Transient well test analysis provides valuable information about reservoir characteristics such as permeability and the hydraulic diffusivity coefficient. It is based on the solution of diffusivity equation, which describes mass transfer in porous media. On the whole, analytical solutions are used for interpreting well test data. However, all these solutions were obtained under the condition of reservoir homogeneity. In a heterogeneous reservoir with spatially variable permeability, the exact analytical solutions are not known. The heterogeneous permeability field can be represented as the sum of two terms. The first term is a constant mean permeability value and the second one is a random function with known statistical properties. The second term can be considered a perturbation. The possibility of evaluating geostatistical parameters from well test analysis was considered by various authors and is still a challenging problem. In a randomly heterogeneous reservoir, a flow equation is formulated for the pressure, which is averaged over all the permeability realizations. It can be solved using Green’s function techniques, where the ensemble-averaged pressure is represented as an infinite perturbation series. This series can be represented graphically using Feynman diagrams and its summation can be performed following the rules that are well known in the quantum theory of solid state. For the first time, this framework was introduced to reservoir simulation in King (J. Phys. A: Math. Gen. 20, 3935–3947, 1987), where the stochastic pressure equation was solved for the steady-state case. In this study, we use diagram approach to obtain the solution of the time-dependent stochastic pressure equation which is derived for a lognormal random permeability field under the assumption of the Gaussian correlation function. The expression for transient ensemble averaged pressure is obtained with respect to high-order corrections of permeability variance. In the limit of sufficiently small variance, analytical expressions for the pressure correction are presented. The two limiting cases were considered: (i) the distance between wells is much bigger than the permeability correlation length; (ii) the opposite case where the correlation length is the smallest length parameter. The resulting solution can be used for the analysis of drawdown, build-up, and interference tests in stochastic porous media. The possibility of estimating the parameters of a random permeability field based on well test data is discussed.

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费曼图解法在随机多孔介质中试井分析中的应用

瞬态井测试分析提供了有关储层特征的有价值的信息，例如渗透率和水力扩散系数。它基于扩散率方程的解，该解描述了多孔介质中的传质。总体而言，分析解决方案用于解释试井数据。但是，所有这些解决方案都是在储层均质的条件下获得的。在具有空间可变渗透率的非均质油藏中，确切的解析解是未知的。非均质渗透率场可以表示为两个项之和。第一项是恒定的平均渗透率值，第二项是具有已知统计特性的随机函数。第二项可以看作是扰动。许多作者都考虑过通过试井分析评估地统计参数的可能性，但这仍然是一个具有挑战性的问题。在一个随机的非均质油藏中，为压力公式化了一个流量方程，该方程在所有渗透率实现中平均。可以使用格林的函数技术来解决它，其中集合平均压力表示为无穷微扰级数。该系列可以使用费曼图以图形方式表示，并且其求和可以按照固态量子理论中众所周知的规则进行。该框架首次在King（J.Phys.A：Math.Gen。为压力制定了一个流动方程，该方程在所有渗透率实现中取平均值。可以使用格林的函数技术来解决这一问题，其中合计平均压力表示为无穷微扰级数。该系列可以使用费曼图以图形方式表示，并且其求和可以按照固态量子理论中众所周知的规则进行。该框架首次在King（J.Phys.A：Math.Gen。为压力制定了一个流动方程，该方程在所有渗透率实现中取平均值。可以使用格林的函数技术来解决这一问题，其中合计平均压力表示为无穷微扰级数。该系列可以使用费曼图以图形方式表示，并且其求和可以按照固态量子理论中众所周知的规则进行。该框架首次在King（J.Phys.A：Math.Gen。该系列可以使用费曼图来图形表示，并且其求和可以按照固态量子理论中众所周知的规则进行。该框架首次在King（J.Phys.A：Math.Gen。该系列可以使用费曼图以图形方式表示，并且其求和可以按照固态量子理论中众所周知的规则进行。该框架首次在King（J.Phys.A：Math.Gen。20（3935-3947年，1987年），其中求解了稳态情况下的随机压力方程。在这项研究中，我们使用图方法来获得时间相关的随机压力方程的解，该方程是在高斯相关函数的假设下针对对数正态随机渗透率场得出的。关于渗透率变化的高阶校正，获得了瞬时整体平均压力的表达式。在足够小的变化范围内，给出了用于压力校正的解析表达式。考虑了两个极限情况：（i）井之间的距离远大于渗透率相关长度；（ii）相关长度是最小长度参数的相反情况。所得解决方案可用于分析压降，堆积，和随机多孔介质中的干扰测试。讨论了基于试井数据估算随机渗透率场参数的可能性。