History matching is a critical step in reservoir numerical simulation algorithms. It is typically hindered by difficulties associated with the high-dimensionality of the problem and the gradient calculation approach. Here, a multi-step solving method is proposed by which, first, a Fast marching method (FMM) is used to calculate the pressure propagation time and determine the single-well sensitive area. Second, a mathematical model for history matching is implemented using a Bayesian framework. Third, an effective decomposition strategy is adopted for parameter dimensionality reduction. Finally, a localization matrix is constructed based on the single-well sensitive area data to modify the gradient of the objective function. This method has been verified through a water drive conceptual example and a real field case. The results have shown that the proposed method can generate more accurate gradient information and predictions compared to the traditional analytical gradient methods and other gradient-free algorithms.

Reliable reservoir models can accurately reproduce the complete reservoir developing process and provide convincing guidance for production management decision-making [

Generally speaking, there is a close correlation between reservoir physical properties and well production performance, and history matching algorithms are mainly used to calculate the gradient and to direct the countermeasures. To the best of our knowledge, the widely used algorithms for history matching mainly include gradient-class method [

In order to reduce the dimensionality of variables in the history matching, several parameterization methods were proposed. The overall idea is to convert the original high dimensional optimization problem to a low dimensional problem without changing the main features. The parameterization methods mainly include principal component analysis (PCA) method [

The pseudo correlation between gradients can be weakened by dividing the single-well sensitivity area and then modifying the calculated gradient [

This paper presented a two-step history matching method by integrating the FMM and the PCA. The brief calculation scheme of history matching is shown in

The pressure wave diffusion equation (

where, _{t} is the comprehensive compressibility coefficient,

where,

where,

where, _{x}, _{y} and _{z} are the propagation velocity in the _{x}, _{y} and _{z} are the grid tracking time in the

(1) Set the initial start grid and then mark it as a freezing grid (red point in

(2) Search for adjacent grids and calculate the flight time

(3) Find the grid to which the minimum flight time is required (“1” point in

(4) Search for adjacent grids surround the freezing grid. Then calculate the flight time and set a new freezing grid (“2” point in

(5) Repeat Steps 2 to 4 until all grids are marked as freezing grids.

In order to reduce the calculation cost, we set the time threshold of single-well sensitive area to a preliminarily defined tracking range. In this paper, we arranged 1/

In this section, we first briefly described the mathematical model of history matching and the process to do parameter dimensionality reduction. Then, we explained how to use the single-well sensitive area data to correct the gradient of the objective function.

History matching is a typical inverse-problem solving process, which aims to generate the maximum posteriori estimate of the reservoir model by matching the observation data. The correction between the actual observation data and the model parameters is:

where, _{obs} is the observed data, such as water content, oil production rate, bottom hole pressure, etc.; g( ) refers to the numerical simulation process; _{d} is the measuring error. According to the Bayesian framework, the conditional PDF of _{obs} can be written as:

where,

where, _{M} is the correlation matrix of model parameters; _{D} is the covariance matrix of measuring errors. Thus, in the history matching process, the following formula should be minimized:

where, _{prior} is the prior estimation vector of reservoir model. In the history matching process, the prior model was often used as the initial value of the continuous iteratives. Since _{prior} often follows the multivariate Gaussian distribution in practical applications, the average values of these prior model parameter that represented the prior geological features would be optimized as the initial value. In this paper, _{prior}:

In general, the actual reservoir parameters often have high dimensionality, making it pretty difficult to directly calculate the _{M}. In the past, many methods such as K-L decomposition method, discrete cosine transform method and gradual deformation method have been applied to reduce the dimensionality of the parameters in history matching. In this paper, efficient SVD decomposition method in PCA was introduced to parameterize the reservoir model and to deconstruct the _{M} to make _{w} non-zero singular values. _{M} was calculated as following:

where,

The objective function can be approximately converted to:

The maximum posteriori estimation _{MAP} could be inversely calculated after the maximum posteriori estimation _{MAP} was obtained by minimizing

where,

In the numerical simulation process, the production data generated by the reservoir model based on the model parameters

where,

where, _{w} is the sensitivity coefficient matrix of

where, α_{l} is the step size in search. In the previous study, the sensitivity matrix _{w}. Herein, we defined ρ(

where, _{d} is the number of the observation data. After the single-well sensitive area was obtained by FMM, the sensitivity gradient beyond the sensitivity area was defined as

where, F_{j} is the sensitive area of well _{ref} is the maximum flight time within the sensitive area of well

Then we defined ρ(

and

According to the chain rule to solve the partial derivatives:

where,

A two-dimensional heterogeneous reservoir model with three reservoir fluids, oil, gas and water was first constructed to verify the proposed method. The size of this model was set to be 20 × 30 × 1. The grid size is, Dx = Dy = 80 m, and Dz = 30 m. We selected twenty prior reservoir realizations and a real model that provided the observation data. The real permeability field of the model was shown in ^{3}/d. The liquid flow rate of the production wells, 100 m^{3}/d. A total of 1200 parameters including permeability and porosity were to be estimated by matching the actual production data.

Properties | Value | Properties | Value |
---|---|---|---|

Initial reservoir pressure, psi | 3503 | Water viscosity, cp | 0.5 |

Initial water saturation | 0.2 | Gas viscosity, cp | 0.012 |

Initial gas saturation | 0.1 | Water density, lb/ft^{3} |
62.428 |

Rock compressibility | 3.1 × 10^{−4} |
Oil density, lb/ft^{3} |
56.93 |

Water compressibility | 3.74 × 10^{−6} |
Gas density, lb/ft^{3} |
0.0516 |

In the history matching process, the FMM method was firstly used to calculate the flight time. Taking the injection well in the average model as an example (

Subsequently, the PCA method was used to reduce the dimensionality of the parameters included in the reservoir model. In order to verify the effectiveness of the proposed method to weaken the pseudo-correlation, the finite difference method (FDM), simultaneous perturbation stochastic approximation (SPSA) [^{th} and the 70^{th} iteration step. Among them, the gradient calculated by FDM marked by red circles was considered as the true gradient. It was easy to notice that the LAGM gave a closer correlation with the true gradient than AGM and SPSA. For the calculated gradient correlation between FDM and the other three algorithms at the 10^{th} iteration step, the SPSA marked by yellow line was 0.095, the AGM marked by gray line was 0.181, and the LAGM marked by blue line was 0.417. Similarly, at the 70^{th} iteration step, the SPSA algorithm was 0.064, the AGM was 0.105, and the LAGM was 0.325. The results implied that the LAGM could effectively eliminate the pseudo correlation of the gradient.

The final permeability estimates calculated by the three algorithms were presented in

Algorithms | Accuracy |
---|---|

LAGM | 95.64% |

AGM | 92.15% |

SPSA | 86.31% |

The Brugge oilfield was developed by TNO as a benchmark for closed-loop reservoir management test. ^{3}/day and the production rate was 600 m^{3}/day. In the history matching process, the variance of the measuring error was a fixed value of 0.05 over the observation data. The grid planar permeability was set to be the estimated parameter. Similar to the previously discussed example, 40 reservoir parameters were applied.

Then, FMM was introduced to calculate the flight time and to determine the single-well sensitive area. The sensitivity area of the first layer of the average model was shown in ^{th} iteration step. LAGM and AGM gave very similar results in the MAP estimates, while SPSA gave more prominent heterogeneous characteristics.

Algorithms | Accuracy |
---|---|

LAGM | 92.35% |

AGM | 86.23% |

SPSA | 83.25% |

In this paper, we firstly reviewed the theorem of FMM and briefly introduced the sensitive area division method. Subsequently, we established a history matching mathematical model based on Bayesian framework. A parameterization method was used to reduce the dimensionality of original model parameters. By integrating the sensitive area information, the gradient of history matching could be better modified. The proposed approach was actually a two-step history matching method and no numerical simulation runs were required in the pretreatment process. To evaluate the feasibility of the presented method, a conceptual case of water drive in a heterogeneous two-dimensional reservoir and a real field case were tested. The results showed that the presented approach had higher accuracy and computational efficiency as compared to non-localized gradient method and gradient free algorithm. Moreover, the presented idea could also be used in other history matching algorithms, such as ENKF, ES-MDA, and so on. It was worth noting that parameterizing reservoir model and then modifying gradient may be a good choice for heavy reservoir history matching problem.

As an approximate-gradient iterative optimization algorithm, SPSA can ensure that the optimization direction is always the uphill direction through the simultaneous disturbance of components of variable. For the

where

where α_{l+1} is the iteration step size with the fixed value of 0.5; the average gradient is defined by

where