Inverse modeling of a radial multistep outflow experiment for determining unsaturated hydraulic properties

Modeling flow and solute transport in the unsaturated zone on the basis of the Richards equation requires specifying values for unsaturated hydraulic conductivity and water potential as a function of saturation. The objectives of the paper are to evaluate the design of a transient, radial, multi-step outflow experiment, and to determine unsaturated hydraulic parameters using inverse modeling. We conducted numerical simulations, sensitivity analyses, and synthetic data inversions to assess the suitability of the proposed experiment for concurrently estimating the parameters of interest. We calibrated different conceptual models against transient flow and pressure data from a multi-step, radial desaturation experiment to obtain estimates of absolute permeability, as well as the parameters of the relative permeability and capillary pressure functions. We discuss the differences in the estimated parameter values and illustrate the impact of the underlying model on the estimates. We demonstrate that a small error in absolute permeability, if determined in an independent experiment, leads to biased estimates of unsaturated hydraulic properties. Therefore, we perform a joint inversion of pressure and flow rate data for the simultaneous determination of permeability and retention parameters, and analyze the correlations between these parameters. We conclude that the proposed combination of a radial desaturation experiment and inverse modeling is suitable for simultaneously determining the unsaturated hydraulic properties of a single soil sample, and that the inverse modeling technique provides the opportunity to analyze data from nonstandard experimental designs.     

Laminar hydromagnetic flow in an annulus with uniformly porous walls of different permeability

Summary The problem of steady laminar hydromagnetic flow in a porous annulus of different permeability in the presence of a magnetic field is considered and solved by a method of perturbation. The case when the injection or the withdrawal rate at the outer wall, depending on the sign of the normal velocity at this wall, is greater than or equal to such rate at the inner wall is given in details. Some numerical results regarding the friction coefficients at the walls and the axial pressure coefficient and velocity distribution have been appended.     

A multiscale adjoint method to compute sensitivity coefficients for flow in heterogeneous porous media

A multiscale adjoint (MSADJ) method is developed to compute high-resolution sensitivity coefficients for subsurface flow in large-scale heterogeneous geologic formations. In this method, the original fine-scale problem is partitioned into a set of coupled subgrid problems, such that the global adjoint problem can be efficiently solved on a coarse grid. Then, the coarse-scale sensitivities are interpolated to the local fine grid by reconstructing the local variability of the model parameters with the aid of solving embedded adjoint subproblems. The approach employs the multiscale finite-volume (MSFV) formulation to accurately and efficiently solve the highly detailed flow problem. The MSFV method couples a global coarse-scale solution with local fine-scale reconstruction operators, hence yielding model responses that are quite accurate at both scales. The MSADJ method is equally efficient in computing the gradient of the objective function with respect to model parameters. Several examples demonstrate that the approach is accurate and computationally efficient. The accuracy of our multiscale method for inverse problems is twofold: the sensitivity coefficients computed by this approach are more accurate than the traditional finite-difference-based numerical method for computing derivatives, and the calibrated models after history matching honor the available dynamic data on the fine scale. In other words, the multiscale based adjoint scheme can be used to history match fine-scale models quite effectively.     

Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels

We present a new approach to reservoir simulation that gives accurate resolution of both large-scale and fine-scale flow patterns. The method uses a mixed multiscale finite-element method (MMsFEM) to solve the pressure equation on a coarse grid and a streamline-based technique to solve the fluid transport on a fine-scale subgrid. The MMsFEM is based on the construction of special approximation velocity spaces that are adaptive to the local properties of the differential operator. As such, MMsFEM produces a detailed subgrid velocity field that reflects the impact of the fine-scale heterogeneous structures. By combining MMsFEM with rapid streamline simulation of the fluid transport, we aim towards a numerical scheme that facilitates routine reservoir simulation of large heterogeneous geomodels without upscaling. The new method is applied to two different test cases. The first test case consists of two (strongly) heterogeneous quarter five-spot problems in 2D. The second test case is a 3D upscaling benchmark taken from the 10th SPE Comparative Solution Project, a project whose purpose is to compare and validate upscaling techniques. The test cases demonstrate that the combination of multiscale methods and streamlines is a robust and viable alternative to traditional upscaling-based reservoir simulation.     

Necessary conditions for inverse modeling of flow through variably saturated porous media

Non-unique solutions of inverse problems arise from a lack of information that satisfies necessary conditions for the problem to be well defined. This paper investigates these conditions for inverse modeling of water flow through multi-dimensional variably saturated porous media. It shows that in order to obtain a unique estimate of hydraulic parameters, along each streamline of the flow field (1) spatial and temporal head observations must be given; (2) the number of spatial and temporal head observations required should be greater or equal to the number of unknown parameters; (3) the flux boundary condition or the pumping rate of a well must be specified for the homogeneous case and both boundary flux and pumping rate are a must for the heterogeneous case; (4) head observations must encompass both saturated and unsaturated conditions, and the functional relationships for unsaturated hydraulic conductivity/pressure head and for the moisture retention should be given, and (5) the residual water content value also need to be specified a priori or water content measurements are needed for the estimation of the saturated water content.For field problems, these necessary conditions can be collected or estimated but likely involve uncertainty. While the problems become well defined and have unique solutions, the solutions likely will be uncertain. Because of this uncertainty, stochastic approaches are deemed to be appropriate for inverse problems as they are for forward problems to address uncertainty. Nevertheless, knowledge of these necessary conditions is critical to reduce uncertainty in both characterization of the vadose zone and the aquifer, and prediction of water flow and solute migration in the subsurface.     

用油田生产测井资料确定产层动态渗透率的方法研究

目前,我国处于开发中后期的油田平均产水率高达80%以上.因此对产层进行评价,确定地层动态渗透率等参数及剩余油的平面分布,由此研究其特点具有十分重要的意义.在储层径向流动方程分析的基础上,研究了储层渗透率与产液量/吸水量和油并内流压变化等生产测井资料之间的关系,提出了利用油田生产测井资料估算各产层的动态渗透率的方法.该方法不需关井,计算过程相对简单、可以节约成本,还可以避免常规压力恢复或压力降落试井造成低产井降产甚至停产的可能性.计算结果能较好地反映各小层的产液性质.     

基于离散余弦变换的集合Kalman滤波方法对河流相油藏渗透率场的反演

针对河流相油藏参数场的非高斯性及油藏生产数据与地质模型之间的非线性关系,提出了耦合离散余弦变换的半迭代集合Kalman滤波方法对河流相油藏进行生产历史拟合及反演.将河流相油藏非高斯渗透率场利用离散余弦变换变为类似高斯分布的DCT(Discrete cosine transform)系数,利用能量压缩特性对DCT系数进行...     

裂缝-孔隙介质储层渗透率表征及其影响因素分析

含裂缝多孔介质渗透率预测是非常规油气资源勘探开发的一个紧迫问题.现有多孔介质岩石物理模型通常利用圆形孔管模拟宏观岩石孔隙空间,难以定量描述软孔隙/裂缝在压力作用下的闭合情况,缺乏裂缝/孔隙间流量交换的连通机制.本文提出含三维裂缝/软孔隙网络多孔介质模型,将储层岩石裂缝/软孔隙表示为椭圆截面微管,建立了周期性压力作用下微...     

Electromagnetic holographic sensitivity field of two-phase flow in horizontal wells

Electromagnetic holographic data are characterized by two modes, suggesting that image reconstruction requires a dual-mode sensitivity field as well. We analyze an electromagnetic holographic field based on tomography theory and Radon inverse transform to derive the expression of the electromagnetic holographic sensitivity field (EMHSF). Then, we apply the EMHSF calculated by using finite-element methods to flow simulations and holographic imaging. The results suggest that the EMHSF based on the partial derivative of radius of the complex electric potential φ is closely linked to the Radon inverse transform and encompasses the sensitivities of the amplitude and phase data. The flow images obtained with inversion using EMHSF better agree with the actual flow patterns. The EMHSF overcomes the limitations of traditional single-mode sensitivity fields.     

A multiscale method for subsurface inverse modeling: Single-phase transient flow

High-resolution geologic models that incorporate observed state data are expected to effectively enhance the reliability of reservoir performance prediction. One of the major challenges faced is how to solve the large-scale inverse modeling problem, i.e., to infer high-resolution models from the given observations of state variables that are related to the model parameters according to some known physical rules, e.g., the flow and transport partial differential equations. There are typically two difficulties, one is the high-dimensional problem and the other is the inverse problem. A multiscale inverse method is presented in this work to attack these problems with the aid of a gradient-based optimization algorithm. In this method, the model responses (i.e., the simulated state data) can be efficiently computed from the high-resolution model using the multiscale finite-volume method. The mismatch between the observations and the multiscale solutions is then used to define a proper objective function, and the fine-scale sensitivity coefficients (i.e., the derivatives of the objective function with respect to each node’s attribute) are computed by a multiscale adjoint method for subsequent optimization. The difficult high-dimensional optimization problem is reduced to a one-dimensional one using the gradient-based gradual deformation method. A synthetic single-phase transient flow example problem is employed to illustrate the proposed method. Results demonstrate that the multiscale framework presented is not only computationally efficient but also can generate geologically consistent models. By preserving spatial structure for inverse modeling, the method presented overcomes the artifacts introduced by the multiscale simulation and may enhance the prediction ability of the inverse-conditional realizations generated.     

Threshold pressure gradient of fluid flow through multi-porous media in low and extra-low permeability reservoirs

After analyzing many studies of fluid flow theory of multi-porous media in low and extra-low permeability reservoirs and the numerical simulation of non-Darcy flow, we found that a negative flow rate occurs in the existing non-Darcy flow equation, which is unreasonable. We believe that the existing equation can only be considered as a discriminant to judging Darcy flow or non-Darcy flow, and cannot be taken as a fluid flow governing equation of multi-porous media. Our analysis of the experimental results shows that the threshold pressure gradient (TPG) of low and extra-low permeability reservoirs is excessively high, and does not conform to fluid flow through multi-porous media in the actual reservoir situation. Therefore, we present a reasonable TPG ranging from 0.006 to 0.04 MPa/m at the well depth of 1500 m and oil drainage distance of 500 m. The results of our study also indicate that the non-Darcy flow phenomenon will disappear when the TPG reaches a certain value. In addition, the TPG or non-Darcy flow in low and extra-low permeability reservoirs does not need to be considered in the productivity prediction and reservoir numerical simulation. At present, the black oil model or dual-porous media is suitable for simulating low and extra-low permeability reservoirs.     

Use of the Fourier-Laplace transform and of diagrammatical methods to interpret pumping tests in heterogeneous reservoirs

Advances in computer power and in reservoir characterization allow simulation of pressure transients in complex reservoirs generated stochastically. Generally, interpretation of these transient gives useful information about the reservoir hydraulic properties: a major goal is to interpret these transients in the stochastic context. First we ensemble average the pressure over all the random permeability field realizations to derive an equation which drives the ensemble averaged pressure. We use the Fourier transform in space and the Laplace transform in time, in conjuction with a perturbation series expansion in successive powers of the permeability fluctuations to obtain an explicit solution. The Nth order term of this series involves the hydrodynamic interaction between N permeability heterogeneities and after averaging we obtain an expansion containing correlation functions of permeability fluctuations of increasing order.Next, Feynman graphs are introduced allowing a more attractive graphical interpretation of the perturbation series. Then series summation techniques are employed to reduce the graph number to be summed at each order of the fluctuation expansion. This in turn gives useful physical insights on the homogenization processes involved. In particular, it is shown that the sum of the so-called ‘one-particle irreducible graphs’ gives the kernel of a linear integro-differential equation obeyed by the ensemble average pressure. All the information about the heterogeneity structure is contained in this renormalized kernel, which is a limited range function.This equation on its own is the starting point of useful asymptotic results and approximations. In particular it is shown that interpretation of pumping tests yields the steady-state equivalent permeability after a sufficiently long time for an infinite reservoir, as expected.     

A Neumann expansion approach to flow through heterogeneous formations

A stochastic approach is used for the study of flow through highly heterogeneous aquifers. The mathematical model is represented by a random partial differential equation in which the permeability and the porosity are considered to be random functions of position, defined by the average value, constant standard deviation and autocorrelation function characterized by the integral scale. The Laplace transform of the solution of the random partial differential equation is first written as a solution of a stochastic integral equation. This integral equation is solved using a Neumann series expansion. Conditions of convergence of this series are investigated and compared with the convergence of the perturbation series. For mean square convergence, the Neumann expansion method may converge for a larger range of variability in permeability and porosity than the classic perturbation method. Formal expressions for the average and for the correlation moments of the pressure are obtained. The influence of the variability of the permeability and porosity on pressure is analyzed for radial flow. The solutions presented for the pressure at the well, as function of the permeability coefficient of variation, may be of practical interest for evaluating the efficiency of well stimulation operations, such as hydraulic fracturing or acidizing methods, aimed at increasing the permeability around the well.     

Estimating flow parameter distributions using ground-penetrating radar and hydrological measurements during transient flow in the vadose zone

Methods for estimating the parameter distributions necessary for modeling fluid flow and contaminant transport in the shallow subsurface are in great demand. Soil properties such as permeability, porosity, and water retention are typically estimated through the inversion of hydrological data (e.g., measurements of capillary pressure and water saturation). However, ill-posedness and non-uniqueness commonly arise in such non-linear inverse problems making their solutions elusive. Incorporating additional types of data, such as from geophysical methods, may greatly improve the success of inverse modeling. In particular, ground-penetrating radar (GPR) methods have proven sensitive to subsurface fluid flow processes and appear promising for such applications. In the present work, an inverse technique is presented which allows for the estimation of flow parameter distributions and the prediction of flow phenomena using GPR and hydrological measurements collected during a transient flow experiment. Specifically, concepts from the pilot point method were implemented in a maximum a posteriori (MAP) framework to allow for the generation of permeability distributions that are conditional to permeability point measurements, that maintain specified patterns of spatial correlation, and that are consistent with geophysical and hydrological data. The current implementation of the approach allows for additional flow parameters to be estimated concurrently if they are assumed uniform and uncorrelated with the permeability distribution. (The method itself allows for heterogeneity in these parameters to be considered, and it allows for parameters of the petrophysical and semivariogram models to be estimated as well.) Through a synthetic example, performance of the method is evaluated under various conditions, and some conclusions are made regarding the joint use of transient GPR and hydrological measurements in estimating fluid flow parameters in the vadose zone.     

高煤级煤储层渗透性与应力耦合模型及控制机理

煤储层应力敏感性是影响煤层气井产能的地质因素,以鄂尔多斯盆地东南缘高煤级煤储层为对象,通过煤样的应力敏感性实验和现场测试,建立了高煤级煤储层渗透性与应力之间的相关关系和模型;探讨了渗透性变化的控制机理.研究结果表明, 煤储层渗透率随应力的增加按负指数函数规律降低.在应力小于5 MPa时,煤储层渗透率随应力增加快速下降,应力敏感性最强;应力在5~10 MPa时,渗透率随应力增加而较快下降,应力敏感性较强;而当应力大于10 MPa后,渗透率随应力的增加下降速度减缓,应力敏感性减弱.与沁水盆地南部高煤级煤样实验结果对比认为,鄂尔多斯盆地东南缘山西组2煤层应力敏感性要小于沁水盆地南部山西组3煤层的应力敏感性.煤储层渗透性是在应力作用下煤储层中裂隙产生压缩(压密)变形,裂隙开度急剧减小的结果.因此在煤层气开发过程中控制排采速度,尤其是排采早期降液速度,对于防止煤储层应力敏感性,提高采收率具有实际意义.     

渗透率场敏感系数的数值计算

由渗流微分方程定解问题，利用格林互易定理从理论上导出了渗透率场敏感系数 的计算公式，并借助于数值积分和差分方法给出了渗透率场敏感系数的离散形式. 由Peacem an方程建立了井压渗透率场敏感系数与网格压力渗透率场敏感系数的关系. 对理论模型利用 三维不均匀非稳定渗流场的压强数值解计算了井压对渗透率场的敏感系数，并用敏感系数的 直接计算方法进行了验证，结果表明本文的计算方法与直接计算方法相吻合.     

随钻斯通利波测井反演地层渗透率的理论、方法及应用

针对孔隙渗透地层的随钻声波测井问题,用Biot-Rosenbaum孔隙弹性波测井理论推导了孔隙地层的随钻井孔声场表达式.据此考察了随钻条件下井中斯通利波的波形、相速度频散、衰减以及相速度对渗透率的灵敏度,并与电缆测井中的情况进行了对比.数值模拟结果表明,随钻条件下斯通利波对地层渗透率的灵敏度相对于电缆测井有明显增加,更有利于用来反演地层渗透率.为快捷有效地处理现场测井数据和反演计算,采用简化Biot-Rosenbaum理论和钻铤的等效模型,对随钻斯通利波的频移和时滞进行联合反演.结果表明,随钻斯通利波反演的渗透率与核磁渗透率和岩心覆压测试渗透率符合较好,并且与常规测井曲线所反映的储层性质具有较好的一致性,证明了利用随钻斯通利波评价地层渗透率的有效性.     

利用阵列感应测井进行储层渗透率评价

钻井过程中储层受到泥浆侵入影响的程度与储层岩性有着密切关系,其中储层渗透率对侵入深度有着较大影响,因此若可以获知泥浆侵入深度,则有望对储层渗透率进行评估.本文首先建立含泥饼增长的泥浆侵入数值模型,然后建立阵列感应测井数值模型,两者的联合正演模拟显示泥浆侵入对地层的影响可以反映在阵列感应测井响应上,利用阻尼最小二乘法对阵列感应测井响应进行反演可以得到侵入深度.对侵入深度和储层渗透率的关系进行分析发现:在渗透率为1～100mD(1mD=0.987×10~(-3)μm～2)数量级的储层中,渗透率的变化可以在侵入深度上得到反映.以储层和井数据进行二维数值模拟发现:利用阵列感应测井响应反演出来的侵入深度曲线反映了渗透率在地层上的变化趋势,采用解释图版的方法可以对储层各层段的渗透率进行粗略估算.     

Upscaling of Forchheimer flows

In this work we propose upscaling method for nonlinear Forchheimer flow in heterogeneous porous media. The generalized Forchheimer law is considered for incompressible and slightly-compressible single-phase flows. We use recently developed analytical results (Aulisa et al., 2009) [1] and formulate the resulting system in terms of a degenerate nonlinear flow equation for the pressure with the nonlinearity depending on the pressure gradient. The coarse scale parameters for the steady state problem are determined so that the volumetric average of velocity of the flow in the domain on fine scale and on coarse scale are close. A flow-based coarsening approach is used, where the equivalent permeability tensor is first evaluated following streamline methods for linear cases, and modified in order to take into account the nonlinear effects. Compared to previous works (Garibotti and Peszynska, 2009) [2], (Durlofsky and Karimi-Fard) [3], this approach can be combined with rigorous mathematical upscaling theory for monotone operators, (Efendiev et al., 2004) [4], using our recent theoretical results (Aulisa et al., 2009) [1]. The developed upscaling algorithm for nonlinear steady state problems is effectively used for variety of heterogeneities in the domain of computation. Direct numerical computations for average velocity and productivity index justify the usage of the coarse scale parameters obtained for the special steady state case in the fully transient problem. For nonlinear case analytical upscaling formulas in stratified domain are obtained. Numerical results were compared to these analytical formulas and proved to be highly accurate.     

Ekman circulation and downwelling in narrow lakes

An analytical solution is obtained for the wind-driven steady flow developing under the action of the Coriolis acceleration in a closed basin of elongated shape. Different from the traditional Ekman approach, which determines the velocity distribution along a water column given the free surface shear stress and pressure gradient, here the flow field is solved in the whole cross-section considering the lateral transfer of momentum due to the horizontal eddy viscosity. The solution is derived exploiting a perturbation method, whereby the inverse of the Ekman number is assumed small, and imposing a wind aligned with the main axis of the lake. In the central part of the lake a secondary circulation develops producing downwelling along the right hand side (in the northern hemisphere) and upwelling along the opposite side, whose intensity is modulated by the turbulence anisotropy. The modification of the primary flow is considered as well. The solution, which is also compared with numerical results, is obtained for simplified conditions, but the extension to more general cases is discussed.