前可能需要大量的叠代。
因为这些标准的接受每一种都有其自身的缺点,并因为这些缺点增加了产生可接受的解所必须的计算资源量,因而希望提供一种对诸如储油层系统建模的方法,该方法要能可靠地产生对于这些系统精确的实际的解。
发明内容
以上概述的一个或多个问题可这样来解决:
使用计算机系统对其中存在多相液流的物理系统形成表示的方法,该方法包括:使用计算机系统生成一个包含多个系数的初始矩阵,所述多个系数中的第一组系数包含该矩阵的对角线上的元素,所述多个系数中的第二组系数包含该矩阵的不在对角线上的元素,所述多个系数中的每个系数都对应于并且代表液体容器的一个位置,并与容器中的液流相关联;调节所述初始矩阵的系数,将矩阵元素限定为可在物理上实现的值,并通过调节所述多个系数中的每个系数来跟踪物理系统的行为,以根据液流穿越每个元素的方向对每一元素的结点进行加权,使得每个对角线上的元素最小为零,并且每个不在对角线上的元素小于或等于零;将调节后的系数存储在所述计算机系统的计算机系统存储器中;并且使用该计算机系统利用调节后的系数生成对该物理系统的表示。
使用计算机系统预测液体容器中的液流的方法,该方法包括:对其中存在多相液流的液体容器的模型离散化;产生对应于模型的有限元矩阵,其中根据与矩阵元素相关联的液流方向,通过有选择性地对有限元矩阵的矩阵元素进行加权,矩阵被设置为产生单调的并保持线性的输出;以及使用计算机系统利用该矩阵产生液流预测。
本发明的技术可以提供胜过先有技术各种优点。例如,因为本发明技术更为精确,产生可接受结果所需的叠代数量可以被降低,因而节省了计算资源。本发明的技术还能够使得解更为可靠地收敛,因为它们被限制在实际的结果范围,且不会按传统的有限元技术求解那样相同的方式振荡。因而本发明技术一般比先有技术更快,更精确,较少的资源强度和更为可靠。
本发明的一个实施例包括使用有限元技术对被建模的系统产生解的方法,其中的系统在有限元网格上被离散化,且其中每一结点对离散化的贡献是基于液流穿越每一元素的方向被加权。贡献被加权是为了着重对作为各个元素的其它结点的上游的结点。以这种方式对结点贡献的加权使得所得的矩阵基本上是对角形的。这就是说,位于矩阵对角线上的元素被强调,而不在对角线上的元素不被强调。一旦矩阵按已这样方式被修改,就使用传统技术求解该矩阵。
本发明的另一实施例包括对储油层中多相流体建模的一种方法。在这实施例中,构造了对应于储油层的四维有限元网格。通过在网格结点中分配有限元算子的残数对网格产生解,并然后使用传统有限元技术求解系统。在这实施例中,分配有限元算子的残数包括对每一元素计算用于每一结点的有限元算子的值,把这些值划分为一正值集合和一负值集合,向结点分配残数,以便强调对角线元素,并且去强调非对角线元素,然后对矩阵计算正切算子。
本发明的另一种方式的实施例是在计算机可读介质中被实施的软件产品。该软件产品包括多个指令,这些指令被设置为引起计算机或任何其它数据处理器执行根据本公开方法的步骤。计算机可读介质可以包括任何数目的不同介质,包括但是不限于CD ROM,DVD,软盘,计算机存储器,磁带等。又本发明的另外的实施例包括计算机系统,该系统设置为执行根据本公开的方法的步骤。可对计算机系统编程以便通过包含在硬件,固件或软件中的指令执行这种方法。
可以有各种各样另外可代替的实施例。
附图说明
在参照附图阅读以下详细说明时本发明的其它目的和优点将显而易见。
图1是表示二维和三维中的单纯形的图示。
图2是表示物理系统实际值和使用传统有限元技术及根据本发明的技术产生的解之间的差别的图示。
图3是表示穿越第一二维单纯形元素的液流的图示。
图4是表示在与图3所示相反方向穿越第一二维单纯形元素的液流的图示。
图5是表示穿越第二个二维单纯形元素的液流的图示。
图6是表示在与图5所示相反方向穿越第二个二维单纯形元素的液流的图示。
图7是表示时-空域的二维投影的图示。
图8是表示根据本发明的一个实施例的方法的流程图。
虽然本发明可以有各种变形和替代的形式,但是其特定的实施例在附图和伴随的详细说明中是通过例子的方式表示的。然而应当理解,附图和详细的说明并不是要限制本发明为所述特定的实施例。本公开是要覆盖所有属于所附权利要求定义的本发明的精神和范围的变形,等价物和改型。
具体实施方式
以下说明本发明的一个优选实施例。应当注意,以下所述这个和任何其它实施例是示例性的,并是为了示例说明本发明而不是限制。
一般来说,本发明包括用于求解有限元模型的方法,其中保证解是单调的并满足线性保持。这保证了解是精确的和实际的。通过产生用于求解有限元模型的一矩阵,使得矩阵对角线上的元素为非负而不在对角线上的元素为非正,可保证解的单调性和线性。然后使用传统的有限元技术能够对这矩阵求解,以便产生所希望的,精确的实际的解。
本方法提供了数个优于传统的有限元和有限差分技术的优点。如上所指出,本方法提供了传统有限元技术的精确性,而没有非实际,振荡解的缺点。类似地,本方法提供了传统有限差分技术的稳定性,同时提供了大得多的精确性。
本方法所提供的精确性和实际性导致了计算机资源使用中附加的优点。例如,本方法可以允许更快速地产生物理系统的解。这产生有几方面的结果。首先,由于解是物理上实际的,它不会振荡,并因而趋向更快速地收敛。进而,因为使用本方法产生解一般要求比传统的有限元技术较少的叠代并较快,因而产生解需要较少的CPU时间。与传统的技术相比本方法还可以使用较少的存储器。因为,本方法一般比传统的有限差分技术更为精确,因而没有必要使用很多结点对物理系统建模。于是,只需较少的存储器即可存储模型(以及基于这些结点计算解需要较少的CPU时间)。本方法还可以比传统技术更为可靠。使用传统有限元技术产生的解的振荡,在某些情形下,会完全妨碍解的收敛。如以上所述,本方法提供了实际的解(例如,它们不会在实际解与不可能的解之间振荡)因而解将会可靠地收敛。
在有限元建模中,被分析的区域被分解为称为元素的子区域。划分区域为子区域的这一过程可称为离散化或网格产生。区域由定义在每一元素上的函数表示。这产生了数个局部的函数,这种函数比要求表示整个区域的函数简单得多。下一步是分析对每一元素的响应。这是通过建立定义区域内各元素的性质的矩阵,以及定义对域中每一元素作用的残数的向量而实现的。一旦所有的元素矩阵和向量已经生成,它们可被组合到一矩阵方程式中。在施加边界条件之后,能够求解矩阵方程式而获得未知的结点的响应。使用在每一元素上定义的函数可从结点值内推出元素内的响应。
以下本方法一个实施例的说明涉及储油层的建模。这一实施例目的是作为示例,但应注意,本方法可用于各种物理系统,诸如航空器上的气流,天气预报,通过内燃引擎的流,以及其它工业液流。本方法潜在应用的这一列表只是示例性的,而不是限制。
以下将陈述一个优选实施例的细节。然而,首先定义一些术语可能是有帮助的。
结点是空间中的一点。在有限元建模中,结点形成被建模的元素的顶点。结点还形成结点与边的网格的部分,边定义被建模的空间中元素之间的边界。
边是两个形成元素顶点的结点之间的线,边形成定义被建模的空间中元素之间的边界的网格的部分。
单纯形是由维数等于或大于n的空间中n+1个点确定的一个n维的空间构形。换言之,一个单纯形是在给定的维数下具有封闭一空间所必须的最小边界点数的几何空间元素。例如,在二维空间中,单纯形包括一个三角形并同时包括以三角形为边界的内部区域(见图1)。两个点不足以形成二维空间中的单纯形,因为没有区域能以两个点(以及互相连接这两点的线)为边界。虽然四个点足以范围一个二维区域,但是它们包括的不是二维区域被范围的最小数的边界点。在三维中,一个单纯形包括由四个顶点所范围的一个四面体(见图1)。在四维中,一个单纯形包括有五个顶点的超四面体(有时称为hypertet)。
网格是填充一个空间的元素的一个集合。这些元素是存在于该空间中的一个系统的一种表示。由于每一元素可由多个结点和/或这些结点之间的边定义,因而网格又可认为是结点和/或它们之间的边的集合。在本公开各处,视术语所使用的上下文,“网格”将交替用来指元素或结点/边的集合。这里网格还可以指称有限元模型或简单地就是模型。
在有单液相建模系统和有多液相的建模系统之间存在着基本的差别。当有多液相时,存在对流和相的潜在混合。(假设存在压力驱动液体并引起它们之间的前沿被传送)。因而有限元模型必须包含对流算子或成分。如果只有一个液相被建模,或如果多个液相被完全混合或乳化,这些算子就不是必须的。
在具有多液相的系统中,系统的特性可能非常突然地改变。这种情形示于图2。图2是表示物理系统实际的值与使用传统有限元技术和根据本发明的技术所产生的解之间的差别。可以看到,实际(“实际世界”)的值11基本上形成一个阶梯函数,在达到距离d1时立即从第一值V1变到第二值V0。在这例子中,值V0和V1定义了物理上可实现的值的范围限制。
当传统的有限元技术用来对这种系统建模时,解一般将趋向振荡。在图2中可以看到,使用这些技术产生的解12在正在对其建模的实际的物理值的上和下振荡。这种解的两个最重要的问题是明显的不精确性(解没有紧密跟踪实际值)以及这样的实事,即在某些点解大于最大物理可实现值或小于最小物理可实现值。因而发生在使用传统有限元技术产生的解中的振荡产生了不实际的结果,或甚至可能完全不收敛。
然而根据本公开的方法不会有这种缺陷,因为解被约束而成为实际的。通过保证控制解向量的矩阵(即Jacobian矩阵)满足单调性线性的保持性质而实现这种约束。以更实际的术语来说,即这在于保证矩阵对角线上所有的系数为非负的,而所有不在矩阵对角线上的系数为非正的。当矩阵具有这些特性时,解在所有的点保持为实际的,并如图2中曲线13所指出,更精确地跟踪物理系统的行为。因而模型更可能产生精确的结果并可靠地收敛。
Jacobian矩阵是基于Darcy定律从描述通过多孔介质的多相流的偏微分方程推导出来的,并表示系统中质量和动量的守恒。这些方程式被离散化为有限元网格,而方程式的系数形成矩阵。取决于方程式的梯度为正还是为负(即液流为正或负),每一系数将为或者正或者负。如果有限元技术用于微分方程,则产生一大组代数方程式。这组代数方程式是非线性的。使用牛顿方法(这涉及取方程式的一阶导数)能够求解这组方程式。一阶导数就是Jacobian矩阵。这矩阵的系数被加权,而获得在其对角线上有非负项而不在对角线上有非正项的矩阵。项的适当的加权保证了矩阵的单调和线性性质。
Jacobian矩阵的系数基本上是对有限元网格中元素的每一结点的加权因子。在传统有限元技术中,有限元的结点被均匀加权。例如,在二维元素中,每一组三个结点将按1/3加权。如果使用三维元素,则每一组四个结点按1/4加权。在本方法中,对结点加权反映了被建模的系统中液体的流动。这参照图3-6被示出。
图3表示单个的二维元素(一个三角形)。相对于元素的液体流由箭头指示,箭头示出液体从上,即图2的右手部分,向下,即图的左手部分流动。使用传统的有限元技术,每一结点21-23将被均匀地加权(即1/3,1/3,1/3)。使用本技术,结点不是被均匀加权。因为液体总的是从结点22和23向结点21流动,每一结点22和23按1/2被加权,而结点21按0被加权。参见图4,可以看到,当液流的方向改变时,结点的加权不同。在图4中,液流总的是从下,即将图的左手部分,向上,即图的右手部分。因而,结点21按1加权,而22和23按0加权。
参见图5和6,可以看到,结点的加权还受到对应元素的形状的影响。在图5中,液流几乎垂直于元素的边。这种情形类似于图3的情形,其中使元素的向前的边终止的每一结点按1/2加权。这种情形下,元素其余的结点是从元素向前的边上的中点的下游。图5中,其余的结点不是从第一到达的结点之一的直接下游。这种情形下,作为对最后的结点33的上游的结点32按2/3加权,而其它上游结点31按1/3加权。在图6中,有一单一的上游结点33。结点31和32都在元素的末尾边上,于是它们都按0加权。另一方面,上游结点33按1加权。
这些情形的每一情形中,结点相对于液流方向的加权有效地强调了矩阵的对角线元素,同时降低或消除了不在对角线元素的效果。这又保证了矩阵具有单调性和线性的性质。
虽然以上足以使得有限元分析领域中一般的专业人员能够实现本技术,以下还是对改项技术进行简短的技术讨论。
考虑将要对时间建模的物理系统,首先构成有限元网格(即系统在空间-时间域被离散化)。参见图7,示出时间-空间域的二维投影。
Q=Ωspacex[O,T]是被建模的空间-时间域,其中
Ω被划分(分割)为多个非重叠子域Ωe。每一这些子域Ωc和
包括四维单纯形(超四面体)。线性有限元基是在每一元素(Ωe)上形成的,并在每一元素上定义试验函数和测试函数。
多孔介质中的多相液流由以下方程式控制:
其中:
i=1,2,…,n相
Vi=-K(Pi+ρigZ)
Pi=P
Pi=P-PCi,i=2,…,n相
λi=Kri/(μiBi)
μi=第i相的黏滞性
Bi=第i相的形成体积因子
ρi=第i相的密度
φ=岩石孔积率
Z=深度的负值
Si=第i相的饱和度
对每一元素,此多相流方程式的标准的Galerkin有限元离散化,给出:
使用本技术的基本方法是按λ- I被定义的方式定义的。因而本方法定义如下:
参见图8,示出表示根据本发明的一个实施例方法的流程图。如以上所指出,当微分方程式在有限元上被离散化时,它们基本上从微分方程被转换为结点值之间的代数关系式。这是在假设结点之间存在对应的性质(例如,饱和度,压力,渗透性,prostate等)的线性变换而实现的。这限制了所有可能解的空间为只是线性的解。当使用这一假设时,对于在每一元素上为线性的系数的值,从微分方程式推导出一组解。当这样作出时,每一元素以一定的离散化量贡献给其结点的每一个。在每一结点处求合的贡献是残数。希望使得残数等于0,以便求解非线性代数方程。
为了分配有限元算子的残数(rj i),首先对4-tet(超四面体)的每一结点计算Kj。所得Kj被划分为正(Kj +)及负(Kj -)值。然后残数γj分配给结点,并计算正切算子(Tjk)。然后按已经使用的普通有限元技术那样对系统进行组合并求解。(由于有限元系统组合和求解过程是众所周知的,这里不再对其详述)。
以上就特定实施例说明了本发明可以提供的好处,其它的优点和对问题的解决。这些好处和优点,以及任何可能引起它们发生或更多被宣称的因素和限制并不构成为任何或所有权利要求的关键的,必须的或基本特征或要素。作为这里所使用的术语“包括”,“包括”或其任何其它的变形,要被解释为非排他性的包含跟随这些术语之后的元素或限制。于是包括要素列表的过程,方法,文章或设备并不仅仅包含那些要素,而是可以包含其它没有明显列出的或所述过程,方法,文章或设备固有的要素。
虽然已经参照特定的实施例对本发明进行了说明,但应当理解,实施例是示例性的,而本发明的范围不限于这些实施例。许多对以上所述实施例的变化,改型,附加和改进是可行的。特别地,这些变化可以包含计算机或其它数据处理装置,包含软件,固件或其它实施上述方法编程的计算机可读介质(诸如软盘,CD-ROM,DVD-ROM等)存储装置,计算机存储器等。这些变化,改型,附加和改进被认为属于本发明的范围。
Claims (11)
1.使用计算机系统对其中存在多相液流的物理系统形成表示的方法,该方法包括:
使用计算机系统生成一个包含多个系数的初始矩阵,所述多个系数中的第一组系数包含该矩阵的对角线上的元素,所述多个系数中的第二组系数包含该矩阵的不在对角线上的元素,所述多个系数中的每个系数都对应于并且代表液体容器的一个位置,并与容器中的液流相关联;
调节所述初始矩阵的系数,将矩阵元素限定为可在物理上实现的值,并通过调节所述多个系数中的每个系数来跟踪物理系统的行为,以根据液流穿越每个元素的方向对每一元素的结点进行加权,使得每个对角线上的元素最小为零,并且每个不在对角线上的元素小于或等于零;将调节后的系数存储在所述计算机系统的计算机系统存储器中;并且
使用该计算机系统利用调节后的系数生成对该物理系统的表示。
2.权利要求1的方法,其中根据液流穿越每一元素的方向对每一元素的结点进行加权包括确定液流穿越一个元素的方向,并如果该结点是元素其它结点的上游,则对每一这样的结点加较重的权,如果结点是在元素其它结点的下游,则对其加较轻的权。
3.权利要求2的方法,其中如果元素较大的部分对该结点比对元素其它结点来说是下游,则每一这样的结点较重加权,并如果元素较小的部分对该结点比对元素其它结点来说是下游,则每一这样的结点较轻加权。
4.权利要求1的方法,其中初始矩阵对应于其中至少有两个液相的系统。
5.权利要求1的方法,其中初始矩阵对应于其中至少有三个或更多液相的系统。
6.权利要求1的方法,其中初始矩阵对应于四维有限元模型。
7.权利要求1的方法,其中物理系统对应于储油层。
8.权利要求1的方法,其中矩阵被设置为产生在任何时候都能在物理上实现的解。
9.权利要求1的方法,其中所述表示被设置为产生非振荡解。
10.权利要求1的方法,还包括对系统模型的离散化以产生有限元网格,并基于该有限元网格产生初始矩阵。
12.使用计算机系统预测液体容器中的液流的方法,该方法包括:
对其中存在多相液流的液体容器的模型离散化;
产生对应于模型的有限元矩阵,其中根据与矩阵元素相关联的液流方向,通过有选择性地对有限元矩阵的矩阵元素进行加权,矩阵被设置为产生单调的并保持线性的输出;以及
使用计算机系统利用该矩阵产生液流预测。
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